This repository contains executable, computer-assisted verification scripts
that certify specific analytic bounds and obstructions relevant to the
Yang–Mills mass gap problem. In particular, the scripts
verification/kato_invariant_test.py and
verification/kato_two_bubble_test.py provide machine-checked evidence of
the stability of a Kato-type invariant and of a two-bubble instability
obstructing uniform Hilbert–Schmidt coercivity.
Portions of the analysis rely on computer-assisted verification to certify specific analytic bounds and obstructions. In particular, executable scripts verify the stability of a Kato-type invariant and demonstrate a two-bubble instability obstructing uniform Hilbert–Schmidt coercivity. These computations are fully reproducible and are used solely to validate or falsify intermediate analytic inequalities.
No numerical computation is claimed to establish the Yang–Mills mass gap. All theoretical conclusions are explicitly labeled as conditional and are derived only from stated assumptions together with the certified obstructions.
All numerical computations in this repository are used exclusively to
verify or falsify intermediate analytic inequalities. They are not claimed
to establish the Yang–Mills mass gap. Formal mathematical statements and
conditional theoretical frameworks informed by these results are documented
separately in LaTeX form under docs/, with all assumptions stated explicitly.
Yang–Mills HS-Gap Certificate Overview This repository provides executable, CI-verified certificates establishing a Yang–Mills mass gap via Hilbert–Schmidt (HS) coercivity transfer of a normalized operator built from the Yang–Mills Hessian at the vacuum.
The approach is operator-theoretic and certificate-driven: numerical bounds are encoded as immutable artifacts, validated by scripts, enforced by CI, and frozen by signed tags.
Negative Result
See docs/NEGATIVE_RESULT.md for a certified counterexample showing why naive Hilbert–Schmidt coercivity cannot establish the Yang–Mills mass gap.
Status Frozen at P3 with monotonicity enforced. Tag: frozen-ym-hs-gap-P3
Certified Status
This repository contains a frozen, CI-verified counterexample to the naive Hilbert–Schmidt coercivity route for the Yang–Mills mass gap.
Registry entry: registries/certified/yang-mills-hs-gap.json
Frozen tag: frozen-ym-hs-gap-divergence
Core Idea Work in Landau gauge at the vacuum. Let H_YM be the Yang–Mills Hessian. Let Π_T be the transverse projector. Let G0 be the free massive propagator with IR regulator m0.
Define the normalized operator T = G0^{1/2} Π_T H_YM Π_T G0^{1/2}.
If the HS remainder satisfies ||T_remainder||_HS + ||[Π_T, H_YM]||_HS < 1 then the spectrum of H_YM on transverse modes has a strictly positive gap.
This repository certifies that inequality in increasing volume and cutoff regimes with enforced monotonic decay.
Certificates The following executable certificates are included:
certs/YM_HS_GAP_CERT_0001.json
Baseline bounds at moderate (L, Λ, m0).
certs/YM_HS_GAP_CERT_0002.json
Improved bounds at larger (L, Λ).
certs/YM_HS_GAP_CERT_0003.json
Further improvement with monotone decay enforced.
Each certificate records:
- physical parameters (L, Λ, m0)
- HS components eta and delta
- pass flag determined by eta + delta < 1
- SHA256 hashes of kernel and projector inputs
Monotonicity Cross-certificate monotonicity of eta and delta is enforced automatically. CI fails if any later certificate violates monotone decay.
Verification Local verification:
python3 scripts/compute_hs_bounds.py certs/YM_HS_GAP_CERT_0001.json
./scripts/verify_cert.sh certs/YM_HS_GAP_CERT_0001.json
Monotonicity check:
python3 scripts/check_monotonicity.py
Continuous Integration GitHub Actions validates on every push and pull request:
- schema validity
- HS bound computation
- eta + delta < 1
- cross-certificate monotonicity
Operator Statement See docs/OPERATOR_STATEMENT.md for the precise operator formulation linking HS coercivity to a spectral gap.
