DRSN I: A Unified Geometric Framework for Matter, Torsion and Dark Energy (De Rerum Spectrale Natura, Report I, Version 2.0)

DRSN I: A UNIFIED GEOMETRIC FRAMEWORK FOR MATTER, TORSION AND DARK ENERGY De Rerum Spectrale Natura series REPORT I (Version 1.1) 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. CONTENTS I. Introduction 6 II. Operators, Similarity and Spectral Structure 7 A. Closed and self-adjoint operators 8 B. Similarity transformations 8 C. Relative boundedness and drift perturbations 9 D. Holomorphic dependence (real drift case) 9 E. Differentiation of the drift 10 F. Summary 10 III. Lichnerowicz Identity with Drift 10 ∗jp[email protected] 3 A. Standard Lichnerowicz formula 11 B. Expansion of the drifted operator 11 C. Computation of the quadratic term 11 D. Computation of the linear term 12 E. Drifted Laplace-type decomposition 12 F. Consequences for the spectral coefficients 12 IV. Spectral Action with Drift 13 A. Drifted spectral action 13 B. Heat-kernel coefficients under drift 14 C. Structure of the drifted action 14 D. Emergence of the universal quartic structure 15 E. Outlook 15 V. The Universal Spectral Potential 15 A. Drifted Lichnerowicz Formula 16 B. General Form of a2(D2 s)16 C. General Form of a4(D2 s)17 D. The Geometric Potential 17 E. Internal Contributions 18 F. Torsion Contribution 18 G. Total Potential 18 VI. Internal Drifted Models: SDM-I and SDM-II 19 A. Internal Dirac operator of the Standard Model (SDM-I) 19 B. Internal Dirac operator of the Pati–Salam model (SDM-II) 20 C. Summary and role in the full Spectral Drift Model 21 VII. Drifted Einstein–Cartan Models: SDM-III and SDM-IV 21 A. The Torsion Contribution in Einstein–Cartan Geometry 21 B. The Drifted Einstein–Cartan Model (SDM-III) 22 C. The Unified Drifted Model (SDM-IV): EC + Standard Model 23 D. Interpretation and Significance 23 VIII. FRW Cosmology of the Spectral Drift 24 A. Motivation 24 B. Effective action in a FRW background 24 C. Equation of motion for the drift 25 D. Friedmann equations with the drift sector 25 4 E. Dynamical regimes of the spectral drift 26 F. Stability of the condensate 26 G. Geometric dark energy from the spectral drift 27 IX. Complex Drift and Irreversibility 27 A. Motivation 27 B. Holomorphic family of type (A) 28 C. Spectral properties for complex drift 28 D. Complex spectral action and its Taylor expansion 29 E. Real and imaginary parts of the complex spectral action 29 F. Dissipative dynamics and time asymmetry 30 G. Conceptual interpretation and limitations 30 X. General Conclusions 31 XI. Analytical Foundations of the Spectral Drift 33 A. Closed and self-adjoint operators 33 B. Similarity by bounded, invertible operators 34 C. Spectral invariance for real drift 34 D. Relative boundedness of [D, φ]35 E. Holomorphic families of type (A) 35 F. Flow equation for the real drift 36 G. Vanishing of the second-order BCH term for scalar multipliers 36 H. Summary 37 XII. Drifted Lichnerowicz Formula 37 A. Expansion of the drifted operator 37 B. The quadratic term W= [D, φ]238 C. The linear term Z={D, [D, φ]}38 D. Combination with the Lichnerowicz identity 38 E. Drifted connection and potential 39 XIII. Explicit computation of ˜a2(s)39 A. General formula for a240 B. Effective drift dependence of ˜a2(s)40 C. Summary 41 XIV. Explicit Computation of the Coefficient a4(s)41 A. Laplace-type operator and general formula for a441 B. Expansion of E2 sand identification of quartic terms 42 5 C. Trace and final expression for a4(s)42 D. Implications for the spectral potential 43 XV. Torsion in Einstein–Cartan Geometry 43 A. Purpose and overview 43 B. Einstein–Cartan connection and torsion 43 C. Dirac operator with torsion 44 D. Lichnerowicz formula with axial torsion 44 E. Contribution to the Seeley–DeWitt coefficient a245 F. Torsion-induced quadratic term in the spectral action 45 G. Summary 46 XVI. Cosmology of the Spectral Condensate 46 A. Purpose and setting 46 B. Effective action and energy–momentum tensor 47 C. Equation of motion for the condensate 47 D. Friedmann equations with the spectral drift 47 E. Evolution in terms of e-folds 48 F. Regimes of the condensate dynamics 48 G. Stability of the spectral condensate 49 XVII. Complex Drift and Holomorphic Spectral Action 49 A. Holomorphic family of drifted Dirac operators 50 B. Holomorphic spectral action and Taylor expansion 50 C. Real and imaginary parts of the spectral action 51 D. Dissipative contributions and time asymmetry 51 E. Scope and limitations 52 XVIII. Physical Input: Masses, Yukawas and Parameters 52 A. Fermion masses of the Standard Model 52 1. Quarks 52 2. Charged leptons 52 3. Neutrinos 