RESULTS: numerics lane (oob_envelope)

2026-09-28: the L = 0.8 bound and the L = 1.19 stage B bound, both in the even sector, passed independent referee review (referee/REVIEW.md, sections "L = 1.19" and 5). The odd-sector bound at L = 1.19 (the "Odd" sections near the end) passed its own independent referee review the same day (referee/ODD_REVIEW.md, verdict PASS); that is recorded below in "Grade update, 2026-10-21", written for the landing of both odd lanes. Every grade uses the AGENTS.md ladder.

Item 6. L = 1.19 (support 2.38), stage B: candidate, hardened by two independent routes

For real even f with supp f ⊆ [−1.19, 1.19] and H = sine:16:

λ_min(R_H(500)) ≥ 5.71789230595e-48, so Q(f) ≥ 5.71789230595e-48 ‖f‖²

given Q ≥ R_H. Route: Arb GL-96 assembly of the N = 500 leading block (orders 0..998, 384 bits) with the Bernstein-ellipse radius ε_Q ≤ 6.9e-66 in every entry; midpoint Cholesky of A − λ₀I snapped to exact dyadics and its residual enclosed in Arb (r ≤ 1.5e-113); entry uncertainty e ≤ 2.3e-63 by row sums; Zhu (13) with ε_D ≤ 1.8e-184, ε_B ≤ 1.5e-87. λ_min(R_H) ≥ min(λ₀ − r − e, β* − ε_D) − ε_B, only the Arb lower endpoint reported. stage_b_modal.py, stage_b_result.json, ledger in RUNS.md.

Grades, weakest first:

• Q ≥ R_H: ordinary derivation, referee PASS with stated scope (the reduction is L-independent; the referee reviewed it at L = 4/5).
• The numerical step: enclosure-carrying, one implementation (this lane's GL radii, which the referee did not audit at L = 4/5 either). The same code on Modal reproduces the reviewed L = 4/5 bracket (≥ 1.15822443198e-17 at λ₀ = 0.9999 λ_meas, referee 1.1579e-17).
• Independent reproduction: the referee's own Clenshaw-Curtis-192/Arb route, with its own quadrature and entry bound and a shift fixed in advance, proves lambda_min(R_H) > 5.7179e-48 on the same 500-mode subspace, and recomputes eps_D < 1.490e-184, eps_B < 1.553e-90 (referee/REVIEW.md, "L = 1.19", verdict lines 1 to 5, PASS).
• So the composite is a candidate positivity bound at support 2.38, hardened by two independent implementations (sharing python-flint), resting on a refereed ordinary derivation; even sector only (the odd sector at this window is bounded separately, at the same two-route grade, with the complex-f consequence in "Grade update, 2026-10-21" below), not kernel-checked, pending external verification.
• Controls (RUNS.md "Negative controls"): at L = 4/5, T# = 60, where the form is indefinite, the step returns no bound; a planted in-band constant fires. Dropping or flipping H does not discriminate at L = 4/5.
• Consistency (one-sided, K1 analogue): 5.7e-48 ≤ λ*(1.1) ≤ 2.78e-38 (Zhu Table 3). N-convergence, measured: λ_meas at N = 500, GL-96 equals stage A's N = 360, GL-64 value to 12 digits.

