DRSN V: Drifted Compactifications and Spectral Moduli Dynamics (De Rerum Spectrale Natura, Report V, Version 2.0)

DRSN V: DRIFTED COMPACTIFICATIONS AND SPECTRAL MODULI DYNAMICS De Rerum Spectrale Natura series REPORT V (Version 2.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 •Compactification arises from spectral factorisation of a drifted Dirac operator, without geometric truncation. •Internal moduli appear as zero–order spectral sectors generated by drifted Laplace–type operators. •Drift produces a universal quartic spectral potential governing moduli dynamics and stability. •G2,SU(3) and Spin(7) compactifications are treated within a unified drifted spectral framework. •Effective four–dimensional gravity and scalar dynamics emerge directly from the reduced spectral action. Spectral Compactifications from Eleven Dimensions: Drift, Moduli and Effective Four-Dimensional Physics J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We develop a fully spectral framework for compactifications from eleven to four dimensions based on drifted Dirac operators. Starting from an eleven–dimensional drifted spectral geometry, we show that dimensional reduction, moduli dynamics, and effective four–dimensional physics emerge from the spectral action without the introduction of auxiliary fields or phenomenological truncation ansätze. The drift deformation induces a universal quartic potential, providing a spectral mechanism for moduli stabilisation, supersymmetry breaking patterns, and cosmological dynamics. We analyse compactifications based on G2 , SU (3)–structure, and Spin (7) geometries, and derive the corresponding effective four–dimensional theories in a unified operatorial language. The resulting framework offers a conceptually economical and analytically controlled route from eleven–dimensional geometry to four–dimensional physics, preparing the ground for brane dynamics, holography, and phenomenological applications within the DRSN programme. Keywords: Spectral Action; Drift Geometry; M–theory Compactifications; Special Holonomy; Moduli Stabilisation; Noncommutative Geometry; Effective Four–Dimensional Physics; Cosmology. ∗jp[email protected] 3 CONTENTS I. Introduction 5 II. Drifted Compactification Framework 6 A. Geometric Setup and Spectral Decomposition 6 B. Heat Kernel Factorisation and Mode Expansion 7 C. Effective Four–Dimensional Spectral Action 7 D. Spectral Origin of Moduli and Stabilisation 8 E. Remarks on Consistency and Scope 8 III. Drifted G2Compactifications 9 A. G2Geometry and Spectral Data 9 B. Zero Modes, Supersymmetry, and Drift 9 C. Effective Potential and Moduli Stabilisation 9 D. Remarks and Comparison 10 IV. Drifted SU(3)-Structure Compactifications 10 A. SU(3)-Structure Geometry and Spectral Data 10 B. Half-Flat Structures, Torsion, and Drift 11 C. Effective Four–Dimensional Theory 11 D. Remarks and Relation to Mirror Symmetry 11 V. Drifted Spin(7) Compactifications 12 A. Spin(7) Geometry and Spectral Encoding 12 B. Supersymmetry Breaking and Zero Modes 13 C. Effective Potential and Stability 13 D. Remarks and Comparison 13 VI. Effective Four–Dimensional Cosmology from Drifted Spectral Geometry 14 A. Spectral Reduction to Homogeneous Cosmology 14 B. Friedmann Equations and Drift Dynamics 15 C. Inflationary and Dark–Energy Regimes 15 D. Spectral Interpretation of Vacuum Energy 16 E. Remarks and Outlook 16 4 VII. Conclusions and Outlook 16 A. Summary of Results 16 B. Comparison of Compactification Classes 17 C. Conceptual Implications 17 D. Relation to the DRSN Programme 18 E. Outlook 18 Appendices 19 A. Spectral Decomposition under Compactification 19 B. Reduction of the Heat Kernel and Effective Action 19 C. Semigroup and Resolvent Control 19 D. Effective Condensate s∗in the Unified Einstein–Cartan–Standard Model 20 Effective condensate equation 20 Independent constraint family 20 Constraint intersection 21 Outcome for the EC+SM model 21 Cross-report linkage 21 References 23 5 I. INTRODUCTION Compactification has traditionally been treated as a separate dynamical mechanism, implemented through dimensional truncations, flux backgrounds, or effective field–theoretic assumptions. While phenomenologically successful, such approaches often obscure the geometric origin of low–energy degrees of freedom and require substantial model–dependent input. Within spectral geometry, by contrast, physical information is encoded directly in the spectrum of a Dirac–type operator, suggesting that compactification itself should admit a purely spectral formulation. In the preceding reports of the DRSN series, we introduced drifted Dirac operators as controlled similarity deformations preserving spectral and analytic structure, and showed that gravitational, matter, and cosmological sectors arise from the spectral action. In particular, Report IV established an eleven–dimensional drifted