52 B. Yukawa couplings 53 1. Quark Yukawas 53 2. Lepton Yukawas 53 C. Mixing matrices: CKM and PMNS 53 1. CKM matrix 53 2. PMNS matrix 53 6 D. Internal traces and contributions to αint and βint 53 E. Gauge couplings and Higgs parameters 54 F. Recommended initial conditions for numerical simulations 54 XIX. Numerical Implementation and Pseudocode 55 A. Equations of motion 55 B. First-order system formulation 56 C. Initial conditions 57 D. Runge–Kutta 4 integration scheme 57 E. Example pseudocode 57 F. Stability considerations 59 References 59 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 D through 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 7 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. 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. a. 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 present report focuses on the external Dirac geometry, the emergence of a universal drift-induced 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. II. OPERATORS, SIMILARITY AND SPECTRAL STRUCTURE In this section we review the analytic properties of the drifted Dirac operator Ds=esφD e−sφ,(II.1) where φ∈C∞ ( M )is a real, bounded multiplication operator. Throughout, H denotes a complex Hilbert space and all operators are complex-linear. “Self-adjoint” always refers to the usual complex adjoint, and 8 “real spectrum” simply means σ ( Ds ) ⊂R . No statements in this chapter involve the complex drift Dz introduced later in Sec. IX. The goal is to establish the operator-theoretic foundations required for the spectral computations performed in later chapters. Most results follow from standard functional analysis (Kato, Pazy, Reed–Simon), but we collect them here for self-containment and to emphasise their applicability to geometric Dirac operators. A. Closed and self-adjoint operators Let D : Dom ( D ) ⊂ H → H be a densely defined operator. It is called closed if its graph is closed in H×H, and self-adjoint if D=D†with identical domains. The Dirac operator on a spinor bundle over a compact Riemannian manifold is closed, self-adjoint, has compact resolvent, and possesses discrete real spectrum. These properties are essential for defining the Spectral Action and for controlling perturbations arising from drift. We emphasise that all operators in this chapter remain self-adjoint; the complex drift Dz , which is no longer self-adjoint, will only appear in Chapter IX and Appendix XVII. B. Similarity transformations Let S∈B(H)be a bounded, invertible operator. The similarity transformation of Dby Sis defined as DS:= SDS−1.(II.2) The crucial case for the drift is Ss = esφ , where φ is a real bounded multiplication operator. Then Ss is bounded, positive, and invertible, with S∗ s=Ss,∥Ss∥≤es∥φ∥∞. Theorem 1 (Domain stability).If Dis closed, then Dom(Ds)=SsDom(D), and Dom(Ds)is dense in H. Theorem 2 (Preservation of self-adjointness).If D is self-adjoint and Ss is bounded, invertible and positive, then Ds=SsDS−1 s is self-adjoint. Theorem 3 (Isospectrality).For all s∈R, σ(Ds)=σ(D), 9 and their resolvents satisfy (Ds−z)−1=Ss(D−z)−1S−1 s. Thus, the drift does not change the eigenvalues of the Dirac operator, but it modifies geometric quantities that enter the heat kernel coefficients, which is precisely what generates the spectral potential V(s). C. Relative boundedness and drift perturbations Let Bbe an operator defined on Dom(D). It is D-relatively bounded if there exist a, b ≥0such that ∥Bψ∥≤a∥Dψ∥+b∥ψ∥ for all ψ∈Dom(D). The commutator [D, φ]=γµ∂µφ is a bounded operator on Hand therefore has relative bound a= 0. This leads to: Proposition 4. For s∈R, the operator D+s[D, φ]is self-adjoint on Dom(D). Since Ds=D+s[D, φ] exactly (no BCH quadratic terms arise because [ φ, [ φ, D ]] = 0), the drift may be viewed as a smooth self-adjoint perturbation of D. D. Holomorphic dependence (real drift case) Although the complex drift is analysed in Chapter IX, it is useful to state a related fact here: even for real s, the family {Ds}is analytic in sin the strong sense of Kato. Theorem 5 (Analytic family of type (A)).The family s7→ Ds is a holomorphic family of type (A); that is: •the domain is independent of s; •for each ψ∈Dom(D), the map s7→ Dsψis analytic. This property will be used in Chapter IV, when deriving variations of the heat kernel coefficients. 16 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 XII, the drifted Dirac operator satisfies the identity D2 s=−gµν ∇(s) µ∇(s) ν+Es,(V.3) 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, 17 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. 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 + 5R2−2Rµν Rµν + 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. 