Five-line graded summary

1. Envelope constants S = Σ_p M_p with P_L − H ≤ S for all real t, at L = 0.8, 1.0, 1.19 (e.g. 1.5552528 and 3.8634773 with the sine kernel, D = 16, against A_L = 2.9420 and 7.0750). Enclosure-carrying (Arb, two independent proofs per prime). envelope.json, commit ca41ce2.
2. Controls, all measured. K3: every H frequency is out of band and ∫|F|²H = 0 on random window functions to 5e-13; in-band plants and a moved-in-band lesion break it by the predicted amounts (k3.json). Sign gate: Ψ + H stays above the envelope through the matrix code path; flipped-H and S → S_opt lesions fire (envelope_check.json). K2 on Davenport-Heilbronn at L = (log 47)/2, where the form is negative (independent enclosure-grade witness): the valid pipeline returns no bound (inconclusive, as required); a forced-β lesion finds λ = −0.288 beamed at the off-line ordinate (k2_dh.json).
3. Reduced form at L = 0.8, T# = 100, H = sine:16, N = 96: the leading block has λ_min(A) > 1.158e-17 (Arb LDL of A − λ₀I, quadrature radius in every entry) and provably < 1.1585e-17; with Zhu's two-block bound (13), λ_min(R_H) ≥ min(λ₀, β* − ε_D) − ε_B = 1.158e-17 − ε_B, ε_B < 3.031e-44, on all of L²_even[−0.8, 0.8]. Outward-safe endpoint: λ_min(R_H) ≥ 1.1579e-17 (the referee's figure; any value up to 1.157999999999e-17 is also safe). Enclosure-carrying (harden_L08_T100_sine16*.json, 58795ee). Calibration: the same pipeline in Zhu's configuration (H = 0, T# = 200, N = 200) gives λ_min(R) ≥ 1.02e-17 − ε_B with ε_B < 1.5e-102, outward-safe ≥ 1.0199e-17, and λ_min(A) < 1.028e-17; the measured R_150 floor 1.3564e-18 matches his 1.356e-18 (harden_L08_T200_none.json). Correction (referee REVIEW §5): an earlier version quoted λ_min(R_H) ≥ 1.158e-17 and ≥ 1.02e-17, which dropped ε_B; the code subtracted it but the quoted figure was the 10-digit display midpoint, and the JSON lower_bound field was a round-to-nearest float. harden.py now stores the exact dyadic lower endpoint and a decimal rounded down (edited, not yet rerun).
4. Composite at L = 0.8: Q(f) ≥ 1.1579e-17 ‖f‖² for real even f with supp f ⊆ [−0.8, 0.8]. An independently reviewed ordinary derivation (Q ≥ R_H, theory lane, referee PASS) combined with a hardened numerical step (this lane's Arb/GL bound, reproduced by the referee's independent Clenshaw–Curtis/Arb implementation at two resolutions, both ≥ 1.1579e-17). Not kernel-checked; pending external verification. Referee verdict: hunts/oob_envelope/referee/REVIEW.md at b65ef69 on branch teal-sea/oob-cert-referee. Scope: even sector only; complex f would also need the odd-sector bound. Not independently audited: this lane's GL entry radii (the referee used its own quadrature instead). K1 holds (below Zhu's 2.27e-17 upper bound and the measured window floor 1.656e-17). Same support as Zhu (1.6); a sharper constant (his 8.9e-18) with half the matrix, not a new window.
5. L = 1.19 (support 2.38), stage A: on the N = 360 leading block of R_H (sine:16), a 384-bit midpoint LDL has one negative pivot at T# = 320 and none from T# = 350 to 525 (λ ≈ 5.78e-48 at T# = 500). Measured only: no quadrature, tail or coupling bound, Arb interval LDL undecided, nothing claimed about the whole form. Modal, ≈ 0.85 core-hours (stageA_L119_reduced.json, ledger in RUNS.md). Stage B (enclosure-carrying) not approved.

L = 1.19 (support 2.38), stage A: measured only

Modal, 9 units, ≈ 0.85 core-hours (stageA_L119_reduced.json). Envelope sine:16 (S = 3.8634773, enclosure-carrying), N = 360, GL-64, 384 bits, no quadrature, tail or coupling bound. On the N = 360 leading block, a plain 384-bit midpoint LDL has one negative pivot at T# = 320 and none at T# = 350, 400, 450, 500, 525; inverse iteration gives λ = 1.65e-48, 4.15e-48, 5.17e-48, 5.776e-48, 6.00e-48 there, monotone as R_{T#} must be. Arb interval LDL is undecided (304 to 306 of 360 pivots) at every checkpoint, so even the block's sign is measured, not enclosed, and nothing is claimed about the whole form. External consistency, one sided only: λ* is nonincreasing in L (nested windows), so Zhu's Table 3 upper bound at L = 1.1 gives λ*(1.19) ≤ λ*(1.1) ≤ 2.78e-38, and the measured 5.8e-48 is consistent with λ_min(R_H) ≤ λ*(1.19). His bound at L = 1.2 (9.98e-49) bounds λ*(1.2) ≤ λ*(1.19) from above and so says nothing about λ*(1.19); no table value brackets it. (Correction, supervisor review: an earlier version claimed the two upper bounds bracket the L = 1.19 floor. They do not.) Zhu's valid-certificate estimate at this support needed N ≈ 1.4 to 2 × 10^4 (his §7); stage B would need N = 500. This is not a positivity bound at support 2.38.

