DRSN I: A Unified Geometric Framework for Matter, Torsion and Dark Energy (De Rerum Spectrale Natura, Report I, Version 3.0)

DRSN I: A UNIFIED GEOMETRIC FRAMEWORK FOR MATTER, TORSION AND DARK ENERGY De Rerum Spectrale Natura series REPORT I (Version 3.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro August 2025 •Drift deformation induces a universal spectral potential V(s) = αs2+βs4with β > 0. •Einstein–Cartan torsion ensures α<0, producing drift-induced condensation. •The condensate s∗yields a spectral geometric origin for dark energy. •Cosmological evolution emerges naturally: stiff →radiation →matter →DE. •Complex drift introduces a spectral arrow of time and CP asymmetry. • A geometric TOE: dark energy, condensation and cosmological eras arise unavoidably from the spectral structure of the drifted Dirac operator. The Spectral Drift Model: A Geometric Origin for Dark Energy and Time Asymmetry J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We develop a unified geometrical framework based on Noncommutative Geometry and the Spectral Action Principle, introducing a spectral drift of the Dirac operator Ds = esφDe−sφ , which preserves self-adjointness for real s and generates a universal spectral potential V ( s ) = αs2 + βs4 with β > 0. We analyse four drifted spectral models, including Einstein–Cartan geometry with torsion and the Standard Model internal Dirac operator, and show that the torsion contribution produces αtot < 0, leading to a non-trivial spectral condensate s∗ and a positive vacuum energy V ( s∗ ) > 0interpreted as dark energy of geometric origin. A cosmological analysis in FRW backgrounds reveals an early stiff phase, followed by standard radiation and matter eras, and late-time relaxation to the condensate, reproducing a ΛCDMlike background expansion. We also construct the complex drift Dz = ezφDe−zφ and its holomorphic spectral action, whose imaginary part induces dissipative effects and provides a geometric mechanism for time asymmetry and CP violation at the spectral level. Altogether, the Spectral Drift Model furnishes a mathematically rigorous and physically meaningful bridge between spectral geometry, dark energy, and irreversibility. Keywords: Spectral Action; Dirac operators; Einstein–Cartan geometry; torsion; Standard Model; dark energy; cosmology; noncommutative geometry; spectral drift; time asymmetry. ∗jp[email protected] 3 CONTENTS I. Introduction 8 II. Operators, Similarity and Spectral Structure 10 A. Closed and self-adjoint operators 10 B. Similarity transformations 10 C. Relative boundedness and drift perturbations 11 D. Holomorphic dependence (real drift case) 12 E. Differentiation of the drift 12 F. Summary 12 III. Lichnerowicz Identity with Drift 13 A. Standard Lichnerowicz formula 13 B. Expansion of the drifted operator 14 C. Computation of the quadratic term 14 D. Computation of the linear term 15 E. Drifted Laplace-type decomposition 15 F. Consequences for the spectral coefficients 16 IV. Spectral Action with Drift 16 A. Drifted spectral action 17 B. Heat-kernel coefficients under drift 17 C. Structure of the drifted action 18 D. Emergence of the universal quartic structure 18 E. Outlook 19 V. The Universal Spectral Potential 19 A. Drifted Lichnerowicz Formula 19 B. General Form of a2(D2 s)20 C. General Form of a4(D2 s)21 D. The Geometric Potential 21 E. Internal Contributions 22 F. Torsion Contribution 22 G. Total Potential 22 4 VI. Internal Drifted Models: SDM-I and SDM-II 23 A. Internal Dirac operator of the Standard Model (SDM-I) 23 B. Internal Dirac operator of the Pati–Salam model (SDM-II) 24 C. Summary and role in the full Spectral Drift Model 25 VII. Drifted Einstein–Cartan Models: SDM-III and SDM-IV 26 A. The Torsion Contribution in Einstein–Cartan Geometry 26 B. The Drifted Einstein–Cartan Model (SDM-III) 27 C. The Unified Drifted Model (SDM-IV): EC + Standard Model 28 D. Interpretation and Significance 28 VIII. FRW Cosmology of the Spectral Drift 29 A. Motivation 29 B. Effective action in a FRW background 29 C. Equation of motion for the drift 30 D. Friedmann equations with the drift sector 31 E. Dynamical regimes of the spectral drift 31 F. Stability of the condensate 32 G. Geometric dark energy from the spectral drift 33 IX. Complex Drift and Irreversibility 33 A. Motivation 33 B. Holomorphic family of type (A) 34 C. Spectral properties for complex drift 34 D. Complex spectral action and its Taylor expansion 35 E. Real and imaginary parts of the complex spectral action 36 F. Dissipative dynamics and time asymmetry 36 G. Conceptual interpretation and limitations 37 X. General Conclusions 37 Appendices 43 A. Analytical Foundations of the Spectral Drift 43 1. Closed and self-adjoint operators 43 5 2. Similarity by bounded, invertible operators 44 3. Spectral invariance for real drift 45 4. Relative boundedness of [D, φ]46 5. Holomorphic families of type (A) 46 6. Flow equation for the real drift 47 7. Vanishing of the second-order BCH term for scalar multipliers 47 8. Summary 48 B. Drifted Lichnerowicz Formula 48 1. Expansion of the drifted operator 49 2. The quadratic term W= [D, φ]249 3. The linear term Z={D, [D, φ]}49 4. Combination with the Lichnerowicz identity 50 5. Drifted connection and potential 50 C. Explicit computation of ˜a2(s)50 1. General formula for a251 2. Effective drift dependence of ˜a2(s)51 3. Summary 52 D. Explicit Computation of the Coefficient a4(s)53 1. Laplace-type operator and general formula for a453 2. Expansion of E2 sand identification of quartic terms 53 3. Trace and final expression for a4(s)54 4. Implications for the spectral potential 55 E. Torsion in Einstein–Cartan Geometry 55 1. Purpose and overview 55 2. Einstein–Cartan connection and torsion 56 3. Dirac operator with torsion 56 4. Lichnerowicz formula with axial torsion 57 5. Contribution to the Seeley–DeWitt coefficient a257 6. Torsion-induced quadratic term in the spectral action 58 7. Summary 58 6 F. Cosmology of the Spectral Condensate 59 1. Purpose and setting 59 2. Effective action and energy–momentum tensor 59 3. Equation of motion for the condensate 60 4. Friedmann equations with the spectral drift 60 5. Evolution in terms of e-folds 61 6. Regimes of the condensate dynamics 61 7. Stability of the spectral condensate 62 G. Complex Drift and Holomorphic Spectral Action 63 1. Holomorphic family of drifted Dirac operators 64 2. Holomorphic spectral action and Taylor expansion 64 3. Real and imaginary parts of the spectral action 65 4. Dissipative contributions and time asymmetry 66 5. Scope and limitations 66 H. Physical Input: Masses, Yukawas and Parameters 66 1. Fermion masses of the Standard Model 67 a. Quarks 67 b. Charged leptons 67 c. Neutrinos 68 2. Yukawa couplings 68 a. Quark Yukawas 68 b. Lepton Yukawas 68 3. Mixing matrices: CKM and PMNS 69 a. CKM matrix 69 b. PMNS matrix 69 4. Internal traces and contributions to αint and βint 70 5. Gauge couplings and Higgs parameters 70 6. Recommended initial conditions for numerical simulations 71 I. Numerical Implementation and Pseudocode 71 1. Equations of motion 71 2. First-order system formulation 72 7 3. Initial conditions 73 4. Runge–Kutta 4 integration scheme 74 5. Example pseudocode 74 6. Stability considerations 76 J. Effective Spectral Condensate and Constraint Intersection 76 1. Definition of the effective condensate 76 2. Independent constraint family 77 3. Constraint intersection and discrimination 77 4. Status of s∗in the drifted models 78 5. Role of this appendix in the DRSN series 78 References 79 8 I. INTRODUCTION The search for a unified understanding of the fundamental structures of nature has led, over the past century, to two complementary but conceptually distant frameworks. General Relativity interprets gravitation as geometry, while quantum field theory describes matter and gauge interactions through the language of operator algebras and local