CORRECTIONS: session 2026-08-10, local positivity attempt

One documentation defect found in the repository, one new hunt to file, and one process note. Ordered by what needs a decision from you.

1. Λ ≤ 0.2 is misattributed in two docs

Status: real defect, fix is one line each.

docs/05-de-bruijn-newman.md §3 is correct and is the authority: it states both bounds with primary citations:

Polymath15 pushed it to Λ ≤ 0.22 (Res. Math. Sci. 6 (2019), paper 31), and feeding Platt–Trudgian's verification of RH up to 3·10^12 back into the same machinery gives the current record Λ ≤ 0.2 (Bull. Lond. Math. Soc. 53 (2021)).

zeta.heatflow.lambda_facts() agrees, listing them as separate entries with separate authors, and flags both as non-strict, the only known strict upper bound is Ki–Kim–Lee's Λ < 1/2 (Adv. Math. 222 (2009)).

Two downstream docs disagree with each other, and one attributes the record to the wrong people:

Line 330 is the substantive one. The phrasing "the Polymath15 machinery ... gives Λ ≤ 0.2" credits the sharpened bound to the collaboration; it is Platt and Trudgian's, in their own 2021 paper. docs/05 §3 says this correctly, so the error is local to docs/12.

Suggested edits, minimal:

• docs/08 line 318, after "Lambda <= 0.22", add: "(sharpened to Λ ≤ 0.2 by Platt–Trudgian, 2021, see docs/05 §3)".
• docs/12 line 264, after "Λ ∈ [0, 0.2]", add: "(Rodgers–Tao below; Platt–Trudgian above, docs/05 §3)".
• docs/12 line 330, replace "into the Polymath15 machinery is what gives Λ ≤ 0.2" with "into the same machinery is what let Platt–Trudgian sharpen Polymath15's Λ ≤ 0.22 to Λ ≤ 0.2".

Why it is worth fixing rather than tolerating. It is a citation defect in exactly the class ROADMAP.md exists to prevent, a fact stated three times at three precisions, where the most-read summary (docs/08 §6, the "what is tractable" list a newcomer starts from) carries the stalest number. It also propagated: my own first draft of the route map inherited the error from docs/12 and had to be corrected. That is the failure mode of a fact with no single source of truth in the tree; docs/05 §3 should be cited from both sites rather than paraphrased.

2. New hunt: hunts/local_positivity/

Status: complete, self-contained, filed as a hunt and not a department.

An ontology attempt in the sense of docs/09 §4, run to its wall. Deliverables:

• localpos.py, the module. Standalone: it imports zeta.epstein only for kappa and for the cross-check, and reproduces every reference row itself.
• 24-the-local-positivity-attempt.md: the record, in docs/ house style, filed against docs/09 §5.1's pseudo-solution taxonomy.
• localpos_gate.png, three panels: the kernel at p = 2, thresholds across places, measured detector power.

Why a hunt and not a department. Its rivals are Davenport–Heilbronn and the disc −23 Epstein forms, the zeta department's rivals. Per harness/README.md and the dossier/ precedent in ROADMAP.md, a department whose battery is another department's battery is not a department. It borrows the zeta battery, which is the correct relationship. Suggested placement:

hunts/local_positivity/
    MISSION.md          <- section 1 of doc 24
    localpos.py
    results.json        <- reference_table() + lesion_sweep() + null_distribution()
docs/24-the-local-positivity-attempt.md
figures/localpos_gate.png

Corrected 2026-10-21 ("Later correction" at the end of this file). The closed form below does not give the converse of the equivalence it is offered for, which now rests on a separate derivation that has not been independently reviewed. The example under "The honest boundary" is not in the Selberg class, and its 65.24 is a truncation of a divergent series. The 60-angle control under "Controls" holds for its margins, but the gate as it now stands FAILs 22 of those 60 angles.

What it establishes. The place-local kernel K_p^{(d)}(θ) = d + 2Σ λ_m p^{−m/2}cos(mθ) has the closed form Σ_j (1−|α_j|²/p)/|1−α_j p^{−1/2}e^{iθ}|², so c_p ≤ d is exactly the local bound |α_j| ≤ √p, a decision procedure, not a heuristic, computed from coefficients alone. Measured: ζ's threshold is exactly 2/(√p+1) to 12 digits with truncation bounded below 3.4e-14 (an elementary floating-point bound that carries no enclosure at any step, so it makes no claim in the ball-arithmetic regime zeta/rigor.py owns, and does not use that regime's reserved word); the closed form pins the failure point at √p to 8 digits.

