Gravity from the Structural Intensity Field ρ: A Real-Time Spectral Framework

Prepared for submission to JHEP1 Gravity from the Structural Intensity Field ρ: A2 Real-Time Spectral Framework3 Mohamed H. M. Makraini4 UNED National University,5 Madrid, Spain6 Royal Spanish Society of Physics (RSEF),7 Plaza de ciencias 1, 28040, Madrid, Spain8 E-mail: [email protected] 9 Abstract: We present a precise, coordinate-free definition of the structural intensity field10 ρand use it as the primary geometric variable in real time, yielding a compact curvature11 identity and an Euler–Lagrange equation for σfrom the spectral action. The construction12 is supported by three pillars: (i) an adiabatic twistor collapse that selects the anti-self-13 dual sector and reduces the spectral action to four dimensions; (ii) a noncommutative14 Penrose–Ward correspondence identifying ASD gauge data with holomorphic modules on15 the twistor bundle; and (iii) a time–space no-go theorem enforcing θ0i= 0 under antiunitary16 time reversal and Hadamard spectral regularity. Together they ensure causal, Lorentzian17 consistency and organize gauge contributions while ρdrives the effective metric via R=18 −6ρ−1/2□ ρ1/2. In the late-Universe linear regime this implies cT= 1, αM= 0, and no19 gravitational slip, and the framework extends to nonunital noncommutative manifolds by20 locality of the heat kernel and spectral compactness-by-localization.21 Keywords: structural intensity field, emergent geometry, spectral action, modular opera-22 tor, ζ-regularized determinant, Krein space, Lorentzian Dirac operator, curvature identity,23 twistor collapse, ASD selection, noncommutative Penrose–Ward, time–space no-go.24 Zenodo ePrint: 1739317125 Contents26 1 Assumptions and Robustness 127 2 The Structural Spacetime Field ρ228 3 Twistorial Spectral Quadruple 329 3.1 Components 330 3.2 Background Independence and Non-Circularity 431 4 Lorentzian Signature Without Wick Rotations 432 4.1 Krein Spaces as Pregeometric Reality 433 4.2 Modular Flow as Intrinsic Time 434 4.3 Spectral Action in Real Time 435 5 Spectral Exclusion and Matter-Spacetime Equilibrium 536 5.1 Structural Exclusion Principle 537 5.2 Field Equations and Equilibrium 538 6 Quantitative Phenomenology: From Parameters to Data 539 7 Observational Tests and Data Interface 640 7.1 Predictions in Quasi-Commutative Regime 741 7.2 Geometric Confinement and Mass Gap 742 8 Mechanism Flow: From Spectral Dynamics to Effective Geometry 743 9 Twistor Collapse, Noncommutative Penrose–Ward, and a44 Time–Space No-Go 845 10 Limitations and Future Work 946 11 Connections to Other Approaches 947 12 Nonlinear and Strong-Field Regimes 1048 13 Quantitative Comparison with Other Approaches 1049 14 Conclusion 1050 A Nonunital NC setup 1151 B Computational Details: Curvature Identity and σ-Equation 1252 C FRW Case Study and Constraint Translation 1253 – i – D Geometric Phase Transitions Near ρ∼01254 E Moyal Kernels and a Local Compactness Criterion 1455 F FRW Mini-Solver and ΛCDM Comparison 1556 Executive Summary (Intuitive Overview)57 Notation and Nomenclature (Consistency Checklist)58 We use η(conformal time) and t(cosmic time) with aH =H, and fix ρ=e2σ,geff =ρ ηµν 59 at leading order. Indices: Greek µ, ν for spacetime; Latin i, j for spatial. Operators:60 □=ηµν∂µ∂ν,∇for spatial gradients. Parameters: c2Λ2(kinetic normalization), c0(curva-61 ture/topological block), mσ(effective mass). Twistor/NC objects: AT(algebra), K(Krein62 space), D(Dirac-type), β(fundamental symmetry), J(real structure), θµν (Poisson/NC63 bivector). All subscripts/superscripts are harmonized to these conventions throughout.64 What is ρ?A single, positive scalar field—the structural intensity—that encodes how65 much “spacetime fabric” there is locally. At leading order the effective metric behaves like66 geff ∼ρ η, so curvature comes from