18 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) 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. 19 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 DF is 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). 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) 20 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 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) 21 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 non-trivial 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: •internal degrees of freedom provide a stabilising quartic term; • the negative quadratic term required for symmetry breaking and condensation must come from the geometric sector. In particular, the Einstein–Cartan torsion contribution analysed in SDM-III will supply a universal negative term in α , and its combination with the internal quartic term will produce the non-trivial condensate s∗ of SDM-IV. We now turn to these geometric models in the next section. VII. DRIFTED EINSTEIN–CARTAN MODELS: SDM-III AND SDM-IV In this section we analyse the drifted spectral models in which the gravitational sector is described by Einstein–Cartan (EC) geometry, where the spin connection admits an axial torsion component. Unlike purely Riemannian geometry, EC geometry generically introduces a negative contribution to the potential derived from the Spectral Action. This feature is precisely what allows the spectral condensate to form in the unified drifted model (SDM-IV). We begin with the EC-only scenario (SDM-III) and then incorporate the internal Dirac operator of the Standard Model to obtain the full model SDM-IV. A. The Torsion Contribution in Einstein–Cartan Geometry Let D denote the Dirac operator associated with a spin connection that includes an axial torsion field Tµ . The Lichnerowicz formula in Einstein–Cartan geometry takes the form D2=∇∗∇+1 4R−3 4∥T∥2+3 2(∇µTµ)γ5,(VII.1) 22 where ∥T∥2 = gµν TµTν . The divergence term does not contribute to the integrated Seeley–DeWitt coefficients on compact manifolds without boundary. Thus the relevant modification induced by torsion is the universal, negative scalar potential term −3 4∥T∥2.(VII.2) This term plays a key role in the quadratic part of the drifted spectral potential. The Seeley–DeWitt coefficient a2contains ∆ators 2=−3 (4π)2ZM∥T∥2dµ, (VII.3) so that torsion contributes negatively to the coefficient αin the universal potential V(s)=αs2+βs4.(VII.4) This observation is central: torsion provides the only universal geometric mechanism available in the drifted Spectral Action that can make α < 0. B. The Drifted Einstein–Cartan Model (SDM-III) Consider now the drifted operator Ds=esφDe−sφ,(VII.5) where D is the EC Dirac operator. As shown previously, the drift deformation introduces additional contributions to the zero-order term in the Lichnerowicz formula: Es=E0+s∆φ+s2∥∇φ∥2,(VII.6) with E0=1 4R−3 4∥T∥2.(VII.7) The effective (truncated) spectral action then yields VEC(s)=(αtors +αgeom)s2+βgeoms4,(VII.8) where αtors <0,(VII.9) αgeom ≥0,(VII.10) βgeom >0.(VII.11) Hence SDM-III already possesses a non-trivial minimum: s∗=s−αtors +αgeom 2βgeom .(VII.12) Although SDM-III does not incorporate matter fields, it demonstrates that the spectral condensate is a gravitational effect. The torsion in Einstein–Cartan geometry makes the drift instability inevitable. 23 C. The Unified Drifted Model (SDM-IV): EC + Standard Model We now incorporate the internal finite Dirac operator DF associated with the Standard Model. The total Dirac operator is Dtot =DEC ⊕DF,(VII.13) and the drift deformation acts component-wise: (Dtot)s=DEC,s ⊕DF,s.(VII.14) The total potential is Vtot(s) = (αtors +αgeom +αint)s2+ (βgeom +βint)s4.(VII.15) From earlier computations: βgeom >0,(VII.16) βint >0,(VII.17) αtors <0.(VII.18) The internal geometric contributions satisfy αgeom ≥ 0and αint may be of either sign but is typically small compared to αtors. Thus, for SDM-IV we obtain the universal hierarchy αtot <0, βtot >0.(VII.19) Consequently, SDM-IV exhibits a stable spectral condensate: s∗=r−αtot 2βtot .(VII.20) The associated vacuum energy density is ρΛ=Vtot(s∗)=−α2 tot 4βtot >0,(VII.21) providing a purely geometric origin for dark energy without invoking additional fields or parameters beyond the Spectral Action. D. Interpretation and Significance In this unified drifted model, the emergence of dark energy is not a phenomenological addition but an intrinsic consequence of the spectral geometry. Torsion supplies the essential negative quadratic contribution, while the drift generates universal stabilising quartic terms. Internal degrees of freedom refine but do not alter the existence of the condensate. SDM-IV thus provides a consistent, geometrically motivated explanation for a positive cosmological constant and establishes a direct link between spectral geometry, spacetime torsion, and the large-scale dynamics of the Universe. 