Deterministic Nuclear Structure, Fission, and Fusion from Curvature Dynamics in Trembling Spacetime Relativity

Deterministic Nuclear Structure, Fission, and Fusion from Curvature Dynamics in Trembling Spacetime Relativity A unified account without probabilistic tunneling, shell closures, or pairing fits Nico F. Declercq George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, USA IRL 2958 Georgia Tech – CNRS, Georgia Tech Europe, Metz, France [email protected] Abstract Conventional nuclear models, from liquid-drop parametrizations to shell and densityfunctional theories, describe many global trends in binding and decay but remain probabilistic and heavily parameterized. They do not explain, from first principles, persistent anomalies such as deep sub-barrier fusion hindrance, odd-even staggering in fission yields, or the long half-life of 14C. This work extends Trembling Spacetime Relativity Theory (TSRT) into the nuclear domain as a deterministic geometric framework for structure, stability, and transformation. In TSRT, each nucleus is a localized trembling-curvature eigenmode whose stability results from sustained suppression of intrinsic spacetime curvature. Decay is a causal relaxation of this suppression and is modeled as curvature reconfiguration rather than a stochastic transition. Binding, fission, fusion, and decay thus emerge from the same geometric dynamics. Using a single global normalization fixed once on 60Co, the TSRT formulation reproduces experimental half-lives across beta decay, alpha decay, and spontaneous fission over more than twenty orders of magnitude, with a mean logarithmic deviation below one part in a million. No shell closures, pairing corrections, empirical preformation factors, or per-nucleus tuning are invoked. The 14C anomaly follows from its geometric beta-path curvature without adjustable hindrance. All Q-values, barrier actions, and emission rates follow deterministically from the spacetime metric and its curvature energy, and TSRT provides a causal mechanism for mass–energy conversion as curvature redistribution. TSRT also reproduces the emergence of neutron magic numbers as geometric minima of the curvature–stiffness map, matching the conventional magic sequence without quantum postulates. Deep sub-barrier fusion hindrance arises from geometric suppression of trembling-mode overlap; odd-even staggering in fission yields originates from phase-locked neck eigenmodes at scission; and electron-screening shifts reflect renormalization of nearfield electromagnetic curvature. These effects are consistent with the trembling-spacetime geometry that also underlies atomic structure, photon emission, and gravitational redshift. Quantitatively, TSRT reproduces absolute nuclear half-lives from milliseconds up to about ten quintillion years, with mean logarithmic deviation below one millionth, using only three global constants fixed once per mode. This accuracy surpasses conventional microscopic and empirical models—which typically yield mean logarithmic deviations between one tenth and one hundredth—by roughly ten thousand times, showing that nuclear decay is a deterministic geometric phenomenon rather than a stochastic process. All physical quantities are expressed in absolute SI and nuclear units, with TSRT calibration constants tied to fundamental curvature and lifetime anchors without empirical scaling. TSRT also reproduces absolute fission energetics, yielding a total energy release of 170.0 MeV for thermal-neutroninduced U-235 (n,f) without any per-observable tuning. By deriving nuclear binding, decay, reaction timescales, and mass–energy conversion from a single geometric principle, TSRT establishes a unified, predictive, and reproducible description of nuclear stability and transformation. It bridges microscopic nuclear structure with macroscopic relativistic consistency, showing that mass–energy balance and decay kinetics are natural manifestations of curvature redistribution in trembling spacetime. 1 2 Contents 1 Introduction 8 2 Core Notation, Conventions and Metric Signature 15 3 The Genesis of Spacetime and Elementary Particles in TSRT 16 4 TSRT Nuclear Foundations: Effective Curvature Potential and Motion 18 4.1 Single-nucleon motion in a trembling nuclear field . . . . . . . . . . . . . . . . . . 19 5 Binding Energy Systematics and Radii from Trembling Geometry 21 5.1 Saturation and the B/A curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 5.2 Charge radii and scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 6 Deformation Landscapes and Fission Barriers 25 6.1 Geometric deformation coordinate and barrier formation . . . . . . . . . . . . . . 25 6.2 Fission barriers from curvature action . . . . . . . . . . . . . . . . . . . . . . . . 28 6.3 Actinide benchmarks: 235U and 239Pu ........................ 29 7 Fusion Barriers, Proper-Time Tunneling, and S-Factor Trends 30 7.1 Fusion rates and astrophysical S-factor . . . . . . . . . . . . . . . . . . . . . . . . 31 7.2 Curvature-enhanced Coulomb barrier . . . . . . . . . . . . . . . . . . . . . . . . . 32 7.3 WKB-like tunneling from proper-time action . . . . . . . . . . . . . . . . . . . . 34 8 TSRT Description of Strong and Weak Nuclear Forces 37 8.1 The Strong Interaction as Curvature Saturation . . . . . . . . . . . . . . . . . . . 37 8.2 The Weak Interaction as Curvature Reconfiguration . . . . . . . . . . . . . . . . 39 8.3 Complementarity of Strong and Weak Forces . . . . . . . . . . . . . . . . . . . . 40 9 Stability and Instability of Atomic Nuclei 40 9.1 Geometric Origin of Nuclear Stability . . . . . . . . . . . . . . . . . . . . . . . . 43 9.2 Deterministic Lifetime Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 9.3 Deterministic Lifetime Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 9.4 Mode-SpecificFactors ................................. 45 9.5 Calibrations and Global Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 46 9.6 Representative Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 9.7 Predictive Criteria for Nuclear Lifetimes . . . . . . . . . . . . . . . . . . . . . . . 49 9.8 A Deterministic βMatrix Element in TSRT and the Case of 14C......... 50 9.9 Geometric Picture of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 9.10 Fusion-driven instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 9.11 TSRT Lifetime Model and Calibration . . . . . . . . . . . . . . . . . . . . . . . . 53 10 Nuclear Fission in TSRT 55 10.1 Curvature Stress in Heavy Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 10.2 Deterministic Cleavage of the Nuclear Mode . . . . . . . . . . . . . . . . . . . . . 59 10.3 Energy Release in Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 10.4InducedFission..................................... 61 10.5 Comparison with Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 10.6 Odd–Even Staggering (OES) in Fission Fragment Yields . . . . . . . . . . . . . . 63 10.7 OES Model: Two Bessel Packets with Localized Neck Bias . . . . . . . . . . . . . 64 10.8 Geometric Synthesis of Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 3 11 Mass–Energy Conversion in Fission and the TSRT Explanation 67 11.1 Conventional View of Mass Defect . . . . . . . . . . . . . . . . . . . . . . . . . . 67 11.2 TSRT Interpretation: Curvature Redistribution . . . . . . . . . . . . . . . . . . . 68 11.3 Geometric Equivalence to E=mc2.......................... 69 11.4 Distribution of Released Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 11.5 Comparison with Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 11.6 Geometric Significance of Mass–Energy Conversion . . . . . . . . . . . . . . . . . 72 12 Nuclear Fusion in TSRT 72 12.1 Conventional Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 12.2 TSRT View: Curvature Concentration . . . . . . . . . . . . . . . . . . . . . . . . 74 12.3 Barrier and Determinism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 12.4 Energy Release Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 12.5 Numerical Example: Deuterium–Tritium Fusion . . . . . . . . . . . . . . . . . . . 78 12.6StellarFusionChains.................................. 80 12.7 Proper-Time Action and the Low-Energy Exponential . . . . . . . . . . . . . . . 81 12.8 Electron–Screening Shifts in Ultra–Low–Energy Fusion . . . . . . . . . . . . . . . 83 12.9 Case Study: Deep Sub-Barrier Fusion Hindrance . . . . . . . . . . . . . . . . . . 85 12.10Geometric Synthesis of Fusion in TSRT . . . . . . . . . . . . . . . . . . . . . . . 88 13 Comparison with Quantum Nuclear Models 89 13.1 Standard Quantum Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 13.2 TSRT vs. Liquid-Drop Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 13.3TSRTvs.ShellModel ................................. 91 13.4FusionandTunneling ................................. 92 13.5 Decay Channels and Deterministic TSRT Lifetimes . . . . . . . . . . . . . . . . . 93 13.6NumericalBenchmarks................................. 93 13.7GeometricAdvantages................................. 94 14 Broader Implications and Future Work 95 14.1 Integration into Fundamental Physics . . . . . . . . . . . . . . . . . . . . . . . . . 95 14.2 Astrophysical and Cosmological Connections . . . . . . . . . . . . . . . . . . . . . 95 14.3 Experimental and Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . 95 14.4 Transport mapping: recoil, Doppler, gravitational . . . . . . . . . . . . . . . . . . 96 14.5 Charge-Sensed Electromagnetic Curvature Transport . . . . . . . . . . . . . . . . 96 14.6 Future Theoretical Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 14.7Perspective ....................................... 97 15 Calibrated Anchors to Absolute Units in TSRT (Deuteron to Fission Chain) 97 15.1 Energy mapping: absolute by construction . . . . . . . . . . . . . . . . . . . . . . 97 15.2 Curvature strength: fixed once and reused . . . . . . . . . . . . . . . . . . . . . . 98 15.3 Lifetime scale: a single physical anchor . . . . . . . . . . . . . . . . . . . . . . . . 98 15.4 On the trembling amplitude hAi: not an independent fit . . . . . . . . . . . . . . 98 15.5 Gamma–ray normalization (E2) and internal conversion . . . . . . . . . . . . . . 98 15.6Practicalsummary ................................... 99 15.7 Outlook: removing anchors entirely . . . . . . . . . . . . . . . . . . . . . . . . . . 99 16 Conclusions 99 17 Acknowledgments 101 18 Appendices 103 © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 4 A Methods: Numerical Procedures, Parameters, and Calibrations 105 B Extended Tables and Data Comparisons 108 B.1 Binding-energy comparison along representative chains . . . . . . . . . . . . . . . 108 B.2 Charge radii fits and residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 B.3 Actinide fission-barrier benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . 109 B.4 Electron screening: gamma-ray lines and beta spectra . . . . . . . . . . . . . . . 109 B.5 Deep sub-barrier hindrance: universal slope pair . . . . . . . . . . . . . . . . . . . 109 B.6 Odd–even staggering (OES) data and TSRT envelope . . . . . . . . . . . . . . . . 109 B.7 Lifetime tables and stability-map excerpts . . . . . . . . . . . . . . . . . . . . . . 110 C TSRT Nuclear Geometry: Definitions and Notation 111 C.1 Metric Convention and Proper Time . . . . . . . . . . . . . . . . . . . . . . . . . 111 C.2 Trembling Deviation, Curvature Measures, and Variance . . . . . . . . . . . . . . 111 C.3 Binding and Stability Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 C.4 Action Decomposition and Stability Maps . . . . . . . . . . . . . . . . . . . . . . 112 C.5 Units and definitions used throughout . . . . . . . . . . . . . . . . . . . . . . . . 113 Units and definitions used throughout . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 D Geometric Origin of Nuclear Magic Numbers in TSRT 114 D.1 Whatisbeingcomputed................................114 D.2 Curvature eigenmodes in a finite trembling domain . . . . . . . . . . . . . . . . . 114 D.3 Degeneracy and cumulative occupation . . . . . . . . . . . . . . . . . . . . . . . . 115 D.4 How we populate the table (roots, degeneracies, cumulative) . . . . . . . . . . . . 115 D.5 Why a small correction is needed (spin–geodesic splitting) . . . . . . . . . . . . . 116 D.6 How to reproduce the numbers (scripts and steps) . . . . . . . . . . . . . . . . . 118 D.7 Link to main text and other appendices . . . . . . . . . . . . . . . . . . . . . . . 118 D.8 Summaryforthereader ................................119 E Calibration, Physical Constants, and Normalizations 120 E.1 ConstantsandUnits ..................................120 E.2 Energy normalization used in computations . . . . . . . . . . . . . . . . . . . . . 121 E.3 MATLAB: Constants and Normalization . . . . . . . . . . . . . . . . . . . . . . . 121 F Numerical Discretization and Integration Schemes 123 F.1 Spatial Grids and Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 F.1.1 MATLAB: Grid and Quadrature Setup . . . . . . . . . . . . . . . . . . . . 124 F.2 Convergence and Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 F.2.1 MATLAB: Convergence Test Routine . . . . . . . . . . . . . . . . . . . . 126 F.3 MATLAB: Grid + Integration Utilities . . . . . . . . . . . . . . . . . . . . . . . . 127 G Trembling Fields for Nucleons and Composite Nuclei 129 G.1 Parametric scalar baseline for nucleons . . . . . . . . . . . . . . . . . . . . . . . . 129 G.1.1 MATLAB: Nucleon Trembling Field Generator . . . . . . . . . . . . . . . 129 G.2 Composite configurations and phase locking . . . . . . . . . . . . . . . . . . . . . 130 G.2.1 MATLAB: Composite nucleus field builder . . . . . . . . . . . . . . . . . . 131 G.3 MATLAB: Field Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 H Curvature Measures and Energy Functionals 132 H.1 From ξµν to a scalar measure K............................132 H.2 Binding, fission, and fusion energetics . . . . . . . . . . . . . . . . . . . . . . . . . 133 H.3 MATLAB: Curvature and Energies . . . . . . . . . . . . . . . . . . . . . . . . . . 133 © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 5 I Stability Maps and Lifetime Estimators 135 I.1 Curvature-Slope Stability Map (Deterministic ∆K2Analysis) . . . . . . . . . . . 135 I.2 Mode-aware Q-values from TSRT bindings . . . . . . . . . . . . . . . . . . . . . . 136 I.3 Energetics and curvature suppression . . . . . . . . . . . . . . . . . . . . . . . . . 137 I.4 Mode-aware Q-values: implementation path . . . . . . . . . . . . . . . . . . . . . 137 I.5 Per-mode lifetime predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 I.6 Pipelineandfiles....................................137 I.7 Numericalsafeguards..................................137 J Kinematic Transport (Recoil, Doppler, Gravity) 138 J.1 Two–bodyrecoil ....................................138 J.2 Doppler shift (source or observer motion) . . . . . . . . . . . . . . . . . . . . . . 139 J.3 Gravitational redshift (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 J.4 Ordering and practical application . . . . . . . . . . . . . . . . . . . . . . . . . . 140 K Charge-Sensed Electromagnetic Curvature Transport (CSCT) 141 K.1 Setupanddefinitions..................................141 K.2 Closed-form near-field model and scaling . . . . . . . . . . . . . . . . . . . . . . . 142 K.3 Implementation recipe (used in figures/tables) . . . . . . . . . . . . . . . . . . . . 144 K.4 Workedexamples....................................144 K.5 Notesoncalibration ..................................144 L Binding-Energy Benchmarks Across the Valley of Stability 146 L.1 Methodology ......................................146 L.2 Representative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 L.3 Discussion........................................148 L.4 Reproducibility.....................................148 L.5 MATLAB: Binding Plot Producer and Core Routine . . . . . . . . . . . . . . . . 148 M Fission: Surface Curvature Gradient and Bifurcation 151 M.1 Computing ∆Ksurface ..................................151 M.2 Bifurcation Path and Energy Release . . . . . . . . . . . . . . . . . . . . . . . . . 152 M.3 Neutron-capture increment and threshold crossing . . . . . . . . . . . . . . . . . . 153 M.4 MATLAB: Fission Routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 N Derivation of the Neck-Mode Model for Odd–Even Staggering 157 N.1 Geometry, notation, and the OES indicator . . . . . . . . . . . . . . . . . . . . . 157 N.2 Cylindrical neck modes and boundary locking . . . . . . . . . . . . . . . . . . . . 157 N.3 From neck modes to an OES amplitude . . . . . . . . . . . . . . . . . . . . . . . 158 N.4 Finite neck extent and Gaussian damping . . . . . . . . . . . . . . . . . . . . . . 158 N.5 Parity locking and action gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 N.6 A localized curvature “bias” and why it is geometric . . . . . . . . . . . . . . . . . 159 N.7 Collecting terms: derivation of the working model . . . . . . . . . . . . . . . . . . 159 N.8 Connection to experiment and reproducibility . . . . . . . . . . . . . . . . . . . . 160 O Fusion: Geodesic Overlap Criterion and Energy 161 O.1 Deterministic geodesic overlap: definition and match to the main text . . . . . . 161 O.2 From overlap to Veff and the transmission exponent . . . . . . . . . . . . . . . . . 162 O.3 Fusion energy from curvature concentration: definition, match, and numerics . . 162 O.4 Reproducibility, grids, and convergence . . . . . . . . . . . . . . . . . . . . . . . . 163 O.5 MATLAB:FusionRoutines ..............................164 © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 6 P TSRT Evaluation of Astrophysical S-Factors 166 P.1 From cross section to S-factor: definition and derivation . . . . . . . . . . . . . . 166 P.2 TSRT origin of the exponential and mapping to S(E)...............167 P.3 Asymptotics and the TSRT invariants Ξ1,Ξ2....................168 P.4 Calibration of the universal slopes (α, β).......................168 P.5 Numerical setup and link to the TSRT action . . . . . . . . . . . . . . . . . . . . 169 P.6 Representative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 P.7 Discussion........................................170 P.8 Reproducibility.....................................171 P.9 MATLAB: TSRT S-factor Script . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Q Deep Sub-Barrier Fusion Hindrance: TSRT Implementation and Validation 175 Q.1 Theory-to-Algorithm Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Q.2 Numerical Details and Determinism . . . . . . . . . . . . . . . . . . . . . . . . . 176 Q.3 MATLAB: Fusion Hindrance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 R Secondary Targets: TSRT Advantages Beyond Hindrance 180 R.1 Odd–Even Staggering (OES) Ranking . . . . . . . . . . . . . . . . . . . . . . . . 180 R.2 MATLAB: OES from Experimental Data . . . . . . . . . . . . . . . . . . . . . . 180 R.3 Electron–Screening in TSRT: Derivation, Data, and Code . . . . . . . . . . . . . 183 R.3.1 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 183 R.3.2 Derivation of the energy-dependent screening . . . . . . . . . . . . . . . . 183 R.3.3 Notation and Units for Screening Analysis . . . . . . . . . . . . . . . . . . 186 R.3.4 Sommerfeld parameter and TSRT enhancement ratio . . . . . . . . . . . . 187 R.3.5 Emergent trembling modulation and Bessel form . . . . . . . . . . . . . . 188 R.3.6 Data table (Pd host d(d, p)t) .........................190 R.3.7 MATLAB listing and workflow . . . . . . . . . . . . . . . . . . . . . . . . 190 S Environment, Determinism, and Validation Protocols 191 S.1 MATLAB Version and Toolboxes . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 S.2 DeterminismandSeeds ................................191 S.3 ValidationChecklist ..................................192 T Reproducibility: Figure and Table Generation 194 T.1 Environment, constants, and units . . . . . . . . . . . . . . . . . . . . . . . . . . 194 T.2 Masterbuild ......................................194 T.3 Artifactmap ......................................194 T.4 Odd–Even Staggering figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 T.5 Screening enhancement figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 T.6 Hindrancefigure ....................................195 T.7 Binding curves and tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 T.8 FusionQtable .....................................195 T.9 Fission comparison table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 T.10 S-factor slopes and system curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 T.11 Convergence plots (optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 T.12 Determinism, filenames, and audits . . . . . . . . . . . . . . . . . . . . . . . . . . 196 U Sensitivity and Uncertainty Analyses 197 U.1 ParameterScans ....................................197 U.2 MATLAB:ScanSkeleton ...............................198 © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 7 V Units, Conversions, and Sign-Convention Notes 200 V.1 UnitsandConversions.................................200 V.2 Physical Constants Used in Computations . . . . . . . . . . . . . . . . . . . . . . 201 V.3 Metric Sign and Reader Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 W Experimental Data Tables 203 W.1 Comprehensive Data Provenance Map (All Results) . . . . . . . . . . . . . . . . . 203 W.2 Binding Energies: Small Verbatim Excerpt (for Reproducibility) . . . . . . . . . . 204 W.3 Charge Radii: Small Verbatim Excerpt (for Reproducibility) . . . . . . . . . . . . 204 W.4 Sensitivity Summary (Representative) . . . . . . . . . . . . . . . . . . . . . . . . 204 W.5 Calibration Anchor: γ(E2) in 156Gd .........................205 W.6 Odd–Even Staggering in 235UFissionYields.....................205 W.7 Screened d(d, p)tS-factor(Pdhost)..........................205 W.8 Hindered Heavy–Ion Fusion: Derivation of the TSRT Asymptotic Form . . . . . . 205 X TSRT Parameters and Constants Used in All Figures and Calculations 208 Y MATLAB Procedures and Reproducibility Workflow 210 Y.1 MATLAB Code Index and Thematic Map . . . . . . . . . . . . . . . . . . . . . . 211 Y.2 Global Constants and Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Y.3 Convergence and Robustness Scans . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Y.4 Binding-Energy and Radius Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . 218 Y.5 Electron Screening in d(d, p)t(Pdhost) .......................223 Y.6 Heavy-Ion Fusion Hindrance (S-factor)........................225 Y.7 Odd–Even Staggering (OES) of 235U(nth,f)Yields..................231 Y.8 E2 Gamma Calibration and Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Y.9 Lifetime Pipeline and Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Y.10 Diagnostics and Validation Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . 240 Y.11 Stability Maps and Diagnostic Lines . . . . . . . . . . . . . . . . . . . . . . . . . 241 Y.12 Numerical Helpers (Quadrature) . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Y.13 Core Field Generators (support) . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Y.14 CSCT / Near-Field Electromagnetism Tables (optional branch) . . . . . . . . . . 254 Y.15 Fission Energetics and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Y.16 Fission Path Correlations (optional) . . . . . . . . . . . . . . . . . . . . . . . . . 274 Y.17 Parameter Scan Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Y.18 Master Orchestrator (Run-All) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 Index of MATLAB Files 348 © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 8 1 Introduction This work is dedicated to the luminous legacy of Sir Isaac Newton, Albert Einstein, and Lise Meitner, visionaries who, from states of solitude, confinement, and exile, discerned the fundamental principles that bind the cosmos, from the celestial to the nuclear.1 The stability of atomic nuclei has long been treated as an emergent statistical property of the strong and weak interactions.2In standard nuclear models, decay probabilities are introduced through barrier-penetration formalisms and stochastic transition rates. Within the Trembling Spacetime Relativity Theory (TSRT), however, decay is a deterministic geometric phenomenon: it represents the relaxation of curvature stress in a localized trembling-spacetime configuration. Unlike empirical relations such as Viola–Seaborg [10] or modern macroscopic–microscopic decay models [11], which typically deviate from experimental half-lives by 10−1–10−2in log10 scale, TSRT reproduces measured lifetimes across all decay channels with a mean logarithmic deviation below 10−6. This precision is achieved without any nucleus-specific fitting or channel-dependent parameters: each mode (β−,α, and spontaneous fission) follows from the same deterministic curvature dynamics. The contrast in predictive power marks a decisive improvement over existing lifetime systematics and validates TSRT as a fully quantitative, parameter-free description of nuclear decay. A direct verification of this absolute scaling is provided by thermal-neutron-induced fission of 235U, for which the TSRT pipeline yields ETSRT fiss = 170.028 MeV, in quantitative agreement with the experimental total energy release (170 ±3 MeV; Table 42 (p. 257)). This result 1The history of physics is often of clarity emerging from obscurity, and few embody this truth more profoundly than Newton, Einstein, and Meitner. Their greatest insights were not born in comfort but were forged in the crucible of personal and professional adversity, reminding us that the path to fundamental truth is often walked alone. Sir Isaac Newton †(1643–1727), following the death of his father and the estrangement of his mother’s remarriage, turned his solitude into a profound inward journey. From this seclusion, he shaped our understanding of the universe, laying the very foundations of physics and calculus [1,2]. Albert Einstein †(1879–1955), grappling with the silent grief of his daughter’s loss and the professional confinement of a patent clerk’s office, held steadfast to his pursuit of truth [3, 4]. There, he refined Newton’s legacy and uncovered the deep unity of mass and energy [5], a principle that would become a cornerstone of the nuclear age. Lise Meitner †(1878–1968), forced into exile from Nazi Germany and stripped of her position, carried the puzzle of nuclear fission in her mind. In a foreign country, with only theoretical tools, she provided the first correct interpretation of the process, calculating the immense energy released and giving physical reality to Einstein’s famous equation [6]. Together, their lives form a powerful narrative of insight wrested from adversity, proving that the deepest truths of nature are often revealed to those who persist against the current. As with the nuclear forces they helped unveil, first harnessed for destruction [7], the deeper lesson is not one of annihilation, but of balance and potential. From the fire of fission, we are guided toward life-sustaining applications [8] and the promise of fusion [9]. The power inherent in nature challenges humanity to choose wisdom over devastation. During her Berlin years, Meitner worked alongside James Franck †(1882–1964), in the closely knit physics community; through Franck, and the subsequent supervisory line descending via Egon A. Hiedemann †(1900– 1969), and Mack A. Breazeale †(1930–2009) to the present author, the academic genealogy of this work traces back to that same generation of atomic pioneers. Their experimental approach was passed on when Breazeale introduced the author to experimental physics in the early 2000s. It was in Breazeale’s laboratory that the author first encountered a remarkable optical lens, salvaged from a decommissioned U.S. spy plane—a relic of Cold War tensions over the very nuclear arsenal Meitner’s science had unlocked, now repurposed to sharpen the focus of peaceful inquiry. There, too, he first worked within a Faraday cage, its silence severing all cell phone and radio links; the eerie quiet was strangely familiar, a direct echo of his childhood, tuning his shortwave radio to the crackling voices from behind the Iron Curtain and wondering at the secret world they conveyed. Although the author’s own research has relied chiefly on experimental methods in engineering physics and ultrasonics, his enduring fascination has remained with theoretical physics, to which the present work belongs; in this respect, it also continues a second supervisory line, through Oswald J. Leroy †(1936–2022), in an academic genealogy that includes Henri Poincaré †(1854-1912). By further uncovering the secrets of the nucleus, this work seeks to honour their tradition: to pursue clarity where the way is narrow, and to trust that nature, when approached with resolve and ethical courage, reveals its coherence even from the deepest obscurity. 2For TSRT curvature-based stability criteria see Appendix C. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 9 follows automatically from the curvature-based energy mapping and the one-time scission-width calibration (Listings 44 (p. 297) and 29 (p. 263)), without any per-observable adjustment. Global liquid-drop fits, macroscopic–microscopic maps, and shell/DFT approaches reproduce many trends but remain probabilistic and parameter-heavy; on representative lifetime benchmarks their mean deviations are typically at the tenth-to-hundredth-of-a-decade level in log space. The TSRT law, calibrated once on a single weak anchor and a single scale per decay mode, achieves a mean absolute deviation of order one-millionth of a decade across β,α, and SF cases presented here, without per-nuclide adjustments. TSRT is a deterministic geometric framework in which spacetime carries a small, causal, locally correlated trembling of the metric. Concretely, dynamics proceed along future-directed timelike geodesics of gµν =ηµν +ξµν, where the deviation ξµν encodes the trembling field and respects proper-time causality (dτ2=gµνdxµdxν>0). Matter corresponds to localized trembling eigenmodes, and the familiar interactions arise as regimes of curvature: gravity from global curvature, electromagnetism from long-range oscillatory trembling, the strong interaction from curvature saturation under overlap, and the weak interaction from curvature reconfiguration when local symmetry fails to be maintained. Quantum-signature phenomena (spectra, blackbody law, interference, and entanglement) follow not from probabilistic postulates but from deterministic correlations among geodesic congruences. Observables are computed from curvature-based action and energy functionals with a single calibration that propagates across domains [12–17]. "While this work uses the framework of TSRT to propose a new unification of nuclear physics, it does so with the deep humility born from contemplating the monumental insights of the past century. We are but latecoming travelers on a path carved by giants, and our view is vast only because of the heights to which they carried us." This article applies TSRT’s causal trembling geometry to nuclei,3using the same structures defined in the foundational papers (metric decomposition, proper-time causality, emergent action scale, and corpuscular energy relations) [12–18]. Classical nuclear models (liquid drop, shell, mac–mic) are referred to only as benchmarks for comparison. The motivation for this work has both scientific and personal origins and develops the nuclear consequences of TSRT.4 Within the unified geometric TSRT picture, a single once-calibrated curvature strength, one lifetime anchor, and exact SI conversion factors connect TSRT directly to measured nuclear 3Disclaimer: This open fundamental research article develops Trembling Spacetime Relativity Theory (TSRT) at a fundamental and theoretical level. All results concern the mathematical and physical foundations of spacetime geometry and its explanatory scope for nuclear phenomena. The content is limited to general scientific theory and does not present, propose, or imply any technology or technological application. 4The author most humbly observes a parallel between the genesis of this work and that of Pablo Picasso’s Guernica. For Picasso in his Paris studio, the catalyst was the bombing of Guernica, an atrocity that transformed a formal commission for the 1937 World’s Fair into a raw, urgent act of witness. Similarly, for the author, a formative period of intellectual dissatisfaction during his studies at the Catholic University of Leuven, while engaging with the courses of KUL Professor J. Coussement and the foundational textbook of Ghent University Professor K. Heyde, revealed a field seemingly constrained by fragmented models and a lack of a coherent, unified theory. This initial disappointment, born of a student’s search for deeper answers that remained elusive, was later crystallized by a visit to Hiroshima in 2004. That catalytic shock transformed a general intellectual frustration into the specific, burning desire to understand the atomic nucleus that underpins this research. Just as Picasso could not have foreseen the legacy of Guernica, the author did not anticipate that this intellectual journey, born of both solemn witness and a resolve to address past uncertainties, would culminate in what he now regards as his most definitive contribution to the field. This work, therefore, is presented not merely as a finding, but as a personal scientific horizon, a representation of the limit of the author’s own capacity for a problem whose potential for deeper understanding remains, as always, infinite. The author hereby invites the physics community and, in particular, nuclear physicists, to continue this work and to apply it to specific topics and research problems in their fields. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 16 We adopt the (+,−,−,−)metric signature throughout.7Proper time and the trembling metric decomposition are given by dτ2=gµν dxµdxν, gµν =ηµν +ξµν, dτ2>0(timelike).(4) All global normalization and calibration constants appear once in Appendix A. These include the curvature–energy conversion constant CE(Equation (6)), the TSRT radius scale rTSRT 0 (Equation (14)), and the global lifetime scale Cλ(Equation (2)). No constant is introduced locally in any section without cross-reference to this master list. Unless stated otherwise, the speed of light cis retained explicitly to maintain dimensional transparency. Numerical work follows a consistent unit system based on MeV and MeV/c2 (1 u c2= 931.49410242 MeV). For compactness, accompanying expressions in natural units (c= 1) may be shown alongside full SI/relativistic forms; these are provided purely for algebraic clarity and are never used for numerical substitution. Table 7: Master notation employed throughout the manuscript. Topic-specific refinements appear in Table 19 (p. 113). All expressions use the metric signature (+,−,−,−). Complete unit definitions: Appendix V. Symbol Meaning Z, N, A Proton number, neutron number, mass number (A=Z+N) τProper time (monotonic in TSRT once causal orientation is fixed) uµFour-velocity along a (future-directed) geodesic Kµν Trembling-curvature (curvature-stress) tensor used in TSRT energy/action functionals hK2iLocal curvature variance (TSRT) ∆K2(Z, N)Curvature-suppression measure (stability diagnostic) SiTrembling action of an isolated nucleon eigenmode Sint Leading overlap (curvature-saturation) contribution to the action δScorr Higher-order interference corrections (multipole/phase) RcCurvature/overlap (locking) scale; effective strong-range indicator Asat Saturation amplitude of trembling modes at stability threshold ℓEM, ℓCElectromagnetic / color trembling coherence lengths λBRelative de Broglie scale recovered by TSRT action thresholds κneck Local scission-neck curvature (controls OES amplitude) ∆(3)(Z)Three-point odd–even staggering indicator on charge yields ∆Seven–odd Proper-time action gap between neighboring even/odd fragment geodesics O[ΞA,ΞB]Overlap functional between trembling configurations ΞA,ΞB Ocrit Critical overlap threshold for fusion onset R⋆Separation at which O=Ocrit ∆UeTSRT geometric screening shift (deterministic EM trembling polarization) ηSommerfeld parameter Z1Z2e2/(~v) ~TSRT TSRT action unit (equals ~in the low-curvature coarse-grained limit) 3 The Genesis of Spacetime and Elementary Particles in TSRT TSRT advances a single, deterministic geometric framework from which spacetime, particles, and interactions emerge coherently [12–18]. The starting point is a causally oriented metric with a small, local, deterministic “trembling”: once a time orientation is fixed, proper time τ increases monotonically along future-directed timelike geodesics. On cosmological scales, the coarse-grained imprint of these local correlations yields an effective global flow that looks like Hubble expansion; the same mechanism that stabilizes microscopic modes thus sets the largescale structure of the observed 4D spacetime [16]. In this view, “dark” phenomenology reflects geometric bookkeeping of trembling curvature rather than additional matter sectors. 7When comparing with sources using (−,+,+,+), adjust signs in line elements, curvature scalars, and action densities accordingly. TSRT’s use of proper-time causality follows [12]. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 17 Elementary particles are modeled not as point objects but as localized eigenmodes of trembling curvature. Their mass, spin, and charges are geometric invariants of the mode’s structure; families arise from a finite catalog of admissible topologies and symmetries [12,17]. Composite systems (atoms and nuclei) are bound configurations of such modes whose stability is diagnosed by reductions in appropriate curvature measures (see Section 4). Photons correspond to propagating electromagnetic trembling modes [13,18]; neutrinos appear as neutral geodesic fragments permitted by curvature reconfiguration constraints [12,15]. Within this geometric picture the four interactions are not postulates but regimes of the same trembling curvature. Gravitation is global curvature accumulation (long-range coherence); electromagnetism is long-range oscillatory correlation carried by curvature waves (photons) [13]; the strong interaction is curvature saturation, where local overlap of nucleonic modes suppresses curvature variance and locks a tightly bound configuration; and the weak interaction is reconfiguration, where locking symmetries fail and the system relaxes deterministically along causal geodesics, exporting charge and lepton number via escaping fragments. Phenomena often split between “quantum” and “relativistic” domains are thereby unified: superposition and tunneling postulates are replaced by causal geodesic correlations and action-minimizing reconfigurations [12,15,18]. Mass–energy and binding follow from the same curvature accounting. The effective inertial mass of a mode is a functional of its trembling curvature, and the binding energy of a composite is the reduction of that measure when isolated modes lock coherently. For nuclei, the empirical mass–energy defect is recovered as the difference between the sum of isolated nucleon eigenmasses and the mass of the bound configuration (Equation (9) in Section 4). The global systematics of nuclear binding used later are summarized by B(A, Z) = Bgeom(A, Z) + ∆Bshell(A, Z) + ∆Bpair(A, Z),(5) introduced formally in Section 5 (Equation (10)). Here Bgeom captures saturation from curvature locking, while shell8and pairing9terms arise from symmetry/phase corrections to the underlying trembling modes. For orientation, we write Afor mass number and Bfor total nuclear binding energy; B/A is the binding per nucleon. In both data and TSRT, B/A peaks in the iron region, A≃56–62 (Section 5).10 Geometrically, this maximum is the balance point where short-range curvature saturation is strongest while long-range Coulomb curvature remains modest; below it the assembly is under-saturated, and above it surface and Coulomb costs erode the gain. 8In conventional nuclear structure, a “shell” denotes a set of single-particle orbitals in a mean-field potential, grouped by quantum numbers (e.g. nℓj) and separated by energy gaps. Magic numbers arise where large gaps yield enhanced stability. In TSRT, a “shell” corresponds instead to a family of localized trembling eigenmodes whose curvature patterns maintain coherence under saturation. The TSRT analogue of a shell gap is a geometric gap: a discrete jump in the curvature-locking pattern that locally suppresses mode rearrangement. Both viewpoints identify special nucleon numbers with enhanced stability, but TSRT attributes the effect to curvature topology rather than quantized orbitals. The numerical shell corrections in Equation (5) therefore map directly onto geometric coherence thresholds rather than quantum level spacings. 9In conventional theories, “Pairing” refers to the empirical tendency of like nucleons (pp or nn) to form correlated J= 0 spin-singlet pairs, lowering the energy of even–even nuclei and producing odd–even staggering in masses and yields. In TSRT, pairing reflects a curvature-coherence effect: when two like trembling modes overlap with opposite spatial/phase orientation, their local curvature stresses partially cancel, reducing the effective action. Even–even nuclei minimize curvature stress geometrically; odd–Aand odd–odd systems retain uncompensated curvature. The phenomenology (enhanced stability of even–even nuclei, staggering with parity of Zand N) is shared in both views, but TSRT explains it as a deterministic geometric cancellation rather than a quantum-correlation energy. Thus ∆Bpair in Equation (5) plays the same role numerically while having a curvature-based origin. 10When the text mentions “B/A @A≈60” or colloquially “B/A = 60,” it is shorthand for the peak of B/A occurs near mass number A≈60 (iron group). Numerically, B/A at the peak is ∼8–9MeV per nucleon in conventional units. If TSRT normalized units are used elsewhere, the conversion is given in Appendix A together with the normalization constant. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 18 The channels most relevant for this paper follow immediately. Protons and neutrons are the nucleonic trembling eigenmodes whose overlap builds nuclei; photons encode released electromagnetic curvature in radiative relaxations; and electrons with (anti)neutrinos are the geodesic fragments produced in weak reconfigurations during βprocesses. In what follows we use this dictionary consistently: strong saturation and the effective curvature potential (Section 4); deformation, barriers, and fission (Section 6 and Section 10); and the fusion program, barriers, proper-time transmission, and near-field screening (Section 7 and Section 12). This brief recap provides the conceptual bridge from first principles to the nuclear results developed in the remainder of the paper. 4 TSRT Nuclear Foundations: Effective Curvature Potential and Motion Consistency with implementation: The effective curvature energy and locking constructions described here are implemented by tsrtnucleonfield.m,tsrtcompositefield.m, tsrtcurvaturetensor.m, and tsrtenergyfromcurvature.m. No fitted, per-nucleus potential is introduced in code; all terms originate from the TSRT functionals summarized in Appendix H and constants in Appendix E. Supporting appendices: Appendix C (definitions and locking), Appendix H (energy functionals), Appendix F (schemes), and Appendix E (calibrations). Core MATLAB: tsrtnucleonfield.m,tsrtcompositefield.m,tsrtcurvaturetensor.m, tsrtenergyfromcurvature.m. In TSRT,11 every fundamental particle is described as a localized trembling eigenmode of spacetime curvature12 [12,17]. Within this framework, trembling spacetime enforces causality by prohibiting backward evolution of τ.13 The four fundamental interactions then emerge not as external postulates but as distinct geometric constraints of trembling curvature: electromagnetism as long-range oscillatory correlation, gravitation as global curvature accumulation, and the strong and weak interactions as short-range curvature-binding and curvature-reconfiguration mechanisms [12]. From this perspective, an atomic nucleus is not a collection of independent nucleons bound by an externally defined potential.14 Instead, it is a self-consistent geometric configuration of trembling modes in which the local curvature fields of protons and neutrons interlock to form a stable causal structure. The strong interaction emerges as a direct consequence of curvature 11In TSRT: (1) Particles are localized trembling eigenmodes of spacetime curvature, respecting the (+,−,−,−) metric and causal orientation fixed throughout; (2) nuclear binding emerges when curvature–overlap saturates a TSRT locking criterion (Appendix C, X); (3) weak processes are deterministic curvature reconfigurations along proper time, not stochastic transitions; (4) a single calibration constant (Appendix A) sets absolute scales and is never re-fit per nucleus or channel. All derivations follow these assumptions and are cross-referenced to the appendices for full reproducibility. 12Throughout this work we adopt the (+,−,−,−)metric signature. This choice is purely conventional: one could equally well use (−,+,+,+), provided all definitions are adjusted consistently. The two signatures yield identical physical predictions, differing only in the sign patterns that appear in the mathematical expressions. In either convention, proper time τadvances monotonically along future-directed timelike geodesics once a causal orientation is fixed. What the (+,−,−,−)convention makes explicit is that τremains positive along causal evolution, while in the opposite convention the same physical behavior appears with inverted algebraic signs. Thus the metric choice does not alter the underlying physics, but only the symbolic form by which we represent it. 13Regardless of signature in Equation (4). 14The locking criterion and curvature-overlap mechanics are reviewed in Appendix C and Appendix X © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 19 saturation:15 when two or more nucleonic trembling modes are brought within a critical proximity, their local oscillatory geometries overlap, leading to a collective minimum in the effective trembling action. This geometric resonance enforces a tightly bound configuration analogous to a potential well, but fundamentally deterministic and metric-based. The weak interaction, in turn, arises from permissible reconfigurations of trembling geodesics when curvature symmetry conditions are not perfectly met.16 For example, beta decay corresponds in TSRT to a local trembling instability in which a nucleonic mode relaxes by emitting a causal geodesic fragment, identified at macroscopic scales as an electron or neutrino [15]. Thus, nuclear stability is regulated not by probabilistic tunneling, but by the deterministic geometry of curvature-bounded trembling paths. We will repeatedly use the curvature-energy functional E[Ξ] = CEZV Kµν(Ξ) uµuνdV, (6) with CEfixed by the calibration protocol in Appendix A. The mass–energy link we employ later (e.g., in Equation (108)) is given in Section 11.2, Equation (119). The tensor Kµν and its scalar norm are defined in Appendix C (Units and definitions). 4.1 Single-nucleon motion in a trembling nuclear field We model single-nucleon dynamics by the trembling-deformed geodesic equation d2xµ dτ2+ Γµ αβη+ξnucldxα dτ dxβ dτ = 0,(7) where xµ(τ)denotes the spacetime worldline of the nucleon parameterized by its proper time τ, and Γµ αβ[η+ξnucl]are the connection coefficients associated with the local metric gµν =ηµν + ξnucl,µν.17 Here, ηµν represents the background Minkowski metric and ξnucl,µν the trembling deviation field generated by the collective nuclear curvature. The field ξnucl is constrained by proper-time causality and energy–momentum conservation, following the same geometric postulates as in References [12,14]. Equation (7) expresses the causal motion of a nucleon as a free geodesic in a locally deformed spacetime whose curvature oscillates at the trembling frequency determined by the nuclear configuration. Physically, the term Γµ αβ[η+ξnucl]acts as an effective internal force describing how the nucleon’s worldline responds to local curvature oscillations rather than to external potentials. The resulting effective potential Veff therefore emerges geometrically from the averaged curvature field and naturally incorporates both bulk curvature (governing binding) and local shell–curvature corrections (governing level structure). These contributions are derived explicitly in Appendix A. Conceptually, Equation (7) extends the standard geodesic equation of General Relativity by replacing the purely gravitational metric perturbation hµν with a deterministic, multiscale trem15Quantitative expressions and parameter definitions appear in Appendix X. 16The geometric decay-rate diagnostic is derived in Appendix A. 17The bracket [η+ξnucl ]does not denote an index contraction or summation. It indicates that the Christoffel symbols Γµ αβ are computed from the metric gµν =ηµν +ξnucl,µν . In fully explicit form, Γµ αβ[η+ξnucl] = 1 2gµλ (∂αgβλ +∂βgαλ −∂λgαβ), gµν =ηµν +ξnucl,µν .(8) The only implied summations are the standard Einstein sums over repeated upper/lower indices (e.g. over λ above). Thus the bracket is simply a functional argument—“the connection built from this metric”—and introduces no additional summation or contraction beyond the usual GR notation. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 20 bling field ξnucl,µν that couples to all forms of energy, not only mass–energy.18 This tremblingdeformed form first appeared in its covariant form in Reference [12], where it described elementary corpuscle motion in vacuum curvature, and was later adapted to atomic curvature potentials in Reference [14]. In the nuclear context, the same geometric principle yields the internal binding mechanism, allowing nucleon motion, stability, and decay processes to be derived from curvature dynamics rather than from phenomenological nuclear forces. In this geometric picture, nuclei are bound states of curvature. Each proton and neutron is a localized trembling eigenmode of the spacetime metric with a characteristic amplitude and phase structure. A nucleus is a stationary, causally locked arrangement of such modes: the near-field curvature patterns interlock by resonance so that phase mismatches are minimized and the total trembling action is reduced. Binding energy is not an externally imposed potential, but the decrease in local curvature variance when isolated modes reorganize into a joint configuration (formalized later in Equation (59)). This connects directly to the mass–energy defect observed [36] experimentally: ∆E=Zmp+Nmn−Mnucleusc2,(9) where Zand Nare the numbers of protons and neutrons, mpand mntheir individual trembling eigenmasses, and Mnucleus the collective bound-state mass.19 In TSRT, ∆Eis precisely the curvature-suppressed trembling energy released when free nucleonic eigenmodes are fused into a causal nuclear geometry. This interpretation provides a natural resolution to the duality of stability and instability in nuclei. Stable nuclei correspond to configurations where trembling resonance yields a local minimum in the geometric action, preventing any further reconfiguration without external perturbation. Unstable nuclei correspond to curvature arrangements where additional trembling modes can be shed or rearranged to lower the total action, producing fission or decay pathways. Thus, the nucleus is not an exception to atomic determinism but an extension of it: just as electron orbits in TSRT arise from curvature-guided trembling geodesics [14], nuclear configurations arise from curvature-bound assemblies of nucleonic trembling modes. This geometric interpretation prepares the groundwork for a causal and quantitative analysis of nuclear stability, fission, and fusion in the following sections. The same trembling-spacetime formalism that governs atomic emission and blackbody radiation also determines nuclear decay. Once the local curvature derivatives ˙ ∆K2are known, they fix the emission rate λwithout additional postulates. This confirms that TSRT provides a unified geometric foundation for both atomic and nuclear temporal behavior. For practical calculations, curvature measures must be converted into physical energies. The calibration procedure, including the definition of the normalization constant Cnorm and its fixing relative to a reference nucleus (56Fe), is described in Appendix A. The resulting constants and parameter values are summarized in Appendix E, and the full numerical tables used in the comparisons are collected in Appendix B. 18In General Relativity, motion is governed by d2xµ dτ2+ Γµ αβ[g]dxα dτ dxβ dτ = 0 with gµν =ηµν +hµν , where hµν represents a weak gravitational perturbation. In TSRT, the analogous perturbation ξµν originates from causal trembling of spacetime itself, as developed in Section III of Reference [12] and extended to atomic bound states in Section II of Reference [14]. The present nuclear form ξnucl follows the same principle but with curvature amplitudes enhanced by the collective mass density and short-range correlation of nucleons. 19In TSRT, “mass” denotes the effective inertial mass obtained from the curvature–energy functional, meff ∝ RVKµν uµuνdV (Equation (119); see Section 11.2 and [12, 13]). For an isolated proton or neutron, mpor mnis the value of the same functional on the single trembling eigenmode with the appropriate asymptotic boundary conditions. For the assembled nucleus, the collective bound-state mass Mnucleus is the value of the same functional on the causally locked multi-mode configuration, including both strong-mode saturation and electromagnetic contributions. Because curvature saturation suppresses the local variance hK2iin the bound geometry, one has Mnucleus < Zmp+Nmn; the corresponding energy difference ∆Ein Equation (9) is the measured mass defect [36]. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 21 5 Binding Energy Systematics and Radii from Trembling Geometry Consistency with implementation: Binding energies and radii follow from the curvature energy functional (Appendix H), evaluated on fields built by tsrtnucleonfield.m/tsrtcompositefield.m; the numerical assembly and outputs are produced by tsrtenergyfromcurvature.m and compared to evaluated data via Appendix B. Supporting appendices: Appendix L (benchmarks), Appendix H (binding functional), Appendix B (tables). Core MATLAB: tsrtenergyfromcurvature.m,tsrt_stability_lines.m, tsrt_get_exp_half_life.m. A concise set of anchor binding comparisons is presented early in Table 3 (p. 11); the corresponding AME overlay and isotopic trends are shown in the left panel of Figure 1 (p. 10). This section develops the TSRT analogue of the semi-empirical mass formula, where binding is not postulated but derived from curvature-saturation and interference. The total binding energy (Equation (10)) splits naturally into a geometric bulk term, shell contributions, and pairing interference. We show that this geometric law reproduces the global B/A saturation curve and the local shell-driven kinks, in parallel to what liquid-drop+shell models achieve phenomenologically. 5.1 Saturation and the B/A curve One of the earliest empirical discoveries in nuclear physics is that the binding energy per nucleon, B/A, rises rapidly with mass number Aup to the iron region and then saturates near ∼8MeV.20 Traditional liquid–drop models reproduce this behavior through volume and surface terms, supplemented by phenomenological shell and pairing corrections.21 Within TSRT, the same pattern follows directly from the geometry of trembling curvature fields. When nucleonic trembling eigenmodes overlap, their local curvature fields interlock into a joint configuration. The reduction in curvature variance per added nucleon diminishes once a critical density is reached, leading to saturation without the need for external parametrization. The master relation is written compactly as: B(A, Z) = Bgeom(A, Z) + ∆Bshell(A, Z) + ∆Bpair(A, Z),(10) 20Here Bis the total nuclear binding energy and A=Z+Nis the mass number; B/A measures binding per nucleon, a standard indicator of average nuclear stability. 21The traditional liquid–drop model treats the nucleus as an incompressible charged fluid droplet, assuming that each nucleon interacts predominantly with its nearest neighbors. Its binding energy is expressed as a semiempirical mass formula B(A, Z) = aVA−aSA2/3−aCZ(Z−1)A−1/3−aA(A−2Z)2/A +δ(A, Z), where each coefficient aiis adjusted phenomenologically to fit data rather than derived from first principles. Key assumptions include: (1) nucleons are homogeneously distributed within a spherical volume of constant density; (2) nuclear forces are short-ranged and saturating, mimicking molecular cohesion; (3) surface energy reduces binding near the nuclear boundary analogously to classical surface tension; (4) electrostatic repulsion is described by a uniformly charged sphere; (5) asymmetry energy penalizes proton–neutron imbalance according to Fermi-gas statistics; and (6) pairing terms account empirically for even–even, odd–odd, and odd–even differences. Although this model captures gross binding-energy trends, it lacks a geometric or dynamical foundation: each correction term represents a separate empirical adjustment rather than a consequence of a unified principle. Consequently, its descriptive success comes at the cost of theoretical fragmentation, underscoring the need for a single causal framework capable of explaining both large-scale and small-scale structure formation without resorting to ad-hoc parametrization. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 22 where Bgeom represents the curvature-driven bulk binding, ∆Bshell encodes corrections from geometric resonance at closed trembling shells, and ∆Bpair reflects the enhanced stability of curvature-symmetric even–even systems. The geometric term Bgeom saturates for a simple reason rooted in TSRT’s finite curvature– overlap range. Let Rbe the nuclear radius (Section 5.2) and Rcthe curvature–overlap (locking) scale from Section 4. In the bulk (r≪R) each nucleonic trembling mode finds essentially the same number of neighbors within Rc, so the curvature-suppression gained by adding one more nucleon is approximately constant. Near the surface, however, a fraction of would-be neighbors lies outside the nucleus; those “missing neighbors” reduce the incremental suppression. This yields the volume–minus–surface structure for the geometric binding: Bgeom(A, Z)≃Cnormhαvol A−αsurf A2/3i−CEM Z2 R,(11) with R=rTSRT 0A1/3from Equation (14) (see further). Dividing by Agives the leading TSRT envelope for the binding per nucleon: B A(A, Z)≃a0−asA−1/3−aC Z2 A4/3+∆Bshell A+∆Bpair A,(12) where a0=Cnormαvol,as=Cnormαsurf, and aC=CEM/rTSRT 0. The shell and pairing terms in Equation (10) ride on this envelope: ∆Bshell produces magic-number kinks, and ∆Bpair gives odd–even staggering (cf. Figure 3 (p. 24) in Section 6). To see why the maximum of B/A occurs near iron, set κ≡Z/A along the valley of stability22 (nearly constant in medium-mass nuclei) and maximize the smooth envelope of Equation (12) with respect to Aat fixed κ. Ignoring the small A–dependence of the asymmetry term in this narrow region, one finds d dA B Aκ = 0 =⇒A⋆≃as 2aCκ2.(13) With the single calibration on 56Fe in Appendix A fixing the ratio as/aC(and with κ≃Z/A ≃ 0.46–0.50 along the stability valley), Equation (13) yields A⋆≈56–60. Thus, in TSRT the peak near A∼60 is a direct geometric consequence of (i) bulk curvature locking (volume term), (ii) missing-neighbor penalty (surface term), and (iii) the long-range electromagnetic curvature cost which grows with Z2/R. Bis the total binding energy B(A, Z);A=Z+Nis the mass number; the envelope a0− asA−1/3−aCZ2/A4/3encodes the deterministic curvature balance. Shell and pairing corrections then produce the observed local structure on top of this global trend. The experimental signatures of this structure are well known: local kinks at magic numbers23 and odd–odd–even staggering.24 Within the liquid–drop framework, these features are accommodated not by the base volume/surface terms but by empirically fitted add-ons, i.e., shell corrections for magic-number kinks and a pairing term for odd–even staggering, whose roles (and TSRT reinterpretation) are detailed in Section 6 (Equation (10)). In TSRT these are not ad hoc add-ons but follow from the deterministic geodesic–correlation rules. A detailed benchmark appears in Section 6, Figure 3 (p. 24), where the parity–Bessel two-packet model derived from trembling geometry quantitatively reproduces the staggering observed in 235U fission yields. For representative numbers and how the global trend compares to data, see Table 3 (p. 11) (with the full, table in Appendix B, Table 25 (p. 147)). 22The valley of stability is the locus of nuclides stable (or longest-lived) against radioactive decay in the N–Z plane. Away from this valley, isotopes tend to undergo β±decay (or other modes) to reduce their energy. 23Magic numbers are specific proton or neutron counts (2,8,20,28,50,82,126, . . .) at which large shell gaps yield extra stability. They produce visible “kinks” in separation energies, radii, and other systematics. 24Odd–even staggering (OES) is the alternating pattern in nuclear observables (masses, fragment yields) where even-Z/even-Nnuclei are favored over odd neighbors, commonly attributed to nucleon pairing correlations. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 23 The liquid-drop model (LDM) [23,37] famously fits B(A, Z)with a volume – surface Coulomb – asymmetry pairing decomposition. TSRT obtains a closely related structure from first principles: the volume term arises from bulk curvature locking (coherent overlap of trembling modes); the surface term from missing neighbors within the finite overlap radius Rc; and the Coulomb cost from long-range electromagnetic curvature scaling like Z2/R with R=rTSRT 0A1/3(Section 5.2). These ingredients produce the TSRT envelope in Equation (12), which explains, without phenomenological saturation constants, why B/A peaks near A≃60 (Equation (13)) and then levels off. LDM gets the right shape because its fitted terms mimic the same geometric scalings that TSRT derives deterministically from curvature integrals. But LDM needs distinct fitted coefficients and additional patches (e.g., shell and pairing terms) to capture kinks and odd–even effects, whereas in TSRT: (i) magic-number kinks arise from discrete changes in geodesic resonance structure (the shell–curvature corrections ∆Bshell in Equation (10)), and (ii) odd–even staggering follows from causal phase-matching of trembling modes (Section 6, Figure 3 (p. 24)), not from a purely statistical pairing ansatz. Likewise, phenomena that strain LDM’s barriertunneling picture, such as deep sub-barrier fusion hindrance, are naturally accommodated by TSRT’s curvature-based barrier and proper-time transmission (Section 7, Figure 6 (p. 86)). In short, LDM’s successes are explained in TSRT as consequences of finite-range curvature locking plus long-range Coulomb curvature, while TSRT additionally provides the causal mechanism and predictive corrections that LDM must fit separately. 5.2 Charge radii and scaling Charge radii provide an independent probe of nuclear structure by quantifying the spatial extent of the nuclear charge distribution. Operationally, the (root-mean-square) charge radius rch is defined from the second moment of the charge density ρch(r)as rch =phr2iand is extracted from elastic electron scattering, isotope-shift spectroscopy, and muonic-atom spectroscopy.25 Empirically, radii follow an approximate saturation law R≃r0A1/3(with mild departures near shell closures,26 along deformation chains, and across isotopic skins). In TSRT, this A1/3scaling is not imposed but emerges from the causal packing of trembling eigenmodes in a finite curvature domain: the leading volume contribution fixes the A1/3trend, while subleading shell–curvature and deformation terms account for systematic deviations (derived in Appendix A). The TSRT prediction for charge radii at leading order is the TSRT radius law R(A) = rTSRT 0A1/31 + δdef(A),(14) where this form is derived (not assumed) by minimizing the trembling action for a finite curvature domain containing Anucleons; the A1/3scaling follows from the volume part of the curvature integral, while subleading surface/shell effects appear as multiplicative corrections (see Appendix A). The constant rTSRT 0is the fundamental length scale fixed by the normalization of nuclear curvature and therefore determined once the TSRT constants are specified (Appendix E); it is not an adjustable LDM-style fit parameter. The factor δdef (A)encodes small, nucleus-dependent departures from sphericity and local shell–curvature effects, vanishing near spherical closed shells, changing sign with prolate/oblate deformation, and scaling like a surface correction at large A. In what follows we use Equation (14) as the working form, with 25For clarity: the charge radius differs from the matter radius (which weights all nucleons) and from the point-proton radius rpp (the proton centers-of-mass distribution). Standard extractions relate these via r2 ch = r2 pp +r2 p+ (N/Z)r2 n+r2 DF +r2 so +···, where r2 pand r2 naccount for intrinsic nucleon charge structure, and rDF, rso denote small Darwin–Foldy and spin–orbit corrections, respectively. We keep the presentation in terms of rch to align with experimental systematics; details of how these corrections enter TSRT fits are deferred to Appendix A. 26Shell closure denotes a filled major shell for protons or neutrons (at a magic number), typically producing reduced collectivity, smaller radii changes, and enhanced stability. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 24 35 40 45 50 55 60 65 Z -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Odd-Even Staggering Experiment TSRT Figure 3: Odd–even staggering in 235U(nth,f). The vertical axis shows the third finite difference ∆(3)ln Y(Z) = ln Y(Z+1) −2 ln Y(Z) + ln Y(Z−1), quantifying the alternating pattern of logarithmic fission yields Y(Z)as a function of fragment charge Z(horizontal axis). Points denote experimental values from Table 17 (p. 110); the solid curve represents the TSRT parity–Bessel two-packet model with localized neck bias [Equation (315)], as discussed in Section 10.7. All arrays and MATLAB code used to generate the figure are provided in Appendix R.2. Model derivation: Section 10.7; parameter values in Table 41 (p. 209); numerical workflow in Appendix R.2. explicit expressions for δdef (A)and the determination of rTSRT 0provided in Appendix A and Appendix E, respectively. Unlike traditional models that assign r0empirically, TSRT relates rTSRT 0directly to the curvature amplitude of a single nucleon eigenmode and calibrates it deterministically against a reference nucleus such as 56Fe. This establishes the nuclear length scale as a measurable manifestation of spacetime curvature rather than an adjustable geometric constant, thereby grounding the radius law in first principles of the theory. Furthermore, the same trembling-induced polarization mechanism that modifies charge radii also governs electron screening in low-energy nuclear fusion. In TSRT, polarization denotes the causal displacement of local curvature modes relative to their mean geodesic position due to the presence of neighboring mass–energy oscillations. This produces an induced curvature dipole field that shifts charge distributions and effective potentials, in exact analogy to how electric polarization arises from charge displacement in a dielectric medium, but here the effect acts on the spacetime metric itself rather than on an external field. In nuclei, such curvature polarization leads to minute radius modifications; in condensed matter environments, it governs the enhancement of tunneling probabilities by reducing the effective Coulomb barrier. This causal connection is demonstrated in Section 7, Figure 2 (p. 12), where TSRT polarization with trembling and dynamic dephasing successfully reproduces experimental screening enhancements © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 25 in Pd-host d(d,p)t reactions.27 In summary, both binding-energy saturation and charge-radius scaling emerge in TSRT as direct consequences of causal trembling geometry. What appear in phenomenological models as separate empirical terms (volume, surface, asymmetry, pairing, r0constants) are here unified as manifestations of a single geometric principle: the deterministic organization of nucleonic trembling modes in curved spacetime. Full numerical tables supporting this section are provided in Appendix B, with calibration settings and code entry points documented in Appendix A. Representative charge radii and binding residuals are summarized in Table 12 (p. 108) (Appendix B), and polarization/electronscreening comparisons are shown in Figure 2 (p. 12) (with code listings in Appendix R.3). 6 Deformation Landscapes and Fission Barriers Consistency with implementation: Barrier curves and deformation energetics are computed by tsrtfissionenergy.m (energy surfaces) and tsrtfissionactionfull.m (action evaluation), with OES neck-mode details in Appendix N. Parent/fragment geometry assembly uses tsrt_build_parent_U235.m, and tsrt_build_daughters_scission.m. Supporting appendices: Appendix M (surface curvature and bifurcation), Appendix N (OES neck mode), Appendix F (discretization). Core MATLAB: tsrtfissionenergy.m,tsrtfissionactionfull.m. The consolidated U-235(n,f) observables table is placed early as Table 4 (p. 12) for quick reference; the OES figure appears in Figure 3 (p. 24). 6.1 Geometric deformation coordinate and barrier formation In conventional nuclear models, deformation is introduced phenomenologically by parameterizing the nuclear surface R(θ, φ)with multipole expansions, most commonly using the parameterization developed by Bohr and Mottelson [38]: R(θ, φ) = R0 1 + ∞ X λ=0 +λ X µ=−λ αλµYλµ(θ, φ) ,(15) where R0is the radius of the spherical nucleus, αλµ are the deformation parameters,28 and Yλµ are spherical harmonics. The quadrupole deformation parameters29 (α2µ), which describe ellipsoidal shapes, are of paramount importance for describing nuclear ground states and fission pathways. 27The notation “Pd-host d(d,p)t” refers to deuterium–deuterium fusion reactions occurring within a palladium host lattice, where two deuterons (d) fuse to form a triton (t) and a proton (p). Such reactions are studied in low-energy condensed-matter fusion experiments to investigate screening effects: the surrounding electron cloud and lattice potential effectively lower the Coulomb barrier between the reacting nuclei. In the present paper, the TSRT-based treatment of this system appears in Section 7, where the same trembling-induced curvature polarization mechanism that governs charge-radius corrections is applied to quantify the observed enhancement in reaction rates. 28Deformation parameters αλµ quantify deviations from a sphere via spherical harmonics Yλµ:λ= 2 (quadrupole) controls elongation/flattening, λ= 3 (octupole) pear-shapes, λ= 4 hexadecapole, etc. 29Quadrupole deformation (often summarized by β2) describes ellipsoidal shapes: prolate (rugby-ball, β2>0) or oblate (disk-like, β2<0). It dominantly controls low-lying collective spectra and fission pathways. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 32 Evaluation pipeline and results. For each reaction system the calculation proceeds as follows: 1. Construct Cβ(β)and Mβfrom the TSRT deformation geometry (Section 6.1; code listings in Appendix F.3). 2. Evaluate the proper-time action STSRT(E)over the classically forbidden region for an energy grid Ei. 3. Compute the transparency TTSRT(Ei)via Equation (34). 4. Form the fusion cross section σfus(Ei)(Equation (36)) and S-factor STSRT(Ei)(Equation (35)). 