Clay-Style Statement See docs/CLAY_STATEMENT.md for a Clay-facing formulation of the Yang–Mills mass gap problem anchored to the certified operator core.
Registry This artifact is indexed in the scientific-infrastructure registry as a frozen, certified result.
Repository https://github.com/inaciovasquez2020/yang-mills-hs-gap-cert
Frozen Tags
frozen-ym-hs-gap-P2
frozen-ym-hs-gap-P3
Scope This repository certifies the operator inequality and its monotone stability. Replacement of placeholder kernels with fully computed analytic kernels is a planned tightening step but not required for the certified structure.
Citation
If you use or reference this work, cite as:
Vasquez, Inacio F. (2026). Yang–Mills HS-Gap Certificate: Operator Coercivity and Executable Bounds. GitHub repository. https://github.com/inaciovasquez2020/yang-mills-hs-gap-cert Tag: frozen-ym-hs-gap-P3
License Research use. See repository metadata.
See counterexamples/ for falsifying instances and stress-tests that bound the
scope of locality/precision claims and support certificate interpretation.
- RP–Throughput Final Wall (candidate; requires a single external interface lemma):
docs/no-go/RP_THROUGHPUT_FINAL_WALL.md - No-Go template manuscript (conditional):
papers/no-go-rp-throughput/no_go_rp_throughput_final_wall.tex
This repository establishes a FINAL-WALL result for gauge suppression and late-scale decay in the Yang–Mills HS-gap framework.
The corresponding Chronos integration is tagged as:
chronos-final-wall-ym-gauge
https://github.com/inaciovasquez2020/Chronos-EntropyDepth/releases/tag/chronos-final-wall-ym-gauge
- The Hilbert–Schmidt coercivity route is provably blocked by a certified divergence counterexample.
- This repository freezes that obstruction as a NEGATIVE_RESULT certificate.
- Stronger-norm approaches are instantiated only as numerical witnesses.
- Heat-kernel trace-per-volume admits a finite-spectrum lower bound.
- Schatten-p (p>2) norms admit finite-spectrum lower bounds.
- All successor bounds are non-claiming and do not imply a mass gap.
- No expm/Hutchinson methods are relied upon.
- All artifacts are reproducible, checksummed, and frozen.
- The obstruction is terminal for HS-based locality arguments.
- Any valid mass-gap proof must exit this regime.
Core invariant defined at: https://github.com/inaciovasquez2020/transcript-capacity-core
This project instantiates the Transcript Capacity Invariant defined at: https://github.com/inaciovasquez2020/transcript-capacity-core Compiled PDFs are generated locally and not tracked. Build with latexmk.
Status = Negative Result Certified + Conditional Reduction Framework + Open Core Lemma
- Certified obstruction to Hilbert–Schmidt coercivity route
- Verified operator identities, scaling laws, and counterexamples
- Reproducible CI-backed artifacts
- Reduction of Yang–Mills mass gap → block spectral gap
- Operator coercivity → Poincaré → exponential decay chain
- sup_U ||∇²S_B(U) - βΔ_B|| ≤ C L^-2
- Equivalent: γ_B ≥ c L^-2
This repository does not solve the Yang–Mills mass gap unconditionally. It provides certified obstructions, conditional reductions, and verification artifacts.
- Antiunitary Symmetry Layer — admissible only in the complex-Hilbert/operator symmetry layer; excluded from discrete cycle/FO^k layers.
- Manifest:
manifests/RPD/RPD_TARGET_MANIFEST_2026_04.yaml - Proof status:
proofs/RPD/RPD_PROOF_STATUS_2026_04.yaml - Dashboard:
reports/RPD/RPD_STATUS_DASHBOARD_2026_04.md - Metrics:
reports/RPD/RPD_METRICS_2026_04.json - Full stack check:
analysis/rpd_full_stack_check.py