Spectral Vacuum Mechanism – Part XVIII Continuum Trajectory and Low-Energy Self-Consistency under SU(3) Truncation

Abstract

This work establishes the foundational diagnostic framework for the Spectral Vacuum Mechanism (SVM) in SU(3) gauge theory. It provides the rigorous operator-theoretic justification for representation truncation (j_max), which serves as the essential basis for all subsequent developments in the SVM series.

This document presents a rigorous operator-theoretic framework for assessing truncation quality in Hamiltonian formulations of SU(3) lattice gauge theory. Working at fixed lattice spacing a, finite volume V, and truncation parameter j_max, we derive precise bounds on how high-energy (high-Casimir) states influence low-energy observables.

The key insight is that absolute decoupling of high-energy states is not necessary for controlled truncation. What matters is decoupling relative to a diagonal energy barrier Δ_H separating low and high sectors. We prove Theorem T1, which bounds the effective Hamiltonian error by ‖B‖²/Δ_H, where B is the coupling operator between sectors. This motivates our central diagnostic: the dimensionless mixing parameter 

ε_mix = ‖B‖²/Δ_H.

When ε_mix ≪ 1, virtual high-energy corrections are perturbatively suppressed and truncation is controlled. When ε_mix ≳ 1, corrections are non-perturbative and truncation artifacts dominate. We provide complete numerical protocols for computing Δ_H (via diagonal energy gaps) and ‖B‖ (via power iteration), with detailed warnings about common pitfalls such as Frobenius norm overestimation.

A worked example on a 3×3 lattice at g²a=1.5 demonstrates all diagnostics, yielding ε_mix=1.42 (uncontrolled). We show that using Frobenius norm would overestimate ε_mix by 133×, highlighting the importance of proper implementation. Sensitivity analysis reveals that the choice of effective Casimir definition (max-link vs sum-over-links) affects results by 63%, while shell treatment strategy causes ~20% variation.

The document includes: (1) Complete mathematical framework with rigorous proofs. (2) Extensive discussion of when Assumption A3 (Casimir monotonicity) fails and what to do. (3) Three figures showing scatter plots, convergence behavior, and scaling roadmaps. (4) Four tables documenting numerical results and sensitivity studies. (5) Production-ready pseudocode. (6) Discussion of outliers, long-term strategies, and approaches for ε_mix > 0.1 regimes.

We do NOT claim to prove continuum limit existence or provide optimal truncation trajectories. Is a diagnostic framework for finite-dimensional truncated theories, allowing practitioners to quantitatively assess whether their calculations are trustworthy.

Keywords: SU(3) Lattice Gauge Theory, Hamiltonian Formalism, Spectral Vacuum Mechanism, Hilbert Space Truncation, Mixing Parameter ($\epsilon_{mix}$), Operator-Theoretic Diagnostic, Theorem T1.

Other works by the author on this topic:

Spectral Vacuum Mechanism - Part XIV Spectral Confinement as a Necessary       Condition for Quantum Field Theory Confinement Gate-Induced Spectral Localization and Dimensional Constraints, Zenodo. DOI: 10.5281/zenodo.18140235 (2026) 

Spectral Vacuum Mechanism - Part XV Unification of the mass formula in SVM particles of the Standard Model, Zenodo. DOI: 10.5281/zenodo.18207487 (2026) 

Spectral Vacuum Mechanism – Part XVI Spectral Confinement under Truncated SU(2) Gauge Embedding: Preservation of the Spectral Confinement Class, Zenodo. DOI: 10.5281/zenodo.18225421 (2026) 

Spectral Vacuum Mechanism – Part XVII Spectral Confinement under Truncated SU(3) Gauge Embedding: Toward a Constructive QCD‑like Framework, Zenodo. DOI: 10.5281/zenodo.18280887 (2026)