Reference table (degrees read off each object's own gamma factors, never chosen):

Controls, all in the module:

• Decoy: swapped coefficients move the verdict by 15 orders of magnitude (a_{p^k}=2^k → excess +9.6e14; random ±1 → +8.4e3; ζ → −0.172). This is the control whose absence made the Imposter Gauntlet vacuous (docs/15).
• Null: against 300 random period-5 sequences, 100 % fail, median excess +5.88; DH sits at the 6th percentile, i.e. an unusually mild failure.
• Lesion: interpolating ζ → DH, the blindness threshold is ε* = 0.184. A PASS means "no violation above ~18 % of the way from ζ to DH at the tested places", and nothing stronger.
• PASS side not vacuous: across 60 Satake angles the genuine degree-2 family keeps margin ≥ 0.343.

Where it dies, and this is the load-bearing part. The prime side decomposes place by place as −Σ_p log p (Q_p(f) − ‖f‖²) with Q_p = (1−1/p)‖Φ_p f‖², a genuine norm at every place: Requirement C achieved locally, from prime data only, with no zeros in any definition. Reconstruction agrees with zeta.weil.explicit_formula_sides to 22 digits. But Q_p − ‖f‖² is not of definite sign (52 of 60 places positive, 8 negative), so the local norms do not assemble into a global one, and local positivity is compatible with either sign of W. Files under docs/09 §5.1 taxonomy item #5, finite approximants.

The honest boundary, stated so nobody overclaims it. The gate is not a test for "has an Euler product": a genuine degree-2 product with α = 2.3, 1/α, legitimate in the Selberg class, violating Ramanujan, is rejected at p = 5 with c_p = 65.24. It tests the local Selberg bound, and localpos.scope() says so in the module rather than only in prose.

3. Process note: three bugs, all caught by a reference case passing

Worth recording because it is evidence for a rule the repo already has.

Four errors occurred while building this, and three were caught not by a counterexample failing but by a reference case behaving wrongly:

1. L(χ₅) scored negative, my diagonal normalisation was ζ-specific, not forced. A genuine Euler product must pass; it did not, so the statistic was wrong.
2. A genuine degree-2 factor scored negative: I had taken a real part that broke the Satake structure.
3. Davenport–Heilbronn passed: I had let the degree float instead of reading it off the functional equation. This is the dangerous one: the gate looked like it worked, and the failure was silent.
4. A Gram-block discriminator scored ζ negative and was discarded outright.

Each was found because the battery was calibrated in both directions. A battery that only ever rejects is indistinguishable from a correct one until you make it pass something it should pass, which is precisely why harness/README.md requires at least one reference claim expected to pass and one expected to fail, and why tests/test_department_conformance.py re-derives verdicts rather than trusting labels. Error #3 would have shipped as a working gate under a rejection-only battery.

Suggested addition to docs/17-the-falsification-harness.md, which currently records five claims refuted by existing controls: this is the complementary case, three instrument defects caught by the calibration requirement, one of which produced a plausible-looking correct verdict for the wrong reason.

4. Standing observation for ROADMAP.md

This is the third independent statistic to hit the same wall from a different direction:

• zeta/factorization.py's D(f): coefficient-side, Gate 4 (docs/18 §6)
• the Fourier quasicrystal separation (docs/18 §4)
• localpos.py's c_p: coefficient-side, local Selberg bound

All three read arithmetic; none can read the position of the critical line.

Superseded 2026-08-11 (docs/25). The proposal in this section must not be adopted, and the reason given for it is wrong. docs/18 §6 says ordinate statistics, not coefficient statistics; ζ(s−δ) has coefficients n^δ a_n, not the same ones; and c_p in fact reads that twist, with threshold exactly δ = ½ (c_p = 2x/(1+x), x = p^{δ−½}: the Selberg-class axiom's θ < ½). What is true is the weaker statement that these three particular statistics are invariant, or nearly so, under the twist. The universal is refuted inside this repository by criteria.py face 1: M(x) = O(x^{½+ε}) ⟺ RH is a criterion in the coefficients of 1/ζ alone (Titchmarsh 14.25(B)/(C)).

That tension is not an artifact of any one attempt, and three independent instruments landing on it is a measurement rather than a coincidence. It belongs in ROADMAP.md under known gaps as a standing constraint on the whole coefficient-side programme: any Requirement-A-compliant statistic is blind to the critical line by construction, so a future candidate that claims to see it must explain where the zero-information enters without violating provenance.