24 A concise comparison of the four drifted spectral models is given in Table I, highlighting the rôle of the torsion sector in driving α < 0and the emergence of a stable condensate in SDM–III and SDM–IV. VIII. FRW COSMOLOGY OF THE SPECTRAL DRIFT A. Motivation In the previous chapters we derived the universal spectral potential induced by the drift deformation of the Dirac operator. In the unified model SDM-IV (Einstein–Cartan geometry plus the Standard Model internal Dirac operator), the drift parameter sis governed by an effective potential V(s)=αs2+βs4, β > 0, α < 0,(VIII.1) so that the total spectral action admits a non-trivial minimum s∗=r−α 2β= 0.(VIII.2) This minimum defines a positive vacuum energy ρΛ=V(s∗)>0,(VIII.3) which is naturally interpreted as a geometric cosmological constant. The aim of this chapter is to investigate the dynamics of the spectral drift in a spatially flat Friedmann– Robertson–Walker (FRW) universe and to show that the system relaxes dynamically to the condensate value s∗ , reproducing a late-time accelerated expansion compatible with a ΛCDM-like cosmology at the background level. B. Effective action in a FRW background We consider a spatially flat FRW metric ds2=−dt2+a(t)2dx2,(VIII.4) where a(t)is the scale factor and tis cosmic time. We assume that the drift parameter sis homogeneous, s=s(t),(VIII.5) in accordance with spatial isotropy and homogeneity. The relevant part of the drifted Spectral Action can be reduced, after integration over space, to an effective one-dimensional action of the form Seff[s] = Zdt a(t)31 2˙s2−V(s),(VIII.6) 25 where dots denote derivatives with respect to cosmic time t and V ( s )is the spectral potential obtained from the Seeley–DeWitt coefficients. The kinetic term is canonical, as dictated by the structure of the leading heat kernel coefficient and the drift-induced corrections. The energy density and pressure associated with the drift are given by the standard scalar field expressions ρs=1 2˙s2+V(s), ps=1 2˙s2−V(s).(VIII.7) C. Equation of motion for the drift Varying the effective action with respect to syields the Klein–Gordon type equation of motion ¨s+ 3H˙s+V′(s)=0,(VIII.8) where His the Hubble parameter, H=˙a a,(VIII.9) and V′(s) = dV ds = 2αs + 4βs3.(VIII.10) Explicitly, the drift evolves according to ¨s+ 3H˙s+ 2αs + 4βs3= 0.(VIII.11) The term 3 H˙s plays the role of a cosmological friction, ensuring that the dynamics of s is strongly damped as the Universe expands. D. Friedmann equations with the drift sector The drift sector contributes to the total energy budget of the Universe via ρs and ps . The Friedmann equations read H2=8πG 3(ρs+ρm+ρr),(VIII.12) ˙ H=−4πG (ρs+ps+ρm+pm+ρr+pr),(VIII.13) where ( ρm, pm )and ( ρr, pr )are the matter and radiation components, respectively, obeying their usual conservation equations. Substituting the expressions for ρsand ps, we obtain ρs=1 2˙s2+αs2+βs4, ps=1 2˙s2−αs2−βs4.(VIII.14) The effective equation-of-state parameter for the drift is ws=ps ρs = 1 2˙s2−V(s) 1 2˙s2+V(s).(VIII.15) 32 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 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 renormalisation-group analysis, and a more detailed exploration of the modular structures associated with the complex spectral action. 33 XI. ANALYTICAL FOUNDATIONS OF THE SPECTRAL DRIFT In this appendix we collect the functional-analytic results used throughout the Spectral Drift Model (SDM). We work on a complex Hilbert space Hwith inner product ⟨·,·⟩ linear in the second argument and anti-linear in the first. All operators are assumed to be linear over C. Our main goal is to justify rigorously the properties of the drifted Dirac operators Ds=esφ D e−sφ, s ∈R,(XI.1) and their complex extension Dz=ezφ D e−zφ, z ∈C,(XI.2) where D is a (geometric) Dirac operator and φ is a real-valued bounded function (multiplication operator) on the underlying spin manifold. We address: (i) stability of domains, closedness and self-adjointness under the drift; (ii) spectral invariance for real s; (iii) relative boundedness of commutators [D, φ]; (iv) holomorphic dependence of Dzin the sense of Kato; (v) the differential equation ∂sDs= [Ds, φ]; (vi) the vanishing of the second-order Baker–Campbell–Hausdorff term [φ, [φ, D]] for scalar multipliers. A. Closed and