5. Save predicted S(E)and generate reaction-specific plots. This is precisely the computational path implemented in Appendix Q.3 and executed by the scripts used for Table 27 (p. 169) and Figure 6 (p. 86). The latter shows the predicted hindrance pattern—including the late-onset downward curvature—emerging naturally from the TSRT proper-time geometry, without the adjustable energy-shift parameters or empirical hindrance thresholds common in quantum-mechanical fits. Observations. TSRT therefore provides a fully deterministic, geometry-based calculation of fusion rates and S-factors. All entrance-channel suppression arises from proper-time curvature integrals, and the observed hindrance systematics follow automatically from the causal structure of the trembling nuclear fields. No phenomenological barrier-reshaping or reaction-dependent tuning is introduced; the same calibrated curvature constants apply across all systems evaluated in this work. 7.2 Curvature-enhanced Coulomb barrier In TSRT, the electromagnetic sector is not an externally postulated field but the long-range manifestation of trembling curvature itself. As established in the foundational article [12] (see Sections III.B–III.C therein), the decomposition of the full trembling curvature tensor Kµν into symmetric and antisymmetric parts produces two complementary sectors: a symmetric (spin-2– like) component governing the strong channel and a trace-free antisymmetric component governing the long-range U(1)-like curvature dynamics that appear macroscopically as electromagnetism. The electromagnetic potential Aµin TSRT thus corresponds to the coarse-grained limit of the local trembling displacement field ξµ, with the field tensor Fµν =∂µAν−∂νAµ←→ 2K[µν],(37) where K[µν]is the antisymmetric component of the trembling curvature tensor. In the weakcurvature (far-field) limit, where geodesic deviations are small and higher-order curvature coupling terms vanish, the governing equations for K[µν]reduce exactly to Maxwell-type field equations derived from the same curvature functional used for the strong channel. This limit reproduces the classical Maxwell equations in vacuum and, for static spherically symmetric sources, yields the familiar Coulomb form of the potential. Consequently, for two well-separated nuclei of charges Z1eand Z2e, the baseline interaction energy emerging from the far-field limit of the TSRT electromagnetic sector is VC(r) = Z1Z2e2 r,(38) © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 33 which corresponds to the leading-order term of the general TSRT electromagnetic potential derived in Reference [12], Section III.D, and reiterated in the atomic-structure formulation [14], Section 7.3, where the same curvature origin of the Coulomb law underlies electron–nucleus binding. At shorter ranges, where the nuclear trembling fields begin to overlap, the effective interaction acquires deterministic curvature corrections that have no counterpart in classical electrodynamics: (i) Curvature polarization of the nuclear surfaces, arising from induced alignment of local trembling modes at the nuclear boundaries, leading to a modification of the surfacecurvature term ∆K2 surf in Equation (99); (ii) Curvature focusing/defocusing, a short-range strong–electromagnetic coupling that alters the entrance-channel potential landscape, leading to explicit corrections in the TSRT effective potential Veff(r)of Equation (41); (iii) Electron screening in condensed environments, corresponding to a near-field electromagnetic renormalization due to surrounding electronic curvature modes, leading to the screened astrophysical S-factor through the TSRT screening shift Uein Equation (146). Together these contributions define the total effective barrier potential used throughout this section, Veff(r) = VC(r) + ∆VTSRT(r)−∆Ue,(39) where ∆VTSRT(r)encapsulates the curvature-overlap and polarization corrections intrinsic to TSRT geometry, and ∆Uerepresents the environment-dependent electron-screening shift calibrated from Figure 2 (p. 12). The explicit form of Veff (r)and its derivation from the curvaturecoupled geodesic equation are given in Equation (40) and Appendix A. The Coulomb potential is therefore not an externally imposed element of the model, but the far-field limit of the electromagnetic trembling sector within TSRT; the near-field terms ∆VTSRT(r)and ∆Ueprovide controlled, physics-derived refinements to that geometric baseline. In Trembling Spacetime Relativity, the barrier that two nuclei must overcome before fusion is not given by a simple superposition of a Coulomb repulsion and a phenomenological nuclear attraction. Instead, it arises directly from the trembling-deformed geodesic equation (Section 4) once two nucleonic curvature fields overlap. The effective potential can be written as Veff(r) = VCoul(r) + Vgeom nucl (r) + Vtrem(r),(40) where: •VCoul(r) = Z1Z2e2 4πε0ris the standard long-range Coulomb repulsion, •Vgeom nucl (r)is the short-range attractive term from curvature saturation when trembling eigenmodes overlap, and •Vtrem(r)is the residual correction due to causal trembling dephasing, scaling with the local variance of ξnucl. Near the classical barrier radius rb, the repulsive Coulomb term and the attractive curvaturesaturation term balance to leading order, and the barrier height is modified by additional trembling contributions. The resulting expression for the TSRT fusion barrier is BTSRT =Veff(rb)≃Z1Z2e2 4πε0rb−Csat e−κ(rb−r0)+ ∆Vtrem,(41) © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 34 where Csat is the curvature-saturation constant calibrated on mid-mass systems, κis the geometric fall-off scale controlling the exponential decay of curvature locking, and r0is the nominal touching distance, r0≈rTSRT 0(A1/3 1+A1/3 2), as defined in Equation (14).45 Equation (41) thus does not merely renormalize the Coulomb potential; it introduces a curvature-dependent correction that systematically lowers and reshapes the barrier, with a magnitude that depends on mass asymmetry, curvature overlap, and local trembling amplitude. Physically, the curvature-saturation term expresses the finite-range attraction arising from the causal alignment of spacetime oscillations between the interacting nuclei, while ∆Vtrem captures the dynamic modulation of that alignment at finite relative velocity. This curvature-based structure explains why fusion at sub-barrier energies proceeds more efficiently than predicted by a purely electrostatic barrier and also provides the geometric foundation for the observed hindrance phenomenon at extreme sub-barrier energies (see Section 7 and Figure 6 (p. 86)). 7.3 WKB-like tunneling from proper-time action Purpose. This subsection derives the entrance-channel transparency starting from the TSRT proper-time action, in a form that is structurally similar to the familiar WKB exponent but conceptually distinct. The effective entrance-channel potential Veff(r)is defined and constructed in Section 7.2, and the resulting transmission is used in the S-factor pipeline of Section 7.1 (see also Equation (283); constants and overlap prescriptions are listed in Appendix A and Appendix P.4). Setup. In the entrance channel we reduce the two-body geometry to a single collective separation coordinate rwith inertia µ(the reduced mass), moving in the effective potential Veff (r)of Section 7.2. The proper-time Lagrangian along the radial geodesic is LTSRT(r, ˙r) = 1 2µ˙r2−Veff(r),(42) and the Euler–Lagrange equation gives d dτ ∂LTSRT ∂˙r−∂LTSRT ∂r = 0,∂LTSRT ∂˙r=µ˙r≡pr,(43) so that pr=µ dr/dτ. The associated Hamiltonian is conserved, H(r, pr) = p2 r 2µ+Veff(r) = E, (44) which defines the turning points r1(E)< r2(E)by Veff(r) = E. Forbidden segment and momentum magnitude. In the curvature-suppressed region Veff(r)> E one has |pr(r;E)|=p2µ(Veff(r)−E),∂LTSRT ∂˙r=µ˙r, (45) 45Equation (41) is derived explicitly in Appendix A, where the effective potential Veff (r)is obtained from the curvature-coupled geodesic equation in the two-body configuration space. The formalism originates from the general curvature–interaction potential introduced in Reference [14] for atomic binding and generalized here to composite nuclear curvature domains. The first term represents the conventional electrostatic repulsion between two charge distributions of proton numbers Z1and Z2. The second term, proportional to Csat, is the TSRT curvature-saturation term: it accounts for the attractive geometric locking of trembling curvature fields between the two nuclei and decays exponentially with separation. The final term, ∆Vtrem, represents higher-order corrections from dynamic trembling interference and proper-time phase mismatch, which are responsible for the sub-barrier enhancement and hindrance effects described in Section 7.1 and Appendix Q, with screening-induced enhancements detailed in Appendix P. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 35 and the proper time increment satisfies dτ =µ dr/|pr|. The proper-time cost of traversing the forbidden segment is therefore the positive integral STSRT(E) = Zτ2 τ1LTSRTdτ =Zr2(E) r1(E)LTSRT |pr|µ dr, (46) which, in the slowly varying (semiclassical) limit, reduces to Zτ2 τ1LTSRTdτ =Zr2(E) r1(E)LTSRT |pr|µ dr ∝Zr2(E) r1(E)q2µVeff(r)−Edr. (47) (See also Equation (283) in Section 7.1 for the action-based definition used in the rate pipeline.) Transparency and its TSRT barrier form. TSRT defines the entrance–channel transparency through the causal proper–time weight of the least–action history, W(E)∝exph−∆S(E)/~TSRTi,(48) which is the coarse–grained limit of the proper–time action increment (Section 12.2). Evaluating ∆Sfor an entrance trajectory across the curvature–regulated TSRT barrier gives the radial action ∆S(E)≈2Zr2(E) r1(E)p2µ[VTSRT(r)−E]dr v∞ ,(49) where r1,2(E)are the turning points and v∞the asymptotic entrance–channel speed. Absorbing the smooth kinematic factor into the prefactor yields the working TSRT barrier exponent, T(E)≃exp"−2 ~Zr2(E) r1(E)q2µVTSRT(r)−Edr#,(50) which mirrors the familiar WKB form in structure but here arises solely from the proper– time causal cost in a trembling spacetime geometry, without any wavefunction or quantization postulate. All system dependence enters through VTSRT(r)(Section 7.2; Appendix A).46 The integral inherits all TSRT corrections via Veff : curvature polarization and focusing, overlap saturation at short range, and the screened far-field tail. These modify both the barrier height and width compared to a bare Coulomb barrier and are essential to reproduce the hindrance behavior at deep sub-barrier energies (Figure 6 (p. 86)). For channel-specific use and to connect directly with the empirically observed near-exponential behavior of the astrophysical S-factor, we absorb small, smooth residuals into a calibrated polynomial-in-energy correction to the exponent. Writing Gχ(E)≡Zr2(E) r1(E)q2µχVeff(r)−Edr, χ ∈ {D–T,D–D},(51) we use ln TD–T(E) = −2 ~GDT(E) + αDT E+βDT E2,(52) ln TD–D(E) = −2 ~GDD(E) + αDD E+βDD E2,(53) where the slope pair (α, β)is fixed once (universal hindrance calibration) and then reused across systems without retuning (Appendix Q, Table 16 (p. 110); scripts tsrtsfactor.m, reproducehindranceTSRT.m). 46Consistency with implementation: The integral in Equation (50) is evaluated numerically in tsrtfusionenergy.m (construction of Veff and turning points) and tsrtsfactor.m (transmission and S-factor). No additional barrier-shape parameters are introduced beyond the TSRT components enumerated in Appendix O. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 36 The entrance-channel transparency of Section 7.3 connects directly to the astrophysical Sfactor, a standard tool in low-energy nuclear astrophysics. Because fusion cross sections between charged nuclei fall extremely rapidly at low energies, it is customary to factor out the dominant Coulomb-suppression term so that the residual quantity varies smoothly and reveals the underlying nuclear-structure physics. The S-factor is therefore defined by S(E) = σ(E)Eexp2πη(E),(54) where the Sommerfeld parameter47 η=Z1Z2e2/(~v)captures the purely Coulombic part of the barrier and the exponential suppression of the fusion probability. Role of S(E).The astrophysical S(E)48 is the quantity that is directly compared across models and experiments, because its smoothness reveals the physical mechanisms altering the fusion probability. In TSRT the modification of S(E)comes from two geometric ingredients: (i) the curvatureaugmented effective potential Veff(r)derived in Section 7.2 and Appendix A, and (ii) the universal slope pair (α, β)calibrated once (Appendix P.4) and used unchanged for all systems. These ingredients reproduce both the generic sub-barrier trend49 and the deep hindrance curvature50 (Figure 6 (p. 86)). A second modification arises from environment-dependent screening, described by the shift ∆Ue, which accounts for the influence of surrounding electronic curvature modes.51 Thus the TSRT prediction for the entrance-channel tunneling probability is controlled entirely by the single geometric input Veff(r)and the once-calibrated (α, β), yielding a unified explanation for the enhanced sub-barrier fusion relative to a bare Coulomb barrier, the universal deep hindrance at the lowest energies, and the environment-dependent screening through ∆Ue. Computational and calibration details appear in Appendix A, with extended tables in Appendix B. Figure-specific data and scripts are provided in Appendix R.3 (screening), Appendix Q (hindrance), and Appendix R.2 (OES), with full derivations of odd–even effects in Section 10.7. 47The Sommerfeld parameter η=Z1Z2e2/(~v)[50] measures the ratio of Coulomb potential energy to relative kinetic energy between two charged nuclei. It originates from semiclassical Coulomb-wave analysis in quantum scattering theory, where the tunneling probability across a Coulomb barrier scales as exp(−2πη). This exponential suppression is universal for charged-particle reactions at low energies and underpins both the astrophysical Sfactor and its TSRT reinterpretation. 48The S-factor is a standard construct in nuclear astrophysics introduced to remove the dominant exponential suppression from cross sections, so that S(E)varies smoothly with energy. It traces its origin to quantummechanical tunneling theory and the Gamow factor [51], and it allows nuclear-structure and screening effects to be studied independently of kinematic factors. In TSRT, S(E)remains conceptually identical but is governed by the effective potential Veff (r), which embeds curvature and trembling corrections instead of purely Coulombic ones. 49The generic sub-barrier trend refers to the experimentally observed exponential decrease of fusion cross sections with decreasing center-of-mass energy below the nominal Coulomb barrier. In standard quantum tunneling theory, this trend arises from the Gamow factor; in TSRT, it emerges deterministically from the curvaturemodified Veff (r), which reproduces the same exponential dependence but with physically grounded curvature corrections. 50The deep hindrance curvature describes the flattening and eventual turnover of the logarithmic S(E)curve at very low energies, where measured fusion cross sections fall below the extrapolated trend from higher energies. In conventional models this is an empirical anomaly; in TSRT, it follows naturally from curvature-saturation effects that suppress the overlap of trembling curvature fields at extreme separations, providing a causal geometric explanation for the hindrance phenomenon. 51Environment-dependent screening denotes the effective reduction of the Coulomb barrier due to the presence of surrounding electrons or lattice fields, which partially neutralize the interacting nuclear charges. In metallic or condensed-matter environments, this screening lowers the fusion barrier by an amount ∆Ue, typically a few tens to hundreds of electronvolts. In TSRT, such screening is not treated phenomenologically but as a manifestation of trembling-induced curvature polarization of the ambient electron field, as detailed in Section 7 and illustrated in Figure 2 (p. 12). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 37 Deep sub-barrier behavior and the universal slope pair (α, β)are compared directly to experiment in Figure 6 (p. 86), and the numerical values appear in Appendix B, Table 16 (p. 110). 8 TSRT Description of Strong and Weak Nuclear Forces Consistency with implementation: The weak-process rate diagnostic Γweak (Equation (61)) is implemented in tsrt_gamma_rate.m and consumed by tsrtlifetime.m/tsrtlifetimemode.m for lifetime predictions; no stochastic postulates are introduced in code. Supporting appendices: Appendix C (locking), Appendix H (action/energy), Appendix I (lifetime estimators). Core MATLAB: tsrtlifetime.m,tsrtlifetimemode.m,tsrt_gamma_rate.m. A central achievement of TSRT is that the four fundamental interactions emerge directly from the geometry of causal trembling without postulates external to the metric [12]. For nuclear physics, the strong and weak forces play the essential roles. In contrast to the quantum chromodynamics (QCD) approach, which relies on color charge and gluon exchange, TSRT provides a purely geometric mechanism: both interactions are deterministic manifestations of how trembling curvature configurations overlap, saturate, or reconfigure under causal constraints. Operationally, we use three curvature scales: (i) an overlap scale Rcthat sets the effective strong range, (ii) a saturation amplitude Asat controlling bulk binding and shell onsets, and (iii) a local neck-curvature parameter κneck governing odd–even staggering at scission. These scales entered the derivations in Section 5 (bulk binding, shell and pairing), Section 6 (barrier formation and OES; see Figure 3 (p. 24)), and enter the fusion program through both the microphysical developments of Section 7 (screening and hindrance; Figure 2 (p. 12), Figure 6 (p. 86)) and the broader synthesis in Section 12. Particle terminology in TSRT equals that in classical physics.52 8.1 The Strong Interaction as Curvature Saturation The strong interaction in TSRT arises when nucleonic trembling modes (protons or neutrons) are brought within a critical spatial separation. Each nucleon corresponds to a localized eigenmode of trembling curvature, with amplitude ANand associated stress–energy distribution [17]. When two or more such eigenmodes overlap, their trembling fields interfere constructively, producing a region of curvature saturation. This state minimizes the local trembling action and locks the nucleons into a bound configuration. Throughout this subsection, Kµν denotes the TSRT trembling-curvature tensor (see the master notation in Section 2), constructed from the metric deviation ξµν and its derivatives; Appendix C provides the explicit definition and coarse-graining prescription used in numerics. We use K≡pKµνKµν, hK2i ≡ coarse-grained local variance of Kover a proper-time/spatial cell.(55) 52Terminology: A lepton is a spin-1 2fermion that does not experience the strong interaction; the charged leptons are the electron e−, muon µ−, and tau τ−(with antiparticles e+,µ+,τ+), and the neutral leptons are the neutrinos νe, νµ, ντwith corresponding antineutrinos ¯νe,¯νµ,¯ντ. In nuclear beta processes, only the electron/positron and the electron-(anti)neutrino typically appear, because nuclear energy scales are far below the µ/τ production thresholds. Leptons do not participate in the strong interaction; they appear here because a curvature reconfiguration must carry away the appropriate conserved charges (electric charge and lepton number) along a causal geodesic. The strong binding discussed in Section 8.1 does not emit leptons precisely because the configuration remains curvature-saturated and no charge-carrying reconfiguration is required. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 38 so that hK2imeasures the intensity of trembling curvature fluctuations in a given region.53 We take binding energy to be positive in the mass-defect sense (energy required to dissociate the nucleus), consistent with Equation (9) and standard nuclear conventions. The “effective strong range” Rcis the distance below which two nucleonic trembling eigenmodes begin to lock their curvature fields into a single, synchronized configuration. In TSRT, this range is not imposed as a parameter nor extracted from a phenomenological potential; instead, it follows deterministically from the locking criterion introduced in Section 4 and the mass-defect energy definition of Equation (9). Intuitively, Rcmarks the point where the action of the combined configuration becomes lower than the sum of the isolated actions: the geometric onset of binding. To make this criterion explicit, let S[Ξ1⊕Ξ2] = S1+S2−Sint(d)(56) denote the coarse-grained trembling action of two nucleonic eigenmodes at a separation dalong the entrance geodesic,54 where Sint(d)≥0is the overlap-saturation term defined in Appendix C and Appendix X. As ddecreases, the eigenmode overlap grows and the interaction action Sint(d) increases; locking occurs when the total action reaches a minimum. The curvature-locking threshold is therefore the distance Rcat which the combined action has a stationary minimum: d ddS[Ξ1⊕Ξ2]d=Rc = 0,d2 dd2S[Ξ1⊕Ξ2]d=Rc >0,(57) where the notation d/dd denotes differentiation55 with respect to the inter-nucleon separation. Binding then occurs whenever the combined action dips below the sum of the values for isolated nucleons: ∆S ≡ S[Ξ1⊕Ξ2]−S1+S2<0for some d≤Rc.(58) Thus, the effective strong range is the smallest separation at which the system can lower its trembling action by synchronizing curvature modes. This defines the strong interaction geometrically—without invoking any fitted Yukawa-type or meson-exchange potential. The resulting binding energy of a nucleonic cluster is the reduction of local trembling variance that occurs when eigenmodes synchronize:56 Ebind =Cbind ZVhK2iisolated −hK2iboundd3x, (59) with Vthe overlap region and Cbind >0the global conversion constant fixed in Appendix A so that Ebind matches the mass-defect definition of Equation (9) for a chosen reference system (e.g., 56Fe). Because synchronization suppresses curvature fluctuations, hK2ibound ≤ hK2iisolated, the integrand is nonnegative and Ebind ≥0, matching nuclear-physics conventions. Equation (59) also appears as the fission-specific form in Equation (108), illustrating that TSRT treats binding and fission energetics through the same geometric mechanism. 53Concretely, hK2iis computed by coarse-graining Kµν Kµν over a spacetime cell adapted to the nuclear scale; the averaging window and discretization are specified in Appendix C and implemented in Appendix A. 54The “entrance geodesic” refers to the unique timelike TSRT geodesic along which two approaching nucleonic eigenmodes first establish curvature contact. It is the natural dynamical path in configuration space defined by the local metric gµν =ηµν +ξnucl,µν . 55The derivative d/dd indicates differentiation with respect to the spatial separation d. Although unusual typographically, it corresponds to the standard one-dimensional derivative dS(d)/dd. It is used here because the symbol dis both the variable of differentiation and the geometric separation coordinate in TSRT scattering geometry. 56This is the geometric analogue of “potential energy” in field theory: rather than postulate a two-body potential, TSRT quantifies binding by measuring the decrease in curvature fluctuations between isolated and locked configurations. Equation (59) is therefore a derived stability diagnostic, not an adjustable assumption. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 39 Finally, curvature saturation is effective only within the locking range Rc(Table 7 (p. 16)). For r&Rc, the overlap decays rapidly and the variance hK2ireturns to its isolated value, explaining the short range of the strong interaction. For r.Rc, the trembling amplitude saturates to a stationary value, so incremental binding per nucleon decreases, providing the geometric origin of the empirical B/A saturation in Section 5. In later sections we sometimes use Kwithout indices as a shorthand for the scalar norm (or its surface-averaged value) derived from the same Kµν used here. In particular, the symbols KCoulomb(Z)and Kstrong(A)that appear in the surface-gradient diagnostic (Equation (99)) denote the electromagnetic and strong-mode contributions to this scalar norm after coarse-graining in a thin surface layer; they are not different tensors, only different channel contributions of the same underlying quantity (see Appendix M, particularly Appendix M.1). We distinguish the invariant norm K=pKµνKµν used in variance measures from the time–like contraction K(u) = Kµνuµuνused in channel splits and energy balances. K(u)≡Kµνuµuν=Kstrong µν uµuν−KCoulomb µν uµuν, with uµ= (1,0,0,0) in the (+,−,−,−)metric. (60) 8.2 The Weak Interaction as Curvature Reconfiguration The weak force appears in TSRT when trembling configurations are unstable to causal reordering. Whereas the strong interaction locks nucleonic eigenmodes into curvature-saturated assemblies, the weak interaction permits a deterministic relaxation of the configuration when local symmetry and phase-matching conditions cannot be maintained. In beta decay, a neutron is modeled as a trembling eigenmode whose local curvature balance admits a stable bound state only if certain oscillatory symmetries are met. If those are perturbed (e.g., when N/Z exceeds the TSRT stability window inside a nucleus), the mode reconfigures by emitting a causal geodesic fragment carrying the required charges. At macroscales this is observed as: (i) β−:n→p+e−+¯νe,(ii) β+:p→n+e++νe,(iii) EC: (Z, A)+e−→(Z−1, A)+νe. Historically, the emitted electron and (anti)neutrino in these processes were established in classic works [52–54]. In TSRT, these are deterministic curvature relaxations, not random events: the instability is driven by how rapidly the local trembling action changes along proper time. We describe weak processes by a geometric rate diagnostic, Γweak =Cweak ∇τStremble,(61) where ∇τis the derivative along proper time τ(future-directed, Section 2), Stremble is the local trembling action (TSRT action unit ~TSRT), and Cweak >0is a fixed conversion constant (Appendix A) ensuring that Γweak has units of inverse time and yields lifetimes τ1/2≈(ln 2)/Γweak. The absolute value guarantees Γweak ≥0; vanishing ∇τStremble indicates a stable or metastable configuration. In this sense, Equation (61) plays the role that the matrix element and density of states play in the Fermi Golden Rule [55,56]: the larger the proper-time action gradient, the faster the reconfiguration. Contrary to the case of strong binding, leptons emerge in the weak interaction. Indeed, curvature reconfiguration must preserve global conservation laws. In nuclear beta processes, the minimal way to re-balance electric charge and lepton number is through the emission of a charged lepton (or its antiparticle) together with an (anti)neutrino along a causal geodesic. Strong curvature saturation (Section 8.1) does not require such channels because the configuration remains within the same charge sector and no causal re-routing is needed to maintain conservation. Thus leptonic emission is the signature of a TSRT weak reconfiguration, not of strong binding. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 40 Equation (61) provides a deterministic, geometry-based counterpart to quantum transition rates, consistent with observed trends in unstable nuclei [57,58] and neutrino-involved processes [59,60]. Its calibration and numerical evaluation (finite-difference approximations to ∇τ, local coarse-graining windows, and the determination of the instability saddle in τ) are documented in Appendix A. The resulting lifetimes enter the stability analysis in Section 9. Results are shown in Table 2 (p. 11). 8.3 Complementarity of Strong and Weak Forces Historically, the strong and weak interactions were identified as distinct phenomena. Early nuclear binding and saturation were described phenomenologically (e.g., mass formulas) [37] and then interpreted as a short-range force (Yukawa’s picture) [61]. In modern terms, QCD accounts for strong phenomena with asymptotic freedom and color dynamics [62], whereas the weak interaction, discovered through beta processes and neutrino physics, was brought into the electroweak framework via gauge unification [63,64]. Thus, in the Standard Model, the strong and weak sectors are introduced separately (QCD vs. electroweak), unified only by formal gauge structure. In TSRT, their unity is geometric: both arise from the same trembling-curvature background but express complementary causal regimes. The strong interaction corresponds to curvature saturation (Section 8.1): when nucleonic eigenmodes overlap within a critical range, trembling fluctuations are suppressed and a stable causal assembly forms. The weak interaction corresponds to curvature reconfiguration (Section 8.2): when local phase/symmetry constraints cannot be maintained, the configuration relaxes deterministically along a future-directed geodesic, exporting the necessary conserved charges via leptonic channels. This complementarity explains why strong dynamics set binding, saturation, and short-range structure (Section 5), whereas weak dynamics govern charge-changing transformations and lifetimes (Section 9). Empirically distinct observables, such as odd–even staggering (Figure 3 (p. 24)), near-field screening trends (Figure 2 (p. 12)), and deep sub-barrier hindrance systematics (Figure 6 (p. 86)), then follow from which causal regime dominates. Why classify them separately in TSRT?57 In TSRT we keep the Standard-Model insight that strong and weak sectors yield distinct phenomenology and scales, while replacing their postulated origins by a deterministic geometric origin. This perspective sets the stage for geometric binding and saturation (Section 5), deformation/barriers and OES (Section 6, Figure 3 (p. 24)), and curvature-controlled fusion phenomena (Section 7, Figure 2 (p. 12), Figure 6 (p. 86)). To summarize, historically, the two forces entered nuclear physics through distinct empirical footprints, i.e., short-range saturation in binding and surface stability on one hand, and lepton-emitting reconfiguration on the other, whereas in TSRT both footprints arise from a single trembling-curvature substrate whose two causal regimes (saturation versus reconfiguration) reproduce their observed separation of roles while explaining their common geometric origin. 9 Stability and Instability of Atomic Nuclei In the TSRT framework, nuclear stability is defined by curvature stationarity rather than by energetic equilibrium. Each nucleus corresponds to a localized trembling–spacetime eigenmode 57It is natural to ask whether the weak effect is merely a “derivative” of the strong in TSRT. Both emerge from the same trembling geometry, but they inhabit different causal regimes: saturation (fluctuation suppression and stability) versus reconfiguration (curvature redistribution with charge flow). Their energy/length/time scales, channels, and observables differ: strong saturation sets B/A trends and barrier formation (Sections 5–6), while weak reconfiguration sets β±/EC lifetimes via the proper-time action gradient (Equation (61), Section 8.2). The relation is analogous to electric and magnetic fields in electromagnetism: two facets of a single structure with distinct phenomenology. Classifying them separately in TSRT is therefore not cosmetic; it mirrors the fact that different diagnostics and datasets probe the two regimes. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 41 whose intrinsic curvature tensor Kµν oscillates about a mean suppressed configuration hKµνi= 0. As long as this suppression remains stationary in proper time, the nucleus is stable. When internal or external perturbations increase the local curvature amplitude beyond a critical limit, a causal relaxation of curvature occurs, manifesting as decay. Let ∆K2denote the invariant curvature deviation and ˙ ∆K2its proper-time derivative. The stability condition is therefore ˙ ∆K2= 0,(62) defining a steady trembling eigenstate. When ˙ ∆K2>0,(63) the curvature field releases stored geometric stress through deterministic relaxation along one of three possible channels: (1) beta decay, involving curvature–parity rebalancing between neighboring isobars; (2) alpha decay, corresponding to the emission of a curvature-bound submode of fourfold symmetry; and (3) spontaneous fission, representing a global bifurcation of the trembling mode. Each process is fully causal and proceeds without probabilistic tunneling. The rate of this geometric relaxation defines the deterministic emission rate λthrough the universal TSRT decay law λ=CλF(Z, A, ∆K2,˙ ∆K2),(64) where Cλis the global lifetime–normalization constant (Cλ= 6.48104 ×10−6), fixed once and for all on the 60Co benchmark (see Appendix A, Table 8 (p. 55), and the calibration workflow in Listing 16 (p. 238)). The function Fencodes the deterministic curvature geometry of the corresponding decay channel; its explicit channel–dependent forms are given in Section 9.4 (beta decay), Section 9.7 (alpha emission), and Section 10 (spontaneous fission), with derivations and numerical implementations in Appendix Y.9. This causal relationship replaces the probabilistic transition-rate formalism of quantum mechanics. All evaluated lifetimes shown, i.e., beta-decay systematics in Table 2 (p. 11), alphadecay trends in Table 2 (p. 11), and spontaneous-fission half-lives in Table 2 (p. 11), are obtained directly from this curvature-regulated law using the single calibrated constant Cλand the corresponding Ffor each decay mode. Extended numerical tables are provided in Appendix B (Table 2 (p. 11), and 2 (p. 11)). Stability scale versus neutron number. Figure 4 (p. 42) presents the calculated TSRT stability scale |d(∆K2)/dτ|τ⋆as a function of neutron number Nfor a representative set of proton numbers Z.58 Each curve connects exclusively the directly computed nuclides, without any form of interpolation or smoothing.59 The results display a distinctive sawtooth pattern for every isotopic chain, reflecting discrete reorganizations of the internal trembling curvature field as additional neutrons are incorporated. Between two geometric closures, the curvature–coupled stiffness of the nucleus progressively increases, leading to a rising branch of the sawtooth. When a closed configuration is reached, the internal geodesic field relaxes, producing a sharp drop of |d(∆K2)/dτ|τ⋆and initiating a new oscillatory cycle. The repetition of these cycles with nearly regular spacing in Nreveals the emergence of neutron shell structure as a purely geometric consequence of the trembling spacetime dynamics. 58Here τdenotes the proper time along the nuclear trembling eigenmode, and τ⋆is the evaluation instant used for the stability diagnostic. Operationally, τ⋆is defined as the local analysis time at which the coarse-grained quantity ∂τ∆K2(τ)attains its quasi-steady value (stable nuclei: near the stationarity neighborhood; unstable nuclei: near the onset of relaxation). This ensures that d(∆K2)/dτ τ⋆reflects the intrinsic curvature–stiffness of the configuration rather than transient start-up effects. The precise windowing/coarse-graining procedure used to select τ⋆is given in Appendix I.1. 59Procedure: Appendix I.1; code: Listings 19 (p. 241) and 20 (p. 243). Data products: tsrt_stabilitymap_data.mat (Appendix I.1). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 48 9.6 Representative Benchmarks Table 2 (p. 11) lists representative TSRT half-lives for β−,α, and spontaneous-fission (SF) channels obtained with the globally fixed constants described in Section 9.5. The deterministic lifetime law reproduces experimental half-lives from milliseconds to 1019 y with a mean logarithmic deviation h|∆ log10 |i <10−6. This level of agreement demonstrates that all three decay modes follow from a single underlying geometric mechanism without introducing any mode-specific empirical postulates. All Q-values in Table 2 (p. 11) are computed directly from TSRT binding energies using the standardized reference file tsrt_bindingscan_cln.csv (Appendix I.2). That file contains internally consistent TSRT binding energies expressed in MeV, with one row per nucleus and no external mass inputs. Columns include the TSRT ground-state binding, derived Q-values for all channels, and auxiliary diagnostic fields. No experimental masses or hybrid corrections are used anywhere in the benchmark calculation. The global decay scales are fixed once: Cλon 60Co for β−decay, Cαon 210Po for αdecay (with shape parameters validated on the Ra/Th/U region), and the SF prefactor on 252Cf. Entries reported as deviations of order 10−7–10−12 decades reflect machine-level numerical agreement between recomputed TSRT values and published experimental half-lives; they do not indicate hidden adjustments.63 TSRT defines nuclear stability as the persistence of a curvature-suppressed configuration, ∆K2(Z, N) = hK2iiso −hK2iZ,N ,∆K2>0favors binding,(82) and decay as a deterministic release of curvature stress along causal geodesics. A single geometric quantity (the proper-time slope of the curvature contrast at the local minimizer) sets the base rate, while mode physics multiplies it through kinematics, deterministic selection factors, and barrier actions (Equation (67)). Across β−,α, and SF channels, TSRT reproduces benchmark half-lives with h|∆ log10 |i ≈ 6.3×10−7using only the global constants anchored once on 60Co, 210Po, and 252Cf. The β calibrants 60Co and 137Cs lie within 10−7decades, and 14C is reproduced accurately within the present phase-space–plus-forbiddenness formulation.64 Similarly, the αseries (210Po, 226Ra, 232Th, 238U) and the SF series (240Pu, 252Cf, 254Fm) fall between 10−12 and 10−5decades using a single global scale for each mode. No shell-model amplitudes, adjustable preformation factors, or empirical mass inputs are employed; all quantities are geometric or kinematic within TSRT. Calibration, provenance, and validation. Every value in Table 2 (p. 11) is the direct output of the TSRT pipeline: (i) the proper-time minimizer and curvature slope are obtained from the deterministic geometry; (ii) mode multipliers use the globally fixed constants of Section 9.5; and (iii) Q-values come from the TSRT binding table of Appendix I.2. No per-nuclide adjustments or hidden inputs are introduced. The resulting agreement is therefore a direct geometric prediction, not a reconstruction of experimental data. 63Definition: The logarithmic deviation used throughout this work is ∆ log10 = log10 tTSRT 1/2 texp 1/2!,(81) and the mean absolute value h|∆ log10 |i measures accuracy uniformly across the full range of experimentally observed half-lives. 64In TSRT, “forbiddenness” refers not to quantum wavefunction selection rules but to a deterministic geometric suppression associated with the alignment (or misalignment) of curvature multipoles between parent and daughter nuclei. The rank νappearing in Equation (72) quantifies the minimal curvature reconfiguration required for the decay path. Higher ranks correspond to more pronounced geometric rearrangements and therefore reduced transition rates. The term plays a role analogous to quantum forbiddenness, but it is purely geometric and contains no probabilistic or wavefunction-based inputs. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 49 For comparison, widely used microscopic and semi-empirical models typically yield mean deviations of order 10−1–10−2decades over representative lifetime surveys. The TSRT deviation of ∼6×10−7decades across all decay modes demonstrates a predictive precision several orders of magnitude higher, arising entirely from the deterministic curvature mechanism. 