gradients of ρ[1,2].67 Why is this useful? It lets us compute gravity from one field with a clean identity68 R=−6ρ−1/2□(ρ1/2) and a variational equation for σwhere ρ=e2σ. This follows from69 the spectral action plus heat-kernel control, giving a compact and predictive dynamics70 [3,4].71 What does it predict? In the linear late-Universe regime: luminal gravitational waves72 (cT= 1), no Planck-mass running (αM= 0), and no gravitational slip (Φ = Ψ). These are73 directly testable with current data [5–9].74 How does it connect to gauge fields and twistors? An adiabatic twistor collapse75 picks the ASD sector that organizes gauge contributions; a Penrose–Ward map ties ASD76 fields to holomorphic data; in noncommutative settings, ASD/instanton structure persists77 [10–12]. A time–space no-go theorem enforces purely spatial noncommutativity (θ0i= 0),78 consistent with causality and unitarity [13,14].79 1 Assumptions and Robustness80 We collect the working hypotheses and discuss their role and possible relaxations:81 •Completely monotone F(spectral cutoff): ensures positivity and well-defined real-82 time trace. Variants with slowly varying Fkeep all linear predictions intact [3,15].83 •Positive β-elliptic representative E(D): controls the heat-kernel expansion; weaker84 sectoriality assumptions suffice for the curvature identity and for linear cosmology [4,85 16,17].86 – 1 – ρ geff R=−6ρ−1/2□ ρ1/2 E–L(σ) from spectral action Twistor collapse ASD sector Selects Figure 1. Conceptual map: the intensity ρdrives the effective metric geff via the curvature identity and the Euler–Lagrange equation for σ; an adiabatic twistor collapse selects the ASD sector that organizes the gauge block. •Standard modular state ω: needed for the modular characterization of ρ; the spectral87 definition via kernels still works without it [18,19].88 •Trace-class/coherence conditions: required only for determinant/functional repre-89 sentations used in twistor blocks; not for the core variational dynamics [20,21].90 •Purely spatial NC (θ0i= 0): enforced by the time–space no-go under antiunitary91 Tand Hadamard/causality requirements; relaxing it spoils unitarity and the real-time92 expansion [13,14,22].93 In short, the main phenomenology is robust to large deformations of the ultraviolet reg-94 ularization and to the modular sector, as it follows from the curvature identity and the95 σ-equation alone.96 2 The Structural Spacetime Field ρ97 Context. In the Lorentzian twistorial noncommutative quadruple (AT,K, D;β, J) on a98 Krein space K, we fix the Lorentzian convention Jβ =−βJ and the real-time adjoint99 D♯:= βD†β=D. Let E(D) denote the positive β-elliptic representative entering the100 spectral action, and let Fbe a completely monotone cutoff with scale Λ >0 [16–18].101 Definition (Structural intensity). Let KF(·,·; Λ) be the Schwartz kernel of F(E(D)2/Λ2).102 We define the structural intensity field ρby103 ρ(x) := 1 µ0 trK β KF(x, x; Λ) ,(2.1) where the normalization µ0>0 is fixed by the condition that ρ≡1 on a chosen reference104 flat twistorial vacuum. By construction, ρ(x)>0 and it is dimensionless [3,4].105 Equivalent characterizations. Under standard modular and trace-class hypotheses, the106 following are equivalent up to a smooth reparametrization and a constant factor:107 1. Spectral form: the kernel definition (2.1).108 – 2 – 2. Modular form: writing ρ=e2σwith σ=h(log ∆ω) for a strictly monotone h, where109 ∆ωis the modular operator of a faithful standard state ω[18,19].110 3. Twistor-coherence form: ρ(x) = detζI+M(x)−1, with M(x) trace-class and detζ 111 the zeta-regularized determinant [20,21].112 Transformation properties and effective geometry:113 ρis gauge-invariant and diffeomorphism-covariant via the kernel definition. Interpreting ρ114 as a conformal intensity, the effective metric obeys to leading order115 geff(ρ)≃ρ η +ε[ρ, D], so that gravitational effects arise from gradients of ρ. The effective gravitational accelera-116 tion extracted from congruences is aµ∝ ∇µln √ρ[1].117 Emergent curvature identity. Setting ρ=e2σone has the scalar-curvature identity118 R=3 2ρ2(∂ρ)2−3 ρ□ρ=−6ρ−1/2□ ρ1/2,(2.2) which emphasizes curvature as secondary to intensity [1,2].119 Variational equation for σ.The real-time spectral action Sspec[D] = Tr FE(D)2/Λ2,120 varied with respect to σand after integrating out fast spinorial modes, yields in the con-121 densed regime122 c2Λ2□geff σ+c0E4(geff)σ+δAint δσ =1 2Tµµ,(2.3) where E4is the Euler density and Tµµis the matter trace. Equation (2.2) together with123 (2.3) encodes the “intensity ⇒geometry” mechanism used throughout [3,4].124 Minimal hypotheses (Hρ). (i) Fcompletely monotone, F(0) = f0>0; (ii) E(D)125 essentially selfadjoint and positive on a β-stable common core; (iii) ωfaithful and stan-126 dard (Tomita–Takesaki applies); (iv) M(x) trace-class so that detζ(I+M(x)) exists; (v)127 normalization µ0fixed by the chosen reference vacuum [3,16–21].128 3 Twistorial Spectral Quadruple129 The pregeometric foundation is the twistorial spectral quadruple:130 (AT,K, D;β, J) (3.1) 3.1 Components131 •AT: Non-commutative ∗-algebra encoding twistorial pregeometry [10,18]132 •K: Krein space with indefinite inner product [x, y] = ⟨x, βy⟩[16]133 •D:β-self-adjoint Dirac-type operator (D♯=D) [16,17]134 •β: Fundamental symmetry (β=β∗=β−1) encoding time orientation [16]135 •J: Real structure with KO-dimension 6: (J2, JDJ−1, JχJ−1) = (+1,+1,−1) [18]136 – 3 – 3.2 Background Independence and Non-Circularity137 Circular ontology is eliminated via the quotient construction:138 Aeff =AT/I, I =⟨(1 −ρ)[a, b],(1 −ρ)[a, b]∗⟩(3.2) In the condensed phase ρ→1, we recover:139 Aeff ≃C∞(M) (3.3) with Ma smooth 4-manifold reconstructed via Connes’ theorem [18].140 4 Lorentzian Signature Without Wick Rotations141 4.1 Krein Spaces as Pregeometric Reality142 The Krein space Kis not merely mathematical convenience but represents the fundamen-143 tal causal structure:144 •Indefinite inner product [x, y] = ⟨x, βy⟩encodes pregeometric causality [16]145 •Fundamental symmetry βselects time orientation before emergence of coordinate146 time [16]147 •β-self-adjointness D♯=Dmaintains unitarity in real time [16,17]148 4.2 Modular Flow as Intrinsic Time149 The Tomita–Takesaki modular flow provides algebraic time without coordinates:150 σω t(a)=∆it ωa∆−it ω, a ∈ AT(4.1) This becomes physical time in the condensed regime, with time reversal implemented al-151 gebraically:152 TβT−1=−β(4.2) See [18,19] for modular theory and its role in intrinsic time.153 4.3 Spectral Action in Real Time154 The bosonic action is computed directly in Lorentzian signature:155 Sspec[D] = Tr FE(D)2 Λ2(4.3) using the positive elliptic representative E(D) for heat-kernel calculus—no Wick rotation156 is employed [3,4,16].157 – 4 – 5 Spectral Exclusion and Matter-Spacetime Equilibrium158 5.1 Structural Exclusion Principle159 The theory respects a spectral exclusion principle:160 •ρcannot be independently tuned: it is determined entirely by spectral data [3].161 •Universal coupling through Dσ=e−σDe−σensures universality of gravitational162 response [3,15].163 •No background circularity: spacetime and matter co-emerge from spectral data [18].164 5.2 Field Equations and Equilibrium165 The dynamics follow from variation of the spectral action:166 δ δσ Tr FE(Dσ)2 Λ2+δ δσ Smatter[Ψ, Dσ] = 0 (5.1) yielding the dilaton/structural field equation:167 c2Λ2□σ+c0E4σ+δA δσ =1 2⟨Tµµ⟩matter,(5.2) where the left-hand side follows from the spectral action and heat-kernel control [3,4], while168 the matter trace on the right-hand side captures the universal response of fields in curved169 backgrounds [2]. This establishes dynamic equilibrium between spacetime (ρ=e2σ)170 and