spectral framework capturing the bosonic sector of M–theory in operatorial form. The present report builds directly on that result, addressing the question of how four–dimensional physics emerges spectrally from eleven dimensions. The central idea pursued here is that compactification is not an additional dynamical ingredient, but a manifestation of spectral factorisation. By analysing the heat kernel expansion of drifted Dirac operators on product geometries, we show that internal geometry, moduli fields, and effective potentials are encoded in the Seeley–DeWitt coefficients of a single operator. This perspective eliminates the need for ad hoc truncations and provides a unified treatment of different classes of special holonomy. We systematically study drifted compactifications based on G2 , SU (3)–structure, and Spin (7) geometries, highlighting both their distinctive features and their shared universal drift sector. A key outcome is the appearance of a universal quartic drift potential, which stabilises moduli and governs cosmological dynamics independently of the detailed topology of the internal space. As a result, inflationary and dark–energy–like regimes arise as calculable consequences of spectral geometry. The structure of this report is as follows. Section 2 introduces the general drifted compactification framework and the spectral reduction mechanism. Sections 3–5 analyse compactifications based on G2 , SU (3), and Spin (7) geometries, respectively. Section 6 derives the effective four–dimensional cosmology. Section 7 summarises the results and situates them within the broader DRSN programme. The operator-theoretic framework of drifted spectral geometry developed in DRSN I–IV establishes the analytic and structural foundations on which the present compactification analysis is built, including bounded similarity deformations, holomorphic drift flows, exact heat-kernel factorisation, supersymmetric unification, and the eleven-dimensional M-theoretic extension [1–4]. 6 About this report. This work constitutes Report V 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, preprints and versioned updates are archived at https://zenodo.org/communities/dsrn/. The present report develops the compactification and internal-geometry layer of drifted spectral geometry. Building on the ten-dimensional supersymmetric framework of Report III and the elevendimensional extension of Report IV, we analyse drifted dimensional reduction and the emergence of geometric, torsional and flux structures on internal manifolds, including SU(3)-holonomy (Calabi– Yau), G2 , SU(4) and Spin(7) settings, together with the associated drift-generated flux towers and moduli sectors. The emphasis is structural and analytic: we formulate the operator-theoretic conditions under which compactified drifted geometries preserve domain stability, self-adjointness, principal-symbol invariance, holomorphicity and heat-kernel factorisation. No new phenomenological claims are made. Subsequent reports in the DRSN series will build on this compactification layer to develop drifted brane spectral geometry and related effective lower-dimensional models. II. DRIFTED COMPACTIFICATION FRAMEWORK A. Geometric Setup and Spectral Decomposition We consider an eleven–dimensional drifted spectral geometry ( A11,H11, D(11) s )as established in DRSN Report IV, and assume a product (or warped product) decomposition of the underlying manifold, M11 ≃M4×X7.(II.1) At the spectral level, this induces a canonical decomposition of the Hilbert space, H11 ≃ H4⊗H7,(II.2) and of the drifted Dirac operator, D(11) s=D(4) s⊗1+ Γ4⊗D(7) s,(II.3) where Γ4denotes the four–dimensional chirality operator. The drift deformation acts by conjugation, D(11) s=esΦD(11)e−sΦ,(II.4) 7 Spectral decomposition of the drifted Dirac operator D(11) s→D(4) s⊕D(7) s FIG. 1. Spectral splitting of the drifted Dirac operator under compactification M11 →M4×X7. with Φa smooth scalar function on M11, allowing a decomposition Φ(x, y) = ϕ4(x)+ϕ7(y),(II.5) up to zero–order corrections. B. Heat Kernel Factorisation and Mode Expansion Since the drift deformation preserves the principal symbol and ellipticity, the square of the operator in (II.3) factorises up to bounded zero–order terms, (D(11) s)2= (D(4) s)2⊗1+1⊗(D(7) s)2+O0.(II.6) Consequently, the heat kernel admits an asymptotic factorisation, TrH11 e−t(D(11) s)2≃TrH4e−t(D(4) s)2TrH7e−t(D(7) s)2,(II.7) up to exponentially suppressed corrections, in the sense of standard heat kernel theory [7,8]. Let {λ(7) n}be the eigenvalues of (D(7) s)2. The spectral trace can then be written as TrH11 e−t(D(11) s)2=X n e−tλ(7) nTrH4e−t(D(4) s)2,(II.8) making explicit the separation between zero modes ( λ(7) n = 0) and massive Kaluza–Klein contributions. C. Effective Four–Dimensional Spectral Action The spectral action principle yields S11 = Tr f (D(11) s)2 Λ2!!,(II.9) with fa smooth cutoff function and Λthe spectral scale [5,6]. Using the heat kernel expansion and integrating over X7 , one obtains an effective four–dimensional action of the form S(4) eff =ZM4 √−g4M2 P 2R4−1 2KIJ (φ)∂µφI∂µφJ−Veff(s, φ)+··· .