RUNS: numerics lane (oob_envelope)

Every run that costs more than a local 10-minute, 2 GB slot is estimated here before launch. Nothing below has been launched remotely.

Local runs so far (Ghost, all under 10 min and well under 2 GB)

Proposed: L = 1.19 (support 2.38) on Modal, two stages

Parameters (chosen from phase 1 and 2, not tuned to a result)

• Envelope sine:16, S = 3.86347732... (enclosure-carrying, envelope.json). Envelope threshold 2π e^S = 299.6. D = 16 rather than 64 because the quadrature bound grows like cosh(b_e · D log 7): at D = 64 the Bernstein bound with GL-96 lands near 1e-52 per entry, too close to the expected floor; at D = 16 it is near 1e-62. D = 64 would only lower T# by 7%.
• T# = 500: β* = log(500/2π) − 1/500 − S = 0.5115. Resolution height T* = 2π e^{2.38} = 67.9, so T# ≈ 7.4 T*. At L = 0.8 the reduced form with H turned positive near 2.1 T* and reached 70% of the window floor by 3.2 T*, so positivity of R_H at T# = 500 is expected but not known: stage A measures it before stage B pays for the bounds.
• Expected λ_min: near Zhu's measured 1e-48 to 1e-46 (his 950-mode run, §7). Working precision 384 bits (115 digits).
• x = T# L = 595. Hardened tail bound (Zhu (12), x^n/(2n+1)!!) needs 2N ≈ 2 x for ε_B ≈ e^-460: N = 640 (orders 0..1278). Measured-only convergence at L = 0.8 needed 2N ≈ 1.15 x: N = 360 for stage A.
• Panels h = 1/2 on [0, 500]: 1000 panels.

Measured unit costs (Ghost, M1, 384 bits, unit_cost.out)

Stage A: measured scout (λ_min(R_H) vs T#, no error bounds)

Approved by the supervisor 2026-09-27 (stage A only). N = 360, q = 64, prec 384, sine:16, panels of 1/2 on [0, 525]. Checkpoints T# = 320, 350, 400, 450, 500, 525 (β* = 0.06 ... 0.56; 300 is dead since 2π e^S = 299.6). Units are t-ranges with checkpoints on their boundaries: [0,80], [80,160], [160,240], [240,320], [320,350], [350,400], [400,450], [450,500], [500,525], so 9 units, the largest 160 panels. Each returns the lower triangles of its partial sums G, C_Ψ, C_H (exact Arb midpoints plus radii) and writes them to a Modal volume on completion. Per node ≈ 45 ms (Bessel K = 719, interpolated from the measured 33 and 151 ms) → 67 200 nodes ≈ 3 000 s; products ≈ 1050 × 3 × 0.2 s ≈ 630 s. ≈ 1.0 core-hour; largest unit ≈ 10 min. The reduction (sums, LDL, inverse iteration at each checkpoint) runs locally, under 10 min.

Stage B: hardened run (only if stage A shows λ_min(R_H(500)) > 0)

N = 500 (harden.py's own tail arithmetic at x = 595: ε_B ≈ 8e-88, against 6.7e+08 at N = 400 and 2e-37 at N = 450), q = 96, prec 384. Bessel cost at K = 999 interpolated ≈ 100 ms per node: 96 000 nodes ≈ 9 600 s; products 1000 × 2 × 0.9 s ≈ 1 800 s. ≈ 3.2 core-hours. Needs a fresh approval after stage A. Plus the final positivity step on one container, < 0.1 core-hour.

Positivity step and endpoint (audited after referee REVIEW §5; stage A showed interval LDL undecided at cond ~1e48, so no interval LDL here).