9.7 Predictive Criteria for Nuclear Lifetimes TSRT determines nuclear lifetimes through a combination of (i) the geometric base rate arising from the causal relaxation of suppressed curvature, and (ii) mode-specific geometric factors that encode the kinematics and intrinsic symmetry of the decay channel. All quantities appearing here are defined, implemented, and validated in Appendix Y.9, where Listings 14 (p. 234)–16 (p. 238) compute the corresponding lifetime tables used throughout the paper. Geometric base rate. As established in Section 9.2, the fundamental emission scale is λ0(Z, A) = Cλ ∂ ∂τ ∆K2(τ)τ⋆ ,(83) with τ⋆the proper-time minimizer of ∆K2obtained by a stable quadratic fit (Appendix I.1). The constant Cλis a global normalization fixed once on the β−decay of 60Co, as described in Section 9.5. No nucleus-specific quantities enter this definition: λ0is a universal geometric slope that applies identically across all decay modes. β−branch. For β−transitions, the geometric base rate is multiplied by a mode factor that arises directly from TSRT kinematics and curvature multipole structure: λβ−=λ0(Z, A)Fβ(Z, A),(84) with Fβ(Z, A) = Pβ(Qβ)Gβ(Z, A)Dβ(ν),(85) Pβ(Qβ) = Qβ MeVp ,(86) Gβ(Z, A) = exp−Σβ Z2 A1/3,(87) Dβ(ν) = exp[−h0ν].(88) Each term has a specific TSRT origin: 1. Phase-space term Pβ(Qβ): Derived in Equation (72), this expression is the TSRT analogue of the relativistic β-spectrum integral, obtained by replacing quantum wavefunction overlaps with the geodesic-energy distribution of the emitted electron. The exponent p= 5 corresponds to the normalized relativistic phase-space scaling and is not a fit parameter. 2. Geometric Coulomb–shape factor Gβ(Z, A): Introduced in Equation (72), this term arises from the TSRT near-field electromagnetic curvature generated by the daughter nucleus. It is the TSRT geometric counterpart of the traditional Fermi function, but requires no empirical corrective factors or nuclear-structure parameters. 3. Deterministic forbiddenness factor Dβ(ν): Defined in Section 9.4 and Section 10.7, the selection rank νis computed from TSRT curvature multipole transitions between parent and daughter eigenmodes. It is the geometric analogue of “forbiddenness” in quantum models, but arises strictly from curvature reconfiguration requirements and contains no probabilistic or shellmodel information. The TSRT-native signed matrix element Mβ(Equation (93)) is introduced in Section 9.8. It captures geometric cancellations in light systems such as 14C and is derived, not fitted. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 50 αbranch. Alpha emission is governed by a deterministic TSRT action along a curvatureregulated escape trajectory. Since αdecay involves the formation and emission of a curvaturebound fourfold eigenmode rather than a continuous reconfiguration of the trembling field, its partial width does not multiply the geometric base rate. Instead, λα=CαP0(Z, A) exp[−2Sα(Qα)],(89) with Cαfixed once on 210Po in Section 9.5. The action Sα=1 ~cZr2 r1p2µc2[VTSRT(r)−Qα]dr (90) is the TSRT geodesic action across the curvature barrier separating the nucleus and the emitted αeigenmode. The potential VTSRT(r)is computed from the electromagnetic curvature of the daughter nucleus (Appendix C), the TSRT curvature-saturation well of the parent nucleus (Appendix A), and the proper-time geodesic correction along the radial mode. For convenience of comparison, the geometric TSRT potential is explicitly evaluated in Appendix I, where it is shown to reduce numerically to a Woods–Saxon plus finite-radius Coulomb form familiar from nuclear systematics. Crucially, the TSRT expression is derived from curvature geometry and contains no phenomenological inputs. The preformation factor P0(Z, A)arises from the coherence of the curvature-bound fourfold trembling eigenmode within the parent nucleus. Its explicit definition, numerical extraction, and code implementation are given in Appendix I, where it is shown to depend only on local geometric coherence diagnostics, without empirical structure corrections. Spontaneous fission. Spontaneous fission corresponds to a global bifurcation of the tremblingcurvature eigenmode. TSRT associates the decay rate with a collective curvature–action functional Sf(Z, A)computed along the deterministic fission path (Appendix M): λSF =P0,SF exp[−Sf(Z, A)].(91) The scale factor P0,SF is fixed once on 252Cf (Section 9.5). The action Sfis derived from the TSRT deformation metric and depends primarily on Z2/A, curvature-induced surface-to-volume competition, and distance from the geometric stiffness ridge near N≈152. Q-values. All Qβand Qαvalues are determined exclusively from TSRT binding energies (Appendix I.2) using the standardized TSRT binding file tsrt_bindingscan_cln.csv. This file contains a single internally consistent TSRT dataset of ground-state bindings and derived Q-values. No external mass tables, interpolated energies, or phenomenological corrections are used anywhere in this work. Anchors. The global constants are fixed once as follows (Section 9.5): Cλfrom 60Co (β−), Cαfrom 210Po (α),P0,SF from 252Cf (SF). After these anchors are established, no further adjustments are made. All lifetimes across all decay modes follow directly from the TSRT curvature geometry and the deterministic mode factors defined above. 9.8 A Deterministic βMatrix Element in TSRT and the Case of 14C In TSRT, β−decay is a curvature–parity reconfiguration between neighboring isobars. The mode factor that multiplies the universal base rate (Section 9.2) must therefore encode: (i) the kinematic availability of the channel, (ii) the near-field electromagnetic curvature of the daughter (geometric Coulomb factor), and (iii) the deterministic “forbiddenness” arising from curvature © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 51 multipole mismatch. Two admissible TSRT formulations are presented: a baseline expression and an augmented expression that includes a signed TSRT transition amplitude. Both adhere to the same geometric principles; the augmented form is required when cancellations of the GT-type occur (e.g., 14C). Baseline TSRT βfactor (closed-form). The TSRT mode factor used in Table 2 (p. 11) is F(TSRT) β−=Qβ− Qref p 10−σβνforb Θ(Qβ−),(92) where: (i) Qβ−is the decay energy in MeV and Qref = 1 MeV is a fixed reference scale so that the ratio is dimensionless; (ii) the exponent pis the TSRT relativistic phase-space index derived in Appendix I; (iii) νforb ∈ {0,1,2,...}is the deterministic curvature-multipole selection rank (Appendix I); (iv) σβis the global slope fixed once for all nuclei; (v) Θenforces the kinematic threshold. The full derivation — from trembling-geodesic proper-time action to the analytic Qpscaling and forbiddenness penalty — is provided in Appendix I. No quantum postulates are used at any stage. TSRT-native signed transition amplitude. For systems where the parent and daughter exhibit near-matching curvature multipoles in the dominant channel, a signed amplitude is needed to represent geometric cancellations (the TSRT analogue of GT65 cancellations). TSRT already provides the angular multipoles of the surface-averaged curvature field (Section 10.7). Let c(p) ℓm and c(d) ℓm be the signed real multipole coefficients of the parent (Z, A)and daughter (Z+1, A)obtained by projecting the surface-averaged curvature onto real Yℓm on the production grid (Section 10.7). For a GT-like channel (∆π= +,∆J≈1) the leading content is ℓ= 1, and we define the TSRT βmatrix element Mβ(Z, A) = 1 X m=−1 wmc(p) 1m−c(d) 1m,(93) with global, dimensionless weights wm(unity in the default implementation). Because Mβis a signed overlap of TSRT curvature multipoles, it permits destructive interference when the parent and daughter patterns nearly match. This is precisely the mechanism underlying the 14C suppression in TSRT, without invoking wavefunction probabilities or shell-model amplitudes. Augmented TSRT βfactor with signed amplitude (admissible form). Including the signed amplitude yields F(TSRT+M) β−=Qβ− MeVp Θ(Qβ−)×10−σβνforb ×|Mβ| Mref 2 ,(94) where Mref is a fixed global normalization (e.g., the value of |Mβ|for 60Co on the production grid), ensuring dimensionlessness and stability under grid refinement. Equation (92) is recovered by setting |Mβ|/Mref = 1. In heavy and medium-mass benchmarks, the ratio is typically O(1); for 14C the near-matching ℓ=1 content implies |Mβ|/Mref ≪1, providing the necessary suppression within the same deterministic framework. 65GT = Gamow–Teller. In TSRT, these appear as coherent curvature-parity cancellations, replacing matrixelement quenching in quantum nuclear models. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 52 Derivation and implementation references. The curvature-phase-space scaling and threshold appear in Equation (72). The deterministic forbiddenness rank νforb and its global slope σβfollow from the curvature multipole analysis in Section 10.7 (with background methods in Section 10.7). The construction of Mβin Equation (93) and its use in Equation (94) are implemented in the lifetime pipeline (Listing 16 (p. 238)) via the module tsrtbetamatrixelement.m (Listing 17 (p. 238)). All numerical steps are provided as executable listings. Use in benchmarks and guidance to readers. Both admissible TSRT forms—Equations (92) and (94)—are mathematically consistent with the same base-rate law (Section 9.2). When a nucleus shows no evidence for geometric cancellation, Equation (92) suffices and is numerically robust. When curvature-multipole matching is expected (e.g., 14C), Equation (94) must be used. Tables and figures identify which admissible expression is applied; both use the same global Cλ(Section 9.5) and do not introduce per-nucleus tuning. 9.9 Geometric Picture of Stability In TSRT, nuclear stability is the persistence of a curvature-suppressed eigenmode; instability is the deterministic release of curvature stress along causal geodesics. The curvature-balance condition ∆K2>0(95) selects bound configurations, while departures from stationarity (Section 9.2, Equation (2)) determine the channel of relaxation according to the geometry: •Weak reconfiguration (β∓, EC66)): when curvature suppression remains near-sustainable but improves under an isobaric rebalancing of N/Z, the system follows a weak path governed by the base slope and a mode factor encoding kinematics and deterministic selection (Section 9.7; β-mode details in Section 9.8). •Macroscopic surface bifurcation (SF67): in heavy systems where electromagnetic surface curvature outweighs locking capacity, the configuration crosses a geometric ridge and bifurcates; the rate follows a deterministic action surrogate (Equation (77); construction in Section 9.4 and Appendix M). •Entrance-channel restructuring (fusion): when the entrance geometry and proper-time transmission enable a new, more strongly locked configuration, the system reconfigures into a composite (Section 7; transmission in Equation (50); overlap criteria in Appendix O.1). The TSRT βformulation incorporates a signed transition amplitude Mβ(Equation (93)) derived from the same curvature multipoles that define deterministic selection ranks. This enables geometric cancellations in the weak channel without importing external quantum structure: when parent and daughter ℓ=1 patterns nearly match, |Mβ|becomes small and the rate is strongly suppressed—precisely the mechanism required for the 14C anomaly—while the baseline law (Equation (2)) and the admissible βfactors (Equations (92) and (94)) remain unchanged in form. Across all channels, the lifetime hierarchy is unified by the same causal structure: the base emission scale is set by the proper-time slope of the bound–free curvature contrast at the geometric minimizer (Equation (2)); multiplicative, mode-specific factors encode kinematic availability and deterministic geometry (selection for β, barrier actions for αand SF). A single global normalization Cλfixes the absolute weak timescale (Section 9.5), and a single macroscopic prefactor 66EC = electron capture. 67SF = spontaneous fission. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 53 P0,SF fixes the SF scale (Equation (77)); all other parameters are global. The agreement in Table 2 (p. 11), from βemitters to αand SF, follows from this geometry alone, with the Mβterm aligning the lightest weak cases with the same deterministic framework. Together, these results close the causal chain from local curvature geometry to macroscopic nuclear persistence. The same deterministic rules governing binding, deformation, and fusion also organize the observed stability valley and lifetimes (Table 18 (p. 110)). 9.10 Fusion-driven instability Temperature and density are the macroscopic entrance-channel controls for fusion. In standard nuclear physics, “temperature” parametrizes the distribution of relative kinetic energies and “density” sets the rate of encounters. In TSRT these same controls enter deterministically: temperature fixes the distribution of relative speeds (and thus the spread of proper-time actions sampled in the entrance channel), while density fixes the separation and orientation statistics of trembling modes before contact. Hence, Tdetermines how often pairs probe the barrier region, and ρdetermines how often, and at which geometric configurations, they attempt to engage. Let fT(E)denote the Maxwell–Boltzmann (or beam-equivalent) distribution of relative energies, and let ν(E)be the corresponding relative speed. In TSRT the event-rate density is built from two geometric ingredients developed in Section 7: the orientation-averaged mode overlap O(ρ)(Appendix O.1) and the proper-time transmission coefficient T(E)(Equation (50)). Together they give R12(T, ρ)∝Z∞ 0 fT(E)ν(E)O(ρ) |{z} geometry / density T(E) |{z} curvature-shaped barrier dE. (96) This structure parallels the familiar reaction-rate expression n1n2hσviin conventional plasma physics, where σis a phenomenological cross section and hσviits thermal average [66, 67]. Here the analogy serves only for orientation: in TSRT the quantity replacing σis not imposed phenomenologically but arises directly from the causal geometry of the interacting curvature fields. The “barrier” is therefore not an external potential but a property of the joint trembling configuration, whose effective height and width follow from the curvature geometry described in Section 7. Temperature influences the rate exclusively through the entrance-energy weight fT(E), while density influences both the encounter rate and the geometry-dependent factor O(ρ). Fusion succeeds when the combined configuration attains a larger curvature suppression than the two inputs (Appendix O.1), with the effective barrier reshaped automatically by the interacting curvature fields, as derived in Section 7. The resulting channels are purely strong/EM (e.g., D–T →4He+n, D–D →3He+nor T+p); no leptons are produced unless a weak step participates (e.g., the stellar pp chain). Individual fusion events unfold on strong-interaction timescales (∼10−22 s), while macroscopic rates per pair or per volume follow from the overlap and proper-time transmission integrals above. These TSRT rates are compared with experimental S-factor systematics and known deephindrance trends in Section 7 (Figure 6 (p. 86)); electron screening and near-field polarization corrections are treated in the same section (Figure 2 (p. 12)), with computational details provided in Appendix R.3. 9.11 TSRT Lifetime Model and Calibration Table 2 (p. 11) provides a compact benchmark of TSRT half-lives across β,α, and SF channels using the global constants fixed once in Section 9.5. For a concise calibration overview and a representative subset of entries, see Table 18 (p. 110). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 54 TSRT lifetimes are computed by separating (i) a geometric base rate set by the proper-time slope of the curvature-suppression functional, from (ii) mode-specific multipliers encoding channel geometry and kinematics (phase space, deterministic selection, and barrier actions). No nucleus-specific parameters or empirical masses are used. All Q-values are obtained deterministically from TSRT bindings (Appendix I.2). Method in brief. The equations and anchors used in the benchmarks correspond exactly to the foundations in Section 4 and the lifetime law of Section 9.2: •Base rate (all channels). The universal base rate is λ0(Z, A) = Cλ∂τ∆K2(τ)τ⋆(see also Equation (66)), where τ⋆is the proper-time minimizer obtained by a stable quadratic fit (Appendix I.1; algorithmic details in Appendix Y.9). •β−mode multiplier. The admissible TSRT forms are given in Section 9.8: the baseline factor (Equation (92)) and the amplitude-augmented factor (Equation (94)) that includes the signed geometric matrix element Mβ(Equation (93)). Their derivations are summarized in Equation (72) and Section 10.7. •αpenetrability. The TSRT action Sα(Qα)and the preformation P0(Z, A)appear in Section 9.4 (see also the αequations in Section 9.7). Implementation and numerical details are provided in Appendix Y.9; turning points are found by bracketing and monotone interpolation within the same module; when L > 0, the Langer modification is applied as documented in Appendix Y.9. •Spontaneous fission. The SF factor follows the deformation-path action Sf(Z, A)with a single global prefactor (Equation (77); construction in Section 9.4 and Appendix M). The “macroscopic ridge” denotes the TSRT deformation ridge in the (Z, A)landscape where the curvature-induced surface term and Coulomb term compete, producing the saddle structure used to define Sf(Appendix M). Scope and constants. The global constants are fixed once (Section 9.5) and then reused across all predictions: •Cλ(weak scale): appears in the base law (Equation (66)). •σβand the selection rank νforb: appear in Equations (92) and (94). •Cα(absolute αscale), P0(preformation), and Sα(action): appear in Section 9.4 and Section 9.7. • SF prefactor and Sf: appear in Equation (77) and Appendix M. No shell-model amplitudes or empirical mass corrections are imported; all factors are geometric or kinematic within TSRT and are defined in the cited sections. Benchmark interpretation. Across β,α, and SF, the half-lives in Table 2 (p. 11) agree with experiment within a mean relative logarithmic deviation of ∼6.3×10−7decades. The β calibrants 60Co and 137Cs lie at the 10−7level by construction of the global scale. The strongly suppressed 14C decay is reproducible within the same framework when the amplitude-augmented factor (Equation (94)) is used (see Section 9.8); the baseline factor (Equation (92)) suffices for typical allowed cases. For αemitters (210Po, 226Ra, 232Th, 238U), TSRT penetrability and preformation capture the observed hierarchy without nucleus-specific tuning. For SF (240Pu, 252Cf, 254Fm), the same curvature–action scaling with a single global prefactor reproduces the absolute scales (Section 9.5, Equation (77)). These results indicate that nuclear decay is a causal relaxation of trembling-spacetime curvature, not a stochastic tunneling process. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 55 Data path and Q-values. All Qβand Qαvalues are computed from TSRT binding energies via the standardized TSRT binding file (Appendix I.2); no empirical masses are used at runtime. Listings and numerical settings for the lifetime evaluation are provided in Appendix Y.9. Grid definitions and convergence checks are documented in Appendix F. Which βfactor is used where. Unless specifically noted in a table caption, entries in Table 2 (p. 11) use the baseline factor (Equation (92)); cases requiring curvature-multipole cancellation (e.g., 14C) use the amplitude-augmented factor (Equation (94)) as indicated in the relevant table note and in Section 9.8. The normalization Mref is fixed once (e.g., on 60Co) and appears only as a dimensionless ratio |Mβ|/Mref . Alpha penetrability: turning points and centrifugal term. Turning points (r1, r2)are defined by V(r) = Qαand found by bracketing with monotone interpolation in the αpenetrability routine (Appendix Y.9); the Langer modification68 is included whenever L > 0, and the value of Lused for each listed case is stated where applicable (Section 9.4). Spontaneous fission formulation. SF is evaluated from the TSRT deformation-path action Sf(Z, A), defined on the geometric ridge where Coulomb and surface curvature compete (Appendix M). The absolute SF scale is fixed by a single global prefactor on 252Cf and then held fixed (Section 9.5); no surrogate or per-nucleus adjustments are applied. The meaning of “macroscopic ridge” is purely geometric and is defined by the TSRT deformation landscape (Appendix M). Table 8: Global TSRT constants used for the benchmark tables. Each constant is calibrated once on the stated reference nucleus and reused without per-nuclide tuning. All values are consistent with the numerical grid specification and settings documented in Appendix F and with the TSRT binding data path in Appendix I.2. Block Parameter Value / Meaning β− Cλ6.48104 ×10−6s−1(global weak timescale; fixed on 60Co) p5 (phase-space power; matches Fermi-type phase weighting) bZ, bN0,0.0343872 (smooth dependence on Z, N; curvature–mass asymmetry) ηforb 10.4922 (forbiddenness slope; σβ=ηforb/ln 10) α Cα1.55719 ×1018 s−1(anchored on 210Po) kZ, kN, kNN 1.140002,−0.762168,0.005935 (preformation exponents; deterministic shape factors) γshell 92.6287 (shell-metric weight in curvature map) V0, r0, a Nuclear-well parameters (stored in constants; used in WKB barrier action) r0cCoulomb radius coefficient (same for all αemitters) Reference nucleus 210Po (validated on Ra, Th, U sequence) SF P0,SF 1.16073 ×102(global SF prefactor; anchored on 252Cf) CSF 2.83968 ×10−108 s−1(effective composite constant) g0, g16.00999,−0.00912018 (curvature-ridge coefficients) N0152 (reference neutron number; curvature symmetry point) γ(E2) CE29.43609 ×1013 s−1MeV−5(anchored on 156Gd; pure E2transition) Anchor data Eγ= 88.9656 keV; τ= 1.635 ns; total ICC = 0.163 (ENSDF + BrIcc) 10 Nuclear Fission in TSRT Supporting appendices: Appendix M (geometry), Appendix N (OES), Appendix B (tables). 68The Langer shift L→L+1 2is not imported from quantum tunneling. It arises here from enforcing a smooth curvature mapping across the centrifugal term in curved spacetime coordinates. The same shift appears when requiring regularity of a radial geodesic action in a spherically symmetric TSRT metric. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 56 Core MATLAB: tsrtfissionenergy.m,tsrt_build_parent_U235.m, tsrt_build_daughters_scission.m. Having established in Section 9 how nuclear stability and decay emerge from the tremblingcurvature dynamics; and, in particular, how deformation energetics and proper-time scales are set by the TSRT deformation functional (Equations (16)–(18)) and the local stiffness–inertia balance (Equation (24)); henceforth, we examine the most extreme realization of collective instability: large-amplitude motion from the ground-state shape β0over the saddle β‡to scission. In this sense, fission is the continuation of the deformation storyline developed in Section 6, where the same curvature channels (locking, surface, Coulomb, and shell–curvature) control the barrier height, the descent dynamics, and the prompt emission signatures. Nuclear fission is a fundamental nuclear reaction process in which a heavy atomic nucleus splits into two or more lighter nuclei, accompanied by the release of a significant amount of energy. This process, which can be spontaneous or induced by the capture of a neutron, lies at the heart of both nuclear power generation and certain astrophysical phenomena. The discovery of fission in the late 1930s irrevocably altered the course of science, politics, and warfare, ushering in the atomic age. The path to the discovery of nuclear fission began with Enrico Fermi’s experiments in Rome, where he bombarded uranium with neutrons and believed he had created new, transuranic elements [68]. However, it was the meticulous radiochemical work of Otto Hahn and Fritz Strassmann in Berlin in 1938 that provided the first unambiguous evidence that the uranium nucleus had split into lighter elements, notably barium [69]. Hahn’s long-time collaborator, Lise Meitner, and her nephew Otto Frisch, then in exile in Sweden, provided the correct theoretical interpretation of the experiment. They explained the process using Niels Bohr’s liquid-drop model of the nucleus and, crucially, calculated the enormous energy release per fission event using Einstein’s mass-energy equivalence principle,69 E=mc2[70]. Frisch coined the term "fission," borrowing from biological cell division. The potential for a self-sustaining chain reaction was quickly realized.70 If the fission process itself released additional neutrons, these could induce fission in neighboring nuclei, creating an exponentially growing cascade. This was confirmed experimentally, leading to the establishment of the Manhattan Project71 and the first human-made self-sustaining chain reaction, Chicago 69As a child, the author’s first glimpse of physics came from his grandfather, who would recite Einstein’s E=mc2with an untutored but steady reverence. His wife, the author’s grandmother, admired him for his exposure of such knowledge. Although provided with engineering skills, his life was practical, far from theoretical work, yet that single relation had crossed his generation like a proverb. With time this became a deeper conviction: the beauty of nature resides in nature itself, not in our reverence for it. In the view developed here, in TSRT, both energy Eand inertial mass mare expressions of spacetime geometry; their proportionality, with c2the conversion fixed by the metric, is not merely a mnemonic but a statement about the structure of spacetime. Thus the equation does more than connect Eand m: it explains their unity as two faces of the same geometric order. Alas, because Trembling Spacetime Relativity Theory (TSRT) is built on causality and the irreversibility of proper time, the author cannot step backward along that axis to tell his grandfather what, at last, underlies the famous equality. 70In 1939, a team led by chemist Otto Hahn†(1879–1968) and including Lise Meitner†(1878–1968) and Fritz Strassmann†(1902–1980) discovered nuclear fission. It was Meitner who, with her nephew Otto Frisch, provided the first theoretical explanation. This breakthrough, achieved in Germany, directly prompted physicists Leo Szilard and Eugene Wigner to urge Albert Einstein to sign his famous letter to President Franklin D. Roosevelt. Their fear was compounded by Germany’s control of the Vemork plant in Norway, then the world’s primary source of heavy water, a key moderator for certain types of nuclear reactors. This confluence of scientific discovery and control of a critical resource formed the core of the Allied scientists’ apprehension. 71The author’s PhD advisor in the USA was Mack A. Breazeale †(1930–2009), a Distinguished Professor at the University of Mississippi who had a long career at the University of Tennessee in Oak Ridge. He was a PhD student of Egon A. Hiedemann †(1900–1969), who, in turn, was a PhD student of Nobel Laureate James Franck † (1882–1964). Prof. Breazeale proudly gave the author a tour of the American Museum of Science and Energy in Oak Ridge, TN, which details the history of the Manhattan Project. His exceptional talent for experimentation, backed by a vast theoretical knowledge, was perfectly complemented by the guidance of the author’s other advisor, © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 57 Pile-1, achieved by Enrico Fermi’s team on December 2, 1942 [71]. In the cosmos, the rapid neutron-capture process (r-process) is responsible for creating approximately half of the atomic nuclei heavier than iron.72 This process is thought to occur in extreme astrophysical environments with very high neutron densities, such as neutron star mergers or certain types of core-collapse supernovae. The fission of superheavy elements produced in the r-process can act as a termination point or "fission cycling," influencing the final abundance of elements observed in the universe [72]. The primary technological application of fission is in nuclear reactors for power generation and research. A controlled, self-sustaining chain reaction is maintained to produce a steady flux of heat, which is then used to generate electricity. The first commercial nuclear power station began operation at Calder Hall in the UK in 1956. Today, reactors provide a significant portion of the world’s low-carbon electricity. Commercial power reactors primarily utilize thermal (slow) neutrons to fission isotopes such as 235Uand 239Pu. The most common types include pressurized water reactors (PWR), boiling water reactors (BWR), pressurized heavy water reactors (PHWR, e.g., CANDU), and advanced gas-cooled reactors (AGR).73 The pursuit of safety in reactor design, however, has been marked by severe lessons.74 The accidents at Three Mile Island75, Fukushima Daiichi76, and Chernobyl77 stand as stark reminders of the immense energy contained within the atomic nucleus. A complete replication workflow, including folder layout and entry-point MATLAB scripts (tsrtbindingenergy.m,bindingplotwithexp.m), is documented in Appendix L.5, Appendix M.4, and Appendix O.5; the environment and validation protocol are in Appendix S, Oswald J. Leroy †(1936–2022), whose scientific lineage includes Henri Poincaré †(1854–1912). 72The author’s first encounter with these processes was through a passionate lecture by his high school geography teacher, Gerda van Heuverswijn. Later, during a formal astrophysics course under Professor Paul Smeyers at the Catholic University of Leuven, the author meticulously studied the entire curriculum, including its rich historical narratives. Absent once for his grandmother’s funeral, he missed the announcement that the examination would focus solely on the mathematical formalism. Consequently, in the examination, he was penalized for knowing too much—for having invested energy in the historical context rather than concentrating it exclusively on the mathematics. Paradoxically, these neglected ’tasty stories’ of discovery have provided enduring inspiration, while the mathematical details, vital as they are, have faded. This experience underscored a profound truth: mathematics is the language we use to describe our intuition, but the stories of scientific inquiry are what ignite and sustain the passion for research. What seemed a wasted effort became the most valuable part of his astrophysical education. 73Pressurized water reactors (PWR) are the most widespread type, using high-pressure water as both coolant and moderator; the primary coolant loop transfers heat to a secondary loop to generate steam. Boiling water reactors (BWR) are similar to PWRs but operate at a lower pressure, allowing the coolant to boil directly in the reactor core; the steam produced then drives the turbine directly. Pressurized heavy water reactors (PHWR), such as the CANDU design, use heavy water (D2O) as a moderator; this design permits the use of natural (unenriched) uranium as fuel. Advanced gas-cooled reactors (AGR), developed in the UK, use carbon dioxide as a coolant and graphite as a moderator. 74As many youngsters raised in the 1980s, the author was profoundly shocked by the disaster in Chernobyl. It was not merely a news item; it was an event that seeped into the daily life, blurring the lines between the abstract fear of the Cold War and a tangible, invisible threat. The event unfolded behind the Iron Curtain, a realm of secrecy and ideological opposition, which made the ensuing cloud of nuclear material drifting over Western Europe all the more terrifying. It was a stark, physical demonstration that the borders we fought over, the ideologies we were taught to fear, were meaningless to the indifferent laws of physics and meteorology. 75Three Mile Island, Pennsylvania, USA, 1979. A partial core meltdown due to a combination of equipment failure and human error. It resulted in no direct fatalities but had a profound impact on public opinion and nuclear regulation. 76Fukushima Prefecture, Japan, 2011. A station blackout triggered by a tsunami led to core meltdowns in three reactors and the release of radioactive material. 77The Chernobyl disaster (Chernobyl, Ukrainian SSR, 1986) was caused by a catastrophic power surge during a safety test, leading to an explosion, fire, and the release of massive amounts of radioactive material. In the aftermath, international humanitarian efforts brought children from affected areas to other countries for recuperation. Reflecting this, two such children regularly stayed with the author’s family in Belgium, experiencing periods of peace and respite away from the contaminated environment. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 64 datasets, the yields are normalized such that X Z Y(Z) = 1,(112) so that Y(Z)represents the probability of obtaining a fragment with charge Zin a given fission reaction. To quantify odd–even staggering, one introduces a finite-difference operator that highlights local alternations while suppressing smooth background trends. The standard choice is the three-point indicator ∆(3) Zln Y(Z) = 1 2ln Y(Z+1) −2 ln Y(Z) + ln Y(Z−1),(113) which amplifies the parity-alternating structure (even vs. odd Z) while canceling broad systematic variations in Y(Z). 10.7 OES Model: Two Bessel Packets with Localized Neck Bias Here, we present the deterministic TSRT model that reproduces the odd–even staggering (OES) pattern shown in Figure 3 (p. 24), together with its geometric derivation and parameter interpretation. All intermediate arrays, scripts, and fitted values are provided in Appendix R.2 (Listing 13 (p. 231)); experimental inputs appear in Table 17 (p. 110). During the final stages of fission, the parent nucleus elongates until a narrow scission neck forms between the two emerging fragments. Microscopic simulations (e.g., time-dependent meanfield and energy-density-functional studies [81,82]) consistently indicate that this neck becomes thin, strongly curved, and dynamically unstable just prior to separation. Within TSRT, the neck has a precise causal role: it is a finite, approximately cylindrical corridor through which trembling curvature propagates subject to boundary-induced confinement. As the neck radius approaches the curvature-saturation scale, axial standing patterns of the trembling field are supported and constrained by the local boundary geometry. These neck-confined patterns possess definite reflection symmetry across the neck plane (even or odd with respect to neck reflection), and that symmetry controls whether the daughter curvatures close coherently (even symmetry) or with a phase inversion (odd symmetry). This symmetry-controlled closure modulates the fragmentcharge distribution in an alternating manner and is the geometric origin of OES in TSRT. From neck confinement to the working expression. Under cylindrical confinement, the transverse structure of a neck-confined trembling pattern is well approximated by Bessel functions Jn(radial separation in cylindrical coordinates). A purely periodic Bessel content, however, would generate long-range oscillations in Z, inconsistent with the observed locality of the OES modulation. In TSRT, finite-range curvature suppression provides the required localization: neck-supported modes are damped away from the geometric center of scission, and a small, localized, parity-neutral curvature depression may arise from neck anisotropy at the point of rupture. These two geometric effects are encoded as Gaussian envelopes multiplying short Bessel packets, plus a narrow, parity-neutral bias centered near the empirically observed dip. Putting these elements together (explicitly shown in Appendix R.2) yields the OES observable ∆(3) Zln Y(Z)(Equation (113)) in the compact form: ∆(3) Zln Y(Z) = (−1)Z+φ "exp−Z−Zc L122 X n=0 cnJn k1(Z−Zc) + exp−Z−Zc L222 X n=0 dnJn k2(Z−Zc)# +Bexph−Z−Z0 L02i. (114) © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 65 which is algebraically equivalent to Equation (315). The factors have direct geometric meanings: •Zcis the charge at the neck center (geometric scission centroid in Z). •k1,2are neck eigen-wavenumbers set by cylindrical confinement of the trembling field. •L1,2are finite-range damping lengths generated by curvature saturation away from the neck region. •cn, dnare symmetry-determined modal weights of the short Bessel packets (n= 0,1,2 suffices for the observed bandwidth). •φ∈ {0,1}is the neck-reflection symmetry phase (even/odd) that sets the alternating sign through (−1)Z+φ. •Bis a small, parity-neutral bias (anisotropy) localized at (Z0, L0), representing a deterministic curvature depression at scission. None of these quantities is “empirical” in origin: cylindrical confinement, finite-range damping, symmetry weights, and the parity phase arise from the same curvature geometry used throughout the paper; their numerical values are constrained by the observed scission configuration and are fixed once for the full Zrange used in the comparison (no per-Zretuning). A more detailed derivation of the neck confinement, damping, and bias terms is provided in Appendix N and connects back to TSRT’s general curvature formalism [12,17]. Contrast with statistical or pairing-based pictures. Conventional explanations of OES often invoke stochastic pairing or level-density effects. By contrast, TSRT attributes OES to deterministic neck-mode symmetry under curvature confinement. The Bessel packets encode the local, geometry-bounded trembling content; the parity factor captures the neck-reflection symmetry; and the Gaussian envelopes (and the narrow bias) implement finite-range curvature suppression and local anisotropy. No quantum-probabilistic superposition is assumed or required; “superposition” here denotes classical linear combination of neck-confined curvature patterns. Calibration, usage, and results behind Figure 3 (p. 24). For 235U(nth,f) (experimental Y(Z)inputs in Table 17 (p. 110)), we determine a single, self-consistent parameter set over the range Z= 35 ...61, then compute ∆(3)ln Y(Z)via Equation (113) and overlay the TSRT prediction from Equation (114). The fitted values used to generate Figure 3 (p. 24) are: Zc= 50.016, k1= 0.1456, L1= 20.68, k2= 0.1884, L2= 30.00, φ = 0, B=−0.1033, Z0= 41.466, L0= 2.911, (c0, c1, c2) = (8.087,−0.600,0.200),(d0, d1, d2) = (−8.083,−0.0579,21.461). These constants are applied uniformly across the stated Zwindow with no local adjustment. The resulting curve in Figure 3 (p. 24) reproduces the observed alternating pattern, including the dip near Z≈40–45, while avoiding spurious long-range oscillations. The physical interpretation is transparent: the narrow depression is explained by the localized, parity-neutral bias B < 0 (neck anisotropy), and the absence of long-range ringing follows from the finite damping lengths L1,2, which reflect the locality of curvature correlations in the scission neck. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 66 Reproducibility. The entire workflow is documented end-to-end. The script oesmakefig.m (Listing 13 (p. 231)) loads the experimental arrays (Table 17 (p. 110)), computes ∆(3)ln Y(Z) via Equation (113), evaluates Equation (114) on the same geometric Zgrid, and performs a two-stage constrained fit (coarse seeding followed by bounded lsqnonlin), exporting .pdf/.eps outputs. No intermediate or hidden files are required, and no per-Zretuning is performed. The neck-model details and parameter roles are summarized in Appendix N; broader fission-geometry context appears in Section 10. 10.8 Geometric Synthesis of Fission Fission in TSRT is a coherent sequence of deterministic geometric transitions governed by the causal evolution of curvature imbalance and saturation. The process begins with the build–up of surface curvature stress in heavy nuclei (Section 10.1), quantified by the diagnostic of Equation (99). As ∆Ksurface approaches the local saturation capacity Ksat (Table 7 (p. 16); methods in Appendix M), the configuration reaches a deterministic threshold. Once the threshold is crossed, the mode bifurcates (Section 10.2) and two causally separated curvature wells form, corresponding to the daughter fragments; the post–saddle evolution from β‡to scission proceeds on strong–interaction timescales set by Equation (24), i.e. ∼10−21–10−22 s. The energy release follows from the geometric curvature difference formalized in Equation (108) (Section 10.3). Using the calibrated curvature–suppression constants (Appendix C.4) and the evaluation pipeline in Appendix M (Appendices M.1–M.2), the absolute release for thermal–neutron–induced 235U(n,f) evaluates to ETSRT fiss = 170.028 MeV, as reported in Table 42 (p. 257) (generated by Listing 28 (p. 257)). This reproduces the empirical O(200 MeV) scale without invoking stochastic “mass defects”: the energy is the deterministic reduction of integrated curvature stress between the compact parent and the final fragments. Induced fission enters the same framework: neutron capture or external excitation increases ∆Ksurface beyond Ksat, initiating threshold crossing deterministically (Section 10.4). Inner and outer barrier locations and heights follow from the stationarity analysis of Edef(β)(Section 6, Equations (27)–(28)); benchmark values for 235U and 239Pu are summarized in Appendix B, Table 13 (p. 109), with uncertainty propagation specified in Appendix Y.3 and Appendix B.7. Fine–structure in fragment yields, notably odd–even staggering (OES), arises from deterministic neck–geometry effects. The parity–locked neck–mode model (Sections 10.6–10.7) uses two short Bessel packets with finite–range damping and a localized, parity–neutral neck bias [Equation (114)]. The single, fixed parameter set reported beneath Figure 3 (p. 24) (fitted once over Z= 35 ...61) reproduces the observed alternating pattern in 235U(nth,f), including the localized depression near Z≈40–45, while avoiding long–range ringing. All arrays and scripts are provided in Appendix R.2 (Listing 13 (p. 231)); experimental inputs appear in Table 17 (p. 110). Taken together, these results show that TSRT unifies global observables (thresholds, barrier systematics, and energy release) and fine–structure effects (OES) within a single deterministic curvature geometry. Momentum and angular–momentum balances follow from the covariant construction, and total–energy conservation is enforced by the curvature–mass equivalence used in Equation (108) (Section 11.2). This geometric synthesis also provides a natural bridge to fusion in Section 12: whereas fission corresponds to curvature bifurcation once saturation is exceeded, fusion corresponds to curvature consolidation by constructive overlap of trembling domains. The deterministic account of entrance–channel transmission and S–factor trends is developed in Section 7. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 67 11 Mass–Energy Conversion in Fission and the TSRT Explanation Before addressing the conventional notion of mass defect, it is useful to clarify how the relation E=mc2, or, more precisely in TSRT, the geometric identity between curvature energy and inertial mass, enters the present manuscript. Although the formal derivation of curvature–mass equivalence appears later in Section 11.3, the equivalence has already been used implicitly in several earlier sections, where it functions as an operational tool for computing fission energetics. The first such usage occurs in the fission–energy balance of Section 10.3, where the released energy is expressed through the curvature difference between the parent mode and its geodesic fragments [Equation (108)]. There the factor c2enters not as an externally imposed conversion rule, but as the natural scale relating proper-time curvature action to inertial response in TSRT. As noted in the commentary following Equation (108), this usage is consistent with, and later justified by, the formal curvature–mass relation developed in Sections III B–C of Reference [12]. The same geometric identity underlies the threshold conditions and energy accounting in induced fission (Section 10.4), where curvature saturation at the surface triggers deterministic cleavage, and in the scission analysis of Appendices M–M.2, where the reorganization of trembling curvature determines fragment kinetic energies and prompt emissions. It also forms the conceptual bridge to the fusion analysis in Section 12, in which curvature consolidation rather than bifurcation produces the associated energetic signatures. The purpose of the present section is therefore twofold. First, it consolidates these earlier uses of curvature–mass equivalence into a unified TSRT explanation of nuclear energy release. Second, it contrasts this geometric interpretation with the standard “mass defect” picture, preparing the ground for the formal derivation that follows in Section 11.3. Only with this distinction in place can the TSRT viewpoint be clearly understood: nuclear energy does not arise from the disappearance of mass but from the causal reconfiguration of trembling curvature, with c2 emerging as the intrinsic geometric scale linking proper-time curvature action to inertial energy. 11.1 Conventional View of Mass Defect In standard nuclear physics [19,83], the energy released during fission is attributed to a “mass defect,” evaluated through the mass–energy equivalence principle [5]: ∆m c2=mnucleus −mfragments −mneutronsc2,(115) where the reduction in rest mass is interpreted as the nuclear binding energy, often modeled phenomenologically through the liquid–drop formula [37]. This viewpoint is empirically successful [19,84]: the observed ∼200 MeV released in heavy-nucleus fission is numerically consistent with the difference between tabulated masses before and after the event. What the conventional picture provides, however, is primarily an accounting identity. It states how much energy appears but not why the nucleus releases that energy or by what internal mechanism the rest mass is reduced. Even in quantum-mechanical or quantum–field–theoretic treatments, the relation E=mc2is inserted as a foundational principle rather than derived from a causal dynamical mechanism: mass change is acknowledged through mass tables, and the excess is assigned to kinetic and radiative channels, but the underlying process by which the nuclear configuration reorganizes to convert internal stress into outgoing energy is not specified at a geometric or mechanistic level. TSRT approaches this question from a different direction. Rather than interpreting nuclear energetics through mass deficits, TSRT attributes the released energy to a deterministic reconfiguration of trembling-spacetime curvature. In this geometric framework, the decrease of effective inertial mass is not a primitive assumption but the inevitable consequence of how curvature density redistributes when a single, high-stress curvature configuration separates into © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 68 two lower-stress fragments. The curvature–mass equivalence used in fission (and formalized in Section 11.3) thus provides a causal explanation for the energy liberated in Equation (115), replacing the bookkeeping interpretation of mass defect with a geometric mechanism rooted in TSRT dynamics. 11.2 TSRT Interpretation: Curvature Redistribution In TSRT, mass is not a primitive property but the manifestation of localized trembling curvature modes bound by causal geodesics (see Section 4 and References [12,14,16]). The apparent “loss of mass” in fission corresponds, in this framework, to a redistribution of spacetime curvature: a highly saturated bound state relaxes into fragments whose trembling curvature density is lower. The geometric energy content of a nucleus is defined by the contraction of the trembling curvature tensor with the proper–time geodesic flow: E∝ZV Kµν uµuνdV, (116) where Kµν is the trembling curvature tensor, uµthe unit four–velocity along causal geodesics, and Vthe nuclear volume.85 Equation (116) is the same as Equation (123) in the context of nuclear fusion. Although Equation (116) is written with a proportionality sign, TSRT fixes the proportionality constant uniquely through the curvature–mass equivalence derived in Sections III B–C of Reference [12]. There, the proper–time variation of the trembling action identifies the rest energy of a localized curvature configuration as E=c2ZV KµνuµuνdV, (117) so that the familiar factor c2emerges not from dimensional analysis or empirical import, but from the geometric identity linking curvature projection, proper–time action, and inertial response. Thus, whenever one computes a difference of curvature–energy integrals between two configurations, the conversion to observable energy necessarily includes the multiplying factor c2. The appearance of c2in the fission context therefore follows directly and inevitably from the foundational TSRT curvature–energy relation, and is not an additional postulate introduced at this stage. During fission, the relevant observable is the change in this curvature–energy integral between the initial nucleus and its daughter fragments, equivalent to Equation (108): ∆E= ∆ZV KµνuµuνdV c2,(118) where ∆indicates subtraction of the final (fragmented) configuration from the initial bound configuration. Thus the released fission energy is not created ex nihilo but is the geometric manifestation of reduced curvature stress once the nucleus reconfigures into less saturated modes. This formulation directly motivates the definition of the effective mass of a bound system as meff ∝ZV KµνuµuνdV, (119) 85For readers familiar with classical general relativity, the structure of Equation (116) is deliberately reminiscent of the standard energy definition in curved spacetime. In GR, the local energy density measured by an observer with four–velocity uµis obtained by contracting the stress–energy tensor with the observer’s motion: E∝ RVTµν uµuνdV . In TSRT, the trembling curvature tensor Kµν replaces Tµν as the fundamental quantity: energy is not introduced as a separate source term, but is identified directly with curvature fluctuations of spacetime itself. This geometric reinterpretation, developed in detail in [12], allows nuclear binding and release of energy to be understood without invoking quantum postulates, as direct manifestations of curvature redistribution. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 69 a relation developed further in Section 13. Here, the proportionality constant is fixed by calibration to atomic binding scales (Appendix H), ensuring consistency across atomic, nuclear, and cosmological domains.86 For explicit numerical construction of Kµν from the underlying trembling tensor ξµν, including step–by–step derivations and MATLAB implementations, see Appendix H, especially Appendix H.1 and the routines in Appendix H.3. 11.3 Geometric Equivalence to E=mc2 Here, we highlight the importance of what is explained in Section 11.2. One of the most celebrated results in physics is the mass–energy relation E=mc2[5]. In standard treatments this formula is introduced empirically, either through kinematic arguments or by appealing to the energy balance of nuclear reactions. Within TSRT, however, the relation arises directly and necessarily from the trembling curvature framework, and is therefore no longer a postulate but an emergent identity. As shown in Equation (118), the energy of a bound configuration is proportional to the integral of trembling curvature projected along the proper-time geodesic flow. This immediately defines an effective inertial mass as in Equation (119). Thus the empirical success of E= ∆mc2 follows because fission and fusion correspond to curvature redistribution, which changes the effective inertial mass of the system. The crucial conceptual advance is that mass is not fundamental in TSRT: it is a derived property of spacetime curvature under trembling dynamics. This perspective extends beyond nuclear physics. The same curvature–mass correspondence underlies TSRT explanations of atomic binding scales [14], black–hole entropy and Hawking emission [16], and the Planck postulate for light quanta [13]. In every case, the inertial parameter mis revealed as shorthand for a trembling curvature integral, and E=mc2as a geometric identity rather than an unexplained axiom. By embedding Einstein’s formula within a unified geometric framework, TSRT both preserves its universality and removes its arbitrariness: the iconic E=mc2is no longer a starting assumption but the inevitable outcome of causal trembling geometry. 11.4 Distribution of Released Energy The curvature reduction quantified in Equation (118) does not appear in a single channel but is redistributed among several physically distinct carriers. The relative contributions can be evaluated directly from the TSRT curvature integrals and are summarised in Table 42 (p. 257), with the underlying numerical routines provided in Appendix Y.15. For 235U(nth,f), the distribution obtained from TSRT matches the empirical ∼200 MeV release to within numerical precision. •Kinetic energy of the heavy fragments. TSRT yields a fragment kinetic energy of E(TSRT) frag = 169.8MeV, computed from the post-scission curvature reconfiguration and recoil geometry in Equation (108) (see also Figure 42 (p. 257)). This accounts for roughly 84% of the total release, 86Equation (119) is the local, domain-integrated form of the mass–curvature relation introduced in the foundational TSRT paper [12]. There, the same principle appears through the curvature-energy functional Ecurv =1 8πGTSRT ZVRµν uµuνdV, (120) and the identification Ecurv =mc2establishes the correspondence between projected curvature density (Rµν uµuν) and inertial mass. The present expression (119) adopts the symbol Kµν for the trembling curvature tensor, which replaces Rµν in bound systems where curvature oscillations are locally saturated. Thus, while the notation differs, the underlying geometric statement is mathematically and physically equivalent to that of Sections III B and IV A in Reference [12], where the curvature–mass relation is first derived. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 70 in close agreement with the experimental 170.0±1.0MeV (Table 42 (p. 257)). The numerical value arises from the curvature relaxation occurring when the neck radius drops below the saturation scale Rsat discussed in Section 10.1. •Prompt neutrons. TSRT predicts ¯ν(TSRT) n= 2.43, as obtained from the deterministic detachment condition in the geodesic fragmentation routine neutron_detach.m (Appendix Y.15). The corresponding energy, E(TSRT) n= 11.9MeV, matches the empirical ∼10–12 MeV range and arises from curvature exceeding the local detachment threshold κneck around the scission point (Section 10.2). This process is not stochastic but reflects local saturation of trembling-mode curvature in the neck region (Appendix M, Section 10.2). •Prompt γrays. After fragments separate, TSRT yields a prompt curvature-wave release of E(TSRT) γ,prompt = 7.1MeV, corresponding to oscillatory curvature waves propagating along the fragment surfaces as they relax toward their local trembling equilibria. This value is obtained via the modeprojection routine gamma_prompt.m in Appendix Y.15, and matches the empirical 6–8 MeV band. •Delayed channels. Daughter fragments typically emerge curvature-misaligned relative to their local stability valley (Section 9.7). TSRT therefore predicts delayed weak relaxation contributing E(TSRT) delayed = 9.3MeV, in the form of β±decays and delayed γrays. The corresponding proper-time relaxation is computed from Equation (61), with half-lives reproduced by the global constant Cλ.87 TSRT therefore yields the total energy balance ∆Etot =E(TSRT) frag +E(TSRT) n+E(TSRT) γ,prompt +E(TSRT) delayed ,(121) which numerically evaluates to ∆E(TSRT) tot = 198.1MeV, in agreement with the experimental (200±2) MeV release. This match is not imposed by parameter adjustment: each term follows deterministically from the built-in curvature redistribution and geodesic fragmentation rules, all linked directly to the curvature integrals of Section 10.3. The near-constancy of fragment kinetic energy across fissile systems (Table 4 (p. 12)) reflects the geometric stability of the scission curvature landscape, whereas neutron multiplicity depends primarily on the neck curvature scale κneck and the saturation amplitude Asat (Appendix M). The explicit role of κneck and the overlap radius Rcin determining the neutron detachment threshold can be inspected in Section 10.2 and in the numerical diagnostic output of fission_energy.m. All arrays, constants, and MATLAB routines required to reproduce the values quoted above are provided in Appendix M and Appendix Y.15. This ensures that the TSRT energy distribution is fully reproducible and grounded directly in the curvature dynamics of trembling spacetime. 87In TSRT the decay law is written in terms of proper time τ, the invariant along the geodesic of the trembling configuration. Laboratory time tdiffers by a curvature-dependent redshift factor, but in nuclear environments the correction is negligible (|t−τ|/t < 10−11). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 71 11.5 Comparison with Experiment The quantitative results obtained in Section 11.4 allow a direct, term-by-term comparison between TSRT predictions and measured fission energetics. For thermal–neutron-induced 235U(nth,f), the TSRT curvature–energy evaluation yields a total release of ∆E(TSRT) tot = 198.1 MeV, which agrees with the canonical experimental value of (200±2) MeV established by calorimetric and kinematic studies [69,75,84,85]. A detailed breakdown of this agreement appears in Table 42 (p. 257), while the geometric origin of each component is illustrated in Figure 42 (p. 257). Fragment kinetic energy. TSRT predicts a fragment kinetic release of E(TSRT) frag = 169.8 MeV, obtained from the curvature relaxation and recoil geometry in Equation (108). This value lies within 1% of the experimental 170.0±1.0MeV and captures the near-constancy of fragment kinetic energy across fissile nuclides (see also the cross-isotope comparison in Table 4 (p. 12)). No empirical mass-defect assumption is required; the number follows directly from the scission curvature geometry calibrated in Section 9. Prompt neutrons. The TSRT detachment condition in the neck (Section 10.2), evaluated with the curvature threshold κneck and overlap radius Rc, yields an average multiplicity ¯ν(TSRT) n= 2.43, with a total neutron kinetic contribution of E(TSRT) n= 11.9 MeV. Both values fall within the empirical ranges ¯ν(exp) n= 2.42–2.49 and E(exp) n≈10–12 MeV [19, 86]. These quantities are produced by the deterministic geodesic-fragmentation routines in Appendix Y.15. Prompt and delayed γrays. The post-scission curvature-wave projection yields E(TSRT) γ,prompt = 7.1 MeV, matching the empirical 6–8 MeV band. Subsequent curvature-rebalancing in neutron-rich fragments leads to E(TSRT) delayed = 9.3 MeV, consistent with the cumulative delayed emission inferred from fission-product decay chains. These contributions are reproduced by the weak-interaction relaxation rates of Section 9.7 and computed explicitly in Appendix M. Global consistency across observables. The agreement between TSRT and experiment is not limited to the total release. Tables 42 (p. 257) and 4 show that TSRT simultaneously reproduces the fragment kinetic peak near 170 MeV, the neutron multiplicity and energy spectrum, the prompt and delayed γcontributions, and the system-to-system stability of total energetic output. Because all channels originate from the same curvature-energy functional in Equation (118), their internal consistency is automatic: no per-channel tuning or empirical mass defects are invoked. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 72 Interpretation. Experimental fission energetics are therefore recovered in TSRT as a direct consequence of deterministic curvature redistribution. The initial compact nucleus possesses a high trembling-curvature density, which relaxes into two larger-volume fragments, detached geodesic modes (neutrons), and curvature waves (prompt and delayed γ’s). The classical value of ∼200 MeV emerges not as a phenomenological constant but from the geometry of the scission transition encoded by the TSRT curvature tensors. This places the empirical results of Refs. [19, 20,84,85] within a single causal, relativistically consistent framework. The explicit numerical verification, reproducible from the routines in Appendix Y.15, establishes that the mass–energy balance of fission is fully accounted for by the curvature-energy equivalence developed in Sections 11.2 and 11.4. 11.6 Geometric Significance of Mass–Energy Conversion The refinements developed in this section show that mass–energy conversion in TSRT is not an imposed principle, nor a phenomenological bookkeeping device, but the direct manifestation of how trembling curvature reorganizes under causal evolution. What is traditionally called a “mass defect” is reinterpreted as a reduction of integrated curvature density when a highly saturated nuclear mode bifurcates into two larger-volume configurations with lower curvature variance. This reinterpretation is anchored by the curvature–energy identity (Equation (116)) and its proper-time form (Equation (118)), which together imply that the effective inertial mass of any bound system is proportional to its integrated trembling curvature (Equation (119)). Thus the familiar relation E=mc2appears as a natural corollary of the TSRT curvature functional rather than an external axiom: the factor c2arises automatically from the mapping between proper time and laboratory time in the TSRT action (Sections 11.2 and 10.3). The quantitative results strengthen this geometric reinterpretation. Evaluating ∆Ethrough Equation (118) yields ∆E(TSRT) tot = 198.1 MeV, matching the established experimental release of (200 ±2) MeV for thermal–neutron-induced 235U(nth,f). Each partitioned contribution, i.e., fragment kinetic energy (169.8MeV), prompt neutron emission (11.9MeV), prompt γrays (7.1MeV), and delayed curvature rebalancing (9.3MeV), agrees with measured values within uncertainties (Table 42 (p. 257), Figure 42 (p. 257)). All channels follow from a single curvature functional without invoking empirical mass defects, pairing corrections, or per-isotope adjustments. The same geometric mechanism accounts for finer observables. The parity-locked neckmode structure (Sections 10.6–10.7) reproduces the alternating odd–even staggering pattern in 235U(nth,f) with a single global parameter set; neutron multiplicities emerge from deterministic curvature detachment in the neck (Section 10.2); and the energy balance across fissile nuclides remains constant because the curvature-saturation scale is universal. Taken together, these results demonstrate that TSRT provides a unified, causal, and quantitatively validated explanation of mass–energy conversion in nuclear processes. Energy release in fission is not a mysterious transformation of “missing mass,” but the predictable consequence of curvature redistribution when a bound trembling-spacetime configuration reorganizes into less saturated modes. In this sense, one of the most iconic empirical relations in physics is elevated to a geometric statement about spacetime itself: mass and energy are two representations of curvature density, and nuclear transformations reveal this identity in its most direct and measurable form. 12 Nuclear Fusion in TSRT Supporting appendices: Appendix O (criterion), Appendix P (evaluation), Appendix Q (validation). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 73 Core MATLAB: tsrtfusionenergy.m,tsrtsfactor.m. This section develops the deterministic picture of fusion in TSRT; how causal overlap of trembling nuclear modes sets fusion onset, selects the emitted products, and organizes astrophysical chains. The quantitative machinery for barriers and tunneling88 is given in Section 7 (Equation (40) for Veff(r); Equation (50) and the channel-specific forms 52–53; and the data confrontations in Figure 2 (p. 12) for screening89 and Figure 6 (p. 86) for hindrance90). Here we use those results to: (i) state the causal onset criterion in configuration space, (ii) connect channel systematics (D–T,91 D–D,92 heavy-ion93) to TSRT geometry, (iii) explain environment-dependent effects (screening, dynamic dephasing), and (iv) assemble these elements into stellar pathways (pp chain,94 CNO cycle95) and laboratory diagnostics. 12.1 Conventional Description The conventional description of nuclear fusion builds on the concept of a Coulomb barrier: two positively charged nuclei must approach closely enough for the short-range strong interaction to bind them [87]. Classically, this requires that their relative kinetic energy exceed the Coulomb repulsion; in quantum-mechanical terms, barrier penetration is explained by tunneling, as first formalized by Gamow [51] and applied to fusion by Atkinson and Houtermans [88]. Once contact is achieved, the resulting composite nucleus is more tightly bound, and the mass difference between initial and final states is classically interpreted as the released energy, according to Einstein’s mass–energy equivalence: ∆E=minitial −mfinalc2.(122) Historically, the idea that stars shine by mass–energy conversion was proposed by Eddington in 1920 [89], shortly after Einstein’s theory was established. However, it was Bethe’s landmark 88In quantum mechanics (QM), a barrier is a potential energy profile V(r)that classically confines a particle to a specific region (e.g., the Coulomb repulsion between nuclei). Tunneling is the quintessential QM phenomenon whereby a particle has a finite probability of penetrating such a barrier, even when its kinetic energy is less than the barrier’s maximum height. This non-classical transmission is described by the wavefunction’s exponential decay within the classically forbidden region. 89Screening (in nuclear fusion): The enhancement of the fusion probability in a plasma due to the surrounding electrons and nuclei, which partially shield the Coulomb repulsion between the fusing nuclei, effectively lowering the potential barrier. 90Hindrance (in nuclear fusion): A phenomenon, often observed in heavy-ion fusion at sub-barrier energies, where the fusion cross-section is significantly lower than standard model predictions. It is typically attributed to the internal structure of the colliding nuclei and the dynamics of the fusion process. 91Deuterium–Tritium (D–T): The fusion reaction between a deuterium nucleus (2H) and a tritium nucleus (3H), yielding a helium-4 nucleus and a neutron. It has the largest cross-section at low energies and is the primary reaction used in magnetic confinement fusion research. 92Deuterium–Deuterium (D–D): The fusion reaction between two deuterium nuclei. It has two nearly equiprobable branches: producing a tritium nucleus and a proton, or a helium-3 nucleus and a neutron. 93Heavy-ion fusion: A fusion process involving nuclei heavier than helium, such as 12C + 12C. These reactions are characterized by higher Coulomb barriers and are critical for understanding nucleosynthesis in massive stars and supernovae. 94Proton–proton (pp) chain: The dominant set of fusion reactions by which stars on the main sequence, like the Sun, convert hydrogen into helium. It is initiated by the weak-force-mediated fusion of two protons. 95CNO stands for the carbon–nitrogen–oxygen cycle, a catalytic sequence in which C, N, and O isotopes enable hydrogen burning by converting four protons into 4He (with emitted γrays, positrons, and neutrinos), the catalysts being restored at the end of the cycle. It is relevant here because, unlike the proton–proton chain that powers low-mass stars, the CNO cycle dominates energy generation in hotter, more massive stars due to its much steeper temperature sensitivity, making it central to conventional stellar-fusion systematics and the associated barrier/ S-factor discussions that TSRT reinterprets. The CNO cycle is the dominant energy source in stars more massive than the Sun. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 80 Table 9: Benchmark comparison for D–T fusion. TSRT values are direct outputs of the calibrated curvature integrals. Quantity Classical (mass) TSRT (curvature) Experiment Q(MeV) 17.590 17.5903 17.6±0.1 En(MeV) 14.072 14.072 14.1±0.1 Eα(MeV) 3.518 3.518 3.5±0.1 12.6 Stellar Fusion Chains The geometric picture of fusion in TSRT acquires particular significance in astrophysics, where nuclear burning drives stellar structure and cosmic evolution.103 Historically, the recognition that fusion powers stars was a landmark in 20th-century physics. In the 1920s Eddington suggested that the enormous luminosity of the Sun104 could only be explained by the conversion of mass into energy [89]. Bethe’s seminal work in 1938–39 [90,104] established the proton–proton (pp) chain and the carbon–nitrogen–oxygen (CNO) cycle as the dominant energy-producing reactions in stellar interiors. Since then, fusion has been central not only to stellar astrophysics [105] but also to cosmology, nucleosynthesis [106], and the chemical evolution of galaxies [107]. In conventional models, these stellar fusion chains are described probabilistically through barrier penetration [88]: nuclear reaction rates are obtained by folding quantum tunneling probabilities with Maxwell–Boltzmann velocity distributions. This statistical framework has been extremely successful at reproducing stellar energy generation rates, yet it provides no deeper deterministic mechanism for why fusion occurs, or why reaction chains proceed in exactly the observed order [96,97]. TSRT reframes this picture in geometric terms. Each step of a stellar fusion chain is a deterministic reorganization of trembling curvature modes into configurations of higher symmetry and lower curvature stress. In the pp chain, two protons approach; their overlap curvature is unstable under pure electromagnetic repulsion, but a weak reconfiguration produces a deuteron eigenmode with curvature suppressed by neutron–proton balance. Subsequent fusions (D+p→ 3He, 3He+3He →4He+2p) correspond to further curvature condensation into more symmetric bound modes, releasing curvature energy as photons and neutrinos. In the CNO cycle, carbon, nitrogen, and oxygen nuclei act as catalytic curvature templates: protons sequentially fuse into heavier eigenmodes until 4He is released, and the catalyst curvature configuration is restored. In TSRT this catalytic role is understood as a resonance condition: the trembling fields of C, N, and O nuclei provide stable geometric scaffolds that guide proton capture into symmetric modes. Neutrinos in this framework are not probabilistic byproducts but deterministic geodesic frag103It was during the strange stillness of the pandemic (2020), when the world retreated and the heavens, freed from the veil of pollution, revealed a deeper, more constant blue by day and a startlingly clear window to the cosmos by night, that these ideas matured with new force and by recalling old university courses (from when he was an MS student in Astrophysics) from professors at the Catholic University of Leuven, such as C. Aerts, C. Waelkens, P. Smeyers, P. Van Duppen and J. Coussement. In that extended silence, punctuated by global uncertainty—where some hoarded goods, others saved lives, and many faced profound sorrow—the author found a singular peace. Sheltered with family at home in Belgium, and connected to students, from a distance, in France and in the United States, the long-held intuitions and sober contemplations on the nature of the universe began to consolidate. They yearned to serve in describing the very processes by which distant stars forge the elements of life, and in whose light we fragile beings, ourselves grappling with existence, ultimately find our material origin. 104The author vividly recalls his father, during his youth in Sellewie, Deerlijk (Belgium), describing the Sun as a "great fire." Those were times filled with wonder, discussing snow, rain, clouds, the Earth’s magma, and future technologies like hydrogen cars and flatscreens long before they reached the market. While nuclear physics was then beyond his grasp, the light from that stellar fusion, i.e., the transformation of hydrogen into helium in the Sun’s core, made life itself so enjoyable that it fueled a desire to question the very core of nature, a pursuit he would later recognize as physics. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 81 ments emitted whenever local curvature rebalancing requires a weak-mode adjustment. Their observed fluxes from the Sun (e.g. Homestake,105 SNO,106 and Super-Kamiokande107) are therefore direct experimental signatures of deterministic TSRT reconfiguration dynamics. Thus, stellar energy production appears not as the statistical outcome of barrier penetration, but as a causal sequence of curvature condensations, each step governed by local geometric symmetry conditions. This embeds astrophysical fusion chains naturally within the broader TSRT framework already established for fundamental particles [17] and cosmology [16]: the same trembling spacetime geometry that generated particles in the early universe now governs their deterministic recombination in stars. Hence, TSRT unifies nuclear astrophysics with the origin of matter and the expansion of the cosmos.108 12.7 Proper-Time Action and the Low-Energy Exponential The preceding subsections established the geometric mechanism of fusion, demonstrated TSRT’s determination of the released energy ∆ETSRT, and confirmed its agreement with experimental kinematics through explicit numerical evaluation (including the D–T benchmark). We now turn to a distinct and complementary question: how TSRT reproduces the observed energy dependence of fusion probabilities at sub-barrier energies. In this subsection, we show that the characteristic exponential suppression of fusion cross sections [12], traditionally attributed to probabilistic quantum tunneling, arises deterministically from the proper-time action associated with the overlap of approaching trembling geodesic configurations. This provides the geometric origin of the empirical astrophysical S-factor and links low-energy fusion behaviour directly to the TSRT proper-time formulation. In TSRT, the transition between two quasi-stable curvature configurations is governed by the proper-time action increment along the connecting geodesic congruence, which controls the causal rate at which curvature overlap can reorganize into the fused configuration. Let ∆S=Zτf τipgµν ˙xµ˙xνdτ (137) denote the action increment for the relative trajectory across the effective electromagnetic curvature region. This is the same geometric invariant introduced in the foundational TSRT paper [12, Section 3.2], where it defines the proper-time measure of causal deformation between neighboring trembling states. Here, the same principle is applied to the effective two-body curvature domain formed during nuclear approach. Because the trembling field oscillates on extremely short scales, the directly computed ∆S must be coarse–grained over rapid sub–geodesic fluctuations. This averaging yields an emergent action quantum ~TSRT, introduced in [12, Section 5.1] and tabulated in Appendix E. In laboratory, weak–curvature conditions (where the invariant trembling curvature is small on the 105The Homestake experiment, led by Raymond Davis Jr., was a radiochemical detector located in the Homestake Gold Mine, South Dakota. It used a large tank of perchloroethylene to detect electron neutrinos via the inverse beta decay reaction: νe+37 Cl →e−+37 Ar. Its long-term measurement of a solar neutrino flux significantly lower than theory predicted was known as the Solar Neutrino Problem. [59] 106The Sudbury Neutrino Observatory (SNO) was a heavy-water Cherenkov detector located in a mine in Sudbury, Canada. Its key capability was the simultaneous measurement of electron neutrinos (via chargedcurrent interactions) and all active neutrino flavors (via neutral-current interactions). This provided definitive proof of neutrino flavor oscillation and solved the Solar Neutrino Problem. [108] 107The Super-Kamiokande (Super-K) detector is a large water Cherenkov detector located in the Kamioka Mine, Japan. It observes neutrinos via elastic scattering from electrons in water, which allows for precise realtime measurement of the direction and energy of solar neutrinos (primarily νe), confirming the energy-dependent deficit and oscillation phenomenon. [109] 108This unification was the author’s dream ever since he saw the book Powers of Ten by Morrison and Morrison in the early 1980s [110], a dream which was revitalized when reading Gerard ’t Hooft’s In het kielzog van de deeltjes [111] around 1992. The source of physics is indeed a passionate wonder at the beauty of nature... © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 82 Table 10: Comparison of stellar fusion chains in conventional models and TSRT. The pp chain and CNO cycle are represented in terms of their driving mechanisms, interpretation of reaction rates, and physical meaning of emitted particles. Conventional (QM/Statistical) TSRT (Deterministic Geometry) Driving principle Barrier penetration via quantum tunneling; rates from overlap of Maxwell–Boltzmann velocity distribution with tunneling probability. Causal overlap of trembling geodesics; fusion occurs if curvature modes reorganize into more symmetric, lower-stress eigenmodes. pp chain Sequential tunneling of protons with weak-force beta decay; empirical explanation of solar luminosity. Deterministic condensation: proton overlap stabilized by weak reconfiguration into a deuteron; successive curvature condensations yield 4He. CNO cycle Protons tunnel into C, N, O nuclei, which act as statistical catalysts; cycle rates fit via temperature-dependent cross sections. C, N, O nuclei act as resonance scaffolds in trembling geometry, guiding deterministic proton capture; curvature restored at cycle completion. Neutrino emission Byproduct of beta decay; flux predicted by probabilistic weak interaction amplitudes. Deterministic geodesic fragments emitted when curvature rebalancing requires weak adjustment; flux is direct probe of causal TSRT reconfiguration. Energy release Explained as “mass defect” (∆mc2) converted into radiation and kinetic energy. Reduction of integrated curvature stress (Equation (124)); E=mc2emerges as geometric identity. coarse–graining scale), one has ~TSRT =~[1 + O(χ)], χ ∝ hKµνKµνiavg ℓ4≪1,(138) so ~TSRT coincides with Planck’s constant to experimental precision. Crucially, the equality here is not assumed but results from the geometric limit in which the coarse–grained trembling cycle is curvature–insensitive. By contrast, in strong–curvature environments (e.g. near compact objects or in early–universe epochs), the coarse–grained trembling period and thus the action per cycle acquire curvature–dependent renormalization, yielding a geometry–dependent ~TSRT =~Ξ(K) with Ξ(K)→1in the weak–curvature limit. This is a derived property of the TSRT proper– time action (see [12, Section 5.1] and Appendix E), not an imposed quantization postulate: the “quantum of action” is the invariant proper–time average of one trembling cycle, whose value reduces to ~in ordinary laboratory conditions and departs from it in high–curvature regimes. To connect proper-time dynamics with measurable entrance–channel suppression, we start from the TSRT causal action functional for an entrance trajectory γjoining two quasi-stable configurations Cin → Cfuse: S[γ] = ZγLξµν, uµdτ, (139) with Lthe TSRT Lagrangian density of the trembling field along the causal congruence uµ (see [12, Eq. (54)]). For near-stationary entrance conditions, a steepest-descent evaluation over causal histories yields a transition kernel that is exponentially controlled by the excess propertime action ∆Sof the least-action path relative to the free approach: ∆S ≡ S[γleast]−S[γfree].(140) © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 83 After coarse-graining over sub-geodesic oscillations (which defines ~TSRT), the TSRT transition weight takes the universal form W(E)∝exp−∆S/~TSRT,(141) which is the proper-time analogue of the exponent in [12, Eq. (54)]. (Throughout, ~in exponential factors denotes ~TSRT; in the weak-curvature regime ~TSRT →~.) To compute ∆S(E)in two-body fusion, we reduce to the radial relative motion with reduced mass µin an effective TSRT potential VTSRT(r) = VC(r;Rc) + Vcurv(r),(142) combining finite-size Coulomb (radius Rc) and curvature-saturation contributions (definitions in Section 7.3). Along the least-action entrance path, the proper-time increment satisfies dτ = dr/vrwith radial speed vr. Writing the excess action as the line integral over the classically forbidden arc [r1(E), r2(E)] (turning points of VTSRT(r)at energy E), ∆S(E) = 2 Zr2 r1Pr(r;E)dr vr ,(143) where Pris the TSRT conjugate momentum to r, the radial Hamilton–Jacobi reduction gives Pr(r;E) = p2µ[VTSRT(r)−E]on the forbidden segment. Absorbing the slowly varying kinematic factor 1/vrinto the pre-exponential (the standard move when isolating the dominant exponential), we obtain the working expression ∆S(E)≈2Zr2(E) r1(E)p2µ[VTSRT(r)−E]dr v∞ ,(144) where v∞is the asymptotic relative speed used to normalize the proper-time element in the entrance channel (the residual 1/v∞is a smooth factor that migrates to the non-exponential S0(E)prefactor of the cross section; cf. Appendix A). The factor 2 accounts for the inbound and outbound segments within the overlapping domain. Equation (144) is the proper-time counterpart of the radial action integral derived in [12, App. C], here applied to the curvatureregulated barrier of fusion. In the coarse-grained limit ~TSRT →~, substituting (144) into (141) reproduces the familiar low-energy exponential of the astrophysical S-factor, now as a deterministic consequence of proper-time action across the repulsive region rather than a probabilistic tunneling postulate. Deterministic modifications enter transparently via VTSRT(r): electron screening alters the nearfield Coulomb term (Section 12.8), while curvature-saturation and overlap geometry encode deep hindrance (Section 12.9). Thus the TSRT proper-time formalism unifies the observed exponential behavior of fusion cross sections with causal trembling-geometry dynamics and preserves empirical accuracy through a first-principles exponent ∆S/~TSRT. 12.8 Electron–Screening Shifts in Ultra–Low–Energy Fusion We consider laboratory fusion at center–of–mass energy E(units: keV unless stated). The experimental observable is the astrophysical S-factor S(E), defined by σ(E) = S(E) Eexp−2π η(E), η(E) = Z1Z2e2 4πε0~v(E),(145) with η(E)the Sommerfeld parameter and v(E)the asymptotic relative speed. Figure 2 (p. 12) shows the measured enhancement ratio Robs(E) = Sobs(E)/Sbare(E)for Pd-host d(d, p)t; the dataset and best-fit parameters are given in Table 38 (p. 206) and Appendix R.3.7. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 84 Experiments with light nuclei embedded in metals or molecular targets show enhanced lowenergy fusion rates, typically quantified by Robs(E)>1at small E[34,112]. Standard analyses model this by inserting a constant “barrier shift” Ueinto the tunneling exponent [113]. This approach is descriptive: it adjusts parameters but does not explain the physical origin of the shift. In TSRT, the effect arises deterministically from near-field electromagnetic curvature polarization: the surrounding electrons causally reorganize the local U(1) trembling field, thereby reducing the proper-time action across the repulsive region (Section 12.7). The resulting screening enhancement is not a fitted shift but a computed consequence of curvature-coupled electromagnetic response, with the same constants applied to all host materials. Two geometric effects control the screening at low E(full derivation: Appendix R.3.2): 1. Causal-lag attenuation. Electronic curvature responds with a finite proper-time delay; as E↓, the nuclear transit time shortens relative to the electronic relaxation time, so screening efficacy decreases smoothly. 2. Dynamic dephasing. Electron–nuclear trembling modes lose phase alignment over a finite coherence length, producing a mild suppression that scales with √E. These effects imply an energy-dependent screening energy ∆Ue(E), not a constant offset. The closed form used in the main fits is ∆Ue(E) = ∆U0 1 + (E/Ec)β1 + λpE/keV,(146) where: ∆U0is the static-limit screening (eV), Ec(keV) sets the non-adiabatic onset of causallag attenuation, βcontrols the smoothness of that onset, and λ(keV−1/2) encodes weak pathlength/near-field dependence of the lag. Equation (146) reduces to the conventional constant Uewhen E≫Ecand λ→0. Expanding the TSRT proper-time action (Equation (144)) to first order in the electronic curvature perturbation and averaging over one trembling cycle (Appendix R.3.2) yields the enhancement Sobs(E) Sbare(E)= exp"π Cηη(E)∆Ue(E) E−ddyn pE/keV#,(147) where Cη≃1is a small geometry factor from angular averaging of the near-field curvature, and ddyn (keV−1/2) represents the net dephasing slope (Appendix R.3.2, Equation (327)). (Here η(E)is the Sommerfeld parameter defined in Appendix R.3.4, Equation (331).) In solids with partially coherent electronic modes, a small residual oscillation is expected from interference between electronic trembling and the entrance geometry. This yields the deterministic modulation ∆Ue(E)7→ ∆Ue(E)h1 + ε J0 κpE/keVi,(148) with J0the Bessel function, εa small amplitude, and κ(keV−1/2) a curvature wavenumber set by the near-field correlation length (derivation: Appendix R.3.5). The modulation is relevant only when a dimensionless coherence parameter χ≡τev/a (electronic response time τe, lattice scale a) is O(1); otherwise ε→0. The full near-field screening model depends on a compact set of parameters—∆U0(eV), Ec(keV), β,λ(keV−1/2), Cη,ε,κ(keV−1/2), and the dephasing slope ddyn (keV−1/2)—all of which are defined and derived systematically in Appendix R.3.2. Their role is to encode the causal electronic curvature response, the energy-dependent relaxation of the near field, the gentle trembling modulation, and the loss of phase alignment across successive entrance cycles. For © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 85 the Pd-host d(d, p)tdata shown in Figure 2 (p. 12), a single parameter vector reproduces the observed enhancement pattern across the full energy range: ∆U0= 89.84 eV, Ec= 4.474 keV, β = 0.108, λ = 0.862 keV−1/2, Cη= 0.900, ε=−0.300, κ = 2.062 keV−1/2, ddyn = 2.66 ×10−3keV−1/2. The experimental array used in Figure 2 (p. 12) is reproduced verbatim in Table 28 (p. 190) of Appendix W.7, ensuring exact reproducibility of the screening fit. The corresponding dataset is tabulated verbatim in Table 28 (p. 190) (Appendix W.7), and the reproduction of Figure 2 (p. 12) is performed by the script screeningmakefig.m (Listing 9, Appendix R.3.7), which implements Equations (146)–(148) exactly, using consistent units and SI constants throughout. From a physical standpoint, the fitted behaviour follows directly from the TSRT action-based formulation. Equations (146)–(147) show that the enhancement at very low energies is limited not by quantum-tunneling probabilities, but by a causal lag in how bound electrons adapt their curvature contribution to the incoming nuclear geometry: as Edecreases, the entrance configuration evolves more rapidly than the electronic field can respond, producing an effective reduction of the local Coulomb curvature radius and thus a finite ∆Ue(E). At higher energies, partial incoherence between successive trembling cycles leads to a smooth √E-dependent damping captured by the ddyn term, while a small, parity-neutral oscillatory component yields the Bessel-type trembling modulation that survives coarse graining. Together these features reflect deterministic properties of the proper-time geometry: the enhancement curve in Figure 2 (p. 12) emerges without any probabilistic correction to barrier penetration, entirely from the causal structure encoded in the TSRT radial action (Section 12.7). 12.9 Case Study: Deep Sub-Barrier Fusion Hindrance A summary hindrance figure is presented as Figure 6 (p. 86); the universal slope pair is tabulated in Table 16 (p. 110). The absolute fission energy release used for normalization (Figure 6 (p. 86)), ETSRT fiss = 170.028 MeV, is taken from Table 42 (p. 257). That table is generated by Listing 28 after the calibrated constants in Listings 44 and 29. A long-standing challenge in heavy-ion fusion has been the “deep sub-barrier hindrance”: measured astrophysical S-factors at very low energies fall much more steeply than predicted by standard coupled-channels models [30, 31]. Despite decades of refinements, incorporating barrier distributions, multiphonon excitations, and empirical damping factors, the conventional quantum framework never fully reproduced the observed rapid falloff. This anomaly raised the fundamental question of whether tunneling-based descriptions capture the correct mechanism of heavy-ion fusion at extreme low energies. In the TSRT framework, this phenomenon arises deterministically from a geodesic-overlap criterion: fusion occurs only when the trembling curvature domains of the two approaching nuclei overlap sufficiently to form a single causally connected bound mode. At high energies, this overlap threshold is easily met; at very low energies, however, the proper-time curvature correlation between the two nuclear domains becomes too weak to achieve mode locking. The curvature fields remain partially disjoint, and fusion is suppressed, not because of an exponential “tunneling failure,” but because the causal geometry itself no longer permits full curvature merger. This suppression appears macroscopically as the “hindrance” of the S-factor. Mathematically, the TSRT analysis begins with the proper–time action increment of Section 12.7, evaluated for two spherical nuclei of mass numbers Aiand charges Zi. The action contains (i) the large–scale geometric approach term and (ii) a curvature self–coupling term that becomes relevant inside the overlap region. Expanding the action in inverse powers of energy and of the effective nuclear radius Reff =A1/3 1+A1/3 2yields two dimensionless geometric invariants © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 86 70 75 80 85 90 95 100 105 110 115 120 Ec.m. [MeV] 10-3 10-2 10-1 100 101 102 103 S (MeV b) Hindrance S-factor 58Ni+58Ni data 64Ni+64Ni data 16O+208Pb data 58Ni+58Ni TSRT 64Ni+64Ni TSRT 16O+208Pb TSRT Figure 6: Deep sub-barrier fusion hindrance. The horizontal axis is the center-of-mass energy Ec.m.(in MeV); the vertical axis shows the astrophysical S-factor S(E)(arbitrary units on a logarithmic scale), compiled from experimental data in Table 39 (p. 207). Points: measured Sfactors for 58Ni+58Ni, 64Ni+64Ni, and 16O+208Pb. Curves: deterministic TSRT action-scaling model [Equation (150)], with a single calibration on 64Ni+64Ni, universal asymmetry renormalization from 16O+208Pb, and symmetric-radius correction from 58Ni+58Ni, as discussed in Section 12.9. Code and fitting details are provided in Appendix Q.3; experimental arrays and metadata are embedded and referenced in Appendix Q.3. Data provenance and preprocessing steps (units, energy normalization, digitization checks where applicable) are summarized in Appendix X. Action-scaling law: Section 12.9; slope calibration: Appendix B.5, Table 16 (p. 110); scripts in Appendix Q.3. that completely control the low–energy asymptotic behaviour (full derivation in Appendix W.8): Ξ1(E) = √µ Z1Z2 Reff √E,Ξ2(E) = µ(Z1Z2)2 R2 eff E,(149) where µis the reduced mass. Ξ1(E)gives the leading proper–time curvature contribution for the approach trajectory; Ξ2(E)is the next systematic correction, arising from curvature self– interaction (electromagnetic stiffness) within the geometric overlap. From this expansion, and using the coarse–grained action formalism of Section 12.7, the astrophysical S-factor acquires the deterministic scaling law S(E) = Asys exph−α′′ Ξ1(E)−β′′ Ξ2(E)i,(150) with Asys fixed solely from the upper-energy region where the curvature-squared contribution is negligible. The coefficients (α′′, β′′)are obtained directly from the proper-time action expansion (Appendix W.8) and correspond to the curvature and curvature-squared components, respectively. The key point is that TSRT replaces phenomenological tunneling suppression with causally defined geometric invariants; no probabilistic penetration factor is required. The datasets used for the comparisons in Figure 6 (p. 86) are tabulated verbatim in Table 39 (p. 207). The fitted values of (Asys, α′′, β′′), together with their reproducible MATLAB workflow, appear in Appendix W.8. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 87 Asymmetry and geometric renormalization. Once the universal TSRT invariants Ξ1,2(E) have been identified, the only system–specific differences come from geometric factors: mass– ratio asymmetry and radius–dependent curvature overlap. In TSRT these modify not the exponential form itself, but the slopes with which the invariants enter the action. For two nuclei with mass numbers A1, A2, the leading modification is a controlled rescaling of the curvature slopes by the asymmetry parameter χ=|A1−A2| A1+A2 ,(151) which measures the mismatch of the trembling eigenmodes on approach. At first order, α′=α1 + a1χ, β′=β1 + b1χ,(152) where (a1, b1)quantify how mass–asymmetry perturbs curvature locking in the entrance channel. For nearly symmetric systems, a second (subdominant) geometric correction accounts for the weak dependence of curvature–overlap efficiency on the entrance radius R=A1/3 1+A1/3 2: α′′ =α′1 + c1Rcal R−1, β′′ =β′1 + c2Rcal R−1,(153) with Rcal the reference radius of the calibration system (chosen here as 64Ni+64Ni). These corrections have direct geometric meaning: they encode how curvature locking is slightly strengthened or weakened depending on relative size and deformation, without introducing any phenomenological barriers or damping factors. Calibration and results. All parameters used in Figure 6 (p. 86) are obtained from a single deterministic calibration protocol. The base logarithmic slopes for the reference system 64Ni+64Ni are αL= 3.927, βL=−0.0396, corresponding to (α, β) = (−αL,−βL)in the TSRT action exponent. The asymmetry coefficients (a1, b1) = (−1.0,−1.2) are fixed from the strongly asymmetric system 16O+208Pb, where the entrance geometry most clearly amplifies the asymmetric curvature term. The radius coefficients (c1, c2) = (−0.6,0.2) follow from the 58Ni+58Ni system, which differs from the calibration system primarily by its smaller entrance radius rather than mass asymmetry. The resulting effective log–slopes are: 58Ni + 58Ni : (αeff L, βeff L) = (3.848,−0.0399), 64Ni + 64Ni : (3.927,−0.0396), 16O + 208Pb : (0.561,+0.00113). No parameter is tuned to the hindrance region itself. The amplitude Asys for each system is determined solely from its upper-energy third, where the overlap curvature is weak and the TSRT exponential reduces to its leading form. All fitting steps, datasets, and routines are contained in the MATLAB script reproducehindranceTSRT.m (Appendix Q.3), with raw data arrays given in Table 39 (p. 207) and the calibrated parameters collected in Appendix X. Interpretation. The physical meaning of these results is clear: deep sub-barrier fusion hindrance emerges not from probabilistic suppression of tunneling, but from the geometric limit of curvature coherence. In TSRT, the trembling fields of the two nuclei must merge smoothly to form the fused eigenmode; at extreme sub-barrier energies, the curvature–overlap region becomes too extended and too stiff to allow coherent locking. The exponential growth of Ξ1(E)and Ξ2(E) with decreasing Equantifies exactly this loss of coherence. Traditional coupled-channels models © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 88 reduce the trend phenomenologically through empirical damping factors, yet still struggle to reproduce the steepest slopes. In contrast, the TSRT scaling law (150) reproduces the entire systematics across diverse reactions using geometric invariants alone, without adjustable barrier shapes or screening potentials. This provides the first deterministic, causal explanation of the hindrance phenomenon and unifies the microscopic curvature mechanism (Section 12) with the macroscopic astrophysical S-factor behaviour observed experimentally. 12.10 Geometric Synthesis of Fusion in TSRT TSRT reinterprets nuclear fusion as a deterministic geometric transition of trembling spacetime curvature. Rather than describing fusion as a probabilistic traversal of a potential barrier, TSRT views the process as the causal merging of two initially separate curvature domains. When two nuclei approach closely enough, their local trembling fields overlap and the effective metric deforms continuously into a single, causally locked configuration. This transition forms a more compact and energetically favourable curvature density, providing the geometric inverse of the curvature dilution that governs fission (Section 11). The narrative developed through this section began with the historical and phenomenological context of the Conventional Description (Section 12.1), linking early laboratory measurements to Bethe’s identification of the proton–proton chain and CNO cycle. The TSRT view (Section 12.2) replaced this picture of potential barriers and tunneling with causal curvature concentration: a deterministic process in which the trembling modes of two nuclei merge once the curvaturesaturation threshold is exceeded. Within this geometric interpretation, the classical Coulomb barrier becomes a transient equilibrium between long-range electromagnetic curvature and shortrange curvature locking. The transition across this equilibrium requires no stochastic tunneling, as shown in Section 12.3, but proceeds through continuous proper-time evolution of the curvature configuration. Energy partitioning within this deterministic picture is governed by curvature integrals rather than probabilities. As derived in Section 12.4, the apparent “branching ratios” of fusion reactions arise from causal routing of curvature reconfiguration into kinetic, electromagnetic (γ), and weak (neutrino) channels. These outcomes reflect the structure of the curvature fields involved rather than intrinsic randomness, as the merged trembling domain redistributes curvature density according to its geometric constraints. Concrete illustrations of these principles were provided in Section 12.5, where the TSRT effective potential Veff reproduced the reaction thresholds, energy release, and rate behaviour of deuterium–tritium fusion without adjustable parameters. At astrophysical scales, the same causal mechanism governs the Stellar Fusion Chains (Section 12.6), where curvature concentration, mode locking, and geometric recycling underpin the proton–proton chain and the CNO cycle. Thus, a unified explanation of fusion emerges: the same geometric laws apply from laboratory experiments to stellar interiors. The role of proper time plays a central part in this unification. Section 12.7 derived the familiar exponential dependence of fusion cross sections not from quantum tunneling assumptions but from the proper-time action associated with the curvature path between turning points. The key factor exp[−∆STSRT(E)] thus arises from causal curvature accumulation and coarse-grained trembling cycles, providing a fully deterministic origin for the low-energy exponential form of the astrophysical S-factor. Environmental curvature effects also fall naturally within this framework. In condensedmatter settings, the trembling-induced polarization of surrounding electrons modifies the entrancechannel curvature field, producing the experimentally observed enhancement of ultra-low-energy fusion rates. Section 12.8 demonstrated that this effect follows from the same TSRT action principle and requires no probabilistic corrections. At the opposite extreme, the deep sub-barrier fusion hindrance of heavy-ion collisions finds its explanation in curvature-saturation limits (Section 12.9). In this regime, the entrance curvature becomes too stiff to reconfigure coherently, © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 89 preventing further action reduction and leading to exponential suppression. Enhancement and hindrance thus emerge as two manifestations of the same geometric mechanism. Taken together, these results yield a coherent and comprehensive synthesis: fusion, across all accessible energy scales, is governed by curvature concentration, proper-time determinism, and causal mode locking. The standard mass–energy relation E=mc2acquires a geometric meaning in this context, expressing the equivalence between curvature density and inertial energy rather than a conversion rule. All quantitative predictions presented in this section use a single, globally fixed curvature calibration to the reference nucleus 16O(Appendix E.3), with no per-system tuning. The present synthesis concerns near-ground-state, even–even nuclei where trembling symmetries are clearest; extensions to odd-Asystems, rotational excitations, and high-energy fusion–fission dynamics may be addressed in future work. Throughout, TSRT preserves energy, momentum, and angular momentum exactly, ensuring consistency with empirical data and with the geometric foundations of trembling spacetime relativity. 13 Comparison with Quantum Nuclear Models Supporting appendices: Appendix B (extended tables) and Appendix L, M, P (domain-specific benchmarks). MATLAB file locations are indexed in Appendix Y. We compare the deterministic TSRT predictions summarized in Tables 5 (p. 13) and 6 (p. 13) with the corresponding features of quantum nuclear models. Results of this paper include: (i) absolute binding energies and radii, (ii) barrier actions and fusion systematics including deep hindrance, (iii) odd–even staggering in fission yields from curvature-locked neck modes, and (iv) a fully deterministic lifetime law reproducing nuclear half-lives over more than twenty orders of magnitude using a single global normalization. These predictions allow for a direct, unified comparison with liquid-drop, shell, tunneling, and Standard-Model descriptions. The contrasts below highlight how empirical corrections in quantum models (pairing, shell closures, tunneling amplitudes, and screening shifts) arise in TSRT from the geometry of trembling curvature without probabilistic postulates or per-nucleus tuning. TSRT reproduces standard nuclear systematics while eliminating phenomenological postulates. Volume/surface saturation and pairing enter via deterministic curvature coherence (Equation (10)); magic-number effects arise as geometric resonances rather than single-particle shell fits (Section 5); barriers follow from action stationarity (Equation (16)), and OES is an interference effect (Figure 3 (p. 24)), not an added pairing term. Sub-barrier fusion trends, screening, and deep hindrance follow from curvature-overlap and proper-time transmission as developed in Section 7 (Figure 2 (p. 12), Figure 6 (p. 86)), while the broader phenomenology is synthesized in Section 12. In all cases, TSRT’s parameters are tied to curvature geometry once and reused, as summarized in Table 6 (p. 13). The modern quantum description of nuclei emerged from a series of crises and empirical puzzles during the first half of the 20th century. The discovery of radioactivity by Becquerel [114] and the early characterization of α-decay posed challenges that classical physics could not address. In 1928, Gamow [51] and, independently, Gurney and Condon [115], introduced the then-radical idea of quantum tunneling through the Coulomb barrier—its first dramatic application to nuclear phenomena. Shortly thereafter, macroscopic parametrizations such as the liquid-drop model [37,87] provided global fits to binding energies and, with extensions, laid the foundation for the first theoretical treatment of nuclear fission [23]. Yet this continuum picture could not account for the striking systematics of nuclear structure: the existence of “magic numbers,” the odd–even staggering of binding energies, or the detailed patterns observed in reaction cross-sections. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 96 TSRT is not merely qualitative. With the calibrated constants of Appendix E and the curvature integrals of Appendix H, the following validations are already achieved: • Binding energies along the valley of stability: Using Equation (154), TSRT reproduces ground-state binding energies of closed-shell nuclei (16O, 40Ca, 56Fe, 208Pb) to within ∼1%, matching the accuracy of liquid-drop fits but with no phenomenological surface or pairing terms. The full comparison table is given in Appendix I. • Sub-barrier fusion suppression: For light-ion systems such as D–D and D–T, evaluation of the proper-time action integral (Section 12.7) yields exponential slopes of the astrophysical S-factor consistent with experimental determinations down to ∼10 keV. Numerical results and code are provided in Appendix O. • Fission fragment staggering: As shown in Section 10.6, the deterministic neck-mode model quantitatively reproduces the amplitude and charge-dependence of odd–even staggering in 235U(nth,f). Appendix N extends this comparison to 239Pu, showing that the same fixedparameter model accounts for both systems. These benchmarks demonstrate that TSRT is already predictive and falsifiable: the same curvature constants calibrated once reproduce a wide set of observables without per-nucleus or per-channel tuning. Future work may extend the validation to heavier odd–Aand odd–odd nuclei, rotational bands, and high-excitation fusion reactions, where additional trembling-mode couplings must be included. Detailed TSRT derivations and numerical benchmarks for binding energies across the valley of stability are provided in Appendix L, while the astrophysical S-factor analysis is developed in Appendix P, together with reproducible MATLAB codes. 14.4 Transport mapping: recoil, Doppler, gravitational We use standard two-body kinematics for recoil, special-relativistic Doppler for source/observer motion, and the usual gravitational redshift where applicable; full implementation details are in Appendix J. 14.5 Charge-Sensed Electromagnetic Curvature Transport In addition to kinematic recoil and ordinary Doppler/gravitational effects (Section 14.4), TSRT predicts a small deterministic transport correction when photons or charged particles traverse the nuclear electromagnetic trembling near field. This is the nuclear analogue of the atomic refinement introduced in [14]: the detected energy differs from the source-frame value by a chargesensed curvature holonomy accumulated along the initial segment of the worldline through the EM trembling geometry. We write Edet =Esrc 1 + δrec +δDop +δgrav +δTSRT EM ,(158) where δrec,δDop, and δgrav are the standard recoil, Doppler, and gravitational terms, and δTSRT EM is the charge-sensed electromagnetic curvature transport correction. For photons, δTSRT EM reproduces the atomic near-field refinement of [14] in the nuclear regime; for charged ejectiles it scales with their charge and the local EM trembling curvature. A full derivation and an implementable closed-form near-field model appear in Appendix K. In the comparisons reported here, δTSRT EM is small (typically .10−3in fractional energy) and does not affect MeV-scale budgets; we include it where line centroids are compared at high precision (e.g., γenergies and conversion electrons) and document its numerical impact explicitly (Appendix K, Tables 15 (p. 109) and 22 (p. 144)). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 97 14.6 Future Theoretical Developments Several theoretical frontiers follow naturally: 1. Multi-nucleon eigenmodes: Classification of collective trembling modes in many-body nuclei, extending the single-particle and few-body cases developed here. 2. Reaction networks: Application of TSRT curvature rules to entire fusion chains, decay cascades, and nucleosynthesis pathways in stars and supernovae. 3. Gravity crossover: Linking nuclear curvature confinement with macroscopic self-gravity, to model neutron stars and collapse into black holes within the same trembling framework. 14.7 Perspective What quantum mechanics once treated as intrinsically probabilistic (tunneling, decay, fusion) emerges in TSRT as deterministic geometry. By unifying atomic, nuclear, and cosmological domains in a single causal framework, TSRT resolves a century-old duality between quantum postulates and relativistic field equations. The broader implication is clear: the same trembling geometry that generates Planck’s constant, atomic spectra, and entanglement also governs nuclear structure and stellar energy production. Future work may refine these predictions, extend them to astrophysical environments, and pursue experimental tests designed to distinguish deterministic curvature dynamics from probabilistic quantum models. TSRT not only unifies nuclear physics with relativity, but elevate all of physics to a single geometric principle of trembling spacetime. 15 Calibrated Anchors to Absolute Units in TSRT (Deuteron to Fission Chain) TSRT predictions are produced from curvature-based functionals evaluated on explicit numerical grids and reported in SI and standard nuclear units (Appendix V). The present normalization chain begins with a single calibrated energy–curvature mapping from the deuteron binding energy and extends consistently up to the fission scale, verified on the 235U(n,f) benchmark (Table 42 (p. 257)). After this step, all reported TSRT energies are absolute and reproducible without further scaling. This section shows that, for the present manuscript, all required mappings from code quantities to physical units are either (i) already absolute through exact unit conversions, or (ii) fixed once by a documented physical anchor with no residual freedom. We summarize these mappings and identify how future work can proceed without any further calibrations. 15.1 Energy mapping: absolute by construction Let Q[K]denote the (dimensionless) curvature functional delivered by the energy pipeline (Appendix Y, Listings 4 (p. 215)–5 (p. 216)). Physical energies are Ephys =CEQ[K], Ephys ∈ {J,MeV}.(159) In this manuscript the energy mapping is already absolute because the pipeline performs a direct SI→MeV conversion using exact constants (Appendix V, Tables 30 (p. 200)–31 (p. 201)): 1 MeV = 1.602176634 ×10−13 J, c = 2.99792458 ×108m/s(exact). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 98 Accordingly, the global factor in Equation (159) is numerically fixed,109 CE≡1(in the code’s MeV reporting) (160) and there is no free energy scale left to adjust. This is empirically verified by the binding overview (Figure 1 (p. 10)), where B/A near A≈60 matches AME2020 values within table precision (Table 3 (p. 11)) without any per-nucleus renormalization. 15.2 Curvature strength: fixed once and reused The geometric curvature-strength parameter used in the nuclear sector is inherited from the atomic TSRT constant and retained here: γ= 4.36 ×1042 m−2(161) as documented in Appendix A. This constant multiplies the dimensionless curvature invariants in the field construction and therefore sets the geometric magnitude of local trembling suppression. Because the energy mapping (160) is absolute, γis not degenerate with CEand is truly fixed once for all sectors. Together, this curvature magnitude and the absolute energy mapping inherited from the deuteron normalization (Section 15.1) define a closed and unit-consistent energy–geometry correspondence used in the fission validation (Table 42 (p. 257)). 15.3 Lifetime scale: a single physical anchor Decay rates are computed from the TSRT base–slope law with per-mode factors (Section 9.7, Appendix B.7). The only global scale entering the half-life predictions is the rate constant Cλ, calibrated once on a standard benchmark and then held fixed: Cλ= 6.48104 ×10−6s−1(162) (see Table 2 (p. 11) and Appendix A). With γfixed by (161) and the absolute energy mapping (160), Cλis the unique rate-scale anchor; no per-isotope adjustments are introduced. 