symmetries. Noncommutative Geometry (NCG), in the sense of Connes [ 1 , 2 ], offers a deep bridge between these paradigms by encoding geometry in the spectral properties of a Dirac operator D acting on a Hilbert space of fermionic states. In this setting, the Spectral Action Principle [ 3 ] asserts that the physical content of a model is determined by the spectrum of Dthrough a functional trace of the form S(D) = Tr f(D/Λ),(I.1) whose asymptotic expansion reproduces gravitational, gauge and scalar sectors with remarkable accuracy. Despite its successes—including the reconstruction of the Standard Model, neutrino masses, Yukawa mixing and seesaw structures—the traditional spectral action is fundamentally static: it is built from a fixed Dirac operator and does not contain an intrinsic dynamical mechanism capable of generating new geometric degrees of freedom, nor does it naturally explain dark energy, vacuum structure, cosmological relaxation, or time asymmetry. In this work we introduce an additional geometric ingredient: a spectral drift of the Dirac operator, Ds=esφDe−sφ,(I.2) where φ is a bounded multiplier in the algebra of the spectral triple. This deformation preserves essential self-adjointness, the real spectrum for s∈R , ellipticity and the principal symbol of D , yet modifies the subprincipal and zero-order structure in a controlled manner. Drift therefore enriches the spectral geometry without altering its fundamental axioms. A key outcome of this modification is a universal quartic potential V(s)=αs2+βs4, β > 0, arising from the drift dependence of the Seeley–DeWitt coefficients. This potential is present for any physically relevant spectral triple and defines a spectral condensate s∗whenever α < 0. 9 The conceptual organisation of these geometric and internal ingredients, and the rôle played by the drift deformation in producing the universal spectral potential and its cosmological consequences, is summarised in Figure 1. We apply this drift mechanism to three established spectral constructions: (i) the Internal Standard Model geometry (SDM–I); (ii) the Pati–Salam extension (SDM–II); and (iii) Einstein– Cartan geometry with axial torsion (SDM–III). The combination of the latter two sectors yields the unified drifted Einstein–Cartan–Standard Model geometry (SDM–IV). A central result of this report is that the torsional contribution in the geometric sector universally drives α < 0, guaranteeing condensation, while the Standard Model sector provides a stabilising quartic term β > 0. The minimum s∗ of the resulting potential contributes a strictly positive vacuum energy V ( s∗ ), offering a purely geometric explanation for dark energy. A cosmological analysis in FRW backgrounds shows that the drift field behaves as a stiff fluid in the early universe, relaxes through radiation and matter eras, and approaches s∗ at late times, thereby reproducing a ΛCDM-like expansion history without introducing additional scalar fields. Finally, we extend the drift to a complex parameter z∈C , producing a holomorphic operator family Dz . Although the real drift is isospectral and self-adjoint, the complex drift Dz acquires complex resonances and induces dissipative dynamics. This provides a geometric mechanism for time asymmetry and CP-violating effects at the spectral level, drawing conceptual connections with modular theory in algebraic quantum field theory. About this report. This work constitutes Report I of the DRSN series (De Rerum Spectrale Natura), a sequence of independent but thematically unified studies on spectral drift geometry. The DRSN series is developed within an open research community on spectral geometry and fundamental physics; related materials and updates are archived at https://zenodo.org/ communities/dsrn/. The present report focuses on the external Dirac geometry, the emergence of a universal driftinduced potential, the Einstein–Cartan–Standard Model synthesis (SDM–IV), and the resulting cosmological and analytic implications. Subsequent reports in the DRSN series will develop the internal, turbulent, Yang–Mills, string-theoretic, black-hole, quantum-informational and arithmetic sectors of drift geometry. 16 F. Consequences for the spectral coefficients The decomposition above shows that: •No cubic terms in sappear in D2 sbecause [φ, [φ, D]] = 0. •The drift modifies only lower-order terms of D2. •The drifted connection contributes linearly in s. • The drifted potential Es contains the universal quadratic term s2∥∇φ∥2 , which produces the quartic contribution to the spectral potential. These structural observations are central for Chapters V–VII, where we compute effective coefficients ˜a2 ( s )and ˜a4 ( s )extracted from the truncated local expansion and derive the universal potential V(s) = αs2+βs4. IV. SPECTRAL ACTION WITH DRIFT The Spectral Action associated with a Dirac operator D on a spectral triple ( A,H, D )is defined by S(D) = Trf(D/Λ),(IV.1) where f is a positive, rapidly decaying function and Λis an energy scale. For large Λ, the action admits an asymptotic expansion in terms of the Seeley–DeWitt coefficients of the operator D2: S(D)∼f4Λ4a0+f2Λ2a2+f0a4+O(Λ−2),(IV.2) where an=an(D2)are the heat-kernel coefficients in dimension 4. In this section we analyse how the Spectral Action changes under the drift deformation Ds=esφDe−sφ, s ∈R,(IV.3) where φ is a real-valued smooth bounded function. As established earlier, Ds is self-adjoint for real s and depends analytically on s . the deformation is isospectral, in the sense that σ ( Ds ) = σ ( D ) for s∈R . While global spectral invariants defined through trace functionals are rigid under this deformation, effective quantities extracted from truncated local expansions may acquire a non-trivial dependence on s. 17 A. Drifted spectral action Replacing Dby Dsin the Spectral Action yields S(s) := S(Ds) = Trf(Ds/Λ).(IV.4) Using the heat-kernel expansion, we obtain S(s)∼f4Λ4a0+f2Λ2˜a2(s) + f0˜a4(s)+O(Λ−2),(IV.5) where ˜an ( s )denote effective coefficients extracted from the truncated local expansion of the drifted operator D2 s , and should not be identified with the integrated Seeley–DeWitt coefficients of the global heat trace. Thus, the s –dependence of the effective (truncated) spectral action is encoded entirely in the effective coefficients ˜a2 ( s )and ˜a4 ( s )extracted from the local expansion of the drifted Lichnerowicz operator. The higher-order terms in (IV.5) follow the same structure but are irrelevant for the leading dynamical effects of the drift. B. Heat-kernel coefficients under drift From the drifted Lichnerowicz formula established earlier, D2 s=−gµν∇(s) µ∇(s) ν+Es,(IV.6) with ∇(s) µ=∇µ+s ∂µφ, Es=E0+s∆φ+s2∥∇φ∥2,(IV.7) we see that the drift modifies only the lower-order terms: the principal symbol remains unchanged. Since the Seeley–DeWitt coefficients depend only on Es and the curvature of ∇(s) , the deformation induces corrections quadratic and quartic in s. The explicit computation of ˜a2 ( s )and a4 ( s )is carried out in detail in Appendices Cand D, and leads to the universal expressions ˜a2(s)=a2(0) + 4 (4π)2s2ZM∥∇φ∥2dµ, (IV.8) ˜a4(s)=a4(0) + 1 2(4π)2s4ZM∥∇φ∥4dµ. (IV.9) Two important remarks follow immediately: 18 •there are no linear or cubic terms in s; •the quartic coefficient is manifestly positive. Thus, for real drift parameters the deformation of the Spectral Action is organised into a purely even function of s. C. Structure of the drifted action Inserting these expressions back into (IV.5), one obtains S(s) = S(0) + f2Λ2As2+f0Bs4+O(s6),(IV.10) where A=4 (4π)2ZM∥∇φ∥2dµ, (IV.11) B=1 2(4π)2ZM∥∇φ∥4dµ. (IV.12) The corresponding effective drift potential is defined as the s –dependent part of a truncated spectral functional obtained from the local operator decomposition of D2 s: V(s)=α s2+β s4,(IV.13) with α=f2Λ2A, β =f0B. (IV.14) Since B > 0for all real φ , we conclude that