D-008 — Numerical sandbox summary (task 0.6)

Date: 2026-08-17 | Session: 001 | Task: 0.6 | Status: complete Experiments: E-002 (zeros), E-003 (finite G̃). Tags: [NUMERICAL-EVIDENCE].

E-002 — first 10⁴ zeros [ALL CHECKS PASS]

mpmath.zetazero, dps 25, 15-way multiprocess (resumable part-files). γ₁ verified to 1e-20; strictly monotone; Riemann–von Mangoldt error at n=10⁴: 0.32 (bound 2 log T ≈ 18); gap stats: min 0.0377, max 6.887, mean 0.986 vs local mean spacing 0.854 at γ₁₀₀₀₀ = 9877.7826540055…. Data: data/E-002-zeros-first-10000.{csv,json} (+ E-002-parts/).

E-003 — finite compression G̃ vs paper predictions [ALL CHECKS PASS]

Construction per D-002 §3 (paper §2) at T ∈ {1000, 2000, 4000}, both windows ψ₀, ψ_MT. Pipeline validated by an independent Lemma 2.1 frame self-test (Σ_k ϕ̂(x−α_k)² vs aL²: max rel err ≈ 1e-5).

Results (ψ₀ / ψ_MT): | T | tr G̃/N | ‖G̃‖²_HS/N | R(ψ) | rel err | 1/L | min eig | |---|---|---|---|---|---|---| | 1000 | 0.9993 | 1.5505 / 1.5593 | 1.3333 / 1.3275 | 16.3% / 17.5% | 19.7% | −1.3e-15 | | 2000 | 0.9997 | 1.5126 / 1.5196 | 1.3333 / 1.3275 | 13.4% / 14.5% | 17.4% | −1.6e-15 | | 4000 | 1.0001 | 1.4872 / 1.4928 | 1.3333 / 1.3275 | 11.5% / 12.5% | 15.5% | −1.5e-15 |

Interpretation:

1. tr G̃ ≈ N(I) to 0.1% at all T — Prop 4.2 confirmed empirically. ✔
2. ‖G̃‖²_HS/N tracks R(ψ) with error shrinking ≈ 1/L exactly as Theorem 5.7's O_χ(L⁻¹) predicts — the method's central analytic input is confirmed structurally (agreement at finite sandbox T is NOT supposed to be better than this). ✔
3. G̃ numerically PSD at all T (min eigenvalue ~ −1e-15 = float64 noise): consistent with Q = 0 when all zeros are on the line — the Bombieri negative-index mechanism is visible in silico. ✔
4. Rank–trace lower bound (4tr−2N−HS²)/N observed 0.44–0.51, below the asymptotic 2−R = 0.667/0.673 — consistent: the finite-L HS² excess directly deflates the bound by the same amount. ✔ (Internally coherent.)
5. Numeric rank ≈ d with ~3% of eigenvalues below 1e-8·max — the compressed frame is near-tight but not tight at these T.

Engineering notes (for future numerics)

• mpmath zetazero: single-core ~2–9 zeros/sec (slower as n grows); 15-way multiprocess part-files + per-row flush + atomic replace = crash-safe and resumable within 10-min execution windows.
• FFT+spline ϕ̂ evaluation: spline domain is |ξ| ≤ πM/L — MUST size M so the domain covers 2T+√T (extrapolation silently corrupts far entries).
• Windows console is cp1252: ASCII only in print() (files are UTF-8).
• RTX 4090 not needed at this scale; V.T@V + eigvalsh at d≈4100 take seconds on CPU BLAS.

What the sandbox establishes for Phase 1

The paper's asymptotics are real and correctly understood; the sandbox is calibrated (we know the size of finite-L deviations). Any NEW certificate we test numerically must beat these baselines AFTER accounting for the same O(1/L) effects — otherwise we risk mistaking finite-size noise for improvement. This calibration is a required control for tasks 1.1–1.2.