1. Ã: the assembled N × N leading block as Arb balls; E_ij := rad(Ã_ij) + ε_Q,ij bounds |A_ij − mid(Ã_ij)| for the exact block A.
2. Measure first: midpoint inverse iteration on the N = 500 block gives λ_meas (adding modes can only lower λ_min, so stage A's N = 360 value is not used). Choose λ₀ = 0.99 λ_meas, an exact dyadic.
3. L̃: 384-bit Cholesky of mid(Ã) − λ₀I, rounded to exact dyadics.
4. Residual in arb_mat: Rres = mid(Ã) − λ₀I − L̃L̃ᵀ, every operand exact, so Rres is a rigorous ball matrix; r := max_i Σ_j |Rres_ij|, taken as an Arb upper bound. Then λ_min(mid(Ã) − λ₀I) ≥ −r (L̃L̃ᵀ ⪰ 0, Weyl).
5. ‖A − mid(Ã)‖₂ ≤ e := max_i Σ_j E_ij (upper bound). Hence λ_min(A) ≥ λ₀ − r − e (Weyl).
6. Full space, Zhu (13): λ_min(R_H) ≥ min(λ₀ − r − e, β* − ε_D) − ε_B.
7. Report only the Arb lower endpoint of step 6, as an exact dyadic and as a decimal rounded down (harden.py now does this). Never a float(), never a display midpoint.

Acceptance: the endpoint of step 6 is positive. Expected sizes from the L = 0.8 run and stage A: r ~ 1e-110, e ~ 1e-60, ε_B ~ 1e-87, against λ₀ ~ 5.7e-48, so each subtraction is 12 or more orders below λ₀; every term is still written out and subtracted. Before any run, the reducer is also tested on the L = 0.8 case (N = 96), on Modal, where it must return an endpoint ≤ the LDL-based 1.158e-17 − ε_B and positive. Units: 50 containers × 20 panels (10 t-units each), ≈ 6.5 min each.

Total both stages ≈ 4.2 core-hours, under the 20 core-hour cap. Wall time with 10 to 50 parallel containers: under 15 minutes per stage. Dollar cost not estimated here (Modal's current CPU rate not checked).

Checkpointing (compute rule 4)

One unit = a contiguous block of panels. Each unit writes, on completion, the lower triangle of its partial sums (stage A: G, C_Ψ, C_H separately so the T# curve can be read at every checkpoint; stage B: the single matrix C_{Ψ+H−β*}) as exact Arb midpoint mantissa/exponent plus radius (≈ 60 bytes per entry, ≈ 12 MB per matrix at N = 640), and its quadrature-bound contribution. A restarted unit starts from zero; no unit exceeds 7 minutes, so preemption costs at most one unit. The reducer sums units in Arb and runs the positivity step. Memory per container < 1 GB.

Owner

The session that launches it watches it to a terminal state (compute rule 5).

Proposed: K2 on Davenport-Heilbronn (k2_modal.py), one container

L = (log 47)/2 = 1.925 (DH form negative there, λ ≈ −0.3, even sector, per hunts/rogue_frontier/weil_trunc/dhneg_log.md). H = 0, valid S_DH = Σ 2|Λ_f(n)|/√n; the job reports the least T# at which any valid β* could be positive (expected astronomically large, so the valid pipeline cannot return a bound). Lesion part: λ_min and LDL inertia of R(T#) at T# = 100, 150 (both past the off-line ordinate 85.7) with β* forced to 0.05, 0.5, 2. N = 150, GL-32, panels 1/2, 256 bits: 9600 nodes × ~8 ms (Bessel K = 299) ≈ 80 s, products and six N = 150 LDLs ≈ 60 s. ≈ 0.05 core-hour, one unit, result written to the volume and to k2_dh.json.

Remote run ledger (Modal, profile teal-sea, 2026-09-27, all apps now stopped)

Totals: stage A ≈ 0.85 core-hours (estimate 1.0); K2 ≈ 0.02 core-hour plus the crash-loop's failed starts (estimate 0.05). Dollar cost not read from Modal billing. Local log k2.log is interleaved: the crash-looping process kept writing at its old offset into the file the relaunch had truncated; it contains the successful run's output, but the authoritative record is k2_dh.json. Future launches use a fresh log file per app and check modal app list for the terminal state instead of the local exit.

