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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.18444149 |
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- <h2>Abstract</h2> <p dir="ltr"><a href="https://docs.google.com/document/d/1pZd2-JxYQHmU7ACARrF2rH74apb0TBLZ/edit#bookmark=id.ofcgfn8pr02f">Spectral Vacuum Mechanism</a> Part XXI presents a comprehensive protocol for operationalizing the <strong>thermodynamic limit V → ∞ at fixed lattice spacing</strong> a within the Gauss-law sector (C=0) for truncated SU(3) Hamiltonian lattice gauge theory calculations. This work establishes v<strong>olume-stability certification</strong> procedures that determine whether declared low-energy observables exhibit convergent behavior as the spatial volume increases, while maintaining a fixed (coarse) lattice discretization.</p> <p dir="ltr"><strong>Scope and non-claims:</strong> This protocol <strong>does not address the continuum limit (a → 0), does not claim confinement, and does not claim a mass gap</strong>. The sole objective is to certify whether a specific set of low-energy observables computed at fixed coarse lattice spacing demonstrates volume stability—that is, whether numerical results converge as V increases along a declared volume ladder.</p> <p dir="ltr"><strong>Methodological framework:</strong> Gauss-law enforcement is treated as a <strong>modular audit layer </strong>utilizing a μ-penalty approach (H_μ = H + μC), optionally complemented by explicit projection or subspace methods. This builds upon the audit primitives and validation techniques introduced in Parts XIX (Gauss-law enforcement as an audit layer) and Part XX (controlled truncation and low-energy reliability). The protocol incorporates <strong>conservative diagnostic thresholds</strong> intentionally biased against false-positive bulk claims, ensuring that volume-stability certifications reflect genuine physical behavior rather than numerical artifacts.</p> <p dir="ltr"><strong>Observable tiers and certification levels: </strong>The protocol defines two observable tiers. The Base tier ( _base) includes energy density e0(V), spectral gap Δ(V), participation ratio (PR) and inverse participation ratio (IPR) tags, and Gauss-law leakage ⟨C⟩_k/V. <strong>The Extended tier ( _ext) </strong>additionally requires at least one infrared-sensitive correlator C(R) with effective mass m_eff(R) computed in a safe window, plus wrap-around diagnostics (ρ_wrap, κ_mid) for periodic boundary conditions. Correspondingly, <strong>conditional certification</strong> is granted when Base tier observables pass all diagnostics, while <strong>full certification</strong> requires Extended tier satisfaction and is mandatory for physics-facing interpretation or extrapolation beyond the ladder ceiling.</p> <p dir="ltr"><strong>Diagnostic framework: </strong>Volume stability is assessed through <strong>tail-drift tests</strong> with conservative default tolerances: energy density drifts must satisfy δ_tail(e0)/|e0| < 3%, spectral gap drifts δ_tail(Δ)/Δ < 10%, and correlator drifts δ_tail(C(R)) or relative logarithmic drifts in m_eff < 2%. <strong>Localization detection</strong> via PR tagging flags states where PR_n/PR_{n-1} < 1.2 under volume ratio V_n/V_{n-1} ≥ 1.5 as LOCALIZED, enabling separation of bulk states from boundary-dominated or non-physical modes. Wrap-around contamination in periodic boundary conditions is quantified via κ_mid = m_eff(R_mid)/m_eff(R_safe); values κ_mid < 0.7 indicate wrap-dominated midpoint behavior unsuitable for correlation length extraction.</p> <p dir="ltr">Correlation length adequacy: The protocol implements a stricter inconclusiveness rule for correlation length ξ(a): if at the ladder ceiling an estimated ξ(a) ≳ L/2.5–3, the outcome is classified as resource-limited inconclusive even if numerical drifts appear small, since safe-window infrared separation is insufficient. The heuristic adequacy condition ξ(a) ≲ L/3–4 ensures that infrared observables are not contaminated by finite-size effects.</p> <p dir="ltr"><strong>Deliverables and reproducibility: </strong>The certification workflow produces <strong>standardized machine-readable outputs:</strong> certification.json containing test point identifiers, volume ladder specifications, tolerance settings, diagnostic summaries, and standardized failure tags (WRAP_AROUND_DOMINANT, LOCALIZED_GHOSTS, LEAKAGE_VOLUME_UNSTABLE, etc.); adequacy tables documenting observable convergence; complete artifact manifests with checksums; and comprehensive reproducibility metadata including software versions, random seeds, hardware specifications, and precision policies.</p> <p dir="ltr"><strong>Design philosophy: </strong>All thresholds and diagnostic rules represent <strong>conservative defaults intentionally biased against false-positive bulk claims.</strong> These may be refined through future protocol demonstrations, accumulated statistics from benchmark models, and community calibration efforts. The protocol prioritizes transparent audit trails and reproducible certification decisions over aggressive claims of volume adequacy.</p> <p dir="ltr"><strong>Keywords</strong></p> <p dir="ltr">Lattice gauge theory, Hamiltonian formulation, thermodynamic limit, volume stability, Gauss-law enforcement, finite-size effects, correlation length, effective mass, participation ratio, localization diagnostics, wrap-around contamination, SU(3) gauge theory, truncation methods, certification protocol</p> <p> </p> <p dir="ltr"><strong>Other works by the author on this topic</strong></p> <p dir="ltr">Nemirovsky M., 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.<a href="https://zenodo.org/records/18140235">18140235</a> (2026) </p> <p dir="ltr">Nemirovsky M., Spectral Vacuum Mechanism - Part XV Unification of the mass formula in SVM particles of the Standard Model, Zenodo. DOI: 10.5281/zenodo.<a href="https://zenodo.org/records/18207487">18207487</a> (2026) </p> <p dir="ltr">Nemirovsky M., Spectral Vacuum Mechanism – Part XVI Spectral Confinement under Truncated SU(2) Gauge Embedding: Preservation of the Spectral Confinement Class, Zenodo. DOI: 10.5281/zenodo.<a href="https://zenodo.org/records/18225421">18225421</a> (2026) </p> <p dir="ltr">Nemirovsky M., Spectral Vacuum Mechanism – Part XVII Spectral Confinement under Truncated SU(3) Gauge Embedding: Toward a Constructive QCD‑like Framework, Zenodo. DOI: 10.5281/zenodo.<a href="https://zenodo.org/records/18280887">18280887</a> (2026) </p> <p dir="ltr">Nemirovsky M., Spectral Vacuum Mechanism – Part XVIII Continuum Trajectory and Low-Energy Self-Consistency under SU(3) Truncation, Zenodo. DOI: 10.5281/zenodo.18415826 (2026)</p> <p dir="ltr">Nemirovsky M., Spectral Vacuum Mechanism – Part XIX Gauss-Law Certificates and Audit Artifacts under SU(3) Truncation, Zenodo. DOI: 10.5281/zenodo.18422292</p> <p dir="ltr">Nemirovsky M., Spectral Vacuum Mechanism – Part XX SU(3) Truncation Removal: Controlled j_max → ∞ at Fixed (a, V) in the Physical Sector, Zenodo. DOI: 10.5281/zenodo.18434530</p> <p> </p>