Volume 36: Robbins–Siegmund Contraction, Fixed-Resolution Entropy Closure (Q₍δ₎; S_traj→0), and Glyph-Quantized Ledger Determinism (SHA-256)

This volume establishes Robbins–Siegmund contraction and fixed-resolution entropy closure in the RSI framework, with a block-diagonal SU(3) kernel, square-summable noise schedules, and a deterministic glyph-quantized ledger. The attested run certifies Xn→0X_n\to 0Xn→0 almost surely and Straj→0S_{\text{traj}}\to 0Straj→0 for a centered mid-tread, odd-bin quantizer, under assumptions S1–S5 and a pinned environment with byte-stable artifacts.

Core certificate results

• Theorems: T1 (Robbins–Siegmund contraction) and T2 (fixed-resolution entropy closure) formally stated and certified under S1–S5.

• Kernel & schedule: JJJ block-diagonal with SU(3) unitary blocks scaled to σ=∥J∥2<1\sigma=\|J\|_2<1σ=∥J∥2<1; ρn\rho_nρn square-summable with exemplar c⋅n−αc\cdot n^{-\alpha}c⋅n−α, α∈(1/2,1]\alpha\in(1/2,1]α∈(1/2,1].

• Decision: PRIMARY_PASS = True; all 17 gates PASS.

• Certificate: sha256 = 3069ac9667…c10 (DATE: 2025-11-05).

• Precision set: float64, float128, and mpmath high precision (≥382-bit mantissa).

• Deterministic ledger: v36/ledger.jsonl with 41 entries; manifest and SHA256SUMS.txt emitted.

🧮 Canonical Constructs

⚙️ Key Modules

• Environment lock and seeds: emit v36/artifacts/v36_env.json; set OMP_NUM_THREADS=1\mathrm{OMP\_NUM\_THREADS}=1OMP_NUM_THREADS=1, OPENBLAS_NUM_THREADS=1\mathrm{OPENBLAS\_NUM\_THREADS}=1OPENBLAS_NUM_THREADS=1.

• Noise-bank builder: deterministic ξn\xi_nξn array, Nbank≥105N_{\text{bank}}\ge 10^5Nbank≥105.

• Multi-precision simulators: float64, float128, mpmath runs with shared ξn\xi_nξn and tail audits.

• Entropy auditor: centered mid-tread dyadic QQQ; tail median and log-slope checks.

• Borel–Cantelli & cross-precision gates: exceedance counts and ρN\rho_NρN-scaled discrepancy.

• Ledger & manifest: append ledger.jsonl; emit manifest.json and SHA256SUMS.txt.

🔬 Findings

• Contraction: tail norms satisfy RS targets across float64/float128/mpmath; theory links by supermartingale margin γ=1−σ2>0\gamma=1-\sigma^2>0γ=1−σ2>0.

• Entropy: fixed-resolution StrajS_{\text{traj}}Straj decreases to zero with non-positive tail log-slope and smallness gate satisfied.

• Cross-precision: ∞\infty∞-norm discrepancy bounded by 3ρN3\rho_N3ρN on the shared ξn\xi_nξn bank.

• Reproducibility: environment pinned; deterministic seeds (ϕ=1729\phi=1729ϕ=1729) and byte-stable ledger.

🔁 Continuity

Carries the cryptographic ledger and artifact practice from the spectral kernel line (V29) and maintains certified, byte-stable replay; extends to SU(3)-structured contraction with entropy closure gates and a cross-precision audit.

Reproducibility & Provenance

• Environment: python=3.12.12, numpy=2.0.2, mpmath=1.3.0; threads pinned; BLAS info recorded. Artifacts include v36_env.json, xi_bank.npy, entropy_def_centered.json, heavy_tail_clip.json, ledger.jsonl, SHA256SUMS.txt, README.md.

• Certificate: DATE 2025-11-05; sha256 3069ac9667…c10; decision PRIMARY_PASS=True with 17/17 gates.

Scope & Limits

Linear JJJ with σ<1\sigma<1σ<1; fixed quantization resolution; heavy-tail validation for pre-scaled, clipped Student-t(2.5). Nonlinear JJJ, adaptive quantization, and broader α\alphaα-stable laws are deferred.