β > 0universally. The sign of α depends on the geometric, torsional and internal contributions detailed in the following chapters. The qualitative behaviour of this universal potential is illustrated in Figure 2, which shows the non-trivial minimum s∗that arises when α < 0and β > 0in the unified SDM–IV model. D. Emergence of the universal quartic structure Equation (IV.13) shows that the drift deformation naturally generates a quartic potential for the drift parameter. The absence of odd powers of sis ultimately a consequence of: •the commutator identity [φ, [φ, D]] = 0 for scalar φ; 19 •the vanishing of RM∆φ dµ on compact manifolds; •the fact that the drift does not modify the principal symbol of D. The universality of the quartic structure is one of the central results of this theory, and it holds for any compact Riemannian spin manifold and for any finite-dimensional internal spectral triple. The sign of α, however, is model-dependent, and its analysis is the content of the next sections. E. Outlook In the next two chapters we evaluate the contributions to αand βfrom: •the internal spectral data of the Standard Model and Pati–Salam model; •the geometric and torsional terms in Einstein–Cartan theory. These ingredients combine in the unified model (SDM-IV), leading to a non-trivial minimum of the potential V(s)and a geometric explanation for dark energy. V. THE UNIVERSAL SPECTRAL POTENTIAL In this section we derive the universal form of the effective potential induced by the drift deformation of the Dirac operator. We show that, for any spectral triple of physical relevance in four dimensions, the deformation Ds=esφDe−sφ, s ∈R,(V.1) generates corrections to the spectral action that organise themselves into a potential V(s)=αs2+βs4,(V.2) with β > 0independently of the model. The coefficient α depends on the specific geometric and internal data and will be central for the analysis of drifted spectral models in Sections VI–VII. A. Drifted Lichnerowicz Formula From Appendix B, the drifted Dirac operator satisfies the identity D2 s=−gµν∇(s) µ∇(s) ν+Es,(V.3) 20 where ∇(s) µ=∇µ+s ∂µφ, (V.4) Es=E0+s∆φ+s2∥∇φ∥2.(V.5) The Laplace-type structure of the drifted Dirac operator is summarised in Figure 3, which highlights how the connection and the zero-order term acquire explicit s–dependent contributions while the principal symbol remains unchanged. Here E0is the potential term in the undeformed Lichnerowicz formula D2=−gµν∇µ∇ν+E0,(V.6) and includes curvature and torsion contributions when present. The drift thus modifies only the lower-order terms in D2, leaving the principal symbol unchanged. B. General Form of a2(D2 s) For a Laplace-type operator F = −gµν∇µ∇ν + E , the Seeley–DeWitt coefficient a2 ( F )in dimension four is a2(F) = 1 (4π)2ZM TrR 6+Edµ. (V.7) Applying this to F=D2 syields ˜a2(s)=a2(0) + 1 (4π)2ZM Tr s∆φ+s2∥∇φ∥2dµ. (V.8) In a compact manifold without boundary, ZM ∆φ dµ = 0, and since Tr(1spin)=4, we obtain the universal expression ˜a2(s)=a2(0) + 4 (4π)2s2ZM∥∇φ∥2dµ. (V.9) There is no linear term in s, and the drift contributes a strictly non-negative quadratic correction to a2. 21 C. General Form of a4(D2 s) The coefficient a4(F)for a Laplace-type operator takes the form a4(F) = 1 (4π)2 1 360 ZM Tr60RE + 180E2+ 30Ωµν Ωµν + 5R2−2RµνRµν + 2RµνρσRµνρσdµ. (V.10) We substitute E = Es . The only terms contributing quartic powers of s arise from E2 s , in particular the component E2 s⊃s4∥∇φ∥4. Thus ˜a4(s)=a4(0) + 4 (4π)2 180 360 s4ZM∥∇φ∥4dµ +··· ,(V.11) where the dots denote lower powers in s. Simplifying, ˜a4(s)=a4(0) + 1 2(4π)2s4ZM∥∇φ∥4dµ. (V.12) Thus the drift induces a strictly positive quartic contribution to the spectral action. D. The Geometric Potential The bosonic spectral action has expansion S(s) = f4Λ4a0+f2Λ2˜a2(s) + f0˜a4(s) + O(Λ−2).(V.13) Extracting the s-dependence yields the geometric potential Vgeom(s)=αgeoms2+βgeoms4,(V.14) with coefficients αgeom =f2Λ24 (4π)2ZM∥∇φ∥2dµ, (V.15) βgeom =f0 1 2(4π)2ZM∥∇φ∥4dµ. (V.16) Since ∥∇φ∥4≥0, we have βgeom >0. 22 E. Internal Contributions Let DFdenote the internal finite Dirac operator. Under drift, (DF)s=DF+sYF,(V.17) where YF= [DF, φF]is a bounded matrix. Then Tr(DF+sYF)2= Tr(D2 F)+2sTr(DFYF)+s2Tr(Y2 F),(V.18) Tr(DF+sYF)4= Tr(D4 F)+···+s4Tr(Y4 F).(V.19) Thus Vint(s)=αints2+βints4,(V.20) where βint =f0Tr(Y4 F)>0. F. Torsion Contribution In Einstein–Cartan geometry, the Lichnerowicz formula contains the torsion term T=−3 4∥T∥2.(V.21) This contributes a negative term to a2, and hence to the quadratic coefficient, αtors <0.(V.22) G. Total Potential Summing geometric, internal, and torsional contributions yields V(s)=αs2+βs4,(V.23) where α=αgeom +αint +αtors,(V.24) β=βgeom +βint >0.(V.25) In the unified drift model SDM-IV, examined in Section VII, we have α < 0, which leads to a non-trivial spectral minimum s∗=r−α 2β.(V.26) 23 This condensate has energy density ρΛ=V(s∗)>0, interpreted as a geometric cosmological constant. This completes the derivation of the universal drift-induced potential. VI. INTERNAL DRIFTED MODELS: SDM-I AND SDM-II In this section we analyse the effect of the spectral drift on the internal part of the Dirac operator, focusing on two models: •SDM-I: the Standard Model (SM) internal Dirac operator; •SDM-II: the Pati–Salam internal Dirac operator. In both cases, the geometric (spacetime) Dirac operator DM is left unchanged, and the drift acts purely on the finite-dimensional internal Hilbert space HFvia DF,s := DF+sYF,(VI.1) where DF is the internal Dirac operator and YF is a bounded operator determined by a choice of internal multiplier φF, YF= [DF, φF].(VI.2) Since DFis a finite matrix, all operators involved are bounded and the spectral analysis is purely algebraic. Nevertheless, the contribution of the internal sector to the spectral action plays a crucial role in the coefficients of the drift potential. A. Internal Dirac operator of the Standard Model (SDM-I) The finite Dirac operator of the Standard Model, DSM F , encodes the Yukawa couplings, fermion masses, mixing matrices (CKM and PMNS) and Majorana mass terms for right-handed neutrinos. It is a finite self-adjoint matrix acting on a 96-dimensional internal Hilbert space (for three generations). In a suitable basis, DSM Ftakes block form DSM F=   0Y Y†0  ,(VI.3) where Yis the matrix of Yukawa couplings (including neutrinos). 24 The drifted internal operator is then DSM F,s =DSM F+sY SM F,(VI.4) with YSM F = [ DSM F, φF ]determined by the choice of internal multiplier φF . For our purposes, it suffices to treat YSM Fas a fixed bounded matrix. The contributions of the internal sector to the spectral action come from the traces Tr (DSM F,s )2,Tr (DSM F,s )4.(VI.5) Expanding in powers of swe obtain Tr (DSM F,s )2= Tr (DSM F)2+ 2sTr DSM FYSM F+s2Tr (YSM F)2,(VI.6) Tr (DSM F,s )4= Tr (DSM F)4+O(s)+O(s2)+s4Tr (YSM F)4.(VI.7) The precise form of the linear and quadratic terms in s is not needed for the universal structure of the drift potential. What matters is that the quartic coefficient is Tr (YSM F)4>0,(VI.8) since it is a sum of fourth powers of Yukawa couplings (dominated by the top quark Yukawa). Inserting these contributions into the spectral action yields an internal potential of the form VSM int (s)=αSM int s2+βSM int s4,(VI.9) with αSM int =f2Λ2Tr (YSM F)2,(VI.10) βSM int =f0Tr (YSM F)4>0.(VI.11) The sign of αSM int depends on the detailed structure of Yukawa couplings and Majorana masses, but βSM int is always strictly positive. Consequently, the Standard Model drifted internal sector alone does not generate a symmetry-breaking spectral condensate: the minimum is at s = 0 unless additional geometric contributions (such as torsion) are present. B. Internal Dirac operator of the Pati–Salam model (SDM-II) The Pati–Salam model extends the internal algebra and symmetry group to SU(2)L×SU(2)R×SU(4) , and its finite Dirac operator DPS F acts on a larger internal Hilbert space. The structure of DPS F is 25 richer, with additional Yukawa couplings and scalar fields arising naturally from the spectral triple construction. The drifted internal operator in SDM-II is defined analogously by DPS F,s =DPS F+sY PS F,(VI.12) where YPS F= [DPS F, φF]is a bounded perturbation. The relevant traces again take the form Tr (DPS F,s)2= Tr (DPS F)2+ 2sTr DPS FYPS F+s2Tr (YPS F)2,(VI.13) Tr (DPS F,s)4= Tr (DPS F)4+O(s) + O(s2)+s4Tr (YPS F)4.