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 Construct Definition S1–S5 σ<1\sigma<1σ<1; ∑ρn2<∞\sum\rho_n^2<\infty∑ρn2<∞; E[ξn]=0\mathbb E[\xi_n]=0E[ξn]=0; Var(ξn)⪯Id\mathrm{Var}(\xi_n)\preceq I_dVar(ξn)⪯Id; ∥X0∥<∞\|X_0\|<\infty∥X0∥<∞. T1 — RS contraction Under S1–S5, Xn+1=JXn+ρnξnX_{n+1}=JX_n+\rho_n\xi_nXn+1=JXn+ρnξn ⇒ Xn→0X_n\to 0Xn→0 a.s. T2 — Entropy closure Fixed Δ and centered mid-tread odd-bin QQQ: Straj(Q(Xn))→0S_{\text{traj}}({Q(X_n)})\to 0Straj(Q(Xn))→0. Borel–Cantelli audit Tail exceedance test An={∥Xn∥>Kρnd}A_n=\{\|X_n\|>K\rho_n\sqrt d\}An={∥Xn∥>Kρnd} with ∑nP(An)<∞\sum_n \mathbb P(A_n)<\infty∑nP(An)<∞. Cross-precision bound Shared ξn\xi_nξn; ∥∥Xn∥(64)−∥Xn∥(128)∥∞≤3 ρN\|\|X_n\|^{(64)}-\|X_n\|^{(128)}\|_\infty \le 3\,\rho_N∥∥Xn∥(64)−∥Xn∥(128)∥∞≤3ρN. ⚙️ 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.