(II.10) 8 4D Term 11D Origin Drift Dependence Interpretation R4a(11) 2none Gravity ∂φ ∂φ a(11) 4indirect Moduli kinetics s2C2 1quadratic Mass term s4(C2 1)2quartic Stabilisation TABLE I. Spectral origin of the main four–dimensional terms after compactification. Crucially, the drift parameter sgenerates a universal contribution Veff(s)=α s2+β s4, β > 0,(II.11) independent of the detailed topology of X7 , in direct analogy with the lower–dimensional analysis of DRSN Report I. D. Spectral Origin of Moduli and Stabilisation The moduli fields φIarise spectrally as parameters controlling •the spectrum of D(7) s, •the internal curvature and flux contributions, •the zero–order endomorphism terms induced by the drift. Their dynamics is therefore fully encoded in the Seeley–DeWitt coefficients of ( D(11) s ) 2 , without introducing additional ad hoc scalar sectors [10]. E. Remarks on Consistency and Scope This framework is: •self–contained: all ingredients are spectral and operatorial, •background–independent: no specific choice of X7is required at this stage, •stable under drift: ellipticity and spectral properties are preserved, •non–redundant: no duplication of the 11D field content is introduced. The present section establishes the universal backbone of drifted compactifications. Specific internal geometries and phenomenological regimes will be analysed in the following sections. 9 G2spectral data under drift φ, ψ −→ D(7) drift −−−−→ D(7) s FIG. 2. G2geometry encoded spectrally and its drift deformation. III. DRIFTED G2COMPACTIFICATIONS A. G2Geometry and Spectral Data We specialise the internal space to a seven–dimensional manifold X7 endowed with a G2 –structure, characterised by a stable three–form φ and its Hodge dual ψ = ⋆7φ . In the torsion–free case, dφ = dψ = 0, the Levi–Civita connection has holonomy contained in G2 , yielding a single covariantly constant spinor. For drifted compactifications, the internal Dirac operator is deformed as D(7) s=esϕ7D(7)e−sϕ7,(III.1) where ϕ7∈C∞ ( X7 ). This preserves the principal symbol and ellipticity, while inducing zero–order endomorphism terms encoding torsion–like contributions at the spectral level. B. Zero Modes, Supersymmetry, and Drift In the absence of drift, the kernel of D(7) is one–dimensional, corresponding to the unique covariantly constant spinor. Under drift, the kernel is preserved as a vector space, but the associated zero mode acquires an effective four–dimensional profile through the factorisation Ψ(x, y) = ψ4(x)⊗e−sϕ7(y)η(y),(III.2) where ηis the G2–invariant spinor. Supersymmetry in four dimensions is therefore preserved at the level of zero modes, while massive Kaluza–Klein excitations are shifted by drift–dependent terms. This realises a controlled partial lifting of degeneracies without breaking the underlying G2structure. C. Effective Potential and Moduli Stabilisation The spectral action induces a four–dimensional effective potential of the schematic form Veff(s, φG2)=α s2+β s4+X I γI(φG2)s2+··· ,(III.3) 16 Regime Dominant term Spectral origin Cosmological role Early universe s4BCH hierarchy Inflation Intermediate s2C2 1term Exit / reheating Late universe V(s∗)spectral vacuum Dark energy TABLE V. Cosmological regimes induced by the drift potential. D. Spectral Interpretation of Vacuum Energy At late times, when ˙s→ 0and moduli are stabilised, the effective cosmological constant is given by Λeff =Veff(s∗, φ∗),(VI.7) where ( s∗, φ∗ )denote the spectral vacuum. Its smallness is protected by the quartic structure of the drift potential and by the absence of linear terms. E. Remarks and Outlook The cosmological dynamics obtained here: •follows directly from the spectral action without extra fields, •unifies inflation and dark energy in a single drift sector, •remains compatible with all compactification classes analysed above, •provides a calculable bridge between high–energy geometry and cosmology. This completes the derivation of the effective four–dimensional physics from drifted spectral compactifications and prepares the ground for brane–based, holographic, and phenomenological extensions. [5–7,9,10] VII. CONCLUSIONS AND OUTLOOK A. Summary of Results In this report we have developed a fully spectral framework for compactifications from eleven to four dimensions based on drifted Dirac operators. Starting from an eleven–dimensional drifted 17 spectral geometry, we have shown that the effective four–dimensional theory emerges from the spectral action without introducing additional degrees of freedom beyond those already encoded operatorially. The central results can be