15.4 On the trembling amplitude hAi: not an independent fit In TSRT, observable magnitudes are governed by the combination of (i) geometric curvature strength γ, (ii) mode-dependent shape/phase (entering the curvature invariants), and (iii) the absolute energy/reporting map. Any putative mean trembling amplitude hAi that rescales all local curvatures would be degenerate with γand CEat leading order. Since γis already fixed by (161) and CEby (160), there is no additional free global amplitude in the present work: hAi is absorbed by γand does not constitute a separate calibration here. (163) Sector-specific shape parameters (e.g. OES neck-parity weights, hindrance slopes) are dimensionless and tied to geometry (Section 6, Section 12.9) rather than a global amplitude. 15.5 Gamma–ray normalization (E2) and internal conversion For electromagnetic decays, the E2block is anchored on the well-characterized 2+ 1→0+ 1transition of 156Gd with Eγ= 88.9656 keV, τ = 1.635 ns, αtot = 0.163, as compiled in Table 36 (p. 205) (ENSDF A=156 and BrIcc). This anchor fixes the E2proportionality CE2used in Appendix Y.8 once and for all; no additional fit parameters are introduced downstream into the lifetime pipeline (Appendix Y.9). 109In practice, the internal code quantity C.norm_energy (see Appendix Y.2, Listing 2 (p. 213)) acts as the SIto-code conversion constant. Its deuteron-matched value, 4.7865×1031 J/(code unit), ensures that the MeV-scale energy reporting is absolute and consistent across all nuclear sectors. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 99 15.6 Practical summary With the above in place, the present manuscript already operates in absolute units: • Energy reporting is SI/MeV exact: CE≡1by construction (Appendix V and Y.2). • The curvature strength is fixed once: γ= 4.36 ×1042 m−2(Appendix A). • The lifetime scale has a single physical anchor: Cλ= 6.48104 ×10−6s−1(Table 2 (p. 11)). • The E2block is locked by the 156Gd anchor (Table 36 (p. 205)), with ICC included via BrIcc. • The absolute energy normalization has been independently verified by reproducing the total fission release of thermal-neutron-induced 235U(n,f) (Table 42 (p. 257)), confirming the deuteron-based constant across the full nuclear scale. No residual global knobs are left within the scope of this paper; the remaining parameters are geometric (dimensionless) and are either universal slopes or mode-shape weights, as detailed in Sections 5, 6, 12.9, and 9.7. This absolute scaling is numerically verified by the fission benchmark (Table 42 (p. 257)), where ETSRT fiss = 170.028 MeV is obtained without any post-hoc adjustment. 15.7 Outlook: removing anchors entirely To make TSRT strictly anchor-free in future applications, one may derive Cλand the electromagnetic proportionalities directly from the trembling–metric Lagrangian and boundary conditions, tying them to (c, ~)and the curvature spectrum of bound modes. The present work already demonstrates that the energy mapping—from deuteron binding up to 235U fission—is absolute, leaving only the rate-scale Cλand radiative proportionalities to be derived from first principles. The present manuscript already eliminates any energy-scale ambiguity (Section 15.1); the remaining pathway is a first-principles derivation of the global rate scale. This may be pursued in forthcoming work. 16 Conclusions This work completes the extension of Trembling Spacetime Relativity Theory (TSRT) into the nuclear domain and shows that nuclear binding, stability, and decay arise from a single causal geometric principle. In TSRT, a nucleus is a localized trembling–curvature eigenmode whose persistence depends on sustained suppression of intrinsic curvature; decay occurs when this suppression relaxes deterministically along geodesic paths. The picture dispenses with probabilistic postulates and tunneling hypotheses and replaces them with curvature overlap and causal relaxation governed by proper time. Within this framework we predict absolute half-lives for beta, alpha, and spontaneous fission channels across more than twenty orders of magnitude using a single once-only normalization fixed on a benchmark nucleus. The agreement with evaluated values is stringent, with a mean logarithmic deviation below one part in a million in the representative set studied. No shell closures, pairing gaps, nucleus–by–nucleus preformation factors, or empirical mass inputs are invoked; all energetics, barrier actions, and rates are obtained from the same trembling-spacetime geometry that also underlies our descriptions of binding and deformation. The notorious longevity of carbon–14 follows naturally as a geometric suppression of its weak transition, without ad hoc hindrance parameters, reinforcing that nuclear decay is a deterministic curvature reconfiguration rather than a stochastic process. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 100 The same causal geometry accounts for phenomena long considered anomalous within standard approaches. Deep sub-barrier fusion hindrance emerges as a deterministic reduction of trembling-mode overlap; odd–even staggering in fission yields reflects phase-locked neck eigenmodes at scission; and electron-screening shifts at ultra-low energies follow from curvatureinduced renormalization of electromagnetic fields. These behaviors do not require additional hypotheses or auxiliary fits: they are direct consequences of the trembling metric that also governs lifetimes. Across β,α, and spontaneous fission, all reported half-lives follow from a single curvaturebased law with global, once-only anchors, and without probabilistic postulates or per-nucleus adjustments. In precise terms, TSRT attains a mean logarithmic deviation below 10−6between predicted and experimental half-lives—four to five orders of magnitude more precise than state-of-the-art quantum or semiempirical models. This level of agreement, obtained from fixed global constants without local tuning, establishes TSRT as the first deterministic framework capable of predicting absolute nuclear decay times directly from spacetime geometry. This unifies nuclear stability and transformation under deterministic spacetime geometry and closes the explanatory loop from curvature suppression to observable lifetimes. The same geometric mechanism that sustains bound configurations also governs their causal relaxation, yielding quantitative agreement over more than twenty orders of magnitude. Quantitatively, the same deterministic law reproduces absolute half-lives from milliseconds to nineteen-power-of-ten years with a mean discrepancy at about one-millionth of a decade in log space, using only three fixed global constants for the three decay modes. This contrasts with the tenth-to-hundredth-of-a-decade dispersion typical of mainstream models on comparable benchmarks, underscoring the explanatory and predictive power of the geometric curvature mechanism. With energy, curvature, and rate scales now fixed in absolute physical units, TSRT achieves a self-contained quantitative bridge between geometry and experiment, paving the way for future anchor-free predictions derived solely from trembling-metric first principles. These results establish not only a quantitative, but also a qualitative unification. Energy conservation and reaction energetics appear as curvature redistribution; nuclear persistence and transformation become facets of the same causal structure that relates atomic spectroscopy and gravitational redshift. The framework is transparent and reproducible: all tables and figures in the paper are generated by the provided MATLAB implementations and datasets, and the calibration steps are documented so that independent users can retrace the workflow end to end. The self-consistent calibration chain, spanning from the deuteron normalization through the curvature-based scission width, now yields the correct absolute energy release for 235U(n,f) (ETSRT fiss = 170.028 MeV), thereby validating the TSRT energy mapping in nuclear-scale units with no residual fitting freedom. Looking forward, the approach is poised for systematic extension to nuclei with competing decay branches, to full network modeling in stellar environments, and to curvature-driven behavior in neutron-star matter and strong-field astrophysics. Refinements of higher-order curvature derivatives and signed geometric amplitudes will further test and expand the predictive reach without departing from the deterministic foundation. In sum, nuclear binding, decay, fission, and fusion, traditionally framed as probabilistic quantum phenomena, emerge here as precise manifestations of trembling spacetime curvature, placing nuclear physics on a single causal geometric footing alongside the broader relativistic universe. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 101 17 Acknowledgments CERN is gratefully acknowledged for providing, through Zenodo,110 an open-access platform that makes this work freely available to the global physics community, written in a style the author prefers. The author111 gratefully acknowledges institutional support from the French Centre National de la Recherche Scientifique (CNRS) through the International Research Laboratory IRL 2958 Georgia Tech–CNRS, as well as Metz Métropole, the City of Metz, and Georgia Tech Europe for hosting him as a Georgia Tech faculty member in France. Infrastructure and mobility support were provided by the Région Grand Est, the French Lorraine Université d’Excellence (LUE impact project I-META ANR-15-IDEX-04-LUE), Institut Carnot ARTS, Conseil Régional de Lorraine, Le Conseil Départemental de la Moselle, Feder-Europe, the Indo-French Centre for the Promotion of Advanced Research (CEFIPRA RCF-IN-0067), Mirabelle Plus, Stellantis, Institut de Soudure, the Association Nationale de la Recherche et de la Technologie (ANRT), Metz Métropole, the Institut Supérieur Européen de l’Entreprise et de ses Techniques (ISEETECH), the Contrat Plan État-Région (CPER-MEPPP05), the Agence Nationale de la Recherche (ANR-09-BLAN-0167-01), the European Union’s Horizon 2020 program (Grant Agreement No. 871260), the Belgian Fund for Scientific Research Flanders (FWO, 2005), the Flemish Institute for the Encouragement of Scientific and Technological Research in Industry (IWT, 2001-13343), and the North Atlantic Treaty Organization (NATO PST.CLG.980315). While these programs provided an invaluable research environment, the conceptual direction and results presented here stem from independent scholarly inquiry developed over decades. The author acknowledges the foundational training in physics, astrophysics, and engineering physics received at the Catholic University of Leuven (KULAK and KU Leuven) and Ghent University, which shaped his interdisciplinary approach. He also thanks the Georgia Institute of Technology and the George W. Woodruff School of Mechanical Engineering for his promotion 110Like an eagle free to fly where it wills, truth is free to wander through the world of physics and face the scrutiny of the entire community at once, for genuine truth has nothing to fear. 111The author was profoundly affected upon visiting the Hiroshima Peace Memorial Museum in 2004 with his friend Dr Filip Windels. The confrontation with the human reality of nuclear weapons stood in stark contrast to their abstract theoretical description. This experience made the subsequent observation of Japan’s postwar journey all the more powerful: a nation that transformed profound tragedy into a testament of resilience, dedicating itself to peace, technological excellence, and international cooperation. A personal encounter with this spirit occurred earlier, in Sri Lanka, following the devastating 1996 floods. There, amidst the recovery efforts, a local family offered a gift of profound generosity: canned tuna, a staple provided by Japanese food aid. This simple act revealed an enduring truth, that the core of a nation’s character is found not in the instruments of its power, but in its consistent choice to extend humanity and compassion, even to distant strangers. Intellectually, the author’s path into nuclear science was shaped by early joys in performing Mössbauer effect experiments as a master’s student, by formal courses in nuclear physics and astrophysics, and by private readings that explored nuclear processes in stellar systems, where the two disciplines met in a unity surpassing the confines of lectures. A prolonged stay in Sri Lanka in 1997 added further depth: there, while studying symmetry groups and a nuclear physics course, the author learned to appreciate the elegance of elementary particles as structured manifestations of mathematical harmony. It was also in Sri Lanka that he first taught mathematics, discovering in its universality a profound truth, that mathematics knows no borders, no race, no color, and thus embodies a common language of humanity. The roots of this perspective reach back further still, to childhood in Sri Lanka in 1985, when public anticipation of Halley’s Comet filled the air. Although the comet itself was not clearly visible at that time and place, the excitement it generated—and a formative visit to the Colombo planetarium—ignited a sense of cosmic wonder. From that period onward, the author found joy in watching shooting stars and contemplating the luminous arc of the Milky Way. In hindsight, these early impressions nurtured the conviction that human inquiry, like the trembling of spacetime itself, is both fragile and immense, anchored in fleeting personal moments yet pointing always toward the universal. The deep respect for the Japanese people and their post-war values, combined with a lifelong fascination for the cosmos, for mathematics, and for nuclear processes at every scale, continues to inspire the ethical and philosophical perspective that underpins this work. Nevertheless, it was in Hiroshima that a desire grew to deeply understand the nucleus of the atom. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 102 to Full Professor, coinciding with his designation as Professor Honoris Causa in Physics by the University of Allahabad, a confluence of recognition that secured the academic freedom necessary to formalize the ideas presented here, which trace their origins to early studies of solar system dynamics (Driesprong, Deerlijk) and electron beam deflection experiments (Koninklijk Atheneum, Waregem). The author also expresses his gratitude to his cousin, Mr. François Berland. While employed at the Doel Nuclear Power Station in Belgium, Mr. Berland donated his complete collection of the scientific journal ’Natuur & Techniek’ to the author during his high school years. At a time when access to advanced scientific resources was limited, these journals, which featured explanations of nuclear physics and relativity theory, among many other interesting scientific topics, served as an invaluable and early source of inspiration and motivation. The author extends his deepest gratitude to colleagues and organizations in acoustics, ultrasonics, acousto-optics, and optics for fostering an intellectually vibrant and collegial environment. Their shared passion for wave phenomena has been a continual source of inspiration, and their engagement with industrial applications always boosted inspiration to seek critical support to sustain the author’s laboratory. He is equally indebted to his current and former students (BS, MS, PhD) and postdoctoral researchers, whose curiosity and fresh perspectives profoundly enriched his work in nondestructive evaluation and physical acoustics. Their enthusiasm reaffirmed that scientific discovery, regardless of discipline, thrives on intuition, relentless curiosity, and intellectual joy, motivating the author’s deeper explorations in fundamental physics beyond conventional working hours. This work stands as a testament to the enduring intellectual and material legacies of Daniel A. Lesage (1911–1988), Pierre F. Vangheluwe (1939–1991) (who triggered the author’s interest in history and academic pursuit in general), Mariette Vanoosthuyze (1925–2009), Godelieve Verborgh (1932–2012), Maurice A. Declercq (1941–2015), and Jeanne M. C. Maesen (1930–2024), whose generosity continues to enable scholarship. The author is equally indebted to Jeanette Verbrugghe and Nelly Vangheluwe for their unwavering support, both moral and logistical, which has been indispensable to this research. The author wishes to offer a personal note of thanks to his goddaughter, Sofie Windels, who is always a beacon of courage, kindness, and integrity. During a formative period in this research, as he navigated the complexities of nuclear structure and the limitations of prevailing models, she provided him with a booklet of quotes by Nikola Tesla. Tesla’s words on the value of seclusion as a secret to invention and the importance of deep, patient thought over immediate results arrived at a critical time, serving as a timely reminder that the solitude demanded by his job and by this work could itself be a source of strength and originality. Furthermore, the author was profoundly charmed when, mere months before the conclusion of this lengthy work, she expressed her pride in his efforts, which was a gesture that provided a final, deeply valued encouragement. Finally, the author expresses profound gratitude to his wife and children for their steadfast presence, boundless encouragement, and enduring patience throughout the many weekends and holidays at home in Belgium devoted to this work. In particular, the author is deeply grateful to his children—Benjamin, Anna-Laura, and Lambert—who graciously lent him their fast computers during the weekends so he could complete the most demanding calculations. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 103 18 Appendices Appendix Guide and Cross-Reference Map The appendices that follow constitute a fully self-contained technical companion to the paper. They are designed so that every numerical result, figure, and table in the main text can be traced to its exact algorithmic and analytical source. The structure is modular: methods and constants first, then derived datasets, followed by thematic derivations that parallel the main sections. Global Appendix Overview •Numerical Foundations and Calibration. Appendix A (p. 105) details procedures, constants, and calibration routines, which are complemented by the geometric definitions in Appendix C (p. 111), the physical constants in Appendix E (p. 120), and the numerical integration schemes in Appendix F (p. 123). These appendices jointly underpin all quantitative work in Sections 5–7. •Nuclear Geometry and Curvature Framework. Appendix C (p. 111) defines curvature fields and coordinate conventions used throughout TSRT nuclear modeling. The companion Appendix G (p. 129) introduces the trembling-field formalism for nucleons and composite nuclei, while Appendix H (p. 132) formulates the corresponding curvature energy functionals. These build the background for the magic-number derivations and deformation energy in the main text. •Magic Numbers and Shell Geometry. Appendix D (p. 114) derives shell closures as stationary curvature resonances. It builds on Appendix C (p. 111) and informs Section 5 (binding-energy kinks), Section 13.3 (comparison with shell models), and Section 6 (odd– even staggering context). •Data Tables and Benchmark Comparisons. Appendix B (p. 108) compiles all extended numerical tables—binding energies, charge radii, fission barriers, screening shifts, and hindrance slopes—each directly tied to the figures and compact tables in the main text. Lifetime benchmarks in Appendix B.7 (p. 110) complete this dataset portfolio, linking to Section 9 and Appendix I (p. 135) for deterministic half-life estimation. •Stability, Lifetimes, and Decay Laws. Appendix I (p. 135) derives the TSRT lifetime estimator and stability-map construction, while Appendix A (p. 105) describes how the absolute lifetime scale Cλis fixed. Together they support the main-text Sections 9.7 and 9. Benchmarks appear in Appendix B.7 (p. 110). •Fusion, Fission, and Hindrance. Appendix M (p. 151) develops the curvature-gradient bifurcation model of nuclear fission; Appendix O (p. 161) formulates the TSRT geodesicoverlap criterion for fusion energy; Appendix P (p. 166) and Appendix Q (p. 175) extend this to astrophysical S-factors and deep sub-barrier hindrance. Their parameters and slope calibrations are linked through Appendix P.4 (p. 168) and Appendix A (p. 105). The combined content supports Sections 7 and 8. •Electromagnetic Screening and CSCT Transport. Appendices K (p. 141) and J (p. 138) provide the curvature-sensed electromagnetic transport formalism (CSCT), used for near-field line shifts in Figure 2 (p. 12). These build on the same constants and grid definitions as the mechanical modules in Appendix A (p. 105). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 104 •Odd–Even Staggering and Neck-Mode Model. Appendix N (p. 157) gives the full derivation of the Bessel-neck model for OES, complementing the data tables in Appendix B.6 (p. 109) and the main-text analysis in Section 6. •Meta-Analyses and Validation. Appendices R (p. 180) through S (p. 191) include validation protocols, environment checks, and performance metrics. Appendix U (p. 197) quantifies uncertainty and sensitivity of key observables. These appendices collectively support the discussions in Section 13. •Reproducibility Infrastructure. Appendix Y (p. 210) provides the complete workflow and run order; Appendix T (p. 194) lists all figure/table generation scripts, followed by the global MATLAB file index at the end of the document. These appendices ensure full transparency and repeatability. Numerical reproducibility preview Before entering the technical appendices, the following cross-map connects headline results to their computational origins: 1. Magic-number closures. Analytical derivation: Appendix D (p. 114); tabulated outputs: Table 20 (p. 116). Supports Sections 5 and 13.3. 2. Binding energies and charge radii. Data: Appendix B.1 (p. 108) and B.2 (p. 108); methods: Appendix A (p. 105); calibration constants: Appendix E (p. 120). 3. Lifetimes and stability maps. Derivation: Appendix I (p. 135); calibration: Appendix A (p. 105); tables: Appendix B.7 (p. 110). Corresponds to Section 9.7 and the main stability diagrams. 4. Fission and fusion barriers. Derivations: Appendices M (p. 151) and O (p. 161); barrier data: Appendix B.3 (p. 109); astrophysical S-factor and hindrance validation: Appendices P (p. 166), Q (p. 175), and B.5 (p. 109). 5. Odd–even staggering (OES). Derivation: Appendix N (p. 157); data: Appendix B.6 (p. 109); figures: Figure 3 (p. 24). Algorithms: Appendix A (p. 105) (OES listings). 6. Electromagnetic screening. Derivation and data: Appendices K (p. 141) and B.4 (p. 109); scripts: Appendix Y (p. 210) (listings 9 (p. 223), 27 (p. 254)); supports Section 7. 7. Sensitivity and reproducibility. Quantitative error bounds: Appendix U (p. 197); global run order and validation environment: Appendices S (p. 191), T (p. 194), and Y (p. 210). How to run the full workflow. For a step-by-step execution order, including filenames and expected outputs, see Appendix Y (p. 210). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 105 A Methods: Numerical Procedures, Parameters, and Calibrations Consistency with implementation. Internal units are SI and are converted for presentation as in tsrtconstants.m. Grids and quadrature follow tsrtmakegrid.m and gaussleg.m; the sin θdθ weight is included once (no double-weighting), matching the notes in Appendix F. Convergence checks use tsrtconvergence.m. Supports Sections 5, 6, 7, 8. MATLAB: tsrtconstants.m,tsrtmakegrid.m,tsrtconvergence.m, gaussleg.m,tsrt_repo_sanity_check.m,tsrt_preflight_checks.m. This appendix documents the numerical path from model definition to plotted result. It is the primary reference for reproducibility. Methods here are invoked in Section 5 (binding/radii), Section 6 (barriers, OES), and Section 7 (screening, hindrance), with entry points listed in Table 6 (p. 13). We state exact parameter values, describe calibration choices (e.g., the 56Fe anchor), and give run-time settings (e.g., grids, tolerances, seeds, platform). Units and conventions. Unless explicitly stated, we retain the speed of light cin all formulas to keep dimensions transparent. Numerical work uses MeV and MeV/c2consistently (1 u c2= 931.49410242 MeV). For algebraic brevity, we occasionally display a companion “natural–units” form by setting c= 1; those compact forms are for orientation only and are never used for numerical substitution. Calibration pipeline 1. Set core constants and units; confirm metric (+,−,−,−)is used consistently (Appendix C). 2. Calibrate the global normalization C.norm_energy to the deuteron binding (2.224 MeV), as implemented in tsrtconstants.m (Listing 2 (p. 213)). 3. Fix the grid via tsrtmakegrid.m (Listing 3 (p. 214)); optional convergence check with tsrtconvergence.m (Listing 6 (p. 217)). 4. For hindrance, extract a single pair (α, β)on 64Ni+64Ni using tsrtsfactor.m and reproducehindranceTSRT.m (Listings 10 (p. 225), 11 (p. 229)); reuse unchanged for predictions. 5. For screening / near-field benchmarks, use screeningmakefig.m and csctmaketables.m (Listings 9 (p. 223), 27 (p. 254)); parameter values are documented inline in each listing. Calibration: Lifetime scale Cλ We fix Cλin Equation (2) by matching one reference decay (or a small set), then use the canonical lifetime driver RunLifetime_All.m (Listing 16 (p. 238)) to generate deterministic predictions under the same coarse-graining and proper-time step. Once fixed, Cλremains unchanged across the chart. All runs use the same coarse-graining window and proper-time step as specified in Appendix M. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 112 C.3 Binding and Stability Functionals The curvature-suppression mechanism underlying nuclear binding is quantified by ∆K2(Z, N) = hK2iisolated −hK2iZ,N ,(169) where hK2iisolated is the sum of the variance contributions from free protons and neutrons, and hK2iZ,N is the variance when they are assembled into a bound nucleus. The binding energy follows as Ebind(Z, N) = Cbind ZVhK2iisolated −hK2ibound(x)d3x, (170) where Cbind is the single global normalization (units conversion). Concretely, Cbind ≡C.norm_energy (Appendix E.3), fixed once from the stated calibration. Equation (170) is the continuum form of Equation (247); both use the same single normalization Cbind ≡C.norm_energy and the same separable quadrature on a common grid. The full computational procedure, including isolated–versus–bound field energy subtraction, is detailed in Appendix I, with implementation in Appendix I. Lifetime estimator. The TSRT instability measure is given by the rate of curvature variance decay in proper time: τ−1 Z,N =β∇τ∆K2(Z, N),(171) with βa proportionality constant fixed against a known benchmark decay (e.g. the free neutron lifetime). This definition provides a deterministic estimator of half-lives, in contrast to the probabilistic postulates of quantum nuclear models. Summary. Equations (164)–(171) define the geometric functionals that underlie all nuclear structure, fission, and fusion calculations. They are the starting point for the explicit numerical evaluations documented in Appendix F. C.4 Action Decomposition and Stability Maps For composite trembling systems such as nuclei, the action is not strictly additive in the number of nucleons. Each nucleon contributes an individual trembling action Si, but overlap of eigenmodes introduces additional terms due to curvature interference. A general decomposition reads Snucleus = Z+N X i=1 Si+Sint(Z, N) + δScorr,(172) where Sint encodes the leading curvature-saturation correction, and δScorr represents higherorder interference contributions. The dominant stability condition corresponds to retaining only Sint, which already accounts for the suppression of curvature fluctuations: ∆K2(Z, N) = hK2iisolated −hK2iZ,N >0,(173) cf. Equation (82). The correction term δScorr includes multipole-phase mismatches and higherorder overlap terms; these are systematically small for ground-state nuclei but become relevant near the drip lines and in high-excitation fission. Appendix I compares curvature-based stability maps with experimental binding trends; the numerical construction of ∆K2(Z, N)and its channel decomposition is detailed in Appendix I. Representative tables are collected in Appendix B. Absolute normalization for lifetimes is fixed © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 113 once via the global weak scale Cλ(Appendix A); representative half-lives appear in Table 2 (p. 11) and are folded into the stability maps in Appendix I. We use the term “slow drift” to emphasize the vast separation between weak and strong timescales (see Section 9). Table 19 (p. 113) is used in Section 6 (actinide barriers) and summarizes values quoted in Table 5 (p. 13). Table 19: Key TSRT stability parameters used in curvature-balance calculations. All quantities are defined geometrically; no stochastic or phenomenological pairing terms are introduced. Parameter Meaning SiTrembling action of an isolated nucleon eigenmode Sint Leading overlap (curvature-saturation) contribution δScorr Higher-order interference corrections (multipole, phase mismatch) ∆K2(Z, N)Net suppression of curvature variance [Equation (82)] κsat Saturation threshold curvature (fixed on the reference set used for Table 19 (p. 113)) CEM Effective EM-to-curvature balance factor (absorbs long-range Coulomb terms) C.5 Units and definitions used throughout • Trembling curvature tensor: Kµν denotes the trembling-curvature (TSRT) contribution obtained from the metric decomposition gµν =ηµν +ξµν and the associated causal geodesic congruence. Its scalar norm is K≡pKµνKµν. • Variance and suppression: hK2idenotes a coarse-grained curvature variance (domain average appropriate to the observable), and ∆K2denotes a variance reduction relative to the isolated-mode baseline (used in stability/lifetime, cf. Equation (2)). • Surface imbalance: ∆Ksurface denotes the channel-decomposed surface-layer imbalance used in fission, ∆Ksurface =KCoulomb(Z)−Kstrong(A),with both terms evaluated on the same scalar norm Kand coarse-grained over a fixed-thickness shell (Appendix M, Appendix M.1). • Dimensions: Kµν carries curvature dimensions; Khas curvature units; hK2iand ∆K2have curvature-squared units; ∆Ksurface has curvature units. The energy integral is implemented in Listing 5 (p. 216) (global normalization C.norm_energy), with the normalization fixed once in Listing 2 (p. 213). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 114 D Geometric Origin of Nuclear Magic Numbers in TSRT Purpose and scope. This appendix gives a complete, step-by-step derivation of magic-number closures in TSRT from elementary ingredients. We avoid quantum-mechanical postulates (e.g., single-particle orbitals in a mean field); instead we derive the relevant curvature eigenmodes of a finite trembling domain and show how their deterministic occupation produces the observed closure sequence 2,8,20,28,50,82,126. TSRT vs. QM language (important terminology). In the conventional shell model, “quantization” comes from a postulated single-particle Hamiltonian and its spectrum. In TSRT, no such Hamiltonian is assumed. The discrete structure arises because a finite nuclear domain enforces boundary conditions on the trembling curvature field. The allowed standing patterns (we call them curvature eigenmodes) are fixed by geometry, not by probabilistic eigenstates. Where QM says “shell,” TSRT says “closed resonance of curvature modes.” We will keep the words ’mode’, ’eigenmode’, and ’closure’ to emphasize this geometric origin. D.1 What is being computed We compute three things: 1. The allowed spatial patterns (eigenmodes) of the trembling curvature field in a nearly spherical nucleus of mean radius RA=rTSRT 0A1/3. 2. The degeneracy (number of independent angular patterns per eigenmode family) implied by geometry and trembling-phase duality. 3. The cumulative occupation, i.e., how many curvature “slots” are filled when eigenmodes are taken in order of increasing stiffness (wavenumber). Discrete closures in this cumulative count are the TSRT analogues of magic numbers. We first obtain the raw cumulative sequence from the ideal spherical boundary problem, and then include a small, well-defined surface-coupling correction (spin–geodesic splitting) that shifts the raw closures to the empirical magic numbers. D.2 Curvature eigenmodes in a finite trembling domain A stable, stationary trembling configuration minimizes the proper-time action (TSRT variational principle) δS= 0,S=Zpgµν ˙xµ˙xνdτ, (174) with metric/signature and conventions as in Appendix C and Section 2. Linearizing the curvature response about a nearly spherical equilibrium nucleus yields a Helmholtz-type equation for the trembling scalar ξ(r, θ, φ)(see also [12]): ∇2ξ+k2ξ= 0,(175) subject to a locking boundary condition that enforces vanishing normal curvature flux at the nuclear surface: ∂rξr=RA= 0 (Neumann/locking).(176) This condition expresses that the surface coherently reflects trembling curvature (no net curvature leakage), which is the correct TSRT analogue of a rigid geometric boundary. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 115 Separation of variables and radial quantization. Using spherical coordinates and the standard separation ξ(r, θ, φ) = Rl(r)Ylm(θ, φ), the regular radial solutions are Rl(r)∝jl(kr) with jlthe spherical Bessel function. Applying (176) gives the radial quantization rule j′ l(knlRA) = 0,(177) so the allowed wavenumbers are knl =xnl RA , xnl =n-th root of j′ l(x) = 0.(178) In TSRT the local phase velocity that ties spatial and temporal structure is denoted ceff (set by the curvature medium; see Appendix A), so the eigenfrequency is ωnl =ceff knl =ceff xnl/RA.(179) D.3 Degeneracy and cumulative occupation For each (n, l), the angular multiplicity is (2l+1). TSRT adds a twofold trembling-phase duality (even/odd phase of ξ, ˙ ξ), giving the geometric degeneracy gnl = 2 (2l+ 1).(180) Ordering mode families by increasing xnl (equivalently knl or ωnl), the raw cumulative occupation after the first Mmode families is N(raw) c(M) = M X i=1 gnili.(181) A closure occurs whenever the next available family lies sufficiently higher in stiffness, yielding a pronounced gap (see discussion below). D.4 How we populate the table (roots, degeneracies, cumulative) To build the main table (Table 20 (p. 116)): 1. Compute the zeros xnl of j′ l(x) = 0 (for small n, modest l) using a standard root finder (e.g., besselzero or any Bessel derivative solver). 2. Sort all families (n, l)by increasing xnl (ties are resolved by the numeric value). 3. For each family, record gnl = 2(2l+ 1) and update the cumulative N(raw) cusing (181). The “Empirical magic number” column then shows the closest observed closure in the experimental sequence, anticipating the surface-coupling correction we derive next. Reading the columns: nis the radial index (1st, 2nd, . .. radial solution at fixed l), lis the angular index (spherical harmonic degree), ’Root xnl’ is the dimensionless zero of j′ l(x), ’Degeneracy’ is the number of independent patterns in that family, ’Cumulative’ is the sum of degeneracies up to that family [Equation (181)]. The last column shows the nearest empirical closure after we include the small TSRT surface-coupling correction below. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 116 Table 20: First few TSRT curvature eigenmode families in a spherical domain with locking boundary condition, sorted by increasing root xnl of j′ l(x) = 0. Degeneracy gnl = 2(2l+ 1) includes trembling-phase duality. The cumulative count N(raw) creflects ideal spherical locking; empirical magic numbers align after the small surface-coupling correction in Table 21 (p. 118). n l Root xnl of j′ l(x) = 0 Degeneracy gnl Cumulative N(raw) cEmpirical magic (for comparison) 1 0 3.1416 2 2 2 1 1 4.4934 6 8 8 1 2 5.7635 10 18 20 (after corr.) 1 3 7.7253 14 32 28 (after corr.) 2 0 6.2832 2 34 — 1 4 9.0950 18 52 50 (after corr.) 1 5 10.4171 22 74 82 (after corr.; see text) 1 6 11.7049 26 100 126 (after corr.; see text) D.5 Why a small correction is needed (spin–geodesic splitting) The ideal spherical locking condition used to build Table 20 (p. 116) is the Neumann boundary ∂rξr=RA= 0 ⇐⇒ j′ ℓ(xnℓ) = 0,(182) with xnℓ ≡knℓRA. In practice, TSRT adds two small but systematic effects that are already calibrated once (and re-used across the paper; see Appendix A): (i) a gentle surface–geometry coupling that modifies the phase of the radial locking, and (ii) a spin–geodesic splitting that correlates angular momentum with the local geodesic orientation of the trembling field. Neither effect introduces per-nucleus tuning; both follow from the same constants used in deformation and OES. 1) Surface–geometry as a Robin-type boundary perturbation. In the idealized limit of a perfectly “free” surface one would impose a Neumann boundary condition,112 j′ ℓ(xR) = 0, which corresponds to zero curvature flux through an infinitely compliant nuclear surface. In TSRT, however, the surface has finite stiffness and couples weakly but deterministically to the interior trembling field, so the effective locking condition acquires a small Robin correction, j′ ℓ(x) + ρℓjℓ(x) = 0,(183) where ρℓis a dimensionless surface–geometry coefficient (weakly ℓ-dependent) that encodes how the curvature flux couples to the nuclear surface stiffness (Appendix A). Linearizing (183) near a Neumann zero xnℓ with j′ ℓ(xnℓ) = 0, j′ ℓ(xnℓ+∆x)≈j′′ ℓ(xnℓ) ∆x, jℓ(xnℓ+∆x)≈jℓ(xnℓ),(184) yields, to first order in ∆x, j′′ ℓ(xnℓ) ∆x+ρℓjℓ(xnℓ)≈0 =⇒∆x(surf) nℓ ≈ −ρℓ jℓ(xnℓ) j′′ ℓ(xnℓ).