self-adjoint operators Let H be a complex Hilbert space. An operator T : Dom ( T ) → H is said to be densely defined if Dom ( T ) is dense in H. It is closed if its graph G(T):={(x, Tx):x∈Dom(T)} ⊂ H×H (XI.3) is a closed subspace. The adjoint T∗is defined in the usual way via ⟨Tx, y⟩=⟨x, T∗y⟩,∀x∈Dom(T), y ∈Dom(T∗),(XI.4) and an operator is self-adjoint if T=T∗and Dom(T) = Dom(T∗). In the spectral triple framework, the Dirac operator D is assumed to be self-adjoint and densely defined. This ensures that: •the spectrum σ(D)⊂R; •the resolvent (D−z)−1exists and is bounded for all z∈C\R; •Dgenerates a strongly continuous one-parameter unitary group eitD on H. 34 B. Similarity by bounded, invertible operators Let S:H → H be a bounded, invertible operator and define TS:= ST S−1,(XI.5) initially on S(Dom(T)). We record the following standard facts (see, e.g., Kato [11], Reed–Simon [13]): Proposition 6 (Domain stability under similarity).If T is closed and densely defined and S∈B ( H )is bounded and invertible, then Dom(TS)=SDom(T),(XI.6) and TSis closed. Proposition 7 (Self-adjointness and similarity).Let T be self-adjoint and S be bounded, invertible and positive (self-adjoint). Then TS:= ST S−1is self-adjoint. The drifted Dirac operators are of the form Ds=esφDe−sφ,(XI.7) with φ a real-valued bounded multiplier. Since esφ is bounded, invertible and positive for all s∈R , Propositions 6–7apply and we have: Corollary 8 (Self-adjointness of Ds ).For all s∈R , the operator Ds is self-adjoint with domain Dom ( Ds ) = esφDom(D). C. Spectral invariance for real drift Let ρ ( T )denote the resolvent set of a closed operator T and σ ( T ) = C\ρ ( T )its spectrum. For similarity transforms TS=ST S−1we have: Proposition 9 (Spectral invariance under bounded similarity).Let T be closed and S bounded, invertible. Then z∈ρ(T)⇐⇒ z∈ρ(TS).(XI.8) Consequently, σ(TS)=σ(T). Proof. If z∈ρ(T), then (T−z)−1exists as a bounded operator. Then TS−z=S(T−z)S−1,(XI.9) and (TS−z)−1=S(T−z)−1S−1∈B(H).(XI.10) The converse is obtained by applying the same reasoning to S−1TSS. 35 Applying this to Dsgives: Corollary 10 (Real spectrum for Ds ).For all s∈R , the spectrum of Ds coincides with that of D : σ(Ds)=σ(D)⊂R. Thus, the real drift does not alter the spectrum, but does change the local geometric content of the operator through lower-order terms in the Lichnerowicz formula. D. Relative boundedness of [D, φ] Let D be self-adjoint. An operator B with Dom ( D ) ⊆Dom ( B )is said to be D -relatively bounded if there exist a, b ≥0such that ∥Bψ∥≤a∥Dψ∥+b∥ψ∥,∀ψ∈Dom(D).(XI.11) The smallest such a is the relative bound. The Kato–Rellich theorem states that if D is self-adjoint and B is symmetric with relative bound a < 1, then D+Bis self-adjoint on Dom(D). In our case, φis a bounded real-valued function on M, so [D, φ]=γµ∂µφ(XI.12) is a bounded operator on H. Therefore: Proposition 11. The commutator [D, φ]is D-relatively bounded with relative bound a= 0. As a consequence, perturbations of the form D+s[D, φ]preserve self-adjointness for all s∈R. E. Holomorphic families of type (A) To study the complex drift Dz , we need the notion of holomorphic families of operators in the sense of Kato. Definition 12 (Holomorphic family of type (A)).Let U⊂C be open and {T ( z ) }z∈U be a family of closed operators on H. The family is said to be holomorphic of type (A) if: •there exists a dense subspace D ⊂ H such that Dom(T(z))=Dfor all z∈U; •for each ψ∈ D, the map z7→ T(z)ψis holomorphic as a H-valued function. Theorem 13 (Holomorphy of the complex drift).Let D be self-adjoint and φ∈L∞ ( M )real-valued and smooth. Then the family Dz=ezφDe−zφ, z ∈C,(XI.13) is holomorphic of type (A). 36 Proof. By Proposition 6, the domain is Dom(Dz)=ezφDom(D),(XI.14) which is independent of zas a set (since ezφ is invertible for all z∈C). For each ψ∈Dom(D), we have Dzψ=ezφDψ +ezφ[D, φ]ψ, (XI.15) and z7→ ezφ is an entire function with values in B ( H ). Since [ D, φ ]is bounded, the map z7→ Dzψ is holomorphic. This gives holomorphy of type (A). F. Flow equation for the real drift We now derive the differential equation d dsDs= [Ds, φ].(XI.16) Differentiating Ds=esφDe−sφ,(XI.17) we obtain d dsDs=φesφDe−sφ −esφDe−sφφ= [Ds, φ].(XI.18) This identity is used in the main text to rewrite D2 s and to understand the contribution of the drift to the Lichnerowicz formula. G. Vanishing of the second-order BCH term for scalar multipliers A formal Baker–Campbell–Hausdorff expansion for the drift would give Ds=D+s[D, φ] + s2 2[φ, [φ, D]]+···.(XI.19) For scalar multipliers φ, the double commutator vanishes: Lemma 14. Let φbe a real-valued smooth function acting by multiplication. Then [φ, [φ, D]] = 0.