K2: Davenport-Heilbronn at L = (log 47)/2

The DH form is negative on this window (independent, enclosure-grade witness in hunts/rogue_frontier/weil_trunc/dhneg_log.md, λ = −0.3163, off-line pair at 85.699). DH normalization pinned to that code (κ to 4e-56, Λ_f(n) to 5e-55). The valid pipeline (H = 0, S_DH = 25.59) needs T# ≳ 1.6e11 for β* > 0, so it returns no bound: inconclusive, as K2 requires. With β* forced (an invalid envelope, lesion only), the matrix code finds λ = −0.288 at T# = 150 with the eigenvector beamed at t = 84.5, matching weil_trunc. k2_dh.json, measured.

The L = 0.8 replication, in full

Reduction (Zhu arXiv:2608.24827v2 Theorem 1.1, eq. (4), with Ψ_L → Ψ_L + H): for even real f with supp f ⊆ [−L, L],

Q(f) = 2F(i/2)² + (1/π)∫_0^∞ |F|² (Ψ_L + H) dt (∫|F|²H = 0) ≥ 2F(i/2)² + (1/π)∫_0^{T#} (Ψ_L + H − β*)|F|² dt + β*‖f‖² = R_H(f),

using, for t ≥ T# ≥ 15/4, Ψ_L + H ≥ log(t/2π) − 1/t − S ≥ β* (Zhu Lemma 3.1 as published, plus P_L − H ≤ S from line 1), β* = log(T#/2π) − 1/T# − S.

Steps of the claim in line 4, each with its own grade:

Error budget of step e: GL-64 on 200 panels of width 1/2; per panel, Trefethen's Gauss bound (h/2)·64M/(15(ρ²−1)ρ^{128}), ρ = 1 + √2, with M bounding the integrand on the rectangle |Re z − c| ≤ h/√2, |Im z| ≤ h/2: |T̂_n| ≤ 2√(Lν_n)e^{L h/2} (Poisson integral), |cos λz| ≤ cosh(λh/2), digamma by a center-plus-derivative bound. Total ≤ 5.8e-44 for the largest entry, added as radius to every entry (asserted in code). Nodes are snapped to exact dyadic Bessel arguments; the shift ≤ 5.5e-75 enters the budget through a Cauchy bound on f'.

Correction, 2026-10-30 (review of #365). The rate ρ^{128} above is Trefethen's for 65 points; the 64-point rule GL-64 has rate ρ^{126} (ρ^{2−2n} for n points, the defect #365 fixed in weil_dh_line and exact_gram). harden.py, stage_b_modal.py and stage_b_odd_modal.py now use ρ^{2q−2}, so every recorded quadrature radius was understated by ρ² = (1 + √2)² ≈ 5.83. Rerun under the corrected bound (harden_L08_T100_sine16_gaussfix.json, harden_L08_T200_none_gaussfix.json): eps_Q_max rises 5.76e-44 → 3.36e-43 and 4.75e-44 → 2.77e-43, and the hardened lower bounds are unchanged, λ_min(R_H) ≥ 1.158e-17 − ε_B and ≥ 1.02e-17. The L = 1.19 Stage B and odd-sector Modal runs were not rerun: their recorded eps_Q_max (6.86e-66, 6.87e-66) become at most 4.0e-65, whose contribution at N = 200 is at most 200 × 4.0e-65 = 8.0e-63 in spectral norm, far below their gaps λ_meas − λ₀ (5.8e-50 and 3.5e-46). That is a margin argument, not a rerun, so those two results hold at the corrected bound by this estimate only.

Measured λ_min(R_H) vs T# (N = 200, no error bounds), sine:32: indefinite (1 to 3 negative eigenvalues) for T# ≤ 60, positive from T# = 65 (2.0e-18), 1.16e-17 at T# = 100, 1.42e-17 at T# = 200. The envelope threshold 2π e^S ≈ 30 is not the operating point; see PROGRESS.md for the table.