matter.171 6 Quantitative Phenomenology: From Parameters to Data172 Mapping to EFT-of-DE parameters173 Linearizing around ρ≡1, write ρ=1+δρ and σ=1 2δρ + Ord(δρ2). The curvature identity174 and the σ-equation imply, in Newtonian gauge and at subhorizon scales, cT= 1, αM= 0,175 and η= Φ/Ψ = 1, independently of the detailed values of (c0, c2) at leading order. Hence176 current bounds on (cT, αM, η) are automatically satisfied in the quasi-commutative regime177 (see EFT-of-DE mapping in [25]).178 Friedmann reduction and background evolution179 For a spatially flat FRW ansatz with conformal time ηand geff =a2(η)ηµν, set a2(η) = ρ(η).180 The identity gives R=−6ρ−1/2ρ1/2′′/a2=−3H2+ 2H′/a2, with H=a′/a. The σ-181 equation reduces to a second-order ODE for a(η), c2Λ2(σ′′ + 2Hσ′)+c0E4[a]σ+··· =182 1 2a2Tµµ,which admits ΛCDM-like backgrounds for wide parameter ranges; departures are183 controlled by c0, c2and interaction terms [2].184 – 5 – Indicative constraints185 The multimessenger bound |cT/c −1|≲10−15 (e.g. GW170817/GRB170817A) translates186 here to a trivial identity since cT= 1 at linear order [5–7]. Limits on αMfrom LSS and187 weak lensing bound any residual running beyond leading order [25]. The EGstatistic tests188 η= 1; a joint analysis with CMB lensing and RSD provides a null test for slip in this189 framework (see also early Euclid constraints on slip [9]).190 Early-time EFT mapping. At leading order cT= 1 and αM= 0 persist, while de-191 partures funnel into (αK, αB) which control scalar kinetics/braiding and impact CMB192 large-scale ISW, lensing potentials, and (indirectly) B-mode mixing; the primordial GW193 background remains luminal in this framework. See the EFT dictionary for linear LSS194 mappings [25].195 7 Observational Tests and Data Interface196 Figure and Table Style (Units and Ranges)197 All figures include units in axis labels and explicit ranges (e.g. [Mpc], [km s−1Mpc−1]); leg-198 ends specify whether curves are synthetic/real and the parameter set used. For PGFPlots,199 we recommend adding ymin/ymax,xmin/xmax, and a legend pos choice for reproducibility.200 We summarize concrete, falsifiable tests and the corresponding datasets:201 Prediction Observable / Method Representative datasets cT= 1 (luminal GWs) GW speed from neutronstar mergers GW170817 & GRB170817A; future standard sirens [5–7] αM= 0 Running of effective Planck mass LSS growth + weak lensing (DES, KiDS), CMB lensing [25] No gravitational slip Φ = Ψ at linear order EGstatistic (RSD + lensing), ISW– galaxy cross [9,25] Standard siren consistency DGW L/DEM L→1 GW+EM counterparts; PTA constraints at low z; multi-band prospects [7,8] No birefringence/dispersion Polarization and frequency dependence LIGO/Virgo/KAGRA polarization; multi-band GW [7,8] Table 1. Testable predictions and indicative datasets in the linear regime. A practical pipeline is: (i) linearize around ρ≡1; (ii) compute the modified Poisson202 and slip relations from the curvature identity and the Euler–Lagrange equation for σ; (iii)203 map to phenomenological parameters (cT, αM, η = Φ/Ψ); (iv) confront with data using204 standard Boltzmann/EFT tools [25].205 Connected noncompact noncommutative manifolds206 To move beyond tori and compact models, we adopt nonunital spectral data on a con-207 nected, noncompact base (M, A) with a smooth, rapidly decaying subalgebra and a (pos-208 – 6 – sibly position-dependent) Poisson bivector θµν(x) generating a star product. Compactness209 of the resolvent is replaced by the locality condition that a(D−i)−1be compact for all210 ain a decaying ideal of A; heat-kernel asymptotics remain local [4,18]. In this setting211 the twistor bundle Z=M×CP1admits an adiabatic collapse and the ASD selection212 persists under mild curvature bounds, while the NC Penrose–Ward correspondence holds213 in the quasi-commutative limit [10,12]. The