numerics PROGRESS (oob_envelope)

HOLD after stage B (approved by Thomas 2026-09-27, run, terminal). Stage B: λ_min(R_H) ≥ 5.71789230595e-48 at L = 1.19, T# = 500, N = 500, enclosure-carrying single route, 1.4 core-hours on Modal (RESULTS item 6, RUNS ledger). Next door: referee's independent route at L = 1.19.

Earlier HOLD text: Stage B was NOT approved; no compute, local tests, push or PR. Referee final REVIEW.md (b65ef69, teal-sea/oob-cert-referee): PASS on the corrected L = 4/5 even-sector bound, λ_min(R_H) ≥ 1.1579e-17, via its own Clenshaw–Curtis/Arb implementation at two resolutions; five Modal units passed; K1, and K2/K3 plus all lesions at measured grade. Composite grade: independently reviewed ordinary derivation + hardened numerical step, not kernel-checked. Not independently audited: this lane's GL entry radii. L = 1.19 unresolved.

Earlier stage B ask, kept for the record. Stage A, measured only: on the N = 360 leading block of R_H at L = 1.19, a plain 384-bit midpoint LDL has one negative pivot at T# = 320 and none at T# = 350 to 525 (inverse-iteration λ 5.78e-48 at T# = 500); Arb interval LDL is undecided, and nothing is claimed about the whole form. Stage B would make T# = 500 enclosure-carrying: N = 500, GL-96, 384 bits, ≈ 3.2 core-hours in 50 units of ≈ 6.5 min (RUNS.md). Its positivity step: midpoint Cholesky of (A − λ₀I) as exact dyadics, the Arb residual of A − λ₀I − L̃L̃ᵀ bounded by a row sum, plus the entry radii and quadrature radii; λ₀ chosen at 0.99× the N = 500 measured value, not fixed in advance (adding modes can only lower λ_min). Payoff if it closes: with theory's step d, a candidate positivity bound at support 2.38, pending referee.

All Modal apps are stopped; ledger with failures in RUNS.md. No Modal run until a separate approval (stage B, or any K2 rerun).

Correction (supervisor, 2026-09-27): commit ecc1491's subject says "R_H positive definite at L = 1.19"; that overstates it. The evidence is a measured midpoint LDL of the N = 360 block, interval LDL undecided.

Hard rule since 2026-09-27 ~18:45Z: no compute on Ghost at all, including tests and scripts that read results. The lexical tests (test_hunt_probe_discipline.py, test_docs_numbering.py) ran before the rule; files added since were grep-checked for the reserved word.

Worker: Claude Code, Opus. Branch teal-sea/oob-cert. Writes only in hunts/oob_envelope/numerics/. Local runs capped at 10 min and 2 GB.

Status

Phase 2 result at L = 0.8, stated with its grade

• K3 (measured, float64, k3.py). For 3 random smooth even f at L = 0.8 and 1.19, every H frequency (all k log p with k > m_p) gives (1/π)∫|F|² cos(λt) ≤ 5e-13 relative to ‖f‖² (the truncation floor), and the whole H gives ≤ 3e-13. Planted in-band frequencies m_p log p reproduce the time-domain autocorrelation g(λ) to 10 digits (nonzero, up to 0.14). Lesion: one H term moved to 0.95·2L breaks the identity by exactly the predicted b·g(0.95·2L). The in-band/out-of-band split itself is decided in Arb (envelope.py, gaps logged per prime).
• Hardened leading block with all Zhu error terms (harden.py). L = 0.8, T# = 100, envelope sine:16 (S = 1.5552528467, enclosure-carrying), β* = 1.2020, N = 96, GL-64 on panels of 1/2, 256-bit Arb: quadrature radius (Bernstein ellipse ρ = 1 + √2, rectangle bounds for Bessel, cosine and digamma) ≤ 5.8e-44 per entry, added to every entry; node-shift term included; tail ε_D ≤ 2.8e-95; coupling ε_B ≤ 3.0e-44 (Schur test); Arb LDL of A − λ₀I passes at λ₀ = 1.158e-17 and provably fails at 1.1585e-17. Hence, by Zhu (13), λ_min(R_H) ≥ 1.158e-17 − ε_B, ε_B < 3.031e-44, on all of L²_even[−0.8, 0.8]; outward-safe λ_min(R_H) ≥ 1.1579e-17 (corrected per referee REVIEW §5; the earlier "≥ 1.158e-17" dropped ε_B). K1 holds: 1.1579e-17 < 2.27e-17, and below the measured window floor 1.656e-17.
• Same pipeline, Zhu's own configuration (H = 0, T# = 200, N = 200): λ_min(R) ≥ 1.02e-17 − ε_B, ε_B < 1.5e-102, outward-safe ≥ 1.0199e-17; λ_min(A) provably < 1.028e-17; his published λ₀ = 9e-18 also passes. Calibration of the hardened pipeline.
• What Q ≥ 1.1579e-17 ‖f‖² additionally rests on: Q ≥ R_H, i.e. Zhu's Theorem 1.1 with A_L replaced by S and H added in band [0, T#]. That is an ordinary derivation (theory lane RESULTS §1, self-reviewed, no referee yet). So the composite statement is a candidate, weakest step an unrefereed ordinary derivation; the numerical steps are enclosure-carrying. It sharpens Zhu's constant (8.9e-18) at the same support 1.6 with half the matrix; it is not a new support.
• Measured-only numbers (no quadrature/tail/coupling bound): the whole λ_min(R_H) vs T# table in the log below and the N scan.