(VI.14) As antes, o termo quartico é estritamente positivo: Tr (YPS F)4>0,(VI.15) devido ao carácter hermitiano de YPS F e ao facto de ser uma soma de valores próprios reais elevados à quarta potência. O potencial interno Pati–Salam tem a forma VPS int (s)=αPS int s2+βPS int s4,(VI.16) com αPS int =f2Λ2Tr (YPS F)2,(VI.17) βPS int =f0Tr (YPS F)4>0.(VI.18) A estrutura é formalmente idêntica à do caso Standard Model, embora os valores numéricos possam ser significativamente diferentes. C. Summary and role in the full Spectral Drift Model In both SDM-I and SDM-II, the internal sector subjected to the spectral drift yields a quartic contribution βints4 with βint > 0, and a quadratic term αints2 whose sign depends on the internal mass spectrum and Yukawa structure. The internal contribution alone does not guarantee a nontrivial spectral condensate: the minimum remains at s = 0 in the absence of additional geometric effects. This observation is crucial for the full Spectral Drift Model. It implies that: 32 b. Intermediate regimes: radiation and matter. As the Universe expands, the friction term 3 H˙s damps the motion of s ( t ), reducing ˙s2 and thus the contribution of the drift sector. The dynamics then becomes dominated by the standard radiation and matter components: ρr∝a−4, ρm∝a−3.(VIII.19) During these eras, the drift behaves as a subdominant component and does not disturb the standard cosmological history. c. Late-time regime: relaxation to the condensate. In late times, when the expansion has sufficiently damped the kinetic term, the dynamics of sis governed mainly by the potential: ¨s+ 3H˙s≈ −V′(s).(VIII.20) Since V(s)has a global minimum at s∗, the field relaxes towards s∗: s(t)−→ s∗, t → ∞.(VIII.21) In this regime, ˙s→0,(VIII.22) and the drift sector behaves as a cosmological constant: ρs→V(s∗), ps→ −V(s∗), ws→ −1.(VIII.23) F. Stability of the condensate The stability of the minimum s∗follows from V′(s∗)=0, V ′′(s∗)=−2α > 0,(VIII.24) since α < 0. Consider a small perturbation δs(t)around s∗, s(t)=s∗+δs(t).(VIII.25) Linearising the equation of motion yields ¨ δs + 3H˙ δs +V′′(s∗)δs = 0,(VIII.26) which describes a damped oscillator with positive restoring force and friction. This ensures that δs decays with time and that the condensate s∗is dynamically stable. 33 G. Geometric dark energy from the spectral drift The asymptotic state of the drift sector is characterised by: ρs→V(s∗)>0, ps→ −V(s∗), ws→ −1.(VIII.27) The spectral condensate therefore behaves as a cosmological constant. Crucially, this constant is not introduced by hand but emerges from the interplay between drift, torsion and internal degrees of freedom in the Spectral Action. From the perspective of the Spectral Drift Model, dark energy is thus a purely geometric phenomenon: it is the vacuum energy associated with a non-trivial minimum of the drifted Spectral Action. No additional scalar fields or phenomenological potentials are required. This provides a conceptually appealing explanation for the observed late-time acceleration of the Universe, while remaining fully compatible with the standard cosmological epochs of radiation and matter domination. In the next chapters and appendices, we further analyse the geometric and analytic structures underlying this phenomenon, including the complexified drift and its relation to irreversibility and time asymmetry. IX. COMPLEX DRIFT AND IRREVERSIBILITY A. Motivation In the previous sections we have analysed the real spectral drift Ds=esφDe−sφ, s ∈R,(IX.1) where D is the Dirac operator of a real spectral triple and φ∈C∞ ( M )is a real-valued function (or its finite-dimensional analogue). For real s , the operator Ds is self-adjoint, has real spectrum, and defines a standard unitary dynamics. The purpose of this chapter is to extend the drift to a complex parameter Dz=ezφDe−zφ, z ∈C,(IX.2) and to analyse the corresponding complexified spectral action. In this regime, Dzis in general no longer self-adjoint, its spectrum may move into the complex plane, and one expects the emergence of dissipative behaviour and an intrinsic geometric mechanism for time asymmetry. 34 This complex drift layer does not affect the real spectral potential V ( s )and the existence of the spectral condensate s∗ ; these are entirely controlled by the real drift. Instead, the complex drift adds a new analytic and conceptual layer relating spectral geometry to irreversibility. B. Holomorphic family of type (A) We recall from the functional-analytic discussion that for φ∈L∞real and smooth, the family Dz=ezφDe−zφ (IX.3) defines a holomorphic family of type (A) in the sense of Kato. More precisely: •The domain is independent of z, Dom(Dz)=ezφ Dom(D),(IX.4) and coincides with the domain of Dtransported by the bounded, invertible operator ezφ. •For each ψ∈Dom(D), the vector-valued map z7→ Dzψ(IX.5) is holomorphic on C. This follows from the explicit identity Dz=D+z[D, φ],(IX.6) valid as an operator identity on Dom(D), with [D, φ]bounded and independent of z. When z = s∈R , Dz is self-adjoint (by bounded perturbation theory), and its spectrum is real. When z acquires a non-zero imaginary part, self-adjointness is lost and spectral values may move into the complex plane. C. Spectral properties for complex drift Writing z=s+iθ, we have Dz=Ds+iθ[Ds, φ],(IX.7) where Ds is self-adjoint. The perturbation iθ [ Ds, φ ]is bounded and skew-adjoint whenever [ D, φ ] is self-adjoint; however, in general Dzis no longer normal. Its spectrum satisfies σ(Dz)⊂ {λ∈C: Im λbounded by C|θ|},(IX.8) for some constant Cproportional to ∥[D, φ]∥. In particular: 35 •for θ= 0,Dzreduces to the real drift Dsand σ(Dz)⊂R; • for θ = 0, eigenvalues may acquire non-zero imaginary parts, leading to exponentially damped or amplified modes under the associated dynamics. The loss of self-adjointness and the appearance of complex eigenvalues suggest that the complex drift provides a natural setting for dissipative phenomena and spectral resonances. This qualitative behaviour is illustrated in Figure 4, which contrasts the purely real spectrum of the self-adjoint drift Ds with the complex resonances generated by the non-self-adjoint operator Dz . D. Complex spectral action and its Taylor expansion We define the complexified spectral action as S(z) = Trf(Dz/Λ),(IX.9) whenever the trace is well-defined (for instance, for sufficiently regular cut-off functions f and large enough Λ). By holomorphy of z7→ Dz , the map z7→ S ( z )is analytic on any domain where f ( Dz/ Λ) remains trace-class. Formally, we may expand S(z)in a Taylor series around z= 0: S(z) = ∞ X n=0 zn n!S(n)(0),(IX.10) with S(n)(0) = dn dznz=0 Trf(Dz/Λ).(IX.11) Using d dz Dz= [Dz, φ],d dz z=0 Dz= [D, φ],(IX.12) and functional calculus, one finds S′(0) = 1 ΛTrf′(D/Λ) [D, φ],(IX.13) and, more generally, S(n)(0) = 1 ΛnTrf(n)(D/Λ) adn φ(D),(IX.14) where adφ(D) = [D, φ]and adn φ(D)denotes the n-fold iterated commutator. For real z = s , the expansion reduces to the real spectral drift studied in previous chapters; for complex z, it provides a natural analytic continuation. 36 E. Real and imaginary parts of the complex spectral action Writing explicitly z=s+iθ, S(z)=SRe(s, θ)+i SIm(s, θ),(IX.15) we have SRe(s, θ) = 1 2S(z)+S(¯z), SIm(s, θ) = 1 2iS(z)−S(¯z).(IX.16) For θ= 0, one recovers the real spectral action and SIm(s, 0) = 0.