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5989 bytes, as of the version from 2026-10-27 01:12 (current). Recorded here, not run.
# Build handoff: T513
Draft: https://numberdb.org/T513
Title: Zeros of the irregular Coulomb wave function $G_\ell(\eta,\rho)$.
The draft contains 336 positive radial zeros, written to 100 significant digits, in the family #210 convention: parameter order `(ell, eta, n)`, DLMF normalization, fixed rational eta, strictly positive roots in increasing order with n=1 first. It holds ell=0,1,2,3, eta=0,+/-1/2,+/-1,+/-2, and n=1,...,12. Couplings are displayed by absolute size, positive before negative. It remains unpublished and has not been offered for review. The controller owns the subsequent critique, repair and offering stages.
The family, local skill and live skill were read. The stricter local argument-selection rules were followed. `already_here`, `already_asked`, three numeric API lookups and both tag pages were checked. The free-case Bessel hits are mathematical equivalences, not duplicate constructions. The three Coulomb siblings are T510, T511 and T512. No duplicate of this table was found; the empty draft was created before the full computation.
## What was decided
The family already settled the function, normalization, parameter domain and finite selection. No mathematical convention was guessed. The builder chose a zero-spacing proof instead of the family's suggested phase-tracking route. The positive regular solution on (0,1/16], the Wronskian, and a certified positive G(1/16) prove the initial interval contains no G zero. On the remaining half-line q<=65, so the Dirichlet Wirtinger inequality gives minimum zero spacing pi/sqrt(65)>1/4. Certified signs on a quarter-step mesh therefore count every simple root. This proves the index as well as the existence of each enclosed zero; `proven` is appropriate.
Final endpoints are exact rationals separated by 2e-120, accepted only after opposite Arb signs and containment in the indexed scan interval. They are enclosed in a Sage interval, then serialized to 100 digits. Final evaluation uses 681 bits. Every interval denoted by the actual decimal spelling was checked separately.
Direct mpmath evaluation very near a free-case zero failed with NoConvergence at 150 digits. The independent calculation instead evaluates Coulomb G and its derivative 1/8 beyond the root and solves its ODE Taylor expansion locally. The 150- and 200-digit calculations use different truncation orders and agree within 1e-140 for all 336 roots. A known cosine-root control validates the method first. This numerical comparison supplements the Arb and indexing proofs.
## Completed validation
- Canonical dry_run.py: 336 entries; exactness, finite enclosures, ring imports and size checks passed. The dry-run block was 59.3 KB; the written values are at most 102 characters each.
- Independent pre-fill checks: all 336 decimal endpoint brackets; DLMF phase representation versus the generator's simplified prefactor; both differential equations; the unit Wronskian; analytic derivatives, derivative recurrences and order recurrences where applicable.
- All 48 free-case roots compared with indexed Bessel Y zeros; the 12 ell=0 free roots compared with (n-1/2) pi.
- All 336 roots compared with mpmath/Taylor calculations at 150 and 200 digits. Largest measured function/derivative ball radius below 1.62e-155.
- Controls: nonzero sine and cosine, the unit Wronskian, deliberately reversed sign, exact decimal interval parsing, a cosine Taylor root, the initial coefficient bound, and six published mpmath function or derivative values.
- Interlacing with T512 checked on all pairs and triples available in the stored ranges.
- Full generator verify(sample=None): 336/336 matched, 0 differing, 0 missing, 0 extra.
- Independent ball identities, decimal endpoint brackets, Bessel checks and interlacing repeated on all 336 values read back from the database. Largest ball radius in that run below 1.58e-155.
- Final authenticated audit: clean, no findings. Its first pass requested moving a cross-reference out of Definition; that prose-only correction was made. The numerical entries were filled once and never repaired.
The reproducible generator, independent checks, canonical check helpers, numerical evidence and logs are attached to the website draft. Sage 10.9, mpmath 1.4.1, numberdb 0.1.10, Python 3.12.3 were used. See dependency-versions.json.
## Links, tags and remaining review
The draft links all three sibling Coulomb tables, the indexed Bessel Y zeros and the Tricomi U table. Bessel matches carry entry-specific equals links. No repeats property is used because this is a theorem relating different named functions. The existing special functions, zero and physics tags are used. The family's proposed quantum mechanics tag remains absent; it would connect T510–T513 with the oscillator tables T404, T406, T408 and T411. No new tag was created.
The separate critique should scrutinize the root-index proof and independent Taylor comparison as well as prose. This build did not perform a critique, publish, offer for review, or change a published table. It did not commit, push, deploy, import a corpus, edit pipeline sources or modify skills.
## Evidence
- generate.py: self-contained generator, full index proof and verification entry point.
- checks.py and check.py: independent computations and shared checking helpers.
- numerical-evidence.json: all 336 root spellings, indexed scan brackets, certified signs and the two mpmath estimates.
- dry-run.log, checks.log, postfill-checks.log: completed calculations.
- document.json and stored-document.json: final authored and read-back documents.
- audit.json, fill-response.json, prose-response.json, attachment-responses.json: API evidence.
- screen.json, lookup-*.json, tags-*.html, family-210.txt: screening and family evidence.
- development-findings.md, probe-mpmath.log, probe-comparison.log: diagnosed prototype/checker failures.
- 20261026T105241Z-build-lessons.md: proposed lessons retained as artifacts, with no shared source edits.