Not done, optional: theory §1.7's sampled B_T at L = 0.8, T = 30..70.

Log

• 2026-09-27. Read MISSION, AGENTS, BRIEF, the seed probes, and Zhu arXiv:2608.24827v2 sections 1 to 5 and 7 (pdf fetched to scratch, not committed). Plan for phase 1: per prime, build the Fejér-smoothed Carathéodory-Toeplitz correction in float, freeze its out-of-band coefficients as exact dyadic rationals, raise M_p by a small margin, and prove Φ_p = M_p − φ_p + h_p ≥ 0 on [0, π] by adaptive Arb evaluation with a second-derivative remainder. The float construction is only a proposal: the enclosure step checks the explicit trigonometric polynomial, so nothing depends on the float roots or weights being accurate.
• 2026-09-27. Phase 1, separable constant: done (enclosure-carrying). envelope.py (134 s, 256-bit Arb), raw data envelope.json. Per prime, Φ_p = M_p − φ_p + h_p ≥ 10^-8 is proved by an exact decomposition into a nonnegative kernel sum plus an Arb-bounded residual, and independently by an adaptive θ-scan with a third-derivative remainder; both pass for every row. Prime sets and m_p decided in Arb. The sine (Fejér-Korovkin) kernel converges like 1/D², the Fejér kernel like 1/D; Fejér D = 64 reproduces the seed probe (1.5493, 3.0273, 3.8385).

Joint LP/SDP (stretch) deferred until phase 2 shows whether the envelope threshold is the operating point at all (the reduced form may go negative well above it).

• 2026-09-27. Phase 2, L = 0.8 curve, N = 200 (measured). assemble.py, 256-bit Arb throughout (no float64 anywhere in the matrix), GL-32 on panels of width 1/2, one pass with cumulative sums. run_L08_N200.json. The Arb radii below are radii of the assembled quadrature-rule matrix: quadrature error, Legendre tail and two-block coupling are not bounded yet, so every number here is measured, not a bound on Q.

Calibration against Zhu (H = 0): λ_min(R_150) = 1.3564e-18 (Zhu 1.356e-18), λ_min(R_200) = 1.0277e-17 (Zhu: ≥ 9e-18 enclosure-carrying). K1 check: every value below is ≤ 1.43e-17 < 2.27e-17, and below the window floor 1.656e-17.

(At T# ≤ 50 the listed λ is the eigenvalue nearest 0, not the minimum; the LDL inertia count is the decisive column.) Reading: the envelope threshold (T# ≈ 30) is not the operating point, as theory warned: R_H is indefinite up to T# = 60 and positive from T# = 65. With H the reduced form at T# = 100 already beats Zhu's T# = 200 floor. Next: K3, then the smallest N that holds λ_min at T# = 65 to 80, then the hardened budget.

Implementation traps met (both fixed, both would have been silent in mpmath): arb's Bessel J at large order needs +512 bits of working precision, and it amplifies an input radius by about e^x, so the Bessel argument is snapped to an exact dyadic and the node shift (~1e-77) is recorded for the error budget.