(XI.20) Proof. We have [ D, φ ] = γµ∂µφ , which is also a multiplication operator composed with the gamma matrices. Thus [φ, [D, φ]]=φ γµ∂µφ−γµ∂µφ φ = 0,(XI.21) since multiplication by φcommutes with multiplication by ∂µφ. 37 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](XI.22) and with the exact expansion of D2 sdeveloped in Appendix XII. H. 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. XII. 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,(XII.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,(XII.2) which is the starting point for the computation of the Seeley–DeWitt coefficients in Section V. A. Expansion of the drifted operator Using the boundedness of [D, φ], we may expand Dsas Ds=D+s[D, φ],(XII.3) 38 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.(XII.4) Define the operators Z:= {D, [D, φ]}, W := [D, φ]2.(XII.5) We analyse each contribution separately. B. 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.(XII.6) Thus Wis a positive, multiplicative operator. C. The linear term Z={D, [D, φ]} We compute Z=D(γµ∂µφ)+(γµ∂µφ)D. (XII.7) Using ∇ν(∂µφ)=∇µ(∂νφ)for scalars and the compatibility of ∇µwith the Clifford action, one finds D(γµ∂µφ) = γν∇ν(γµ∂µφ)=γνγµ∇ν∇µφ, (XII.8) (γµ∂µφ)D=gµν (∂µφ)∇ν+γµν (∂µφ)∇ν.(XII.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.(XII.10) D. Combination with the Lichnerowicz identity For a generalised Dirac operator D (including Einstein–Cartan torsion terms), the Lichnerowicz identity reads D2=∇∗∇+1 4R+T,(XII.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.(XII.12) 39 E. Drifted connection and potential Define the drifted connection ∇(s) µ:= ∇µ+s(∂µφ),(XII.13) so that gµν ∇(s) µ∇(s) ν=gµν ∇µ+s∂µφ∇ν+s∂νφ(XII.14) =gµν ∇µ∇ν+ 2s gµν (∂µφ)∇ν+s2gµν (∂µφ)(∂νφ).(XII.15) Comparing with the expression for D2 s, we identify the drifted Laplace-type form D2 s=−gµν ∇(s) µ∇(s) ν+Es,(XII.16) with potential Es=1 4R+T+s∆φ+s2∥∇φ∥2.(XII.17) This completes the derivation of the drifted Lichnerowicz identity used in Sections IV and V. XIII. 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,(XIII.1) 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 XII, the square of Dscan be written in Laplace-type form D2 s=−gµν ∇(s) µ∇(s) ν+Es,(XIII.2) with ∇(s) µ=∇µ+s ∂µφ, (XIII.3) and Es=E0+s∆φ+s2∥∇φ∥2.(XIII.4) Here E0 denotes the undeformed endomorphism appearing in the Lichnerowicz formula for D2 ,∆ φ = ∇µ∇µφ is the scalar Laplacian, and ∥∇φ∥2=gµν (∂µφ)(∂νφ). 40 A. General formula for a2 For a Laplace-type operator F=−gµν ∇µ∇ν+E, (XIII.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, (XIII.6) where Ris 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. (XIII.7) B. 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. (XIII.8) We identify the undeformed value a2(0) = 1 (4π)2ZM trR 61+E0dµ. (XIII.9) Since Mis compact and without boundary, we have ZM ∆φ dµ = 0,(XIII.10) and hence the linear term in sintegrates to zero: ZM tr∆φ1dµ = tr(1)ZM ∆φ dµ = 0.(XIII.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. (XIII.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. (XIII.13) In particular, for a standard four-dimensional Dirac operator (r= 4), ˜a2(s) = a2(0) + 4 (4π)2s2ZM∥∇φ∥2dµ. (XIII.14) 41 C. 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.(XIII.15) In four dimensions with a standard Dirac operator, tr(1)=4and c2=4 (4π)2.(XIII.16) This term is always non-negative and provides the quadratic drift contribution to the spectral action, entering directly in the coefficient αof the universal potential V(s) = αs2+βs4discussed in the main text. XIV. 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 βof the spectral potential V(s) = αs2+βs4is strictly positive. A. Laplace-type operator and general formula for a4 For an operator of Laplace type F=−gµν ∇µ∇ν+E, (XIV.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 + 5R2−2Rµν Rµν + 2RµνρσRµνρσdµ. (XIV.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,(XIV.3) where, as shown in Appendix XII, Es=E0+s∆φ+s2∥∇φ∥2,(XIV.4) with E0=1 4R+Tthe undeformed potential including torsion. 48 E. Evolution in terms of e-folds For numerical convenience (see Appendix XIX), it is often advantageous to use the number of e-folds N= ln aas time variable. Then d dN=1 H d dt.(XVI.14) Introducing x1=s, x2= ˙s, (XVI.15) we obtain from (XVI.10) the first-order system dx1 dN=x2 H,(XVI.16) dx2 dN=−3Hx2−2αx1−4βx3 1 H.(XVI.17) The Hubble parameter in terms of (x1, x2, ρr, ρm)reads H2=8πG 31 2x2 2+αx2 1+βx4 1+ρr+ρm.(XVI.18) The evolution of ρrand ρmis given by their continuity equations: dρr dN=−4ρr,(XVI.19) dρm dN=−3ρm.