Later correction: the closed form does not establish the local iff

What is false, and where it was stated. The docstring of localpos.satake_kernel said the rational expression Σ_j (1−|α_j|²/p)/|1−α_j p^{−1/2}e^{iθ}|² is nonnegative for all θ exactly when every |α_j| ≤ √p. That is false. For p = 2 and α = (3√2, 0, 0, 0) the expression is 3 − 8/(10 − 6 cos θ), which stays in [1, 5/2] while one parameter exceeds √p. Its defining Fourier series has coefficients of magnitude 3^m and diverges, so the rational expression is not its sum there. The step that read the iff off the closed form (the module docstring, section 2 of this record under "What it establishes", MISSION.md, and docs/24 §2) is therefore not a valid argument for the converse.

What the counterexample does not refute. The iff for c_p as the module defines it. gate() never evaluates the rational expression (satake_kernel is called only by satake_check). On exactly this input threshold() reports regime "divergent" and gate(..., 4, primes=[2]) returns FAIL, both now pinned by test_closed_form.py. The recorded verdicts and results.json, whose scope() line is a statement about c_p, stand as historical measurements; their wording "exact" and "decision procedure" is graded below.

One reading of c_p, used everywhere from now on. c_p is the least diagonal c for which every finite Toeplitz section with diagonal c and off-diagonals s_m = λ_m p^{−m/2} is positive semidefinite. On an absolutely summable sequence this equals the cosine-series minimum the module evaluates (the standard Toeplitz–symbol correspondence). On a growing sequence it is +∞, which is the module's existing "divergent" convention. At |α_j| = √p the two readings part: the cosine series diverges, while the least PSD diagonal can stay finite. For ζ(s − ½) it is exactly 1 = d (measured 1.000000 on every section N = 5, 50, 200, 400), and on that input gate() reports INDETERMINATE; it does not do so at every input on the bound (finding 3 below).

Derivation of both directions (written in this landing, 2026-10-21; not independently reviewed). Assumptions: a degree-d Euler factor at p with real coefficients, so its Satake parameters are real or come in conjugate pairs; write β_j = α_j/√p, so s_m = Σ_j β_j^m is real for m ≥ 1.

1. Forward: every |α_j| ≤ √p gives c_p ≤ d. The section with diagonal d is the sum over j of the Hermitian Toeplitz matrices with entries β_j^{i−k} for i ≥ k and conjugates above the diagonal (the sum matches because s_m is real). For |β_j| < 1 that matrix holds the Fourier coefficients of the Poisson kernel (1−|β_j|²)/|1−β_j e^{iθ}|² ≥ 0, hence is PSD; for |β_j| = 1 it is the rank-one Gram matrix v v* with v_i = β_j^i. Each has unit diagonal, so the sum is PSD with diagonal d.
2. Converse: some |α_j| > √p gives c_p = +∞. A PSD matrix has every 2×2 principal minor nonnegative, so |s_m| ≤ c for all m. But Σ_{m≥1} s_m z^m = Σ_j β_j z/(1 − β_j z) is rational with a pole at 1/β for each nonzero β among the β_j, residue −k/β for multiplicity k. Copies of one β add and distinct β give distinct poles, so nothing cancels. With R = max_j |β_j| > 1 the radius of convergence is 1/R, so limsup |s_m|^{1/m} = R > 1 and s_m is unbounded: no finite c works.

Measured support for step 2, pinned by test_closed_form.py: a single parameter with |β| = 1.05 has least PSD diagonal 1.2806, 77.31 and 1.770e5 on the sections N = 5, 50, 200.

Grade after this correction. The forward direction is derived. The converse is not established by the closed form; for c_p read as the least PSD diagonal it rests on the derivation in step 2, which is elementary but unreviewed, so the iff as a whole is an unreviewed derivation, pending external verification, and the words "exact" and "decision procedure" in the original record should be read at that grade. The c_p = 65.24 recorded for the α = 2.3, 1/α violator is a finite-depth truncation in the "divergent" regime (2.3/√5 ≈ 1.0286 > 1), where the module's own convention is c_p = +∞; the FAIL it records stands, the number is not the value.

Found while landing this correction (2026-10-21). Five findings, measured in the landing worktree on top of origin/main at fb647bfc7. They lie outside the closed-form question but concern the same instrument or the same sentences. Each is pinned in test_closed_form.py, none changes a verdict in the reference table, and this landing does not change gate().

