RESULTS: hunt #147, `dbn_pt_height`

Status: closed at landing, 2026-10-31. No claim promoted, no bound on Lambda stated, nothing here bears on RH (docs/08).

Written at landing by a session other than the lanes' author. Most of this directory was written by an agent whose claims the owner asked to have checked, so this file states only what the committed record supports, says where the landing session re-checked a claim (section 7), and says where it could not. The sources of the record are CHAIN.md, REDUCTION.md, chain.json, sources.json, evaluator/*.md, the JSON and JSONL results in evaluator/, and RUNS.md.

0. How to read this file

Every finding is one of two kinds, or a stated mix of both, and says which.

• Read. A statement found in a published or archived source, at a pinned version, graded as what that source shows. The lab did not check the mathematics behind it. [Q:ID] names a quotation in chain.json; its source's URL, SHA-256 and retrieval time are in sources.json. A digest is written sha256: and its first twelve hex digits.
• Computed. A number or a decision the lab produced, graded on the ladder: measured (one route, float grade), hardened (independent routes agree, or ball enclosures carry every step), kernel-checked (none here). A computation that uses a published lemma takes the lemma's status, and a composite takes the grade of its weakest step.

"CI" below means what runs where the private download cache is absent, which is every CI runner (section 6). "Cache only" means a claim whose only check skips there.

1. Dispositions of the questions in MISSION.md

The four words are those of ALIGNMENT.md section 5: refuted, obstructed, unresolved (attempt unresolved), paused (paused by allocation).

Context for Q4, not a finding of this hunt: hunt #145 records a kernel-checked theorem placing a zero of its Davenport-Heilbronn function off the critical line below height 100 (dh_separation_hundred, pending external verification). This hunt did not check that #145's function is the one pinned in zeta/epstein.py, so the grade here stays measured.

2. Findings, graded, with sources

2a. Read in sources (the chain lane)

Every row here is cache only: on a CI runner its quotations are checked against chain.json, not against the source (section 6). Row R10 was also re-checked at landing against its hashed source, as was the project's barrier log behind E4 (section 7).

2b. Computed by the lab (both lanes)

3. The gap in the published chain, stated exactly

This is a claim about other people's work, so it is stated at the strength the evidence carries and no further.

The claim. In the published and archived record read for this hunt, at the versions listed below, the asymptotic-region hypothesis of Polymath 15's Theorem 1.2 at Table 1 row 2 (X = 5e12 + 194858, t0 = 0.186, y0 = 0.16733) is evaluated only at N = N0 = 630783 and y = y0. The paper says the rest is expected to be verifiable [Q:P12]; a project author wrote in 2019 that it had not been verified carefully, with a heuristic reason [Q:I1]; and the project repository holds no sweep at those parameters [Q:R5]. Platt and Trudgian's Lambda <= 0.2 reads that row [Q:PT3], so their published chain contains one step that its cited source did not carry out.

What it does not say. It does not say the hypothesis is false, that Lambda > 0.2, or that Platt and Trudgian erred in a computation of their own: their Theorem 1, the verification height, is not questioned. The margins reported at N0 (0.0376 for \|E_2 f\|, 0.0292 for \|f\|, against error terms near 1e-7) and the unpublished 2024 check point toward the hypothesis holding. It does not say no verification exists anywhere: three sources that could hold one were not read, namely the published Res. Math. Sci. text (a client challenge blocked it), the published Bull. London Math. Soc. text of Platt and Trudgian (closed access) and Broughan's book chapter (closed access). Terence Tao, who led the project, wrote on his blog in 2020 that the Platt-Trudgian height immediately improves the bound to 0.20 [Q:B9]; that is the contrary-leaning record, and it adds no computation.

Versions read (retrieved 2026-10-21; URL, size and SHA-256 of each in sources.json):

• arXiv 1904.12438 v1 and v2 e-prints, debruijn.tex sha256:3f54d0a68bbe and sha256:560a28fe31be; the arXiv comment calls v2 the final version with the referee's corrections [Q:ABS1];
• arXiv 2004.09765 v1, its only arXiv version, TeX sha256:c6c29099c943;
• km-git-acc/dbn_upper_bound cloned at 5fde84e (all refs), its issue and pull-request list, and the comments of issues 81, 107, 113, 118, 126, 131;
• the Polymath wiki through dated Wayback snapshots (the live wiki returned 404), and the empty GitHub wiki;
• the ninth, tenth and eleventh Polymath15 threads and the 2019 paper post, with their comments;
• the two 2019 referee reports linked from that post, read once in a PDF reader (not re-readable by the test).