(185) Thus, the surface coupling produces a mode-dependent shift governed solely by Bessel values at the known Neumann roots. The sign and magnitude of ρℓ(calibrated once) determine whether a given (n, ℓ)moves slightly up or down in the ordered list of x. 112In standard PDE terminology, a Neumann boundary condition fixes the normal derivative (here j′ ℓ(xR) = 0 at the surface xR), corresponding physically to vanishing normal curvature flux across a perfectly stress-free interface. A Dirichlet boundary would instead fix the value of the field itself. A Robin condition is the most general linear combination, a jℓ(xR) + b j′ ℓ(xR) = 0, and reduces to Neumann or Dirichlet when a= 0 or b= 0, respectively. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 117 2) Spin–geodesic splitting as a tiny angular correlation. TSRT’s trembling geometry induces a small correlation between the local geodesic frame and an internal angular-momentum– like degree of freedom traditionally called ’spin.’113 At the level of the magic-number spectrum, this weak correlation effectively splits each (n, ℓ)family into two nearby branches labeled by j=ℓ±1 2. We parameterize the corresponding dimensionless shift by a coefficient κℓ(again calibrated once; Appendix A) multiplying the standard angular factor hl·si=j(j+ 1) −ℓ(ℓ+ 1) −s(s+ 1) 2=(+ℓ 2, j =ℓ+1 2, −ℓ+1 2, j =ℓ−1 2,s=1 2.(186) We write the induced x-shift as ∆x(spin) nℓj ≈κℓhl·si ℓ+1 2 =       +κℓ 2, j =ℓ+1 2, −κℓ 2 ℓ+ 1 ℓ+1 2 , j =ℓ−1 2, (187) which cleanly separates an overall ℓ-scale from a tiny splitting (κℓ≪1). This preserves total degeneracy while allowing re-ordering among near neighbors. 3) Corrected locking arguments and ordering. Combining (185) and (187), the corrected locking arguments are exnℓj =xnℓ + ∆x(surf) nℓ + ∆x(spin) nℓj , j ∈ℓ−1 2, ℓ +1 2.(188) The algorithm is then: 1. Raw list. Assemble the Neumann roots xnℓ for the desired (n, ℓ)range and sort by increasing x(Table 20 (p. 116) corresponds to this list and its cumulative degeneracy with trembling-phase duality). 2. Apply surface correction. Compute ∆x(surf) nℓ from (185) using the shared ρℓand Bessel values at xnℓ. 3. Resolve spin branches. For each (n, ℓ), create two branches j=ℓ±1 2with ∆x(spin) nℓj from (187). Assign sub-degeneracies consistent with total 2(2ℓ+1) (trembling-phase duality is preserved; the splitting does not change total count, it only redistributes it between the two close-by entries). 4. Resort and accumulate. Resort the list by exnℓj, then recompute the cumulative occupancy Ncby summing the (sub-)degeneracies in that new order. 4) From raw closures to TSRT closures. Because both ρℓand κℓare small, only near neighbors move, but this is precisely enough to bundle the cumulative counts into the empirical closure set. With the single shared calibration (Appendix A), the cumulative sequence becomes N(TSRT) c={2,8,20,28,50,82,126},(189) as quoted in, e.g., Section 9. Table 21 (p. 118) (below) summarizes the mapping from the raw spherical ordering (Table 20 (p. 116)) to the corrected TSRT closures. 113In conventional quantum mechanics, spin is an intrinsic, quantized angular momentum carried by particles and represented by irreducible SU(2) multiplets. In TSRT, we do not postulate such quantum degrees of freedom. Instead, what is usually called “spin” is interpreted as a discrete geometric orientation of the trembling eigenmode with respect to the local geodesic frame. For nucleons this orientation space has the same two-valued structure as a spin-1 2representation, so the familiar labels s=1 2and j=ℓ±1 2can be reused as ’bookkeeping’ for how internal trembling geometry couples to orbital curvature, without importing shell-model wavefunctions or quantum matrix elements. The present paper does not attempt a full TSRT spin theory; it only exploits this minimal geometric structure to organize the small splitting between closely spaced curvature modes. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 118 5) Reproducibility (exact same steps in code). The MATLAB routine magicnumber_calc.m (Appendix Y, Listing 63 (p. 336)) implements precisely the steps above: (i) build the raw Neumann list (n, ℓ, xnℓ); (ii) compute ∆x(surf) nℓ from (185); (iii) build spin branches with (187); (iv) resort by exnℓj via tsrt_magic_correction_order; (v) re-accumulate the cumulative occupancy. The script writes both the raw and corrected sequences to CSV (magic_raw.csv,magic_corrected.csv) and generates a diagnostic plot (raw vs. corrected Nc). No hidden parameters are introduced; ρℓand κℓare the same constants used in deformation/OES fits and documented in Appendix A. Table 21: Mapping from raw spherical closures (cumulative Ncobtained from the Neumann list in Table 20 (p. 116)) to TSRT-corrected closures using the small surface–geometry and spin– geodesic shifts described in Equations (185)–(188). Neighborhood (raw ordering) Correction effect Bundled closure (TSRT) Nc≈18 ↔20 surface +spin split reorders near neighbors 20 Nc≈32 ↔28 re-bundling across (ℓ=3) and (n=2, ℓ=0) 28 Nc≈50 minor local reorder; stable 50 Nc≈82 minor local reorder; stable 82 Nc≈126 minor local reorder; stable 126 The correction is not an ad hoc re-labelling: it is a transparent, first-order perturbation of the locking condition and a tiny spin–geodesic correlation. Together they induce only local re-orderings in the x-spectrum, which are sufficient to align the cumulative degeneracy steps with the empirical magic closures. The procedure is fully deterministic and reproducible with the shared constants. D.6 How to reproduce the numbers (scripts and steps) A minimal workflow uses magicnumber_calc.m (Appendix Y, Listing 63 (p. 336)): 1. Compute zeros of j′ l(x)for l= 0 . . . lmax and small n(e.g. n= 1,2) using a robust Bessel routine; collect (n, l, xnl). 2. Sort by xnl; assign gnl = 2(2l+ 1) and form the running sum N(raw) c. 3. Apply the shared surface/geometry and spin–geodesic correction from Appendix A (exact constants/flags are read from tsrtconstants.m), which slightly reorders near neighbors to produce (189). 4. Emit magic_raw.csv,magic_corrected.csv, and a comparison plot (raw vs. corrected vs. empirical). This path reproduces Table 20 (p. 116) (raw) and Table 21 (p. 118) (corrected mapping) exactly under the current calibration. D.7 Link to main text and other appendices Binding kinks (shell systematics). Gaps ∆ω∝(kn′l′−knl)at closures (Table 20 (p. 116)) generate the magic-number kinks in Section 5 via the shell–curvature contribution in Equation (10). Parity and OES. The angular multiplicity (2l+1) together with the twofold trembling-phase degeneracy explains the even/odd occupancy structure used by the deterministic OES model (Figure 3 (p. 24); Section 10.7). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 119 Shared calibration. The same small surface/geometry and spin–geodesic correction (Appendix A) is used in magic-number closures, deformation stiffness, and OES. No additional parameters are introduced here. D.8 Summary for the reader Magic numbers in TSRT arise from geometric resonance closures of trembling curvature modes in a finite domain with locking boundary conditions. The discrete structure comes from boundaryenforced standing patterns (not from quantized single-particle orbitals). A tiny, universal correction—already present in other TSRT sectors—maps the ideal spherical count to the empirical sequence 2,8,20,28,50,82,126. The entire construction is deterministic, reproducible from the listed scripts, and tightly integrated with the rest of the paper. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 120 E Calibration, Physical Constants, and Normalizations Consistency with implementation. All constants and normalizations are centralized in tsrtconstants.m and established by TSRT_Calib_NormEn_Deuteron.m, TSRT_Calib_Fission_ScissWth_U235.m, and TSRT_Calibrate_GammaE2_156Gd.m. Per-nucleus overrides are not used. Supports Sections 4, 15. MATLAB: tsrtconstants.m,TSRT_Calib_NormEn_Deuteron.m, TSRT_Calib_Fission_ScissWth_U235.m,TSRT_Calibrate_GammaE2_156Gd.m. This appendix specifies all constants, units, and normalization procedures required to make the TSRT nuclear results reproducible. Every numerical calculation in the main article and later appendices refers back to the conventions defined here. All conversions and sign-convention notes are provided in Appendix V, Appendices V.1–V.3. Action scale. We denote by ~TSRT the emergent TSRT action unit that coincides numerically with Planck’s constant ~in the low-curvature, coarse-grained limit. For readability we write ~ throughout, with the understanding that ~≡~TSRT wherever TSRT action thresholds appear. E.1 Constants and Units All values are expressed in SI units unless otherwise noted. For convenience, fundamental constants are listed here with their CODATA 2022 recommended values: c= 2.997 924 58 ×108m/s,(190) G= 6.674 30(15) ×10−11 m3kg−1s−2,(191) ~= 1.054 571 817(13) ×10−34 J s,(192) mp= 1.672 621 923 69(51) ×10−27 kg,(193) mn= 1.674 927 498 04(95) ×10−27 kg,(194) me= 9.109 383 7015(28) ×10−31 kg,(195) e= 1.602 176 634 ×10−19 C,(196) ε0= 8.854 187 8128(13) ×10−12 F/m.(197) Geometric constants vs. computational scale. The computational pipeline uses a single global energy scale, C.norm_energy, to map curvature integrals to Joules. This scale is fixed once by matching the deuteron binding energy (2.224 MeV) as defined in tsrtconstants.m (Listing 2 (p. 213)); see Appendix E.3. For clarity in the theoretical development, we sometimes refer to abstract geometric conversion constants; however, the shipped MATLAB code does not expose separate α, β, γ parameters for energy conversion—the entire mapping is handled by C.norm_energy.S-factor slopes (α, β)in Appendix P are unrelated (they are logarithmic slopes in S(E)). Calibration strategy. A single normalization is used throughout: C.norm_energy is fixed by matching the deuteron binding (2.224 MeV) via TSRT_Calib_NormEn_Deuteron.m and then pasted into tsrtconstants.m. Where lifetimes are discussed, a separate proportionality Cλ appears only in the lifetime estimator (Appendix A); it does not affect energy integrals or any figure/table outside lifetime. No other per-observable rescalings are used. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 121 Normalization policy. The single global scale C.norm_energy is fixed once by the deuteron binding (2.224 MeV) via TSRT_Calib_NormEn_Deuteron.m and used verbatim throughout; see Listing 2. E.2 Energy normalization used in computations All energies are obtained by integrating a scalar contraction K(x)≡Kµνuµuνover volume and applying a single global scale C.norm_energy. In spherical coordinates with separable quadrature, the discrete integral used in the listings is E=C.norm_energy X i,j,k hK(ri, θj, φk)r2 ii(wr)i(wθ)j(wφ)k,(198) where (wr, wθ, wφ)are the one-dimensional weights returned by tsrtmakegrid.m (see Listing 3 (p. 214) in Appendix Y); wθalready integrates the sin θ dθ measure, so no extra sin θfactor is applied. Equation (198) is implemented in Listing 5 (p. 216). The single global factor C.norm_energy is fixed once, within tsrtconstants.m (Listing 2 (p. 213)), to reproduce the deuteron binding energy 2.224 MeV. Workflow. Run Listing 44 (p. 297) once to write C.norm_energy into tsrtconstants_calibrated.mat, then copy the printed line into tsrtconstants.m so all modules (binding, fission, hindrance, lifetimes) consume a consistent normalization. E.3 MATLAB: Constants and Normalization Purpose. tsrtconstants.m (Listing 2 (p. 213)) defines CODATA constants, unit conversions, grid defaults, and the single global normalization C.norm_energy. It is the sole source of truth for constants used by all other scripts. Usage. Call C = tsrtconstants(); once at the start of every script and pass Cdownstream (tsrtmakegrid, curvature, energy). The function throws a clear error if C.norm_energy has not yet been calibrated. Calibration. The global normalization constant C.norm_energy is defined directly in tsrtconstants.m (Listing 2 (p. 213)) to reproduce the deuteron binding energy (2.224 MeV). No other per-observable tunings are used. Sanity checks. Verify conversions (e.g. C.MeV_to_J = 1.602176634×10−13 J/MeV) and basic radii/weights (tsrtmakegrid.m) before production runs. Usage. All other codes in the appendices begin with C = tsrtconstants(); to guarantee reproducibility. No hidden or local constants are allowed in subsequent scripts: any physical or model parameter must come from this struct. Reader guidance. The placeholders for (α, β, γ)shown here are set to unity for clarity. Their actual values are computed in Appendix E and written back into this file when reproducing the tables. This separation ensures that readers can distinguish between (i) fixed physical constants (immutable), and (ii) single-point geometric calibrations (derived once, then reused consistently). The script is fully self-contained and can be copy–pasted directly into MATLAB without modification. All subsequent numerical results in the paper are traceable back to the constants defined here. Code. tsrtconstants.m performs central constant/units setup and defines the global normalization C.norm_energy. The full listing is in Appendix Y, Listing 2 (p. 213). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 128 Determinism and portability. All grids and weights are deterministic (no RNG). simpsonweights is self-contained. gaussleg here is also self-contained; if one instead uses an external lgwt, one must ensure only one GL provider is on the MATLAB path to avoid ambiguity. Validation and convergence. For smooth integrands in this work, Simpson, with (Nr, Nθ, Nφ) = (2048,256,128) yields δQ/Q < 10−3(Appendix Y.3). To verify on the system, one runs tsrtconvergence() and confirms the reported relative changes plateau below one’s target tolerance. Performance notes. Cost scales approximately linearly with each grid dimension for separable quadratures. If memory/runtime are tight, begin with (Nr, Nθ, Nφ) = (512,128,64) and increase until convergence is met. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 129 G Trembling Fields for Nucleons and Composite Nuclei This appendix provides the trembling field profiles used in all nuclear calculations. For the baseline reproduced in code, we use a time-averaged scalar envelope Ξ(x)on the spherical grid (Appendix F). The curvature contraction reduces to a scalar K(x)and, in the listings, is taken as K= Ξ (Listing 4 (p. 215)); energies are then E=C.norm_energy×the weighted integral of K(Listing 5 (p. 216)). Tensor refinements can be added later, but all figures/tables here use this scalar baseline. G.1 Parametric scalar baseline for nucleons Each nucleon is represented by a localized, time-averaged scalar envelope on the spherical grid, Ξ(N)(x) = ANcos(φN) exph−r/r0,N 2i,(213) where r=kxk,r0,N is a species-dependent core scale (defaults ∼0.85 fm for protons and ∼0.90 fm for neutrons), ANis a dimensionless amplitude, and φNis a fixed phase used only as a deterministic multiplier cos φN. This is the exact profile constructed by tsrtnucleonfield.m (Listing 24 (p. 250)); optional ’L2’ normalization enforces P|Ξ|2r2wrwθwφ= 1 using the grid weights. Normalization used in computations. Energies are not set by a per-nucleon rest-energy condition; instead, a single global scale C.norm_energy maps the curvature integral to Joules/MeV and is fixed once by matching the deuteron binding energy (2.224 MeV) directly within tsrtconstants.m (Listing 2 (p. 213)). This constant ensures a unique curvature-to-energy conversion throughout all computations. See Appendix E.2 for the exact numerical integral used. Notes. The baseline is deliberately spherical and time-averaged; any tensorial construction and explicit time dependence are suppressed at this stage and can be layered later without changing the interfaces or the reproducibility of the present results. G.1.1 MATLAB: Nucleon Trembling Field Generator This routine builds the time-averaged scalar profile Ξ(x)for a single nucleon on the common grid. Purpose. tsrtnucleonfield.m returns a deterministic scalar field Xi.profile on Gof size Nr×Nθ×Nφ, with metadata Xi.meta. The envelope is the spherical Gaussian in Equation (213). Inputs and outputs. •Inputs: –G: grid from tsrtmakegrid (must include w_r,w_theta,w_phi if one requests ’L2’ normalization). –Either: ∗Struct form: Xi = tsrtnucleonfield(G, params), with optional fields type=’proton’|’neutron’ (default ’proton’), r0 (core scale, SI), A(amplitude), phi (phase, radians), norm=’none’|’L2’ (default ’none’). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 130 ∗Legacy form: Xi = tsrtnucleonfield(G, C, type, t) — accepted for compatibility; Cand tare ignored (time-averaged profile). •Output: Xi.profile (real Nr×Nθ×Nφarray) and Xi.meta. Usage. C = tsrtconstants; G = tsrtmakegrid(C); Xi = tsrtnucleonfield(G, struct('type','proton','norm','L2')); If ’L2’ is requested, the code uses the grid’s separable weights with the correct measure (no extra sin θfactor; see Appendix E.2). Code. Full listing in Appendix Y, Listing 24 (p. 250). G.2 Composite configurations and phase locking A nucleus with (Z, N)nucleons is modeled as a phase-fixed (e.g. t= 0) superposition of scalar envelopes centered at nucleon positions Ri: Ξ(Z,N)(x) = A X i=1 siAicosφiexph−kx−Rik2/r2 0,ii,(214) where si∈ {+1,−1}is a species/coupling flag (proton vs. neutron; used only to set per-species defaults), and r0,i, Ai, φiare the nucleon’s scale, amplitude, and phase. In code, tsrtnucleonfieldshifted.m evaluates the shifted envelope by radial interpolation of the unshifted profile (Listing 25 (p. 251)); tsrtcompositefield.m sums them (Listing 26 (p. 252)). Phase locking (deterministic). Phases φi∈ {0, π}are chosen to maximize local overlap under the even/odd constraints (even–even pairs lock in-phase; odd unpaired nucleons retain φ= 0). This global t= 0 convention keeps energy differences phase-consistent; RMS factors are absorbed once into C.norm_energy (Appendix E.3). Geometry. The Rimay be chosen by simple symmetric placements (light A) or by approximate shell-like layers (heavier A); in all cases results reported here are time-averaged and use the same quadrature backbone (Appendix F). Phase-locking principle. For stable nuclei, relative phases φiare not arbitrary. Stability requires that destructive interference of curvature variance is minimized across the nuclear volume, i.e.: φi−φj≈0 (mod π),∀i, j, (215) so that oscillations reinforce collective curvature binding. Odd-even staggering in nuclear binding energies corresponds in TSRT to the extra stability provided when all nucleons are fully phasesynchronized. Geometric placement. Nucleons are placed on symmetry-constrained lattices: • Light nuclei (A≤4): tetrahedral/triangular arrangements, • Medium nuclei: shell-like layers of radii ∼RN3 √A, • Heavy nuclei: approximately spherical packing constrained by curvature minimization. This prescription is consistent with the experimental trends in nuclear radii RA≈r0A1/3with r0≈1.2fm. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 131 Curvature cancellation and binding. Equation (214) ensures that local oscillations cancel in the far field if phases are aligned, preventing runaway curvature and leading to a finite binding functional as defined in Equation (170). G.2.1 MATLAB: Composite nucleus field builder Purpose. tsrtcompositefield.m builds the composite scalar field Ξ(nucleus)(x)by summing shifted nucleon envelopes. Inputs and outputs. •Input G: grid from tsrtmakegrid. •Input nuc: struct with types{1..A} ∈ {’proton’,’neutron’},R∈R3×A(centers in meters), and optional per-nucleon fields r0,Aamp,anis,phi (scalar or 1×A). •Output Xi.profile: real Nr×Nθ×Nφarray (sum of constituents). The third argument tis accepted but unused (time-averaged baseline). Implementation notes. The helper tsrtnucleonfieldshifted computes a shifted envelope by interpolating the radial average of the unshifted field; this is adequate for isotropic templates used in all reproduced results. If strong anisotropy is later introduced, upgrade the translation to a full 3D interpolation. Code. Full listing in Appendix Y, Listing 26 (p. 252). G.3 MATLAB: Field Generators These routines define the scalar nucleon envelope and place it at arbitrary centers; they are the building blocks for all composites. Purpose. •tsrtnucleonfield.m: constructs the canonical time-averaged scalar envelope for a proton or neutron on G. Options: amplitude A, core scale r0, phase phi, and optional ’L2’ normalization. •tsrtnucleonfieldshifted.m: places a nucleon envelope at a chosen Cartesian center R (m) by radial interpolation of the unshifted profile. Inputs and outputs. •Inputs: G(from tsrtmakegrid); for tsrtnucleonfield, a params struct as above; for tsrtnucleonfieldshifted,params and center R. •Output: Xi struct with Xi.profile (Nr×Nθ×Nφ)and Xi.meta. Notes. An anis parameter is accepted and recorded in metadata but not used to deform the profile in the baseline reproduced here (spherical Gaussian). Energies are obtained by passing the resulting Xi to tsrtcurvaturetensor →tsrtenergyfromcurvature; see Listings 4 (p. 215) and 5 (p. 216). Code. tsrtnucleonfield.m and tsrtnucleonfieldshifted.m are listed in Appendix Y, Listings 24 (p. 250) and 25 (p. 251). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 132 H Curvature Measures and Energy Functionals Consistency with implementation. The measures Kµν,K≡pKµνKµν, and hK2iare computed by tsrtcurvaturetensor.m, with energy assembly in tsrtenergyfromcurvature.m as used in Sections 4 and 5. Supports Sections 4, 5. MATLAB: tsrtcurvaturetensor.m,tsrtenergyfromcurvature.m. The trembling fields described in Appendix G are mapped to a scalar curvature–like measure K(x), which is then integrated to produce physical energies. This appendix documents (i) the theoretical construction from the metric decomposition gµν =ηµν +ξµν and (ii) the reference implementation used for all figures/tables: a minimal, time-averaged scalar contraction with K≡Ξ.profile on the spherical grid and a single global energy calibration C.norm_energy. All expressions are consistent with the TSRT metric signature (+,−,−,−). H.1 From ξµν to a scalar measure K Theoretical construction (optional; not used in the baseline). Starting from the TSRT metric gµν =ηµν +ξµν (216) ηµν = diag(+1,−1,−1,−1) (217) one may build a Ricci-like contraction Kµν =gαβRµανβ (218) from the Christoffel symbols and Riemann tensor, and then define K(u) = Kµνuµuνwith uµ= (1,0,0,0) in the nuclear rest frame. A variance hK2ican be formed by hK2i=1 VZV [K(u)]2d3x, (219) which is useful for stability diagnostics (Appendix I). Reference implementation (used throughout). For all published results in this manuscript we do not compute derivatives of ξµν. Instead, we adopt a minimal, time-averaged scalar contraction: K(x)≡Ξ.profile(x),(220) as produced by the field generators in Appendix G.3. Energies are then obtained by a weighted spherical integral of Kwith a single global scale, C.norm_energy (Appendix E.2). This guarantees transparency and reproducibility. Discrete integral, as implemented. On the separable spherical grid G(Appendix F), the energy-like integral is evaluated as E=C.norm_energy X i,j,k hK(ri, θj, φk)r2 isin θji(wr)i(wθ)j(wφ)k∆r∆θ∆φ, (221) for the Simpson baseline (Appendix F.3). If Gauss–Legendre is used in θvia x= cos θ, the sin θ factor is absorbed by the change of variables and must not be applied again (see Appendix E.2). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 133 H.2 Binding, fission, and fusion energetics Binding energy, as implemented. Energies are obtained by integrating the scalar measure K(Equation (221)) for isolated nucleons and for the composite nucleus, then taking the difference: Ebind(Z, N) = A X i=1 EΞ(i) iso−EhA X i=1 Ξ(i)i,(222) with a single global calibration C.norm_energy fixed once (deuteron benchmark; Appendix E.2). With this sign convention, a bound system has Ebind >0. This is exactly what tsrtbindingenergy.m implements (Appendix I). Fission energy release. For (Z, N)→(Z1, N1) + (Z2, N2), Efiss =Ebind(Z1, N1) + Ebind(Z2, N2)−Ebind(Z, N).(223) All three bindings are computed by the same integral pipeline on the same grid. Fusion energy release. For (Z1, N1) + (Z2, N2)→(Z, N), Efus =Ebind(Z, N)−hEbind(Z1, N1) + Ebind(Z2, N2)i.(224) Numerical implementation and calibration. All integrals use the grid and weights of Appendix F; convergence is verified per Appendix F.2. The only energy calibration is the global C.norm_energy (deuteron anchor); there is no per-nucleus tuning. Stability diagnostics may additionally use the variance hK2iand ∆K2as defined in Appendix I, but these are not used to map energies in the baseline. H.3 MATLAB: Curvature and Energies These routines map scalar fields to energies with a single global calibration. They form the direct link from the field generators to all tabulated observables. Purpose. •tsrtcurvaturetensor.m: returns the scalar measure Kon the grid. In the reference implementation used here, it simply sets K≡Ξ.profile. Advanced users may replace this by a derivative-based contraction without changing the interface. •tsrtenergyfromcurvature.m: integrates Kover the spherical grid with the correct Jacobian and converts to energy via C.norm_energy. Inputs and outputs. •Inputs: grid G, field struct Xi, constants C. •Outputs: K(3D array), E(Joules; convert to MeV via C.MeV_to_J). Quadrature and determinism. We use separable Simpson weights by default (Appendix F.3); Gauss–Legendre in θis supported if used consistently (see Appendix E.2). Both functions are pure and reproducible. Code. Listings 4 (p. 215) and 5 (p. 216). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 134 What these do. tsrtcurvaturetensor(G,Xi,C) conceptually produces the scalar contraction K≡Kµνuµuνfrom a trembling field Ξ. In the reference implementation used for all results here, it simply returns K≡Ξ.profile (minimal/reference contraction), ensuring every table/figure is reproducible without hidden tensor machinery. tsrtenergyfromcurvature(G,K,C) integrates Kover the nuclear volume using separable quadrature (Simpson weights in r, θ, φ) and the spherical Jacobian, returning energy in SI Joules after applying the global normalization C.norm_energy. Units and normalization. The integral returns SI Joules. Convert to MeV using EMeV = EJ/C.MeV_to_J. The only calibration is the global factor C.norm_energy (Appendix E.3; see also Appendix E.2); once fixed, it is used everywhere (no per-nucleus tuning). Quadrature details, as implemented. With composite Simpson in all three coordinates (default grid from Appendix F.3), the 3D integral is evaluated as Eint ≈X i,j,khK(ri, θj, φk)r2 isin θjiwr,i wθ,j wφ,k ∆r∆θ∆φ, (225) where wr, wθ, wφare the normalized Simpson weights returned by the grid builder, and (∆r, ∆θ, ∆φ)are the uniform steps. This matches the implementation in tsrtenergyfromcurvature. If one switches θto Gauss–Legendre: either one has to integrate directly on [0, π]and keep sin θin the Jacobian, or integrate in x= cos θ∈[−1,1] and drop sin θ(since dx= sin θdθ). One must use one convention consistently. Determinism and reproducibility. Both functions are pure (no RNG, no file I/O). Given the same grid G, field Xi, and constants C, output is identical across runs (up to floating-point roundoff). All published values in this appendix were regenerated with these routines. Validation checks. •Constant field test: set K≡1. The unnormalized integral should equal the geometric volume Rr2sin θdrdθdφ, converging to 4 3πR3when the radial limit is R. •Separable test: use K(r, θ, φ) = f(r). Verify that angular sums collapse to 4π(within quadrature tolerance) and the result reduces to 4πRf(r)r2dr. •Grid refinement: run tsrtconvergence() (Appendix Y.3) and confirm relative changes .10−3with the default grid. Extending the contraction. To implement the full tensor contraction, replace the minimal contraction in tsrtcurvaturetensor by 1. compute spatial derivatives of Ξµν on (r, θ, φ)(finite differences or spectral); 2. construct Kµν from the TSRT curvature–stress definition; 3. contract with a unit time-like vector uµ= (1,0,0,0) in the (+,−,−,−)metric (Appendix V); 4. (optionally) time-average the result over one cycle if Ξcarries an explicit cos ωt. One must keep the interface unchanged so downstream energy integrations and tables remain drop-in reproducible. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 135 I Stability Maps and Lifetime Estimators Consistency with implementation. Lifetime predictions use tsrtlifetime.m, and tsrtlifetimemode.m, with the deterministic rate diagnostic from tsrt_gamma_rate.m (see Section 8, Equation (61)). Supports Sections 9, 8. MATLAB: tsrtlifetime.m,tsrtlifetimemode.m, and tsrt_get_exp_half_life.m. This appendix documents the deterministic pipeline behind the lifetime predictions in the main article. All constants are calibrated once on anchors and stored in tsrtconstants_calibrated.mat. I.1 Curvature-Slope Stability Map (Deterministic ∆K2Analysis) This subsection documents the procedure used to compute and plot the TSRT stability map and the line plots shown in Figure 4 (p. 42), and Figure 5 (p. 43). The method evaluates, for each nuclide (Z, A)present in the bindings CSV, the proper-time curve ∆K2(τ)on a fixed TSRT grid and extracts the local slope magnitude at the proper-time stationary point τ⋆: d(∆K2)/dττ⋆,(226) which we refer to as the stability scale. A larger value indicates increased curvature-coupled stiffness (more rapid deterministic response); local minima correlate with geometric shell closures. Inputs and calibration. The pipeline consumes: (i) the cleaned TSRT bindings table tsrt_bindingscan_cln.csv (columns at least Z, A, and either Ebind or Ebind/A), and (ii) the calibrated constants file tsrtconstants_calibrated.mat, which must contain the global normalization Cλand the grid parameters for reproducible geometry: Cλ, rmax, Nr, Nθ(and implicitly Nφ= max(128,2Nθ)). No empirical masses are injected in these steps; any Q-values elsewhere in the paper (when needed for decay channels) are computed from TSRT bindings via Equation (227). Geometric evaluation. For each (Z, A), we generate the TSRT grid Gvia tsrtmakegrid(C) and compute a robust surrogate of ∆K2(τ)with tsrtdeltaK2longtau(Z,A,C,G,τ)over a uniform τarray. After a small moving-average smoothing to suppress micro-oscillations, we locate τ⋆by the minimum of ∆K2(τ)on the production grid and estimate the local slope by a short least-squares line fit in a symmetric window. This yields the scalar map value S(Z, N) = d(∆K2)/dττ⋆at N=A−Z. Outputs. The main builder writes a compact data file: tsrt_stabilitymap_data.mat ⇒ { ZN,NN,F,Zmin,Zmax,Nmin,Nmax }, where ZN and NN are integer grids of proton and neutron numbers and Fis the stability scale on those grid cells (NaN for uncomputed cells). The diagnostic step additionally writes a ridge polyline CSV (per-Zmaxima and minima for QA): tsrt_stability_ridge.csv. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 136 How to reproduce. From MATLAB: clear functions; rehash toolboxcache stabilitymap_deltaK2_strict('tsrt_bindingscan_cln.csv','tsrt_stabilitymap'); % Optional QA panels and ridge CSV: tsrt_stabilitymap_diagnose('tsrt_stabilitymap_data.mat','tsrt_bindingscan_cln.csv'); % Publication variants: tsrt_stabilitymap_points('tsrt_stabilitymap_data.mat','tsrt_stabilitymap_points'); tsrt_stabilitymap_filled_from_data('tsrt_stabilitymap_data.mat', ... 'tsrt_stabilitymap_filled', 'natural'); % Line plots along isotopic chains: tsrt_stability_lines('tsrt_stabilitymap_data.mat', 'tsrt_stability_lines'); Notes on interpretation. Figure 4 (p. 42) and Figure 5 (p. 43) plot only directly computed points (no interpolation) of S(Z, N)across the nuclide chart, thereby revealing the geometric sawtooth oscillations (shell cycles) along isotopic chains. I.2 Mode-aware Q-values from TSRT bindings We compute all decay Q-values deterministically from the TSRT binding table (tsrt_bindingscan_cln.csv) without injecting empirical masses. At the level of binding energies B(Z, A)(in MeV), the general Q-definition used in the code is: Qmode =Ebind(initial)−Ebind(final)−Eemit,(227) where Eemit is the emitted particle’s (or cluster’s) binding energy contribution appropriate to the channel. For the two channels used in this work: Qβ−(Z, A) = hB(Z+1, A)−B(Z, A)i+ ∆npe,(228) Qα(Z, A) = B(Z−2, A−4) + Bα−B(Z, A).(229) Here ∆npe ≡(mn−mp−me)c2≈0.782343 MeV accounts for the neutron–proton–electron mass difference in atomic conventions, and Bα≈28.295674 MeV is the 4He binding energy. Atomic-mass route equivalent. If atomic masses M(Z, A)are available, the code path is equivalent: Qβ−=M(Z, A)−M(Z+1, A)c2, Qα=M(Z, A)−M(Z−2, A−4) −Mαc2,(230) with Mα= 4.00260325413 u and uc2= 931.49410242 MeV. In practice tsrtQvalues.m uses the bindings route by default and falls back to mass-route identities only if needed. No empirical Q values are injected. Implementation note. The helper tsrtQvalues.m computes only the requested channel’s Q, verifies the presence of the required daughter rows, and returns a finite value when available; otherwise the prediction is marked unavailable upstream (never NaN in tables). © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 137 I.3 Energetics and curvature suppression Binding and stability follow from the trembling-curvature suppression, ∆K2(Z, N) = hK2iiso −hK2iZ,N,(231) with ∆K2>0favorable. The geometric base factor used in βand SF descends from the proper-time slope |∂τ∆K2|τ⋆obtained on the production grid by local quadratic regression. I.4 Mode-aware Q-values: implementation path We compute Qβand Qαfrom a curated TSRT bindings table114 (tsrt_bindingscan_cln.csv) using atomic masses or binding differences (Equation (227)); we never inject experimental masses at runtime. Light systems may provide explicit Qoverrides in dedicated columns, but they are optional. I.5 Per-mode lifetime predictions Per-mode predictions are gathered in tsrtlifetimemode.m: •β−:λβ=Cλ|∂τ∆K2|τ⋆Fβwith Fβgiven by Equation (72). The signed TSRT matrix element Mβ(Section 9.8) is an optional multiplicative factor. •α:λα=CαP0exp[−2Sα(Qα)] with finite-size Coulomb, Woods–Saxon nuclear well, and Langer correction when L > 0;Cαanchored on 210Po. • SF: λSF =P0,SF exp(−Sf)with a macroscopic ridge surrogate; P0,SF anchored on 252Cf. I.6 Pipeline and files The benchmark driver is RunLifetime_All.m: 1. Preflight checks, load constants, grid defaults. 2. βanchor: set Cλon 60Co. 3. Optional: adjust σβon 137Cs. 4. αanchor: set Cαon 210Po; light shape validation on Ra/Th/U. 5. SF anchor: set P0,SF on 252Cf. 6. Emit diagnostics CSV and L A TEX table via tsrt_emit_tables.m. I.7 Numerical safeguards We map invalid rates to t1/2=∞(never NaN), compute mode-aware Qvalues only, and perform two-level resolution checks. Calibrated constants are saved atomically. 114built from TSRT-derived binding energies and used as a single source of truth. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 144 K.3 Implementation recipe (used in figures/tables) 1. Compute the source-frame energy Esrc from TSRT curvature (main text). 2. Apply recoil and, if relevant, Doppler/gravity (Appendix J) to obtain the baseline detected energy. 3. Add δTSRT EM from Equation (245): estimate Reff from nuclear geometry, set LNF =κNFReff with κNF ∈[2,5], choose Zeff (angular average for photons), use (ηγ, ηq)as fixed in Appendix E and carry them unchanged across all lines. The MATLAB in Appendix Y generates Tables 22 (p. 144)–15 (p. 109) from these settings. K.4 Worked examples The CSCT numbers shown in this appendix are produced by the canonical script, csctmaketables.m (Listing 27). For archival completeness and to eliminate any external file dependencies at compile time, we embed the resulting tables explicitly below. No \input from external files is required to compile this manuscript. Table 22: CSCT for representative γlines. Baseline includes recoil; CSCT uses Equation (245) with κNF = 3,Zeff = 0.7Z, and αTSRT EM = 1/137. The coupling ηγis fixed by the 57Fe 14.4keV line (2.00 eV). Transition Esrc (keV) Baseline Edet (keV) δTSRT EM (eV) Fraction (10−4) 57Fe (14.4 keV) 14.4 14.4−recoil 2.00 1.39 137Ba (661.7 keV) 661.7 661.7−recoil 197.94 2.99 Table 23: CSCT for charged ejectiles. Baseline includes two-body kinematics; CSCT uses Equation (245) with κNF = 3,Zeff = 0.7Z,αTSRT EM = 1/137, and ηqfixed by a 300eV shift for a ∼300keV conversion electron. Channel Baseline Edet (MeV) δTSRT EM (keV) Fraction (10−4) Internal conversion e−(0.2–0.5 MeV) 0.2–0.5 0.30 0.6–1.5 β±endpoint (few MeV) 1–3 0.30 0.10–0.03 K.5 Notes on calibration The charge-sensed electromagnetic curvature transport (CSCT) model provides a deterministic, closed-form description of near-field energy shifts for both photons and charged particles departing from a nucleus. The same geometric formalism is applied consistently throughout all tabulated data, ensuring that the CSCT corrections remain transparent, reproducible, and fully parameterized by measurable quantities. Purpose. The numerical generation of Tables 15 (p. 109) and 22 (p. 144) is automated by the CSCT table generator, a deterministic routine (csctmaketables.m, Listing 27 (p. 254)) that computes near-field electromagnetic shifts from the analytical model in Equation (245). The procedure applies the same calibration to all transitions without empirical fitting or per-line adjustment. © Nico F. Declercq DOI: 10.5281/zenodo.17666433 November 21, 2025 [Document text truncated for crawler view.]