(IX.17) For θ = 0, the imaginary part SIm ( s, θ )is typically non-zero; it can be interpreted as encoding dissipative effects in an effective action, in analogy with non-Hermitian Hamiltonians or influence functionals in open quantum systems. In the semiclassical or heat kernel expansion, S ( z )can be expressed in terms of local invariants built out of the drifted Lichnerowicz operator. The imaginary contributions correspond to those parts of the complexified invariants that do not cancel under conjugation. This leads to local imaginary couplings which may be associated with irreversible processes. F. Dissipative dynamics and time asymmetry Given a complex drift Dz, a natural spectral evolution is governed by dψ dt =−iDzψ. (IX.18) Decomposing Dz=H−iK, with H=H∗and K=K∗≥0in suitable regimes, one finds d dt∥ψ(t)∥2=−2⟨ψ(t), Kψ(t)⟩ ≤ 0,(IX.19) indicating norm decay and hence dissipation. Even when K is not positive, the appearance of non-zero imaginary parts in eigenvalues generically leads to exponential damping or growth of modes, breaking time-reversal invariance. From a geometric perspective, the complex drift z7→ Dz selects a direction in the complex plane which can be interpreted as a spectral orientation of time. The imaginary part of the drift encodes a deformation away from unitary evolution, and the imaginary part of the spectral action quantifies the associated entropy production or CP-violating effects. 37 G. Conceptual interpretation and limitations The complex drift layer should be viewed as a geometric and analytic extension of the real Spectral Drift Model. It does not modify the real spectral potential V ( s ), nor does it affect the existence or stability of the condensate s∗ and the geometric origin of dark energy. Instead, it provides: •an analytic continuation of the drift into the complex plane; •a natural framework for non-self-adjoint Dirac operators with complex spectrum; • a source of dissipative terms and time-asymmetric behaviour encoded in the imaginary part of the spectral action; • a possible bridge to modular theory and KMS states, where time evolution is generated by modular flows associated with equilibrium states. It is important to stress that, in this work, the complex drift is treated at a conceptual and structural level: no full phenomenological model is extracted from SIm ( s, θ ), and we do not attempt a complete classification of non-unitary evolutions. Nonetheless, the complex drift shows that Noncommutative Geometry and the Spectral Action naturally accommodate a geometrical mechanism for irreversibility and CP violation, complementing the geometric explanation of dark energy provided by the real spectral condensate. In this sense, the drift deformation of the Dirac operator yields two intertwined but distinct layers: • areal layer, responsible for the emergence of a positive vacuum energy and late-time acceleration in cosmology; • acomplex layer, responsible for dissipative phenomena and an intrinsic arrow of time at the spectral level. Together, they provide a unified spectral picture where geometry, dark energy and irreversibility arise from a single deformation principle. X. GENERAL CONCLUSIONS The Spectral Drift Model (SDM) developed in this work establishes a unified geometrical framework in which gravitational, internal, cosmological and analytic phenomena arise from a single 38 deformation of the Dirac operator. The key idea is the introduction of the drifted operator Ds=esφ D e−sφ,(X.1) where φ is a real scalar multiplier belonging to the algebra of the spectral triple. For real values of s , this operator retains self-adjointness, preserves the spectrum and modifies only the lower-order terms in the Lichnerowicz formula. As a result, the drift affects the local Seeley–DeWitt coefficients of the spectral action without altering the principal symbol of D. By performing explicit computations of the effective coefficients ˜a2 ( s )and ˜a4 ( s )for D2 s , we demonstrated the emergence of a universal spectral potential V(s)=αs2+βs4,(X.2) with β > 0in all relevant geometries. This universality is a consequence of the positivity of the term [ D, φ ] 2 and the vanishing of higher-order contributions in the Baker–Campbell–Hausdorff expansion due to the scalar nature of φ. Four drifted models were analysed in detail: the Standard Model (SDM-I), the Pati–Salam extension (SDM-II), the Einstein–Cartan geometry with axial torsion (SDM-III), and the unified Einstein–Cartan plus Standard Model model (SDM-IV). While SDM-I and SDM-II yield positive quartic coefficients but no condensation, the torsion contribution in SDM-III introduces a universal negative quadratic term. Combining this effect with the geometric and internal contributions in SDM-IV leads to αtot <0, βtot >0,(X.3) which in turn implies the existence of a non-trivial spectral condensate s∗=r−αtot 2βtot ,(X.4) and a positive vacuum energy density ρΛ=V(s∗).(X.5) Thus, dark energy appears in SDM-IV as a purely geometric phenomenon: a stable minimum of the drift-induced potential arising solely from the spectral geometry of the Dirac operator. A cosmological analysis in a flat FRW background revealed that the drift behaves as a stiff fluid in the early Universe, becomes subdominant during radiation and matter eras, and relaxes to the 39 condensate at late times, yielding accelerated expansion compatible with a ΛCDM-like background. This establishes the SDM as a geometrically motivated explanation of dark energy that requires no additional scalar fields or phenomenological assumptions. We further examined the complex extension of the drift, Dz=ezφ D e−zφ, z ∈C,(X.6) which forms a holomorphic family of type (A). For nonzero imaginary part, the operator Dz becomes non-self-adjoint and develops complex resonances. The resulting complexified spectral action acquires an imaginary part that encodes dissipative effects and potential CP-violating contributions. This provides a geometric mechanism for time asymmetry, complementing the real spectral drift that generates dark energy. Taken together, the results presented in this monograph show that the drift deformation of the Dirac operator enriches the Spectral Action with two profound mechanisms: •areal drift producing a stable spectral condensate that acts as a cosmological constant; •acomplex drift introducing dissipative behaviour and a natural geometric arrow of time. All dynamical consequences derived in this report arise from effective, truncated functionals built on spectrally rigid operators. This guarantees full consistency with the invariance of global spectral quantities under bounded similarity transformations and aligns the present construction with the canonical methodology formalised in later reports of the DRSN series. The SDM therefore furnishes a mathematically rigorous Theory of Everything within the spectral framework, in which geometry, matter, dark energy and irreversibility arise from a single universal principle encoded in the behaviour of drifted Dirac operators. Future developments may include phenomenological tests of the model, extensions to non-compact geometries, a full renormalisationgroup analysis, and a more detailed exploration of the modular structures associated with the complex spectral action. 40 Geometry M Dirac DM gµν , R, Tµ Internal space F Dirac DF Yukawas, masses Total Dirac D=DM⊕DF Drifted Dirac Ds=esφDe−sφ Spectral action S(Ds) = Trf(Ds/Λ) Spectral potential V(s)=αs2+βs4 Condensate s∗= 0,V(s∗)>0 Cosmology FRW + drift Dark energy DMDF φ heat kernel Seeley–DeWitt α < 0, β > 0 ρΛ=V(s∗) FIG. 1. Conceptual architecture of the Spectral Drift Model: geometric Dirac DM and internal Dirac DF combine into the total operator D , which is deformed by a spectral drift into Ds . The drifted spectral action produces a universal potential V ( s )with a non-trivial condensate s∗ and a geometric contribution to dark energy. 