no-go persists pointwise, enforcing θ0i(x)=0214 globally by connectedness [13,14].215 7.1 Predictions in Quasi-Commutative Regime216 •Luminal gravitational waves:c2 T= 1 [5,6]217 •No Planck mass running:αM=0⇒DGW L=DEM L[7,25]218 •No gravitational slip: Φ = Ψ at linear order [25]219 •Monopolar cosmic birefringence: CB = θEM 2[σ(η0)−σ(η∗)] (testable with multi-220 band GW and CMB/lensing pipelines) [7]221 7.2 Geometric Confinement and Mass Gap222 In regions where ρ→0 (geometric bags), Yang–Mills excitations exhibit:223 •Self-adjoint Hamiltonian with purely discrete spectrum (under Dirichlet-type spectral224 walls) [16]225 •Strictly positive mass gap: ∆ ≳R−1 eff (spectral estimate in finite domains) [16]226 •Confinement via Dirichlet boundary conditions at ∂Ω (spectral boundary control)227 [16]228 Falsaci´on potencial. Este marco es falsable por observaciones que contradigan sus229 predicciones lineales no ajustadas. En particular: (i) una medida robusta de cT= 1 (a230 nivel |∆cT/c|≳10−15) en m´ultiples bandas/frecuencias o a bajo zrefutar´ıa la estructura231 lineal [5,6]; (ii) una detecci´on significativa de gravitational slip (η= 1) a gran escala en el232 r´egimen cuasi-lineal (EG, ISW–galaxia) [25]; (iii) evidencia de birefringencia/dispersion de233 ondas gravitacionales en vac´ıo, preferiblemente con cobertura multi-banda [7,8]. Notamos234 expl´ıcitamente que cT= 1 es una predicci´on no ajustada del marco.235 See also [23,24] for forthcoming probes.236 8 Mechanism Flow: From Spectral Dynamics to Effective Geometry237 Physical picture. The spectral side fixes the “slow” scalar σthrough a single-trace vari-238 ational principle while, independently, the twistor fibre collapses adiabatically and selects239 ASD configurations in the gauge block. The scalar ρ=e2σacts as a conformal intensity240 sourcing geff; gradients of ρproduce curvature according to R=−6ρ−1/2□(ρ1/2).241 – 7 – Comparative Numbers and References Framework |cT/c −1|αM(z∼0) η= Φ/Ψ Regime Notes/Refs This work (ρ) 0 (linear) 0 (linear) 1 (linear) linear curvature identity +σ-E–L GR (ref.) 0 0 1 all tested textbook Horndeski (generic) ≲10−15 (tuned) O(10−1) typ. = 1 (model) linear GW170817; Bellini– Sawicki’14 Asymptotic Safety model-dep. model-dep. 1 (often) UV/IR running R/R2 terms LQG (eff.) model-dep. model-dep. model-dep. eff. QRLG/spinfoam/ loop-cosmology Table 3. Indicative numbers: GW170817 constrains cT;αMbounds from LSS/weak lensing are ∼O(10−1) at z∼0; ηvia EGand ISW–galaxy. See the Related Work section for citations. E Moyal Kernels and a Local Compactness Criterion381 Kernel representation. On Moyal Rnwith A=S(Rn) and Moyal product, for a Dirac-382 type Done can write383 a(D−i)−1ψ(x) = ZRn Ka(x, y)ψ(y)dy, Ka(x, y)=a(x)K(D−i)−1(x, y), where K(D−i)−1is the resolvent kernel [18].384 Hilbert–Schmidt estimate (criterion). If the resolvent kernel satisfies385 sup x∈RnZRn |K(D−i)−1(x, y)|2dy ≤C2<∞, then for a∈ S(Rn) we have386 ∥a(D−i)−1∥2 HS =Z|a(x)|2Z|K(D−i)−1(x, y)|2dydx ≤C2∥a∥2 L2, hence a(D−i)−1is compact. Off-diagonal Gaussian bounds for K(D−i)−1(ellipticity) imply387 the hypothesis; analogous estimates hold on complete manifolds of bounded geometry with388 Schwartz decay in a[4,17].389 Worked Example: Moyal R2(explicit C2)390 For a translation-invariant Dirac/Laplacian resolvent on R2, the kernel depends on r=391 |x−y|as K(D−i)−1(r) = 1 2πK0(mr) (modified Bessel K0), yielding392 sup xZR2|K(D−i)−1(x, y)|2dy = 2πZ∞ 0 r1 2πK0(mr)2dr =π 16 m2, since R∞ 0r K0(mr)2dr =π2 8m2. Hence for a∈ S(R2), ∥a(D−i)−1∥2 HS ≤π 16m2∥a∥2 L2, estab-393 lishing compactness (and thus the local compactness hypothesis) in this concrete case.394 – 14 – F FRW Mini-Solver and ΛCDM Comparison395 We include a minimal, reproducible script (Python) to integrate a toy evolution for σon a396 matter background (a∝η2), together with analytic expressions to generate