DRSN XVI: Quantum Corrections, Renormalisation-Group Running, and Controlled Phenomenology Within Spectral Geometry (De Rerum Spectrale Natura, Report XVI, Version 1.0)

DSRN XVI: QUANTUM CORRECTIONS, RENORMALISATION-GROUP RUNNING, AND CONTROLLED PHENOMENOLOGY WITHIN SPECTRAL GEOMETRY De Rerum Spectrale Natura series REPORT XVI (Version 1.0) J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 • Analyses quantum corrections and renormalisation-group running strictly at the effective level of truncated spectral actions. • Proves that perturbative corrections do not modify spectral invariants, KO-dimension, or algebraic structure of the spectral triple. • Clarifies the status of Higgs-mass predictions and scalar-sector extensions as phenomenological inputs, not spectral consequences. • Demonstrates that phenomenological viability does not imply spectral uniqueness, in accordance with the DSRN Separation Principle. • Prepares the transition from noncommutative-geometric structure to abstract emergence mechanisms in subsequent reports. DSRN XVI: Corrections, Running, and the Higgs–Singlet Extension Controlled Phenomenology within Spectral Geometry J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal (Dated: December 14, 2025) We analyse the status of quantum corrections, renormalisation-group running, and the Higgs–singlet extension within the noncommutative-geometric formulation of the Standard Model. Emphasis is placed on separating spectrally rigid input from effective, scale-dependent phenomenology, in strict accordance with the DSRN Separation Principle. We show that corrections and running do not modify spectral invariants and that all phenomenological viability arises at the effective level after truncation of the spectral action. Keywords: spectral action; renormalisation group; Higgs mass; singlet scalar; noncommutative geometry; effective field theory; scale dependence. CONTENTS I. Scope and Position within the DSRN 3 II. Quantum Corrections and Spectral Rigidity 3 III. Renormalisation-Group Running 4 IV. The Higgs Mass Problem 4 V. Singlet Scalar Extension 4 VI. What Corrections Can and Cannot Do 5 VII. Summary and Transition to Applications 5 A. Bibliographic Notes 6 References 6 ∗jp[email protected] 3 I. SCOPE AND POSITION WITHIN THE DSRN This report completes the noncommutative-geometric core of the DSRN. It addresses phenomenological issues—quantum corrections, running couplings, and scalar-sector extensions—without elevating them to the level of spectral invariants. It relies on: •spectral rigidity and separation principle (DSRN XI); •heat-kernel invariants (DSRN XII); •spectral action and truncation (DSRN XIII); •real spectral triples and Standard Model structure (DSRN XV). Non-claims. We do not claim that phenomenological success fixes the underlying spectral triple, nor that renormalisation-group running is encoded spectrally. II. QUANTUM CORRECTIONS AND SPECTRAL RIGIDITY Quantum corrections in the noncommutative-geometric approach arise only after interpreting the truncated spectral action as a classical Lagrangian to be quantised. Remark 1.Loop corrections do not act on the spectral triple ( A,H,D,J ). They modify only effective couplings derived from truncation. Proposition 1. Perturbative quantum corrections do not alter: •the spectrum of D, •heat-kernel coefficients of D2, •KO-dimension or algebraic constraints of the triple. Remark 2.Any back-reaction of quantum corrections onto spectral data would require modifying Ditself, which lies outside the controlled framework of the spectral action. 4 III. RENORMALISATION-GROUP RUNNING After truncation of the spectral action at scale Λ, one obtains an effective classical action Seff (Λ) = N X n=0 Fd−nΛd−nan(D2). Renormalisation-group (RG) running describes how the corresponding effective couplings vary with energy scale µ < Λ. Remark 3.RG running is not a spectral phenomenon. It depends on the choice of truncation, regularisation, and renormalisation scheme. Proposition 2. RG flow preserves spectral rigidity: two drift-related Dirac operators Ds produce identical RG equations for effective couplings at the same truncation scale. Proof. By DSRN XI–XIII, truncation coefficients an ( D2 s ) = an ( D2 ). Thus initial conditions for RG flow coincide. IV. THE HIGGS MASS PROBLEM Early implementations of the spectral action predicted a Higgs mass in tension with experimental observations. Remark 4.This mismatch does not signal failure of spectral geometry, but of overly restrictive truncation assumptions in the scalar sector. The Higgs mass prediction depends on: •boundary conditions at the cutoff scale Λ, •RG running to low energies, •assumptions on scalar-sector completeness. All three belong to the effective layer. V. SINGLET SCALAR EXTENSION Including an additional real scalar singlet σ , naturally associated with the Majorana mass sector, modifies the scalar potential and RG flow. 5 Remark 5.The singlet field does not arise as a new spectral invariant. It enters through additional terms in the truncated spectral action. Proposition 3. The Higgs–singlet extension restores phenomenological viability without modifying the spectral rigidity of the underlying triple. Remark 6.The precise values of scalar couplings and masses are scale-dependent and therefore effective. VI. WHAT CORRECTIONS CAN AND CANNOT DO • Corrections can: improve phenomenological agreement, modify RG trajectories, stabilise scalar potentials. • Corrections cannot: change spectral invariants, remove the need for truncation, enforce uniqueness of the Standard Model. Remark 7.Any claim that phenomenological success feeds back into spectral rigidity violates the DSRN Separation Principle. VII. SUMMARY AND TRANSITION TO APPLICATIONS The status of corrections and running within the DSRN is now fixed: •Spectral data remain rigid and drift-invariant. •Quantum corrections and RG running operate only on effective actions. •Phenomenological viability does not imply spectral uniqueness. This closes the noncommutative-geometric core of the DSRN and prepares the transition to DSRN XVII–XVIII, where general mechanisms of emergence are abstracted, and to DSRN XXI–XXIV, where concrete applications are analysed. 6 Appendix A: Bibliographic Notes Key references on the Higgs–singlet extension and RG analysis within spectral geometry include [1,2]. [1] A. H. Chamseddine and A. Connes, “Resilience of the Spectral Standard Model,” JHEP 1209 (2012) 104. [2] W. D. van Suijlekom, Noncommutative Geometry and Particle Physics, Springer, 2015.