• 2026-09-27. K3 passes (details in Status). N scan at L = 0.8, sine:32 (measured): N = 32 reproduces N = 200 to all digits for T# ≤ 70, N = 40 for T# ≤ 100. Hardened bounds need more (2N ≈ 2.4 x) because Zhu's tail bound x^n/(2n+1)!! is crude near n ≈ x.
• 2026-09-27. Hardened L = 0.8 results (Status). Two more implementation traps met: acb digamma returns nan on wide boxes (replaced by a center-plus-derivative bound, checked against a grid), and at L = 1.19 sizes arb's Bessel J needs up to ~1400 bits at order 1259.5 (seed call now retries with more bits until the ball is relatively tight; the L = 0.8 hardened run reproduces unchanged after the fix).
• 2026-09-27. Phase 3: unit costs measured, estimate in RUNS.md, QUESTION at top. Stopped.
• 2026-09-27. Advisor-requested gates added. envelope_check.py: the sign of H in the matrix code path is right (Ψ + H above the envelope at 4000 Arb points and a 1.2M-point grid; flipped-H and S → S_opt lesions fire), and harden.py now asserts the quadrature radius is inside every LDL entry. Stage B shrinks to N = 500 (tail arithmetic). Stage A approved.
• 2026-09-27T18:43:33Z. Stage A launched on Modal (supervisor-approved, stage A only): app ap-mucAZVkQ7RThKhLbUB9CuN, 9 units, volume oob-envelope-stages, tag stageA_L119, log stage_a.log. This session owns it and watches it to a terminal state. Routing rule received: no compute on Ghost from now on; the reducer runs on Modal too.
• 2026-09-27T18:52Z. Stage A units: terminal, all 9 complete, no unit failure (one client heartbeat warning). Unit seconds: [0,80] 273, [80,160] 267, [160,240] 316, [240,320] 401, [320,350] 146, [350,400] 323, [400,450] 425, [450,500] 420, [500,525] 206; sum 2776 s = 0.77 core-hours, plus the reducer container (~2 min). Wall 535 s. Actual ≈ 0.8 core-hours against the 1.0 estimate. First reduction (measured, N = 360, 384 bits): eigenvalue nearest 0 is 3.37e-46 (T# 320), 1.65e-48 (350), 4.15e-48 (400), 5.17e-48 (450), 5.78e-48 (500), 6.00e-48 (525). Arb interval LDL undecided (304 to 306 of 360 pivots) at cond ~1e48, so these do NOT exclude negative eigenvalues; T# = 320 violates the monotonicity of R_{T#}, which points to a negative eigenvalue there. Reducer rerun on Modal with a midpoint LDL inertia (two restarts: a glob matched the reducer's own output file; fixed). K2's first launch failed at container import (path), no compute lost; relaunched.
• 2026-09-27T18:56Z. Stage A reduction, terminal (Modal, 154 s). Midpoint LDL inertia (384-bit, measured) at N = 360:

Monotone in T# as R_{T#} must be. Grade: measured (no quadrature, tail or coupling bound; N = 360 is 2N/x = 1.21, above the 1.15 that converged at L = 0.8, but N-convergence at L = 1.19 is not checked). Total stage A spend ≈ 0.85 core-hours (units 0.77, three reducer containers ≈ 0.08, two startup failures ≈ 0), under the 1.0 estimate.

• 2026-09-27T18:55Z. K2, terminal (Modal, 70 s, ≈ 0.02 core-hour). L = (log 47)/2, N = 150, GL-32, 256 bits.
• Witness that Q_DH < 0 on this window (independent, not this code): weil_trunc Arb LDL inertia at (c, N) = (47, 96) with two negative eigenvalues, λ = −0.3163 at (47, 64), zero-side attribution to the off-line pair 0.8085 + 85.699i (hunts/rogue_frontier/weil_trunc/dhneg_log.md).
• Normalization pin against weil_trunc/galerkin.py: κ to 4e-56, all Λ_f(n), n ≤ 46, to 5e-55, supports identical. Archimedean term Re ψ(3/4 + it/2) − log(π/5) as in weil_trunc SOURCE.md s4 (iii).
• Valid envelope gate (the K2 test): inconclusive, as required. S_DH = Σ 2|Λ_f(n)|/√n = 25.59, so β* > 0 needs T# ≳ e^25.82 ≈ 1.6e11: the valid pipeline returns no bound at all.
• Lesions (β* forced, invalid envelope; these cannot establish validity): at T# = 150, β* forced to 0.5, the matrix code finds λ = −0.288 with |F|² peaked at t = 84.5 and 76% of its mass within ±6 of 85.699, the same beam weil_trunc found (−0.316, peak 84.5). Interval inertia was undecided for all six forced rows and their nearest-0 values are near-null directions, not read.
• 2026-09-27. Correction recorded (supervisor review): RESULTS.md had said Zhu's Table 3 upper bounds at L = 1.1 and 1.2 bracket the L = 1.19 window floor. They do not: λ* is nonincreasing in L, so only λ*(1.19) ≤ λ*(1.1) ≤ 2.78e-38 follows; the L = 1.2 value bounds λ*(1.2) ≤ λ*(1.19) from above and constrains nothing here. Text fixed, no rerun. Supervisor reports the referee's Modal controls unit passed (acceptance_gate true, K3 on 10 functions, all three lesions detected, measured) and its leading-block assembly is in progress. Stage B still unapproved; no Ghost compute.
• 2026-09-27. Referee final REVIEW.md at b65ef69 read (read-only git show). Grades updated in RESULTS.md item 4 and the status table: the L = 0.8 composite is an independently reviewed ordinary derivation plus a hardened numerical step reproduced by an independent implementation; not kernel-checked; even sector only. The referee notes its implementation is independent in construction and quadrature but shares python-flint with this lane, and that this lane's GL entry radii were not audited. L = 1.19 unresolved; stage B unapproved. Holding.