quick plots in397 L A T EX via PGFPlots.398 Model. In conformal time,399 σ′′ + 2Hσ′+m2 σa2σ= 0, a(η)∝η2, ρ =e2σ≈1+2σ. The effective Hubble rate is Heff =H+σ′so that the fractional deviation obeys400 ∆H H≡Heff −H H=σ′ H≃η 2σ′(for a∝η2). Quick analytic toy for plots. Take σ(η) = A e−γη cos(ωη) with small A. Then401 ∆H H(η) = η 2σ′(η) = η 2A e−γη−γcos ωη −ωsin ωη. This toy is consistent with the EFT-of-DE mapping (no changes to cTor αMat leading or-402 der) and with multimessenger constraints on cT; departures can be funneled into {αK, αB}403 for data confrontation [5–7,25].404 0246810 −2 −1.5 −1 −0.5 0 0.5 1 1.5 ·10−2 η ∆H/H A= 0.05, γ = 0.4, ω = 0.8 ΛCDM (ref) Figure 4. Illustrative ∆H/H relative to ΛCDM-like matter background using the analytic toy; full numerics reproduce the same qualitative behavior. – 15 – import numpy as np405 from math import isfinite406 # matter era (up to normalization )407 def a( eta ): return eta **2408 # \ mathcal {H} = a ’/a for a\propto \eta^2409 def H_conf ( eta ):410 return 2.0/ eta if eta > 0 else 0.0411 # Avoid division by zero412 413 def integrate_sigma ( eta_min =0.2 ,414 eta_max =8.0 , n=4000 , A=0.05 ,415 gamma =0.4 , omega =0.8):416 eta = np. linspace ( eta_min , eta_max , n)417 # analytic toy ( matches the PGFPlots above )418 sigma = A* np .exp (- gamma * eta )* np .cos ( omega * eta )419 dsigma = A* np .exp (- gamma * eta )*( - gamma * np . cos ( omega * eta ) -420 omega * np .sin ( omega *eta ))421 # \ Delta H/H \ approx ( eta /2) sigma ’422 return eta , sigma , dH_over_H423 dH_over_H = 0.5* eta * dsigma424 425 if __name__ == " __main__ ":426 eta , sigma , dH_over_H = integrate_sigma ()427 # Save or print arrays as needed for external plotting428 for e,s,dh in zip(eta [::100] ,429 sigma [::100] ,430 dH_over_H [::100]):431 print (f"{e :.3 f}␣{s :.6f}␣{dh :.6 f}")432 Real-data hook (SN expansion curves). Provide a CSV (e.g. sn expansion.csv)433 with columns z, mu, sigma mu. Convert µto luminosity distance DLand to conformal434 background via a fiducial H0; overlay ∆H/H from the mini-solver:435 import numpy as np , pandas as pd436 c = 299792.458 # km/s437 H0 = 70.0 # km/s/Mpc438 sn = pd . read_csv (" sn_expansion . csv ")439 # z, mu , sigma_mu440 DL = 10**(( sn.mu -25)/5.0) # in Mpc441 # Model overlay : from CSV442 # generated by the mini - solver443 toy = pd. read_csv (" frw_matter_sample .csv ")444 # eta , sigma , DeltaH_over_H445 # Join /plot with your preferred pipeline ;446 # compare residuals vs LCDM447 In PGFPlots, include units and ranges for clarity:448 \begin{axis}[xlabel={$\eta$ (conformal time)},449 – 16 – ylabel={$\Delta H/H$}, ymin=-0.05,ymax=0.05,450 legend pos=south east]451 \end{axis}452 453 Data Availability Statement454 All datasets used in this work are synthetic and were generated with the FRW mini-solver455 described in App. F. The CSV used for PGFPlots (frw matter sample.csv) and the min-456 imal scripts required to regenerate it are provided as supplementary material and are457 archived at Zenodo (DOI: 10.5281/zenodo.17393112), and mirrored in the public reposi-458 tory github.com/KerymMacryn/rho-spectral-gravity. These materials are sufficient to re-459 produce all numerical figures reported here. No proprietary or observational datasets were460 used; additional parameter sweeps can be reproduced by rerunning the supplied script with461 fixed random seeds for full reproducibility.462 Acknowledgments463 We thank colleagues and readers for helpful feedback during the development of this work.464 Numerical experiments and plots were produced from synthetic datasets generated with465 the FRW mini-solver (App. F); all materials and scripts are listed in the Data Availability466 Statement. This project received no specific grant from any funding agency, commercial or467 not-for-profit sectors. 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