DRSN XV: Real Spectral Triples, and the Standard Model Structural Derivation and Limits (De Rerum Spectrale Natura, Report XV, Version 1.0)

DSRN XV: REAL SPECTRAL TRIPLES, AND THE STANDARD MODEL STRUCTURAL DERIVATION AND LIMITS De Rerum Spectrale Natura series REPORT XV (Version 1.0) J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 • Introduces real spectral triples as the minimal algebraic extension required to encode fermionic charge conjugation and gauge structure. •Analyses the role of KO-dimension and real structure in constraining admissible noncommutative geometries. • Clarifies which features of the Standard Model are spectrally rigid and which arise only through inner fluctuations and truncation. • Separates structural reconstruction from phenomenological input, in accordance with the DSRN Separation Principle. •Introduces no claim of uniqueness or derivation of Standard Model parameters from spectral data alone. DSRN XV: Real Spectral Triples, Charge Conjugation, and the Structural Limits of the Standard Model J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal (Dated: December 14, 2025) We introduce real spectral triples as the minimal extension of the spectral-triple framework required to encode fermionic charge conjugation and gauge structure. We review the axioms relevant for the noncommutative-geometric reconstruction of the Standard Model and clearly separate which features are spectrally rigid from those arising only after inner fluctuations and truncation. In accordance with the DSRN Separation Principle, we identify precisely where physical interpretation enters and where it remains effective. Keywords: real spectral triples; charge conjugation; KO-dimension; Standard Model; inner fluctuations; spectral action; effective truncation. CONTENTS I. Scope and Relation to Previous Reports 3 II. Real Spectral Triples 3 III. KO-Dimension and Algebraic Constraints 3 IV. Almost-Commutative Geometries 4 V. Structural Reconstruction of the Standard Model 4 VI. Inner Fluctuations and Gauge Fields 4 VII. What Is Rigid and What Is Effective 5 VIII. Summary and Role within the DSRN 5 A. Bibliographic Notes 5 References 5 ∗jp[email protected] 3 I. SCOPE AND RELATION TO PREVIOUS REPORTS This report extends DSRN XIV by introducing real spectral triples. It prepares the ground for DSRN XVI, where phenomenological consequences are discussed in a controlled manner. The present report relies on: •spectral rigidity and drift invariance (DSRN XI); •heat-kernel invariants (DSRN XII); •the spectral action as an effective truncation (DSRN XIII); •minimal spectral triples (DSRN XIV). Non-claims. We do not claim uniqueness of the Standard Model from spectral data alone. We identify structural constraints and isolate additional assumptions explicitly. II. REAL SPECTRAL TRIPLES Definition 1 (Real spectral triple).A real spectral triple is a quadruple (A,H,D,J), where (A,H,D)is a spectral triple and J:H → H is an antiunitary operator (the real structure) satisfying: J2=ε, J D =ε′DJ , JΓ = ε′′ΓJ(if a grading Γexists), with ε, ε′, ε′′ ∈ {±1}determined by the KO-dimension modulo 8. Remark 2.The operator Jimplements charge conjugation at the Hilbert-space level. III. KO-DIMENSION AND ALGEBRAIC CONSTRAINTS The signs (ε, ε′, ε′′)define the KO-dimension of the real spectral triple modulo 8. Remark 3.The KO-dimension is a discrete invariant determined by commutation relations between D,J, and Γ. It is spectrally rigid in the sense that it does not vary under bounded drift. 4 Proposition 4. For a fixed representation of A on H , the KO-dimension fixes the allowed algebraic relations compatible with fermionic reality. IV. ALMOST-COMMUTATIVE GEOMETRIES The geometries relevant for particle physics are almost-commutative: A=C∞(M)⊗ AF,H=L2(M, S)⊗ HF,D=DM⊗id +ΓM⊗ DF, where (AF,HF,DF,JF)is a finite real spectral triple. Remark 5.The spacetime Dirac operator DM contributes infinite-dimensional spectral data, while DFencodes internal (finite) structure. V. STRUCTURAL RECONSTRUCTION OF THE STANDARD MODEL For a specific choice of finite algebra AF=C⊕H⊕M3(C), together with a suitable finite Hilbert space and real structure, one reproduces the gauge group U(1) ×SU(2) ×SU(3) and the fermionic representation content of the Standard Model. Remark 6.This reconstruction uses additional assumptions: •a specific choice of AF, •the order-one condition, •a particular KO-dimension. These are inputs, not consequences of spectral rigidity alone. VI. INNER FLUCTUATIONS AND GAUGE FIELDS Gauge fields arise from inner fluctuations of the Dirac operator: D 7−→ DA=D+A+JAJ−1, with A=Pai[D, bi]. 5 Remark 7.Inner fluctuations are not similarity transformations. They generically change the spectrum and thus lie outside the rigid spectral layer of DSRN. Remark 8.Gauge dynamics extracted from the spectral action depend on truncation and scale choice, and are therefore effective. VII. WHAT IS RIGID AND WHAT IS EFFECTIVE • Rigid: spectrum of D , heat-kernel coefficients of D2 , KO-dimension, algebraic relations fixed by J. • Effective: gauge couplings, Higgs potential, Yukawa parameters, all derived from truncated spectral action. Remark 9.The Standard Model parameters do not follow from spectral rigidity; they arise from effective truncation and scale dependence. VIII. SUMMARY AND ROLE WITHIN THE DSRN Real spectral triples provide the minimal algebraic structure to encode fermionic reality and gauge symmetry. Within the DSRN: •spectral data remain rigid under drift; •physical fields arise from inner fluctuations; •numerical parameters belong to the effective layer. This closes the structural part of the Standard Model derivation and prepares the transition to controlled phenomenology in DSRN XVI. Appendix A: Bibliographic Notes Standard references for real spectral triples and the noncommutative-geometric formulation of the Standard Model include [1–3]. [1] A. Connes, Noncommutative Geometry, Academic Press, 1994. 6 [2] A. H. Chamseddine and A. Connes, “The Spectral Action Principle,” Commun. Math. Phys. 186 (1997) 731–750. [3] W. D. van Suijlekom, Noncommutative Geometry and Particle Physics, Springer, 2015.