Grade. A reading of those sources, with its negative-existence parts bounded by them. Its source ties are cache only (section 6), last run on the lanes' own machine; the landing session could not re-read the arXiv TeX, because it had no cache and downloaded no papers. The observation is original to this lab. It is not claimed novel: the search behind it was the newest 60 de Bruijn-Newman abstracts on arXiv and an author search, abstracts only (CHAIN.md, section 4). CLAUDE.md routes such an observation to a public GitHub issue; none was opened, and whether to raise it outside this repository is the owner's decision. Owner decision, 2026-10-31: not raised outside this repository for now. Its source ties are cache only and have not been re-run on another machine.

4. The Davenport-Heilbronn gate: what it catches, and what rests on it

The gate asks whether the evaluator's barrier rule, run with DH's heat flow in place of zeta's, refuses a box that holds a DH zero. Stage 1 was preregistered (evaluator/DH_GATE_PREREG.md, commit 2ba8cbba3) and run on 2026-10-26 (RUNS.md E7). Stage 2 was preregistered at landing (evaluator/DH_GATE_PREREG_2.md, commit de7cd593a) and run from that commit (RUNS.md E8). Stage 1 is not changed or rerun.

Why stage 1 could not see the margin faults. A principal-argument count goes wrong only where the zeros on one side of a segment subtend more than a half turn. Every stage-1 box holds one zero, and Z1, the one box beside a close pair, is centred on it, so its four-point mesh splits the pair's angle over two segments, about 142.5 degrees each. RUNS.md E7's own reason ("|H| on the boundary is far above any margin") is not the operative one: the clean rule refined on Z1, Z3 and Z4, so the margin did bind (corrected in E8). Stage 2 centres a four-point segment over the pair, about 296 degrees, and no_margin is caught there, as preregistered.

Why no_M2 stays uncaught. Near an isolated zero or a close pair, H is close to quadratic along a short segment, so the endpoint maximum of \|H'\| already bounds \|H'\| on it and the dropped term M2 h/2 changes no decision. A gate that checks outcomes on boxes the truth step can decide is not expected to see it; a per-segment audit of the acceptance inequality would.

Is the gate enough? It is enough for what it shows: with the right H in balls, the barrier rule refuses the five gate boxes that hold a DH zero and decides the three zero-free controls, while a one-corner check (the row-2 pattern of CHAIN.md), a main-terms-only evaluation and a margin-free count are caught. It is not enough, on its own, to license a zeta barrier or sweep:

1. it checks outcomes, not inequalities, so it cannot see no_M2;
2. it runs dh_gate.dh_barrier, a copy of the rule with DH's H and D, not effective.barrier_rectangle, so it tests the rule and not the zeta code path; for zeta the margin's soundness rests on _audit_barrier in evaluator/test_effective.py, which audits two rectangles at finitely many points;
3. it does not exercise Theorem 1.3's error terms (on DH, E = 0) or the Euler-mollified sweep, input 2 of REDUCTION.md part D, which is the step row 2 left open.

What rests on it. Nothing among this hunt's findings. No sweep, no barrier and no Lambda statement was run under its licence, and the row-2 rectangles of E4 were computed on 2026-10-21, before the gate existed, so their grade does not depend on it. Whether a later zeta sweep may proceed on this gate plus _audit_barrier, or needs a per-segment audit and an error-term control first, is the owner's decision.

5. Claims corrected at landing

Each correction is dated 2026-10-31 where it is made; the earlier text is left on record where the file's convention keeps it.

1. The case-log status said the barrier rule refuses every box holding a DH zero. It refuses the gate boxes that hold one (four in stage 1, one in stage 2), which is all a finite gate can show.
2. RUNS.md E7 and a docstring in evaluator/test_dh_gate.py gave the margin faults' invisibility as \|H\| being far above any margin. The clean rule refined on Z1, Z3, Z4 and the development box D0, so the margin did bind; the operative reason is the geometry of section 4 (E8, and the docstring).
3. The case log, CHAIN.md and REDUCTION.md said every quotation is re-read from its hashed source by test_chain.py. That holds only where the download cache exists, which no CI runner has (section 6).
4. The two quotations of this repository's own zeta/epstein.py ([Q:L1], [Q:L2]) were skipped with the rest when the cache was absent, though they need no cache. They now run everywhere.

