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

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.