(XVI.20) These equations form the dynamical system used for numerical simulations of the SDM-IV background cosmology. F. Regimes of the condensate dynamics We briefly classify the regimes according to the relative magnitude of the kinetic and potential contributions of the condensate. a. Stiff regime. When ˙s2≫V(s), the condensate behaves approximately as a stiff fluid, ρs≈ps≈1 2˙s2, ws≈1,(XVI.21) and its energy density scales as ρs∝a−6.(XVI.22) b. Potential-dominated regime. When the potential dominates the kinetic term, ˙s2≪V ( s ), we have ρs≈V(s), ps≈ −V(s),(XVI.23) and hence ws≈ −1.(XVI.24) In particular, for s(t)→s∗at late times, the condensate behaves as an effective cosmological constant. 49 c. Intermediate regime. During the radiationand matter-dominated eras, the condensate energy density is subdominant, and the background evolution is controlled by ρr and ρm . The field s ( t )slowly evolves under the influence of the friction term 3 H˙s and the potential V ( s ), eventually relaxing towards s∗ . G. Stability of the spectral condensate The minima of V(s)satisfy V′(s) = 2αs + 4βs3= 0.(XVI.25) Besides the trivial solution s= 0, the non-trivial minima are located at s∗=±r−α 2β,(XVI.26) provided α < 0and β > 0. The second derivative of the potential at the minimum is V′′(s∗)=−2α > 0,(XVI.27) so the condensate is stable under small perturbations: s(t)=s∗+δs(t),(XVI.28) with δs(t)decaying due to the friction term in the equation of motion. The energy density at the minimum is ρΛ=V(s∗)=−α2 4β>0,(XVI.29) which plays the role of an effective cosmological constant. The SDM-IV model thus yields a late-time accelerated expansion driven by the spectral condensate, without introducing any additional scalar field beyond the drifted Dirac geometry. This completes the technical derivation of the cosmological equations used in Sec. VIII. The relaxation behaviour of the drift field is shown in Figure 5, where s ( t )approaches the condensate value s∗and asymptotically behaves as an effective cosmological constant. A schematic overview of the cosmological eras induced by the drift field is shown in Figure 6, indicating the natural progression stiff →radiation →matter →dark energy. XVII. COMPLEX DRIFT AND HOLOMORPHIC SPECTRAL ACTION In this appendix we present the compact technical formulation of the complex extension of the spectral drift and the corresponding holomorphic spectral action. This formalism complements the real drift analysis developed in the main text and provides the analytic layer required to discuss dissipation, spectral time asymmetry, and potential CP-violating effects from a geometric point of view. 50 A. 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.(XVII.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 Dzis independent of zand for every ψ∈Dom(D)the map z7−→ Dzψ(XVII.2) is holomorphic on C. This follows from the boundedness of the commutator [D, φ]∈ B(H),(XVII.3) and from the identity Dz=D+z[D, φ].(XVII.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. B. Holomorphic spectral action and Taylor expansion The spectral action associated with Dzis defined by S(z) := Trf(Dz/Λ),(XVII.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),(XVII.6) with S(n)(0) = dn dznz=0 Trf(Dz/Λ).(XVII.7) Using the differential identity d dz Dz= [Dz, φ],(XVII.8) and functional calculus for f(Dz/Λ), we obtain, at least formally, d dz f(Dz/Λ) = 1 Λf′(Dz/Λ) [Dz, φ],(XVII.9) 51 so that S′(0) = 1 ΛTrf′(D/Λ) [D, φ].(XVII.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),(XVII.11) where adn φ(D)denotes the n-fold iterated commutator. C. 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),(XVII.12) with SRe(s, θ) = 1 2S(z)+S(z), SIm(s, θ) = 1 2iS(z)−S(z).(XVII.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),(XVII.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. D. Dissipative contributions and time asymmetry Consider the evolution equation dψ dt =−iDsψ−θ[Ds, φ]ψ, (XVII.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. 52 E. 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,(XVII.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 Dswith 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. XVIII. 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 the drifted potential Vint(s) = αints2+βints4.(XVIII.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 DFand test the Spectral Drift Model numerically. A. Fermion masses of the Standard Model We list here the rest masses of quarks and charged leptons, in GeV, in a convenient PDG-compatible scheme. Neutrino masses are much smaller and can be included or neglected as appropriate. 1. Quarks 2. Charged leptons 3. 