Reproduction (from the repo root; about 10 minutes in total)

PY=.venv/bin/python   # from the repo root; python-flint 0.9.0, numpy, scipy
cd hunts/oob_envelope/numerics
$PY envelope.py --D 8 16 32 64 128                 # 134 s, writes envelope.json
$PY k3.py                                          # 1 s, k3.json
$PY envelope_check.py                              # 3 s, envelope_check.json
$PY assemble.py --N 200 --Tmax 200 --prec 256 \
    --checks 40 50 60 65 70 75 80 100 120 150 200 \
    --env sine:16 sine:32 sine:64 --out run_L08_N200.json   # 6 min
$PY harden.py --lam0 1.158e-17 1.1585e-17 --out harden_L08_T100_sine16_control.json   # 30 s
$PY harden.py --env none --T 200 --N 200 --lam0 9e-18 1.02e-17 1.028e-17 \
    --out harden_L08_T200_none.json               # 3 min
$PY summarize.py run_L08_N200.json

Modal jobs (profile teal-sea): modal run stage_modal.py (stage A units then reducer; --reduce-only to rerun the reducer on the stored units) and modal run k2_modal.py.

Not yet done

Stage B (enclosure-carrying L = 1.19, needs approval), N-convergence at L = 1.19, the joint LP/SDP envelope, theory §1.7's B_T scan.

Odd sector validation, 2026-09-28

New odd-specific runner stage_b_odd_modal.py, derived in odd_derivation.md; the even runner is unchanged. The theory file's complex pole calculation and parity row were checked against Zhu v2 Lemma 6.1. Both multiplier and pole cross terms vanish as shown explicitly in odd_derivation.md section 1. Thus the weaker of valid odd and even bounds applies to all complex f.

Zhu section 6.2 reports 9.1183e-15 measured and 8.2065e-15 as its lower bound. The measured ratio is 1.0000042173. Both validations use GL-64, h=1/2, 256 bits. JSON artifacts: odd_zhu_result.json, odd_val_result.json. The full-space steps carry Arb enclosures; the eigenvalue measurements do not. These are one route, pending independent reproduction.

Controls: changing -2ss^T to +2ss^T raises the measured floors to 6.40880126462e-10 and 7.36487811883e-10, respectively. The direction check fires, but a positive-pole lesion is not expected to be rejected by a positivity checker. At 1.01 times each measured eigenvalue, the shifted Cholesky fails. Dropping H or reversing H at T#=100 does NOT discriminate: the measured floors remain positive, 9.40521108856e-15 and 9.28959833914e-15. These are recorded as non-firing lesions.

The T#=40 control (L=4/5, sine:16, N=96) is refused despite a positive beta. Its head has an Arb-enclosed Rayleigh upper endpoint below -1.35054e-7, including quadrature error. This proves that reduced form is indefinite, not that the original Q is negative. All altered control forms at T#=40 also have enclosed negative witnesses. Artifact: odd_val40_result.json.

Odd main result at L = 119/100, 2026-09-28

For real odd f supported in [-119/100,119/100], the computed reduction gives

Q(f) >= R_H(f) >= 3.42853488205e-44 ||f||_2^2.

This is a candidate fixed-window statement. Its numerical step is single-route enclosure-carrying, pending independent reproduction. Its analytic reduction is the ordinary derivation in odd_derivation.md, using the existing sine:16 envelope and Zhu's digamma inequality. It is not kernel-checked and makes no assertion at unbounded support.

Artifact odd_main_result.json stores the exact dyadic endpoint and input SHA-256 values. Parameters: N=500 odd modes, orders 1,3,...,999; first discarded order 1001; T#=500; GL-96, h=1/2, 384-bit Arb. No change to the even-run parameters or to H. Measured head lambda_min = 3.46316654753e-44. The lower shift is 0.99 times the measured value, snapped to an exact dyadic.

The code evaluates min(lambda0-r-e, beta-eps_D)-eps_B in Arb and reports its lower endpoint, then rounds the displayed decimal downward. The error budget is far below the retained head floor. A 1.01-times overshift fails at pivot 179, so the positivity step can reject a proposed lower bound. Validation and other controls are recorded immediately above; the H lesions that did not fire remain non-firing controls, not successful checks.

With the unchanged even result stage_b_result.json, whose downward-rounded endpoint is 5.71789230595e-48, parity and real/imaginary splitting give the combined candidate

Q(f) >= 5.71789230595e-48 ||f||_2^2

for every complex L^2 f supported in [-119/100,119/100], on the finite form domain and with the extended positive value allowed otherwise. Each of the four orthogonal real parity components satisfies its sector's inequality; their squared norms sum to ||f||_2^2. The odd sector is not the limiting one. The combined claim retains the weaker review status of its component steps.