1. gate() misreads oscillating weights, in both directions. threshold() decides summable, divergent or boundary from two weighted terms about a decade apart near the truncation edge, a test added on 2026-08-11 (commit 5eed14344; docs/25 §6; C-LOCPOS-01). For a conjugate pair of Satake parameters the weights oscillate, so that comparison can fall on either side of 1 whatever their envelope does. Inside the bound, α = 0.9·√2·e^{±0.2i} at p = 2 is called divergent and FAILs although c_p ≈ 1.894 < d = 2. Outside it, α = 1.2·√7·e^{±0.4i} at p = 7 is called summable and PASSes although c_p = +∞ (least PSD diagonal above 10^9 on the N = 120 section). On it, α = √2·e^{±0.9i} at p = 2 FAILs although c_p = d = 2 exactly.
2. The 60-angle positive control regressed when that test went in. Section 2 above records that "across 60 Satake angles the genuine degree-2 family keeps margin ≥ 0.343". Re-run on 60 equally spaced angles φ from 0.05 to π − 0.05 (α = e^{±iφ}, the default 25 primes), which reproduces that minimum margin (0.3434), the statement still holds for the margins of the truncated c. But the current gate's verdict on the family is FAIL at 22 of the 60 angles, while the module as first committed (e3e4426b4, 2026-08-10) passes all 60 when re-run. docs/25 §6 and C-LOCPOS-01 report the hunt's recorded verdicts unchanged by the repair. That holds for the verdicts docs/25 §6 lists (ζ, L(χ), Davenport–Heilbronn, both Epstein forms, the violator), and the reference table, the lesion threshold ε* = 0.1841 and the null distribution reproduce exactly, but the 60-angle control was not among them.
3. INDETERMINATE on the bound holds only for steady weights. threshold()'s comments equated its regime test with the bound: a ratio above 1 "is exactly |alpha| > sqrt(p)", and the boundary regime is "|alpha| = sqrt(p) to numerical resolution". That holds when the weighted terms near the edge keep a steady modulus. ζ(s − ½), the single real parameters +√2, −√3 and +√97, and the pair ±√5 all give INDETERMINATE, and for one real parameter the verdict changes within a relative 10^{−6} of √p (PASS below, INDETERMINATE inside, FAIL above; measured at p = 2, 97). For a conjugate pair on the bound the gate can say FAIL (φ = 0.9, p = 2) or PASS (φ = 2.0, p = 2, with the regime called summable), where c_p = d exactly. The comments are now scoped; the code is unchanged.
4. The honest-boundary example is not in the Selberg class. Section 2 above, MISSION.md, docs/24 §5 and the case log called α = 2.3, 1/α legitimate in the Selberg class; the note ramanujan_violator returns, also stored in results.json, said |alpha| > sqrt(p) "is allowed in the Selberg class"; docs/37 and the ramanujan_violator docstring called the factor legitimate. At p = 5 it is excluded twice over: by docs/09 §3's axiom (5) (Ramanujan, a_n ≪ n^ε) and by its axiom (4) (b_n ≪ n^θ with θ < ½, which needs |α| < √5). It is a genuine Euler factor outside the local bound, which is all the honest boundary needs, and the gate rejects it for that reason. results.json keeps its note as the historical record.
5. The violator's 65.24 moves with the truncation depth. Beyond the grade above: the same call reads 22.46 at depth 23 (tol = 1e-8), 65.24 at the default depth 41 and 110.80 at depth 52 (tol = 1e-18), and the least PSD diagonal of its N = 200 section exceeds 10^3.

Two smaller points, both pinned as well:

• An excess the gate cannot resolve reads as PASS. In the summable regime a place whose truncated c exceeds d by less than ten tail estimates is no witness, so the verdict is PASS rather than undecided. For ζ(s − δ) at δ = 0.49, p = 2, the truncated c is 15.99 against d = 1, the tail estimate is 149.9, and the PASS is right only because the true c_p = 0.9965 lies below d.
• A rejected universal still stood in the module docstring. C-SHIFT-01 lists the localpos.py docstring among the places that stated "no coefficient functional can read the position of the critical line". The director run corrected provenance_report(), docs/24 §6 and the case log, but the docstring's "Why the kernel is the thing" section still stated it until this landing.

A repair of the regime test is a deliberate act: it has to re-measure the hunt's recorded numbers and change the tests that pin findings 1 to 3. This landing records and pins the defect and leaves the gate as it is. Nothing here is evidence for or against RH.