6. What CI checks, and what only the cache machine checks

The download cache (<cache>/dbn/) holds third-party documents, so it is not committed, and no CI runner has it. Counts measured in the landing worktree on 2026-10-31, without the cache, slow tests included:

• test_chain.py: 49 pass and 223 skip. Before landing, 43 passed and 225 skipped. Of the six added runs, two are the [Q:L1] and [Q:L2] quotations, which no longer skip; three check the landing's dated notes; one checks this file's citations.
• evaluator/: 146 pass and 2 skip (the comparison with the project's barrier log, and the arXiv-source digest). Before landing, 143 passed; the three added pin stage 2 of the gate.

So 195 tests run in CI for this hunt and 225 skip there.

What the skipped tests pin, and so what is cache only: that each of 167 quotations occurs at its locator in the source whose SHA-256 is recorded; the 46 registered facts that read a source (comment authors, dates and parents, commit dates, commit and issue counts, the outsider status of the 2024 commenter, the 2018 sweep minima); and ten history and bundle checks (the cached digests, a planted misquote, the DOIs, the commit that introduced the expect-to-verify sentence, the absence of a row-2 sweep at every commit, the 0.16733 string, the files naming the 2018 points, the absence of a run of the banked window bound, the record's PDF, and the frozen anchor copy). Each skip says so in its reason.

What CI does check. That the notes agree with the committed record: every number in the notes occurs in chain.json or sources.json, every date and named person sits inside a registered fact, every inline quotation is verbatim in the recorded quotation text, every citation tag exists and sits beside its fact, and the headline verdict and the dependency table are the recorded ones. Also the stated arithmetic, the DH pin and the lexical rules. planted_faults.py, run without the cache on 2026-10-31, had all 17 of its planted prose edits refused, because each breaks agreement with chain.json. What CI cannot see is a quotation in chain.json that is wrong about its source, or an edit made consistently to a note and to chain.json.

So: every row of section 2a, and every date, name and count in CHAIN.md and REDUCTION.md, is checked against its source only where the cache exists. The last such run on record is the lanes' own (RUNS.md C3, 264 passed with the cache), by the author whose claims are under check; the landing session could not repeat it and re-checked two sources by digest instead (section 7). No paper text was added at landing; the quotations already in chain.json are the lanes' own.

7. Spot-checks made at landing (2026-10-31)

• Recomputed independently (mpmath at 50 digits and exact rationals, not the hunt's code): N = floor(sqrt(x/(4 pi) + t/16)) = 630783 at all four corners of the slab x in [X, X + 1], t in {0, 0.186} (the expression is increasing in both); t0 + y0^2/2 = 3999993289/20000000000; X/2 = 2500000097429 and its margin 500175235371; the 2018 points' 19/100 and 1/5 and heights 2500000097429 and 500000023440; the 0.1875 site's 3/16, margin 350479773/2 and N = 690988. All agree with the record.
• Recomputed through the hunt's code: the evaluator lane's tests, slow tier included, pass (section 6), among them the trib2 numerator 0.037617 and the t = 0 rectangle.
• A source re-checked against its digest: the catalogue file constants/21a.md at 46dbf03, fetched from GitHub into an empty directory, has 6986 bytes and sha256:e8b198b124d2, as sources.json records, and its five quotations [Q:C1] [Q:C2] [Q:C3] [Q:C4] [Q:C12] are verbatim at their lines.
• A second source: the project's row-2 barrier log (output/windingnumbers/windnum_nolemma_x5000000194858_y_0.16733_1_t_0_0.186.txt at 5fde84e), fetched the same way, has sha256:340617569529, the digest evaluator/EVALUATOR.md records, and [Q:R3] and [Q:R4] are verbatim at lines 4 and 442.
• Not re-checked: the arXiv TeX quotations, on which section 3 rests most (no paper was downloaded); the commit and issue counts (the GitHub API refused the request from this session).
• Hunt number: #147 is used by no other case-log entry or file on origin/main as of this landing.

8. Why there is no doors section

CLAUDE.md asks a hunt that measures a ceiling to end with its doors. This hunt measures none. The chain lane audits a published argument; the evaluator lane reproduces one printed number, winds one rectangle and runs a refusal gate; no bound was optimized and nothing was driven until it stopped moving. The re-optimisation of (t0, y0) at the Platt-Trudgian height was only costed (RUNS.md E3(b)), never run. What would change the verdict is listed in CHAIN.md section 5: a deposited or published sweep of row 2's right region that this hunt missed, or a ball-arithmetic sweep with a y-direction argument and a tail, either at row 2 or at one of the 2018 points.