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.(XVIII.2) 53 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. B. Yukawa couplings The Yukawa couplings are defined by yf=√2mf v, v = 246 GeV,(XVIII.3) where v is the Higgs vacuum expectation value. We list here the approximate values for quarks and charged leptons. 1. Quark Yukawas 2. 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. C. 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. 1. CKM matrix 2. PMNS matrix 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 DFbut do not alter the sign structure of αint and βint. D. Internal traces and contributions to αint and βint In the internal sector, the drifted Dirac operator is DF,s =DF+sYF,(XVIII.4) 54 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),(XVIII.5) Tr(D4 F,s) = Tr(D4 F)+···+s4Tr(Y4 F).(XVIII.6) The quartic internal contribution to the spectral potential is βint =f0Tr(Y4 F),(XVIII.7) and is strictly positive since YF has non-zero eigenvalues (dominated by yt ). The quadratic internal contribution is αint =f2Λ2Tr(Y2 F),(XVIII.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,(XVIII.9) with frunning over all fermions and the top Yukawa giving the dominant contribution to Tr(Y4 F). E. Gauge couplings and Higgs parameters For completeness we also record the following parameters at the electroweak scale: •Strong coupling: αs(MZ)≈0.118.(XVIII.10) •Electroweak gauge couplings: g1(MZ)≈0.357, g2(MZ)≈0.652, g3(MZ)≈1.218.(XVIII.11) •Higgs mass: mH≈125.10 GeV.(XVIII.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. F. 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,(XVIII.13) 55 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 XIX. 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. XIX. 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. A. Equations of motion We consider a spatially flat FRW metric ds2=−dt2+a(t)2dx2,(XIX.1) with scale factor a(t)and Hubble parameter H(t) = ˙a(t) a(t).(XIX.2) The drift condensate is taken to be homogeneous: s=s(t).(XIX.3) The effective action for the drift condensate is Seff[s] = Zdt a(t)31 2˙s2−V(s),(XIX.4) with spectral potential V(s)=αs2+βs4, β > 0, α < 0,(XIX.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).(XIX.6) The equation of motion for s(t)is the damped Klein–Gordon equation ¨s+ 3H˙s+V′(s)=0,(XIX.7) 56 where V′(s)=2αs + 4βs3.(XIX.8) The total energy budget includes radiation and matter: H2=8πG 3ρs+ρr+ρm,(XIX.9) with ρr∝a−4, ρm∝a−3.(XIX.10) B. First-order system formulation For numerical integration it is convenient to rewrite the system in first-order form. Define x1=s, x2= ˙s. (XIX.11) Then ˙x1=x2,(XIX.12) ˙x2=−3Hx2−2αx1−4βx3 1.(XIX.13) The Hubble parameter can be written as H=s8πG 31 2x2 2+αx2 1+βx4 1+ρr+ρm.(XIX.14) A numerically more stable time variable is the number of e-folds N= ln a(t).(XIX.15) Using d dN =1 H d dt,(XIX.16) we obtain the e-fold system dx1 dN =x2 H,(XIX.17) dx2 dN =−3Hx2−2αx1−4βx3 1 H,(XIX.18) dρr dN =−4ρr,(XIX.19) dρm dN =−3ρm.(XIX.20) 57 C. Initial conditions A typical choice of initial conditions for the drift condensate in the early Universe is x1(N0)=sini, x2(N0) = ˙sini,(XIX.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. D. Runge–Kutta 4 integration scheme The system dX dN =F(N, X),(XIX.22) with X= (x1, x2, ρr, ρm),(XIX.23) can be integrated using a standard Runge–Kutta 4 (RK4) algorithm. The update step from N to N + ∆ N reads k1=F(N, X),(XIX.24) k2=FN+∆N 2,X+∆N 2k1,(XIX.25) k3=FN+∆N 2,X+∆N 2k2,(XIX.26) k4=FN+ ∆N, X+ ∆Nk3,(XIX.27) and the new state is X(N+ ∆N)=X(N) + ∆N 6k1+ 2k2+ 2k3+k4.(XIX.28) E. Example pseudocode Below is pseudocode in a Python-like syntax illustrating a possible implementation of the e-fold system. It assumes units in which 8πG = 1 for simplicity; this can be restored if desired. import numpy as np # Physical parameters (example values) G = 1.0/(8.0 * np.pi) # Newton’s constant (can set to 1/(8 pi)) 64 cosmic time stiff radiation matter dark energy w≃1w=1 3w= 0 w=−1 stiff era radiation era matter era DE era FIG. 6. Cosmological timeline in the Spectral Drift Model. The drift field naturally produces the stiff, radiation, matter and dark-energy eras as it relaxes towards the spectral condensate s∗. 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. Lepton Mass (GeV) e(electron) 0.000511 µ(muon) 0.10566 τ(tau) 1.77686 TABLE III. Charged lepton masses in GeV. 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. Lepton Yukawa yf e2.9×10−6 µ6.1×10−4 τ1.0×10−2 TABLE V. Charged lepton Yukawa couplings. 65 |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. |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.