41 s∗ V(s∗) s V(s) FIG. 2. Schematic shape of the universal spectral potential V ( s ) = αs2 + βs4 , with α < 0and β > 0. The drifted spectral action produces a non-trivial minimum s∗ with positive vacuum energy V ( s∗ ) > 0, interpreted as a geometric contribution to dark energy. Drifted Dirac Ds=esφDe−sφ Square D2 s Drifted connection ∇(s) µ=∇µ+s ∂µφ Drifted potential Es=1 4R+T+s∆φ+s2∥∇φ∥2 Laplace-type operator −gµν ∇(s) µ∇(s) ν+Es FIG. 3. Lichnerowicz formula with drift: the drifted Dirac operator Ds yields a square D2 s that can be written in Laplace-type form −gµν ∇(s) µ∇(s) ν + Es , where both the connection and the potential acquire explicit s–dependent contributions. 48 This shows that, for scalar multipliers, the BCH expansion truncates at first order in s . Nevertheless, it is more convenient to work directly with the exact identity Ds=D+s[D, φ](A.22) and with the exact expansion of D2 sdeveloped in Appendix B. 8. Summary We summarise the main analytic facts established in this appendix: •For each s∈R, the drifted Dirac operator Dsis self-adjoint with σ(Ds)=σ(D)⊂R. •The commutator [D, φ]is bounded and D-relatively bounded with relative bound zero. •The complex drift Dzforms a holomorphic family of type (A) in the sense of Kato. •The real drift satisfies the flow equation d ds Ds= [Ds, φ]. •For scalar multipliers, the second-order BCH term [φ, [φ, D]] vanishes identically. These results provide the analytic backbone for the derivation of the drifted Lichnerowicz formula, the computation of the Seeley–DeWitt coefficients and the analysis of the spectral potential developed in the main text. Appendix B: Drifted Lichnerowicz Formula In this appendix we derive the explicit Laplace-type expression for the square of the drifted Dirac operator Ds=esφD e−sφ,(B.1) where φ∈C∞ ( M, R ) ∩L∞ ( M )and D is the standard Dirac operator (possibly with torsion, as in Einstein–Cartan geometry). The purpose of this appendix is to provide the full technical derivation of the formula D2 s=−gµν∇(s) µ∇(s) ν+Es,(B.2) which is the starting point for the computation of the Seeley–DeWitt coefficients in Section V. 49 1. Expansion of the drifted operator Using the boundedness of [D, φ], we may expand Dsas Ds=D+s[D, φ],(B.3) since the double commutator [ φ, [ φ, D ]] vanishes for multiplicative φ , implying that higher-order Baker–Campbell–Hausdorff corrections disappear. Hence D2 s= (D+s[D, φ])2=D2+s{D, [D, φ]}+s2[D, φ]2.(B.4) Define the operators Z:= {D, [D, φ]}, W := [D, φ]2.(B.5) We analyse each contribution separately. 2. The quadratic term W= [D, φ]2 Write [ D, φ ] = γµ ( ∂µφ ). Using the Clifford relation γµγν = gµν 1+ γµν with γµν antisymmetric, we obtain [D, φ]2=γµγν(∂µφ)(∂νφ)=gµν(∂µφ)(∂νφ)1=∥∇φ∥21.(B.6) Thus Wis a positive, multiplicative operator. 3. The linear term Z={D, [D, φ]} We compute Z=D(γµ∂µφ)+(γµ∂µφ)D. (B.7) Using ∇ν ( ∂µφ ) = ∇µ ( ∂νφ )for scalars and the compatibility of ∇µ with the Clifford action, one finds D(γµ∂µφ)=γν∇ν(γµ∂µφ)=γνγµ∇ν∇µφ, (B.8) (γµ∂µφ)D=gµν(∂µφ)∇ν+γµν (∂µφ)∇ν.(B.9) The antisymmetric γµν terms vanish after contraction with a symmetric second derivative or do not contribute to Laplace-type rewriting. The symmetric part yields Z= 2 gµν (∂µφ)∇ν+ (∆φ)1.(B.10) 50 4. Combination with the Lichnerowicz identity For a generalised Dirac operator D (including Einstein–Cartan torsion terms), the Lichnerowicz identity reads D2=∇∗∇+1 4R+T,(B.11) where Tencodes torsion contributions (zero in the purely Riemannian case). Substituting D2,Zand Winto D2 s, we obtain D2 s=∇∗∇+s2gµν(∂µφ)∇ν+s∆φ+1 4R+T+s2∥∇φ∥2.(B.12) 5. Drifted connection and potential Define the drifted connection ∇(s) µ:= ∇µ+s(∂µφ),(B.13) so that gµν∇(s) µ∇(s) ν=gµν∇µ+s∂µφ∇ν+s∂νφ(B.14) =gµν∇µ∇ν+ 2s gµν(∂µφ)∇ν+s2gµν (∂µφ)(∂νφ).(B.15) Comparing with the expression for D2 s, we identify the drifted Laplace-type form D2 s=−gµν∇(s) µ∇(s) ν+Es,(B.16) with potential Es=1 4R+T+s∆φ+s2∥∇φ∥2.(B.17) This completes the derivation of the drifted Lichnerowicz identity used in Sections IV and V. Appendix C: Explicit computation of ˜a2(s) In this appendix we derive explicitly the dependence of the Seeley–DeWitt coefficient ˜a2 ( s )on the drift parameter s. We work in dimension four and consider the drifted Dirac operator Ds=esφDe−sφ,(C.1) 51 where φ∈C∞ ( M, R ) ∩L∞ ( M )is a real scalar multiplier, and D is the (possibly torsionful) Dirac operator of the underlying geometry. As shown in Appendix B, the square of Dscan be written in Laplace-type form D2 s=−gµν∇(s) µ∇(s) ν+Es,(C.2) with ∇(s) µ=∇µ+s ∂µφ, (C.3) and Es=E0+s∆φ+s2∥∇φ∥2.(C.4) Here E0 denotes the undeformed endomorphism appearing in the Lichnerowicz formula for D2 , ∆φ=∇µ∇µφis the scalar Laplacian, and ∥∇φ∥2=gµν(∂µφ)(∂νφ). 1. General formula for a2 For a Laplace-type operator F=−gµν∇µ∇ν+E, (C.5) on a four-dimensional compact manifold without boundary, the second Seeley–DeWitt coefficient is given by a2(F) = 1 (4π)2ZM trR 61+Edµ, (C.6) where R is the scalar curvature, 1is the identity on the internal spinor space and tr denotes the trace over that internal space. Specializing to F=D2 swe obtain ˜a2(s) = 1 (4π)2ZM trR 61+Esdµ. (C.7) 2. Effective drift dependence of ˜a2(s) Substituting the explicit form of Es, we find ˜a2(s) = 1 (4π)2ZM trR 61+E0+s∆φ1+s2∥∇φ∥21dµ =1 (4π)2ZM trR 61+E0dµ +1 (4π)2ZM trs∆φ1+s2∥∇φ∥21dµ. (C.8) 52 We identify the undeformed value a2(0) = 1 (4π)2ZM trR 61+E0dµ. (C.9) Since Mis compact and without boundary, we have ZM ∆φ dµ = 0,(C.10) and hence the linear term in sintegrates to zero: ZM tr∆φ1dµ = tr(1)ZM ∆φ dµ = 0.(C.11) Thus there is no linear dependence on sin the effective coefficient ˜a2(s). The only non-trivial drift dependence arises from the term s2∥∇φ∥2: ˜a2(s)−a2(0) = 1 (4π)2ZM tr s2∥∇φ∥21dµ. (C.12) If the spinor bundle has rank r (for a 4D Dirac operator on a spin manifold, r = 4), then tr (1) = r , and we obtain ˜a2(s)−a2(0) = r (4π)2s2ZM∥∇φ∥2dµ. (C.13) In particular, for a standard four-dimensional Dirac operator (r= 4), ˜a2(s)=a2(0) + 4 (4π)2s2ZM∥∇φ∥2dµ. (C.14) 3. Summary We conclude that the second Seeley–DeWitt coefficient for the drifted Dirac operator Ds differs from its undeformed value a2(0) by a universal, non-negative quadratic contribution in s: ˜a2(s)=a2(0) + c2s2ZM∥∇φ∥2dµ, c2=tr(1) (4π)2.(C.15) In four dimensions with a standard Dirac operator, tr(1) = 4 and c2=4 (4π)2.(C.16) This term is always non-negative and provides the quadratic drift contribution to the spectral action, entering directly in the coefficient α of the universal potential V ( s ) = αs2 + βs4 discussed in the main text. 53 Appendix D: Explicit Computation of the Coefficient a4(s) In this appendix we compute the drift-dependent contributions to the Seeley–DeWitt coefficient a4 ( D2 s )in four dimensions. We show that the drift induces a universal quartic term proportional to ∥∇φ∥4 , ensuring that the coefficient β of the spectral potential V ( s ) = αs2 + βs4 is strictly positive. 1. Laplace-type operator and general formula for a4 For an operator of Laplace type F=−gµν∇µ∇ν+E, (D.1) the coefficient a4(F)in four dimensions is given by the standard Seeley–DeWitt expression a4(F) = 1 (4π)2 1 360 ZM Tr60RE + 180E2+ 30Ωµν Ωµν + 5R2−2RµνRµν + 2RµνρσRµνρσdµ. (D.2) Here E is the endomorphism term in the Laplace-type expression, and Ω µν is the curvature of the corresponding connection. For the drifted Dirac operator Ds = esφDe−sφ , the squared operator has the Laplace-type form D2 s=−gµν∇(s) µ∇(s) ν+Es,(D.3) where, as shown in Appendix B, Es=E0+s∆φ+s2∥∇φ∥2,(D.4) with E0=1 4R+Tthe undeformed potential including torsion. 2. Expansion of E2 sand identification of quartic terms Write Es=E0+sA +s2B, (D.5) where A:= ∆φ, B := ∥∇φ∥2.(D.6) 54 Then, E2 s= (E0+sA +s2B)2(D.7) =E2 0+ 2s E0A+s2(A2+ 2E0B) + 2s3AB +s4B2.(D.8) We now analyse the drift dependence: • The linear term in A integrates to zero, since RM ∆ φ dµ = 0 on compact manifolds without boundary. • The cubic term integrates to zero as well or reduces to lower-order contributions via integration by parts. •The quartic term is s4B2=s4∥∇φ∥4, providing the universal positive contribution. Substituting E2 sinto the 180E2term of Eq. (D.2) yields the quartic drift contribution: 180 s4∥∇φ∥4.