summarised as follows: •compactification is realised as a spectral reduction rather than a truncation ansatz, •moduli fields arise as parameters of the internal Dirac spectrum, •the drift deformation induces a universal quartic potential, •stabilisation of moduli and vacuum energy follows spectrally, •cosmological dynamics is derived, not postulated. These statements hold uniformly across the classes of internal geometries analysed. B. Comparison of Compactification Classes The three classes of special holonomy considered in this report display complementary features. G2 compactifications preserve N = 1 supersymmetry while allowing controlled moduli lifting. SU (3)–structure compactifications interpolate between Calabi–Yau and torsional regimes, providing a flexible yet spectrally controlled setting. Spin (7) compactifications yield rigid geometries with intrinsic supersymmetry breaking. Despite their differences, all three cases share the same universal drift sector, whose quartic potential governs stability and cosmological behaviour. This universality is a direct consequence of the operatorial nature of the drift deformation and is independent of the detailed topology of the internal space. C. Conceptual Implications From a conceptual standpoint, the results of this report support the view that compactification is not a separate dynamical mechanism but an intrinsic aspect of spectral geometry. The distinction between geometry, matter, and cosmology becomes blurred: all are encoded in the spectrum of a single operator and its controlled deformations. In particular, the drift parameter plays a dual role: it acts both as a geometric deformation at high energies and as an effective scalar degree of freedom at low energies. This provides a natural 18 bridge between ultraviolet spectral data and infrared physics, without invoking additional effective field theory assumptions. D. Relation to the DRSN Programme Within the broader DRSN programme, the present report occupies a pivotal position. It translates the eleven–dimensional spectral framework of Report IV into concrete four–dimensional physics, thereby closing the dimensional reduction sector of the theory. The results obtained here serve as direct input for brane dynamics, holography, and phenomenological applications. In particular: •Report VI will extend the spectral formalism to branes and worldvolume dynamics, •Report VII will address holographic and gauge–gravity correspondences, •subsequent reports will explore quantum corrections and particle phenomenology. The compactification framework developed here provides the common ground for all these extensions. E. Outlook Several directions merit further investigation. First, a detailed analysis of perturbations around the spectral vacuum may yield testable cosmological signatures. Second, the interplay between drifted compactifications and brane embeddings is expected to shed light on non–perturbative sectors of the theory. Finally, the operatorial nature of the construction suggests that quantum corrections should be incorporated at the spectral level, rather than through ad hoc loop expansions. This points towards a genuinely non–perturbative formulation of quantum gravity within the drifted spectral paradigm. In conclusion, drifted spectral compactifications provide a unified, analytically controlled, and conceptually economical route from eleven–dimensional geometry to four–dimensional physics, completing the objectives of the present report. [5–7,9,10] 19 APPENDICES Appendix A: Spectral Decomposition under Compactification We collect here the technical results underlying the spectral factorisation used throughout the report. Let D(11) s be a drifted Dirac operator on a product geometry M11 = M4×X7 . Under the assumptions stated in Section 2, the operator decomposes as D(11) s=D(4) s⊗1+ Γ4⊗D(7) s+O0,(A.1) where O0denotes bounded zero–order endomorphisms. Squaring the operator yields a Laplace–type operator, (D(11) s)2= (D(4) s)2⊗1+1⊗(D(7) s)2+O0,(A.2) ensuring that standard heat kernel techniques apply. The validity of the asymptotic expansion follows from general results on elliptic operators with bounded perturbations [7,8]. Appendix B: Reduction of the Heat Kernel and Effective Action Let K11 ( t )denote the heat kernel of ( D(11) s ) 2 . Up to exponentially suppressed terms, one has the factorisation TrK11(t)≃TrK4(t) TrK7(t),(B.1) where K4 and K7 are the heat kernels of ( D(4) s ) 2 and ( D(7) s ) 2 , respectively. This factorisation justifies the separation of four–dimensional and internal contributions in the spectral action. Integrating over the internal manifold X7yields the effective four–dimensional spectral action, S(4) eff =X n f4−nΛ4−nZM4 a(4) n(x)√−g4d4x, (B.2) where the coefficients a(4) n depend parametrically on the internal spectral data. This is the precise origin