All Modal work finished: 1.509934183 timed core-hours for the complete odd task, and a conservative app-lifetime reservation envelope below 2.856 core-hours, against the 4-core-hour limit. Five owned apps stopped. The unit checkpoints remain on volume oob-envelope-stages; see RUNS.md for commands, app IDs, the pre-run estimate and exact accounting. Nothing pushed.

Grade update, 2026-10-21: the odd sector reproduced independently

Written for the joint landing of this lane's odd work (branch teal-sea/oob-odd, 08ff0491) and the odd referee (branch teal-sea/oob-odd-referee, a5c40f7), both merged with their commits kept. The two odd sections above are kept as recorded on 2026-09-28; their "pending independent reproduction" is superseded by this section. The same phrase heads odd_derivation.md, which is left byte-identical because odd_manifest.json and referee/outputs_odd/comparison.json record its SHA-256.

referee/ODD_REVIEW.md (verdict PASS, 2026-09-28) re-derived the odd reduction independently in referee/ODD_FORMULAS.md: odd orders 2i+1, the negative rank-one sinh pole, first omitted order 1001 with the pole tail charged in eps_D, exact cancellation of H at every frequency of size at least 2L (the boundary included), and the parity and real/imaginary split. It found no gap. Formulas and exact shifts were frozen at 3ce6ab9 and the independent results at cefd6363, before any of this lane's artifacts were read; the brief disclosed this lane's endpoint, and no Python from this lane was read at any point. The referee route is Clenshaw-Curtis-192 with its own quadrature, Bessel and tail bounds, at 768 bits (Zhu case) and 1280 bits (main case): 132 Modal inputs, none failed, 2.348 core-hours of recorded input work (referee/ODD_RUNS.md). Both routes run on python-flint, so the arithmetic library is shared.

The referee's exact dyadic Rayleigh witnesses for the reduced forms, 9.1183384545008638e-15 and [3.463166547529606864e-44 +/- 2.77e-63], agree with this lane's measured head values 9.11833845450e-15 and 3.46316654753e-44 at their displayed precision. Its lower endpoints are smaller because its shifts were fixed in advance (8e-15 and 3e-44), so the referee did not reproduce this lane's tighter 3.42853488205e-44, which stays a single-route value. Both referee controls passed: flipping -2pp^T to +2pp^T changes the form and fails the independently integrated pole identity, and the artificial operator R - 1e-42 I is refused with an enclosed negative Rayleigh value.

Grades after this update, weakest first:

• Q ≥ R_H on real odd f: ordinary derivation, written by this lane and re-derived independently by the referee, no gap found. It uses the out-of-band lemma, the sine:16 envelope and Zhu's digamma inequality as before. Not kernel-checked.
• The odd numerical step at L = 1.19: hardened, two independent implementations sharing python-flint, every error term enclosed on both.
• So for real odd f with supp f ⊆ [−1.19, 1.19], Q(f) > 2.99e-44 ‖f‖² (this lane alone: ≥ 3.42853488205e-44 ‖f‖²) is a candidate at the same grade as item 6's even bound.
• With item 6, for every complex f ∈ L² with supp f ⊆ [−1.19, 1.19], Q(f) ≥ 5.71789230595e-48 ‖f‖². Each route gives this on its own, since on both the even endpoint is the smaller one. Candidate, hardened by two independent implementations sharing python-flint, resting on ordinary derivations, pending external verification. A fixed-window statement, not a route to RH, with no claim at unbounded support.

The L = 4/5 odd rows are validations against Zhu's section 6.2. This record does not combine them with summary line 4 into a complex-f statement at L = 0.8.

The doors

No optimum or ceiling was sought. At this single candidate, parameter range L=1.19, T#=500, N=500, sine:16, the retained head eigenvalue is the limiting term, not beta-eps_D or the error budget. The 0.99 shift deliberately leaves one percent of the measured value unused. No shadow prices were measured. Frozen choices: L is the requested target; T# trades a sharper reduction against quadrature and tail cost; N trades smaller coupling against matrix cost; q=96 and 384 bits trade smaller numerical error against compute; h=1/2 trades panel count against ellipse size; sine:16 trades its envelope constant against frequency growth in the quadrature bound. Adjusting these stays within the same symbol data and information class. Independent reproduction tests this result rather than enlarging that information class. No further run or allocation is proposed by this record.