(D.9) 3. Trace and final expression for a4(s) The spinor trace of the identity in four dimensions is Tr(1) = 4.(D.10) Thus the quartic drift-dependent part of a4is ˜a4(s)−a4(0) = 1 (4π)2 1 360 ZM 180 s4Tr(1)∥∇φ∥4dµ (D.11) =1 (4π)2 1 360 ZM 180 ·4s4∥∇φ∥4dµ (D.12) =1 2(4π)2s4ZM∥∇φ∥4dµ. (D.13) Hence: ˜a4(s)=a4(0) + 1 2(4π)2s4ZM∥∇φ∥4dµ. (D.14) 55 4. Implications for the spectral potential The contribution of a4(s)to the drifted Spectral Action is S(s)⊃f0˜a4(s),(D.15) so that the quartic term in the effective spectral potential is Vgeom(s)⊃βgeom s4, βgeom =f0 1 2(4π)2ZM∥∇φ∥4dµ. (D.16) Because ∥∇φ∥4≥0pointwise, we obtain: βgeom >0.(D.17) This establishes the universal positivity of the quartic coefficient in the spectral drift potential, independently of the geometry, torsion, or internal Dirac operator. The total quartic coefficient is the sum of the geometric and internal contributions, both strictly positive: βtot =βgeom +βint >0.(D.18) Thus, the drift-induced spectral potential is always bounded from below and stabilised at large |s|, a fact crucial for the existence of the spectral condensate in SDM-IV. Appendix E: Torsion in Einstein–Cartan Geometry 1. Purpose and overview In this appendix we show how axial torsion in Einstein–Cartan geometry modifies the Lichnerowicz formula and contributes a universal negative quadratic term to the spectral action. This term is responsible for the torsion-induced contribution αtors <0in the spectral potential V(s)=αs2+βs4, and plays a central role in the existence of a non-trivial spectral condensate in the SDM-III and SDM-IV models. 56 2. Einstein–Cartan connection and torsion Let ( M, g )be a four-dimensional Riemannian (or Lorentzian) manifold endowed with a metriccompatible affine connection ∇EC whose torsion tensor Tλµν := Γλ µν −Γλ νµ does not necessarily vanish. We may decompose the connection as Γλ µν =◦ Γλ µν +Kλµν,(E.1) where ◦ Γis the Levi–Civita connection and Kλµν is the contorsion tensor. In the Einstein–Cartan setting relevant for the Spectral Drift Model, we consider a purely axial torsion, parametrised by a vector field Tµ, such that Tµνρ =ϵµνρσTσ,(E.2) where ϵµνρσ is the Levi–Civita tensor. The corresponding spin connection on the spinor bundle S→Mcan be written as ∇EC µ=∇LC µ+3 2γ5Tµ,(E.3) where ∇LC µis the Levi–Civita spin connection and γ5denotes the chirality matrix. 3. Dirac operator with torsion The Dirac operator in Einstein–Cartan geometry is defined by DEC =γµ∇EC µ=DLC +3 2γµγ5Tµ,(E.4) where DLC =γµ∇LC µ is the Dirac operator of the torsion-free Levi–Civita connection. Thus, the torsion contribution behaves as an axial vector coupling in the Dirac operator. 57 4. Lichnerowicz formula with axial torsion The squared Einstein–Cartan Dirac operator satisfies a generalised Lichnerowicz formula of the form D2 EC =∇∗∇+1 4R−3 4∥T∥2+3 2(∇µTµ)γ5+(terms quadratic in curvature and torsion),(E.5) where: •∇∗∇is the Bochner Laplacian built from ∇EC, •Ris the scalar curvature of the Levi–Civita connection, •∥T∥2:= gµνTµTνis the squared norm of the torsion vector, • the divergence term ( ∇µTµ ) γ5 integrates to a boundary term (and vanishes on compact manifolds without boundary), • the remaining terms (quartic in gamma matrices and curvature/torsion) do not contribute to the quadratic part of the spectral potential and can be grouped in the “higher-order” sector. For the purposes of computing the Seeley–DeWitt coefficient a2 , we can therefore write the effective endomorphism term as E0=1 4R−3 4∥T∥2,(E.6) up to contributions that do not affect the torsion-dependent bulk integral. 5. Contribution to the Seeley–DeWitt coefficient a2 Recall that, for a Laplace-type operator F=−gµν∇µ∇ν+E, on a four-dimensional compact manifold, the second Seeley–DeWitt coefficient is given by a2(F) = 1 (4π)2ZM Tr R 6+Edµ. (E.7) Applying this formula to D2 EC with E0=1 4R−3 4∥T∥2, 64 1. Holomorphic family of drifted Dirac operators Let D be the self-adjoint Dirac operator of a spectral triple and φ a real-valued, bounded, smooth function acting by multiplication. For z∈Cwe define the complex drifted operator Dz:= ezφ D e−zφ.(G.1) As proven in the operator-theoretic appendix, the family {Dz}z∈C is a holomorphic family of type (A) in the sense of Kato: the domain of Dz is independent of z and for every ψ∈Dom ( D )the map z7−→ Dzψ(G.2) is holomorphic on C. This follows from the boundedness of the commutator [D, φ]∈ B(H),(G.3) and from the identity Dz=D+z[D, φ].(G.4) For z∈R , Dz is unitarily equivalent to D and remains self-adjoint, hence has real spectrum and generates a unitary group. For Im z = 0, self-adjointness is lost and the spectrum generally moves into the complex plane. 2. Holomorphic spectral action and Taylor expansion The spectral action associated with Dzis defined by S(z) := Trf(Dz/Λ),(G.5) for a fixed cut-off function f and an energy scale Λ. Under the same conditions that guarantee the asymptotic expansion of the real spectral action, the function S ( z )is holomorphic in a neighbourhood of z= 0. Therefore, it admits a Taylor expansion S(z) = ∞ X n=0 zn n!S(n)(0),(G.6) with S(n)(0) = dn dznz=0 Trf(Dz/Λ).(G.7) 65 Using the differential identity d dz Dz= [Dz, φ],(G.8) and functional calculus for f(Dz/Λ), we obtain, at least formally, d dz f(Dz/Λ) = 1 Λf′(Dz/Λ) [Dz, φ],(G.9) so that S′(0) = 1 ΛTrf′(D/Λ) [D, φ].(G.10) Higher derivatives involve iterated commutators of Dwith φ, schematically of the form S(n)(0) ∼1 ΛnTrf(n)(D/Λ) adn φ(D),(G.11) where adn φ(D)denotes the n-fold iterated commutator. 3. Real and imaginary parts of the spectral action Writing z=s+iθ we can decompose the spectral action into real and imaginary parts: S(z) = SRe(s, θ) + i SIm(s, θ),(G.12) with SRe(s, θ) = 1 2S(z)+S(z), SIm(s, θ) = 1 2iS(z)−S(z).(G.13) For z∈R we have S ( z ) ∈R and hence SIm ( s, 0) = 0, whereas for θ = 0 the imaginary part SIm ( s, θ ) is in general non-vanishing. For small θ, expanding around (s, θ) = (s, 0), we obtain SIm(s, θ)=θ ∂θSIm(s, 0)+O(θ3),(G.14) so that to leading order in θ the imaginary part of the spectral action is linear in the imaginary deformation. From an operator-theoretic point of view, when z acquires a non-zero imaginary part, the generator Dz becomes non-self-adjoint and its evolution no longer defines a unitary group. Instead, it typically generates a semigroup with dissipative features. 66 4. Dissipative contributions and time asymmetry Consider the evolution equation dψ dt =−iDsψ−θ[Ds, φ]ψ, (G.15) with s∈R fixed and small imaginary deformation θ . The first term generates the usual unitary evolution, whereas the second term is anti-Hermitian and can be interpreted as a dissipative contribution, driving the system towards preferred states. At the spectral level, this is reflected by the appearance of complex eigenvalues and resonances corresponding to exponential damping or growth. The presence of an imaginary part in the spectral action S ( z )indicates that the effective action acquires a non-Hermitian component, which may be interpreted as an emergent source of dissipative dynamics, CP violation or time asymmetry. This is conceptually consistent with the structure of modular flows and KMS states in operator-algebraic quantum field theory, where time asymmetry arises from analyticity and the structure of non-trivial states. 