Readback of remote outputs (exact grades)

• stageA_L119_reduced.json (from the volume). N = 360 leading block of R_H, L = 1.19, sine:16, 384 bits, measured (no quadrature, tail or coupling bound). Midpoint LDL (plain 384-bit, measured): one negative pivot at T# = 320, none at 350 to 525. Inverse iteration λ: 1.65e-48, 4.15e-48, 5.17e-48, 5.776e-48, 6.00e-48 at T# = 350, 400, 450, 500, 525. Arb interval LDL: undecided (304 to 306 of 360 pivots) at every checkpoint. So there is no enclosure-grade statement about the sign of even this block, and none at all about the whole form.
• k2_dh.json (volume copy identical). Measured. Valid gate: S_DH = 25.59, β* > 0 needs T# ≳ e^25.82, so no bound (inconclusive, as required). Forced-β lesion rows: interval LDL undecided in all six; shifted inverse iteration at T# = 150, β forced 0.5: λ = −0.288, eigenvector beamed at t = 84.5 (76% of |F|² within ±6 of 85.699).

No further Modal run until a separate estimate and approval (K2 rerun, stage B).

Stage B (approved by Thomas, 2026-09-27: build, validate, execute)

Code: stage_b_modal.py (units + reducer; positivity step 1 to 7 above, error terms are harden.py steps 2 and 3 called verbatim). Plan as estimated above: N = 500, GL-96, 384 bits, T# = 500, sine:16, 50 units of 20 panels, ≈ 3.2 core-hours.

Validation on Modal (config val, L = 4/5, T# = 100, N = 96, GL-64, 256 bits)

App ap-8ektPW0uhAr2FHdT1Tn6RJ, 2 units (12 s, 18 s) + reducer (1.6 s), wall 30 s. Result stage_b_val_result.json (volume readback).

• Residual route, λ₀ = 0.99 λ_meas: λ_min(R_H) ≥ 1.14675686334e-17 (exact dyadic stored). r = 3.3e-76, e = 3.7e-42 (row sum of entry radius
• ε_Q), ε_B = 3.03e-44. λ₀ = 0.9999 λ_meas: ≥ 1.15822443198e-17.
• Consistency: positive, below 1.158e-17 − ε_B (the LDL-proved value), and the tight run lies above the referee's 1.1579e-17, below the measured λ_meas = 1.15834026601e-17 (referee's midpoint Ritz value 1.15834026600857…e-17: agrees). The harden.py interval-LDL route on the same unit data passes at 1.158e-17 (0 negative, 0 undecided).
• Lesion that discriminates: midpoint Cholesky at 1.01 λ_meas fails (negative pivot at index 29) in every row, so the step cannot pass above λ_min(mid A).
• Lesions that do NOT discriminate here (recorded, not claimed as passes): dropping H or flipping its sign still leaves a positive form at L = 0.8, T# = 100 (λ 1.150e-17, 1.141e-17), because H is small in band at this L. The H-sign gate remains envelope_check.py.

Stage B run (L = 119/100, T# = 500), terminal

Total stage B ≈ 1.4 core-hours (estimate 3.2). All three stage B apps stopped in modal app list. Result read back from the volume: stage_b_result.json, log stage_b.log.

Numbers (all in the JSON as Arb strings or exact dyadics): β* = 0.511253705003064…, ε_Q,max = 6.9e-66, node shift ≤ 7.3e-113, e = 2.29e-63, λ_meas = 5.77564879389e-48 (N = 500, GL-96; stage A N = 360, GL-64 gave 5.775648793894e-48, same to 12 digits), λ₀ = 0.99 λ_meas, r = 1.42e-113, ε_D = 1.8e-184, ε_B = 1.44e-87. λ_min(R_H) ≥ 5.71789230595e-48 (decimal rounded down from the exact dyadic lower endpoint). Cholesky at 1.01 λ_meas fails (pivot 209), as it must.

Negative controls (Modal, 2026-09-27, after stage B)

Results read back from the volume: stageBval2_L08_result.json, stageBval60_L08_result.json. Total ≈ 0.02 core-hours.