of moduli fields and drift–induced potentials in the effective theory. Appendix C: Semigroup and Resolvent Control The drift deformation is implemented by similarity transformations of Dirac–type operators. As reviewed in the methodological appendix of Report IV, such deformations preserve self–adjointness 20 and spectral properties. The resolvent admits the Laplace representation (λI −iDs)−1=Z∞ 0 e−λteitDsdt, ℜλ > 0,(C.1) which is valid under the standard hypotheses of semigroup theory [9]. This representation underlies the control of higher–order spectral corrections and ensures that all drift–induced terms remain bounded and well defined. No additional analytic assumptions are required beyond those already imposed in the eleven–dimensional framework. Appendix D: Effective Condensate s∗in the Unified Einstein–Cartan–Standard Model In this appendix we document the logical status and admissibility of the effective spectral condensate s∗ within the unified Einstein–Cartan plus Standard Model (EC+SM) drifted geometry. No new structural results are introduced. The purpose of this appendix is purely methodological: to make explicit how the existence and viability of s∗ follow from the intersection of independent constraints in this specific model. Non-claim. The parameter s∗ is not a spectrally rigid invariant, not a universal constant, and not a fundamental coupling. It is an effective quantity arising after truncation of the spectral action and depends on the adopted geometric and internal sectors. Any numerical value is scheme-dependent and plays no role in the present analysis. Effective condensate equation In the quartic-truncated effective regime of the drifted spectral action, the s -dependent potential takes the universal form V(s)=α s2+β s4, β > 0,(D.1) where the coefficients α and β collect geometric, torsional and internal contributions. A non-trivial stationary point is defined by s2 ∗=−α 2β,(D.2) provided α < 0and β > 0. Independent constraint family Admissibility of s∗ in the EC+SM model is tested against the following independent constraints: 21 1. Existence of a non-trivial minimum: The unified model satisfies β > 0due to both geometric and internal quartic contributions, while axial torsion in the Einstein–Cartan sector provides a universal negative contribution to α, yielding α < 0. 2. Spectral stability (UV/IR): The quartic term ensures that the effective potential is bounded from below and excludes runaway behaviour at large |s|. 3. Dynamical viability: In homogeneous FRW backgrounds, the condensate dynamics exhibits a stiff regime, followed by relaxation towards s∗ under cosmological friction, with s∗ acting as a late-time attractor. 4. Scale consistency: The vacuum energy density V ( s∗ )is positive and finite, arising without fine-tuning and without introducing additional scalar degrees of freedom. 5. Non-fine-tuned regime: The condensate value s∗ is of natural order in the effective description and does not require extreme hierarchies or cancellations between independent sectors. Constraint intersection Each constraint above defines an admissible region in the effective parameter space. For the unified EC+SM model, the intersection of these regions is non-empty. The effective condensate s∗ = 0 therefore exists and is fully admissible within the methodological boundaries of the DRSN programme. Outcome for the EC+SM model Case C (viable condensate). The intersection of independent constraints is non-empty. The unified Einstein–Cartan–Standard Model drifted geometry admits a stable, dynamically viable effective condensate s∗, which underlies the emergence of a positive vacuum energy in this model. Cross-report linkage This appendix instantiates the condensate-selection principle established in the foundational DRSN reports and applies it to the unified EC+SM setting. Subsequent reports may use s∗ as an effective parameter, but no application-level result upgrades its status beyond that fixed here. 22 This appendix instantiates the condensate-selection criterion within the unified Einstein–Cartan– Standard Model setting and does not modify the methodological status of the effective condensate s∗as fixed in the foundational DRSN reports. 23 REFERENCES [1] J. Pinho-da-Cruz, DRSN I: A Unified Geometric Framework for Matter, Torsion and Dark Energy, August 2025. Version 3.0, doi: 10.5281/zenodo.17981231. [2] J. Pinho-da-Cruz, DRSN II: Drift Geometry and the Unified Spectral Operator, September 2025. Version 2.0, doi: 10.5281/zenodo.17982131. [3] J. Pinho-da-Cruz, DRSN III: Supersymmetric Drift Geometry and the Spectral Master Operator, September 2025. Version 2.0, doi: 10.5281/zenodo.17986205. [4] J. Pinho-da-Cruz, DRSN IV: Drifted M-Theory Spectral Geometry, September 2025. Version 2.0, doi: 10.5281/zenodo.17987598. [5] A. 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