DRSN XIV: Spectral Triples, Noncommutative Geometry, and the Foundations of Spectral Rigidity (De Rerum Spectrale Natura, Report XIV, Version 1.0)

DSRN XIV: SPECTRAL TRIPLES, NONCOMMUTATIVE GEOMETRY, AND THE FOUNDATIONS OF SPECTRAL RIGIDITY De Rerum Spectrale Natura series REPORT XIV (Version 1.0) J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 • Introduces spectral triples in their minimal form sufficient for spectral invariants, heat-kernel methods, and the spectral action. •Clarifies which axioms are essential for spectral rigidity and which are optional or model-dependent. • Shows that drifted similarity transformations preserve the spectral content of Dirac-type operators and all derived invariants. • Separates rigid spectral data from effective constructions arising from truncation, projection, or inner fluctuations. •Contains no physical interpretation or phenomenological assumptions. DSRN XIV: Spectral Triples and Noncommutative Geometry Minimal Axioms and Spectral Rigidity J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal (Dated: December 14, 2025) We introduce the minimal noncommutative-geometric framework required by the Drifted Spectral Renormalisation Network (DSRN). We define spectral triples in their weakest form sufficient for spectral invariants, heat-kernel methods, and the spectral action. We emphasise which axioms are essential for spectral rigidity and which additional conditions are optional or model-dependent. In particular, we show that drifted similarity transformations preserve the spectral content of Dirac-type operators and hence all spectral invariants derived from them. No physical interpretation is assumed at this stage. Keywords: spectral triples; noncommutative geometry; Dirac operators; spectral invariants; isospectrality; inner fluctuations; effective truncation. CONTENTS I. Scope and Relation to the DSRN 3 II. Motivation: Why Spectral Triples 3 III. Minimal Definition of a Spectral Triple 3 IV. Spectral Invariants of a Triple 4 V. Inner Fluctuations and Drift 4 VI. Dimension and Regularity 5 VII. Effective Geometry from Truncation 5 VIII. Summary and Role within the DSRN 5 A. Bibliographic Notes 6 ∗jp[email protected] 3 References 6 I. SCOPE AND RELATION TO THE DSRN This report introduces the noncommutative-geometric objects required in later DSRN reports, in particular DSRN XV–XVI. It relies on: •the drifted similarity and isospectrality framework (DSRN XI); •heat-kernel invariants and asymptotics (DSRN XII); •the spectral action as an effective truncation (DSRN XIII). Non-claims. No claim is made here regarding the physical interpretation of noncommutative geometry. All constructions are purely spectral and algebraic. Any physical identification is postponed to later reports and must respect the DSRN Separation Principle. II. MOTIVATION: WHY SPECTRAL TRIPLES Noncommutative geometry reformulates geometry in spectral terms: distance, dimension, and action functionals are encoded in the spectrum of a suitable operator. For DSRN purposes, the central object is the spectrum of a Dirac-type operator, not the underlying algebraic interpretation. III. MINIMAL DEFINITION OF A SPECTRAL TRIPLE Definition 1 (Spectral triple).A (possibly noncommutative) spectral triple is a triple (A,H,D), where: •Ais a unital ∗-algebra represented faithfully on H; •His a separable Hilbert space; •Dis a densely defined self-adjoint operator on Hwith compact resolvent. Remark 2.Compact resolvent ensures a discrete spectrum with finite multiplicities, which suffices for heat-kernel and zeta-function constructions. 4 Remark 3 (Minimality).We do not assume at this stage: real structure, grading, first-order condition, orientability, or Poincaré duality. These additional axioms are model-dependent and are introduced only when required. IV. SPECTRAL INVARIANTS OF A TRIPLE Given a spectral triple ( A,H,D ), all spectral invariants are defined in terms of Spec ( D )or Spec(D2). •Heat trace: Tr(e−tD2); •Zeta function: ζD(s) = Tr(|D|−s)(where defined); •Seeley–DeWitt coefficients of D2; •Spectral action Tr(f(D2/Λ2)). By DSRN XI–XII, all these quantities depend only on the spectrum and are therefore spectrally rigid. Lemma 4 (Spectral rigidity of triple invariants).Let ( A,H,D )be a spectral triple and let Ds = U−1 sDUs with Us bounded and invertible. Then all spectral invariants of ( A,H,Ds )coincide with those of (A,H,D)whenever they are well-defined. Proof. By bounded similarity, Spec ( Ds ) = Spec ( D ). All listed invariants depend only on spectral data. V. INNER FLUCTUATIONS AND DRIFT In noncommutative geometry, one often considers inner fluctuations of the Dirac operator: D 7−→ DA:= D+A+JAJ−1, where Ais a self-adjoint one-form built from A. Remark 5.Inner fluctuations are not similarity transformations in general. They modify the spectrum and hence correspond to genuine (effective) changes of spectral data. Remark 6 (Compatibility with DSRN).Drifted similarity transformations preserve the spectrum, while inner fluctuations generically do not. This distinction is essential: inner fluctuations belong to the effective/dynamical layer, not to the rigid spectral layer. 5 VI. DIMENSION AND REGULARITY The dimension of a spectral triple is defined spectrally via the growth of eigenvalues of |D|. Definition 7 (Spectral dimension).A spectral triple has dimension d if the eigenvalues λn of |D| satisfy λn∼Cn1/d as n→ ∞. Remark 8.This notion of dimension is purely spectral and hence invariant under drift. Additional regularity conditions (smoothness of A under the derivation [ D,· ]) are required for full pseudodifferential calculus, but are not needed for the rigidity statements of DSRN XIV. VII. EFFECTIVE GEOMETRY FROM TRUNCATION While the spectral triple encodes geometry spectrally, any concrete geometric interpretation requires additional steps: •truncation of the spectrum; •projection onto low-energy modes; •finite-order heat-kernel or spectral-action expansions. Each of these operations discards spectral information and produces effective, scale-dependent quantities. Remark 9.According to the DSRN Separation Principle, any emergent metric, curvature, or field content obtained in this way is effective and must not be conflated with spectral invariants. VIII. SUMMARY AND ROLE WITHIN THE DSRN The role of noncommutative geometry within the DSRN is now fixed: •Spectral triples provide a canonical setting for Dirac-type operators. •All spectral invariants of a triple are rigid under bounded drift. •Inner fluctuations and truncations belong to the effective layer. •No physical interpretation is assumed at this stage. This prepares the ground for DSRN XV–XVI, where additional axioms are introduced only insofar as required for concrete models. 6 Appendix A: Bibliographic Notes Foundational references for spectral triples and noncommutative geometry include [ 1 – 3 ]. The present report extracts only the minimal axioms needed for DSRN purposes. [1] A. Connes, Noncommutative Geometry, Academic Press, 1994. [2] J. M. Gracia-Bondía, J. C. Várilly, H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser, 2001. [3] M. Eckstein and B. Iochum, Spectral Action in Noncommutative Geometry, Springer, 2018.