• T# = 60 negative control fires. β* = 0.6845 > 0 there, so the valid gate is open, yet the positivity step returns no bound: the midpoint Cholesky of A itself breaks (pivot 19, −1.07), and the harden.py interval LDL on the same data finds exactly one negative eigenvalue (0 undecided). This matches the measured indefiniteness at T# = 60 in run_L08_N200.json. So the step does refuse a form that is truly indefinite, with every error term in place.
• Planted in-band constant (−2e-17 · G) fires: no bound (pivot 19).
• Planted in-band cos(0.95·2L·t), amplitude ±1e-16: does not discriminate. λ moves in the 12th digit. The amplitude is too small against the form's floor; recorded, not claimed as a pass.
• Main row reproduced bit for bit (≥ 1.14675686334e-17 at 0.99 λ_meas; ≥ 1.15822443198e-17 at 0.9999).

Odd sector, 2026-09-28: authorized four-core-hour task

Owner: this foreground Codex session, branch teal-sea/oob-odd, starting 46c896b0. Thomas explicitly authorized validation and main run, Modal only, 4 core-hours total, local edits and tiny checks only, no push. This supersedes older HOLD text for this odd task alone. Writes remain in numerics: new filenames contain odd; existing RUNS/RESULTS/PROGRESS are append-only.

Runner: stage_b_odd_modal.py. Derivation: odd_derivation.md. The original even runner and theory inputs remain untouched. Every checkpoint records SHA-256 of runner, assemble.py, harden.py and envelope.json; the reducer refuses a mismatch. Odd tags are isolated from even checkpoints.

Pre-validation estimate from the even measurement (N=96, 200 panels, 2 units in 30 core-seconds): N=200, 300 panels scales conservatively as 30*(200/96)^2*(300/200)=196 core-seconds, plus 30 s for the reducer and pole-sign control. Allow 0.10 core-hour for Zhu, 0.05 for envelope validation and low-T control. Actual timings replace these estimates below.

Zhu gate: L=4/5, T#=150, H=0, N=200, GL-64, 256 bits, h=1/2, six units. The source uses 200 odd modes and reports 9.1183e-15 measured, with a lower bound 8.2065e-15. The runner blocks later configurations unless the measured value agrees to 1 percent and the enclosed lower endpoint is positive. Next: val at L=4/5, T#=100, sine:16, N=96; val40 is a prospective negative control, not presumed to be indefinite. Wrong pole sign must raise the measured floor. A failed approximate pivot is followed by a ball-enclosed Rayleigh witness, so a refusal alone is not called proof of indefiniteness.

Main remains N=500, T#=500, GL-96, 384 bits, h=1/2, 50 units. Only after validation, time unit 49 on Modal and multiply by 50, add validation and reducer allowance, then append that estimate BEFORE launching the main job. No detached apps, retries disabled, at most 12 one-CPU unit containers. Each unit checkpoints to oob-envelope-stages; reducer has one CPU.

Tiny local check (1.1 s): first three odd Legendre modes, direct Fourier and sinh integrals, orthonormality, double factorial indices and actual cut ratios pass. Maximum measured direct-integral defect 1.02e-56 at 55 dps. All four configurations have both tail ratios below one (largest 0.08807). The exact checks and measured checks are distinguished in check_odd.py.

Odd Zhu validation, terminal

App ap-uvTBLRYDv5KdLXmzacpTWr, command: modal run hunts/oob_envelope/numerics/stage_b_odd_modal.py --cfg zhu. Six units 111.432 core-seconds, reducer 9.186 s; app wall 48 s. Volume readback odd_zhu_result.json; app confirmed stopped. Measured 9.11833845450e-15, source 9.1183e-15 (ratio 1.0000042173). Lower endpoint rounded down 9.02715506995e-15, above source 8.2065e-15. Wrong positive pole gives 6.40880126462e-10, a measured 70284.75-fold increase. The sign-direction check fires; positivity does not reject this wrong sign, as expected since it adds a positive rank-one matrix. The 1.01-times overshift is refused in both rows.