5. Scope and limitations It is important to stress that the complex drift and its spectral action are not used in any of the spectral computations leading to the real potential V(s)=αs2+βs4,(G.16) nor in the analysis of the four drifted models SDM-I–IV presented in the main body of the text. All physical results concerning the spectral condensate s∗ and the geometric origin of dark energy are derived exclusively from the real drift Ds with self-adjoint Dirac operators and real spectral actions. The complex drift Dz and the holomorphic spectral action provide an additional, intrinsically geometric mechanism for discussing dissipation, CP violation and time asymmetry at the spectral level. Fully developing this idea and its phenomenological implications is beyond the scope of the present work and is left for future research. Appendix H: Physical Input: Masses, Yukawas and Parameters In this appendix we collect the physical input parameters needed to construct the internal Dirac operator DF explicitly and to compute the internal contributions to the spectral coefficients entering 67 the drifted potential Vint(s)=αints2+βints4.(H.1) These include the fermion masses, Yukawa couplings, mixing matrices and gauge couplings at a reference scale. The values below are indicative and compatible with current PDG data; they are organised in a way that allows any investigator to reconstruct DF and test the Spectral Drift Model numerically. 1. Fermion masses of the Standard Model We list here the rest masses of quarks and charged leptons, in GeV, in a convenient PDGcompatible scheme. Neutrino masses are much smaller and can be included or neglected as appropriate. a. Quarks Quark Mass (GeV) u(up) 0.0022 d(down) 0.0047 s(strange) 0.096 c(charm) 1.27 b(bottom) 4.18 t(top) 172.69 TABLE II. Quark masses used for the construction of the internal Dirac operator DF. b. Charged leptons Lepton Mass (GeV) e(electron) 0.000511 µ(muon) 0.10566 τ(tau) 1.77686 TABLE III. Charged lepton masses in GeV. 68 c. Neutrinos Neutrino masses are very small and currently not known with high precision. For spectral purposes one typically uses a normal or inverted hierarchy, for example m1≈0, m2≈8.6×10−3eV, m3≈5.0×10−2eV.(H.2) Their contribution to Tr ( D2 F )and Tr ( D4 F )is negligible compared to the top Yukawa and can be omitted in a first approximation. 2. Yukawa couplings The Yukawa couplings are defined by yf=√2mf v, v = 246 GeV,(H.3) where v is the Higgs vacuum expectation value. We list here the approximate values for quarks and charged leptons. a. Quark Yukawas Quark Yukawa yf u1.3×10−5 d2.7×10−5 s5.5×10−4 c7.3×10−3 b2.4×10−2 t0.995 TABLE IV. Quark Yukawa couplings at the electroweak scale. The top Yukawa dominates internal traces. b. Lepton Yukawas Yukawa couplings for Dirac neutrinos are extremely small, typically of order 10−12–10−13, and can be neglected in the quartic internal trace Tr ( Y4 F )unless one studies specifically the neutrino sector. 69 Lepton Yukawa yf e2.9×10−6 µ6.1×10−4 τ1.0×10−2 TABLE V. Charged lepton Yukawa couplings. 3. Mixing matrices: CKM and PMNS The mixing in the quark sector is encoded in the CKM matrix, while the mixing of neutrinos is described by the PMNS matrix. For the construction of DF , we use the magnitudes of the standard parameterisations. a. CKM matrix |VCKM| 0.974 0.226 0.0036 0.226 0.973 0.042 0.0087 0.040 0.999 TABLE VI. Approximate absolute values of the CKM matrix elements. b. PMNS matrix |UPMNS| 0.821 0.550 0.150 0.355 0.700 0.620 0.450 0.450 0.770 TABLE VII. Approximate absolute values of the PMNS matrix elements. These matrices enter the internal Dirac operator through the diagonalisation of mass matrices in flavour space. For the spectral drift analysis, the leading effect comes from the eigenvalues (masses/Yukawas); mixings are relevant for a fully realistic DF but do not alter the sign structure of αint and βint. 70 4. Internal traces and contributions to αint and βint In the internal sector, the drifted Dirac operator is DF,s =DF+sYF,(H.4) where YF is determined by a chosen internal drift profile. The relevant traces entering the Spectral Action are schematically Tr(D2 F,s) = Tr(D2 F)+2sTr(DFYF)+s2Tr(Y2 F),(H.5) Tr(D4 F,s) = Tr(D4 F)+···+s4Tr(Y4 F).(H.6) The quartic internal contribution to the spectral potential is βint =f0Tr(Y4 F),(H.7) and is strictly positive since YF has non-zero eigenvalues (dominated by yt ). The quadratic internal contribution is αint =f2Λ2Tr(Y2 F),(H.8) whose sign is model dependent but typically positive for realistic Yukawa spectra. Explicitly, in the simplest drift profile one can approximate Tr(Y2 F)≈X f y2 f,Tr(Y4 F)≈X f y4 f,(H.9) with f running over all fermions and the top Yukawa giving the dominant contribution to Tr ( Y4 F ). 5. Gauge couplings and Higgs parameters For completeness we also record the following parameters at the electroweak scale: •Strong coupling: αs(MZ)≈0.118.(H.10) •Electroweak gauge couplings: g1(MZ)≈0.357, g2(MZ)≈0.652, g3(MZ)≈1.218.(H.11) 71 •Higgs mass: mH≈125.10 GeV.(H.12) These values are used to fix the initial conditions for running couplings in more detailed phenomenological analyses of the Spectral Drift Model, but are not essential for the derivation of the universal form of the spectral potential. 6. Recommended initial conditions for numerical simulations Typical initial conditions for the drift field s(t)in a cosmological setting are: s(t0)=sini ∈[10−3,1],˙s(t0)≈0,(H.13) with radiation and matter densities chosen consistently with a given early-Universe temperature. The scale Λand the Yukawa spectrum then determine the numerical values of αint and βint , which can be fed into the cosmological equations described in Appendix I. In summary, this appendix provides the necessary physical input to reconstruct the internal Dirac operator DF and to reproduce the internal contributions to the spectral potential V ( s ), allowing for independent numerical tests of the Spectral Drift Model. Appendix I: Numerical Implementation and Pseudocode In this appendix we present a concrete numerical scheme to solve the cosmological evolution of the spectral drift condensate. The goal is to provide an operational recipe by which one can integrate the drift field s ( t )in a spatially flat Friedmann–Robertson–Walker (FRW) background and reproduce the qualitative behaviour discussed in Sec. VIII. The focus is on the unified Einstein– Cartan + Standard Model drifted geometry (SDM–IV), but the implementation can be adapted to other drifted models. 1. Equations of motion We consider a spatially flat FRW metric ds2=−dt2+a(t)2dx2,(I.1) 72 with scale factor a(t)and Hubble parameter H(t) = ˙a(t) a(t).(I.2) The drift condensate is taken to be homogeneous: s=s(t).(I.3) The effective action for the drift condensate is Seff[s] = Zdt a(t)31 2˙s2−V(s),(I.4) with spectral potential V(s)=αs2+βs4, β > 0, α < 0,(I.5) as derived in Sec. Vand Sec. VII. The energy density and pressure associated with s(t)are ρs=1 2˙s2+V(s), ps=1 2˙s2−V(s).(I.6) The equation of motion for s(t)is the damped Klein–Gordon equation ¨s+ 3H˙s+V′(s)=0,(I.7) where V′(s)=2αs + 4βs3.(I.8) The total energy budget includes radiation and matter: H2=8πG 3ρs+ρr+ρm,(I.9) with ρr∝a−4, ρm∝a−3.(I.10) 2. First-order system formulation For numerical integration it is convenient to rewrite the system in first-order form. Define x1=s, x2= ˙s. (I.11) 73 Then ˙x1=x2,(I.12) ˙x2=−3Hx2−2αx1−4βx3 1.(I.13) The Hubble parameter can be written as H=s8πG 31 2x2 2+αx2 1+βx4 1+ρr+ρm.(I.14) A numerically more stable time variable is the number of e-folds N= ln a(t).(I.15) Using d dN =1 H d dt,(I.16) we obtain the e-fold system dx1 dN =x2 H,(I.17) dx2 dN =−3Hx2−2αx1−4βx3 1 H,(I.18) dρr dN =−4ρr,(I.19) dρm dN =−3ρm.(I.20) 3. Initial conditions A typical choice of initial conditions for the drift condensate in the early Universe is x1(N0)=sini, x2(N0) = ˙sini,(I.21) with sini ∈ [10 −3, 1] (in suitable units) and ˙sini small or zero. The initial radiation and matter densities can be fixed by specifying the energy scale of the initial epoch, e.g. at reheating or at some early time deep in the radiation-dominated era. The spectral coefficients α and β should be taken from the SDM-IV analysis in Sec. VII; they encapsulate both geometric and internal contributions.