Odd envelope validation and negative control, terminal

val: app ap-Lfjwjx0GIZ20PL1zZkL1AI, units 36.802 core-seconds, reducer 1.781 s, wall 34 s; odd_val_result.json. Measured 9.50644182948e-15; lower endpoint 9.41137741118e-15. Wrong positive pole raises the measured floor to 7.36487811883e-10. Drop-H and flip-H stay positive (9.40521108856e-15 and 9.28959833914e-15), so these lesions do not discriminate at this window. The 1.01 overshift fails in all four rows.

val40: app ap-mHTe5wmfzkcIiwJ9nSWS4c, units 11.322 core-seconds, reducer 2.199 s, wall 21 s; odd_val40_result.json. The valid beta gate opens, but the main head is indefinite: a Schur witness has enclosed Rayleigh upper endpoint < -1.35054e-7 after quadrature error. No positive endpoint is returned. All three altered forms also fail, with enclosed negative witnesses. Their witnesses differ, so these values are not eigenvalue comparisons; the JSON pole comparison is unevaluated (null ratio) here. Both apps confirmed stopped. Validation total is below 0.05 timed core-hour.

Next authorized run is ONLY main unit 49 (panels 980:1000). The even run timed that unit at 210 s; reserve 300 s for the odd pilot, then estimate the complete run from its actual duration before proceeding.

Odd main run estimate, written before launch

Pilot app ap-ng9DiaREARkLBjGwtAMuO1 completed and stopped: unit 49, panels 980:1000, 223.455382 core-seconds, 229 s app wall. Checkpoint is already on the volume and will be skipped by the complete run.

Conservative extrapolation: 50 * 223.455382 = 11172.769 core-seconds for all units (including the pilot once), plus 172.723 validation core-seconds, 180 s for the reducer, and 300 s startup/serialization allowance = 3.285 core-hours. This is below the authorized 4.0 core-hours. No parameter changes: N=500, T#=500, sine:16, GL-96, 384 bits, h=1/2. Launching all remaining units and the reducer in one foreground app, watching checkpoints and stopping the app at terminal state. Invocation: modal run hunts/oob_envelope/numerics/stage_b_odd_modal.py --cfg B. No claim yet about the main odd eigenvalue or lower endpoint.

Odd main run, terminal and read back

App ap-OlmaObfN0y2EzyA3R6lFcm, foreground, source runner from 93f1fc6; launch commit 710b7a8. All 50 units present, including skipped pilot unit 49. Units total 5134.263733 core-seconds, including the pilot exactly once; reducer 128.776208 s; main app wall 649 s. Two client heartbeat timeouts were logged, with no unit failure, retry or restart. Longest unit 253.664 s. Result downloaded from volume oob-envelope-stages, path odd_stageB_L119_v1_result.json, to odd_main_result.json. All source/input hashes match the local files. The original even runner is unchanged.

Result: measured lambda_min 3.46316654753e-44; full odd-space lower endpoint rounded DOWN 3.42853488205e-44. The exact dyadic lower endpoint is in JSON. The 1.01-times overshift fails at pivot 179. Numerical step is single-route enclosure-carrying; the complete fixed-window claim is a candidate pending independent reproduction, including its ordinary reduction.

Total task compute: 5435.763058 timed one-CPU function-seconds = 1.509934183 core-hours, versus the 3.285 estimate and 4.0 allowance. This timer excludes checkpoint serialization/commit and container startup. For a conservative budget check, sum each app's entire lifetime times the maximum possible one-CPU containers in that app: 51*7 + 40*4 + 30*4 + 236*2 + 655*14 = 10279 core-seconds, or 2.855277778 core-hours. This even counts idle gate containers and non-overlapping reducers simultaneously. All five owned apps have state stopped and zero tasks, recorded in odd_apps_terminal.json. No billing dollar estimate or invoice was queried.

Post-run checks: 50 distinct panel blocks; source SHA-256 matches for all four reducers; positive decimals do not exceed exact rational endpoints; all positive main rows refuse the 1.01 overshift; even and odd L/H match; odd endpoint exceeds even endpoint. Source/doc/JSON whitespace check passes. Raw CLI logs retain their emitted whitespace. No broad local numerical suite was run: local work was edits, compilation, tiny formula and artifact checks. No push. A computation manifest records the result and artifact hashes.