# Numerical evidence for T636

Build run: 20261031T100952Z-build. This evidence is from the builder, not the required independent critique.

The coordinated skill manifest is `agents/runs/20261031T100952Z-build-skills/manifest.json`, also attached as `skill-manifest.json`. Both archived local and live copies have SHA-256 `a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`; the run verified both byte digests and their equality. The source snapshot's canonical skill has the same digest.

Scope: 11 class labels in family order, every degree 2 through 10, every distinct root. There are 594 rows from 99 squarefree polynomials: 506 real roots (including exact integers -2 and 2), and 88 nonreal roots. The degree-one common zero is stated in prose. No rational or continuous grid is selected. See argument-selection.md for each parameter's family and bounds.

Generator: Sage 10.9, numberdb 0.1.10. Exact integer finite products produce the needed Fourier coefficients; the monic Faber recurrence produces the polynomials. QQbar isolates the roots exactly. The comparison recognizes conjugates before using certified strict real-part separation, with exact algebraic fallback for unresolved ties. Arb converts algebraic roots to enclosures at 429 bits, 96 extra working bits beyond 100 decimal digits. Every nonzero approximate component retains at least 427 bits of relative accuracy. The exact rational roots are recognized algebraically, not from trailing zeros.

The repository's dry_run.py and check.py ran before writing any numbers. All 594 entries passed the exactness/error-bound and prose checks. Raw working-precision values measured 142.5 KB with longest value 307 characters. The final 100-digit spellings have longest value 213 characters. The actual serialized block measurement is recorded in the handoff.

Independent root verification in checks.py does not use QQbar isolation to prove existence. It uses 504 exact rational sign brackets, two exact rational evaluations, two sign brackets on the imaginary axis, and 86 Taylor/Rouché certificates at 768 bits. For the complex certificates, two independent Newton steps choose a rational disk center; Taylor bounds prove that the disk holds exactly one zero and lies inside the written decimal rectangle. Disjoint rectangles and the polynomial degrees prove completeness. Both the full generated enclosures and the independent certificates fit the 100-digit intervals.

Known-answer controls are sqrt(2), the integer 2, i*sqrt(2), and 1+i. Deliberately displaced real and complex rectangles must fail. A nonfinite ball is explicitly rejected. An initial attempt to center every disk at the decimal midpoint failed because the inscribed disk can miss a root inside the rectangle; the final rational Newton-centered disk resolves this geometric issue without changing any generated value.

All 99 exact polynomials agree with the attached T635 snapshot. Laurent multiplication independently checks every principal part through the constant term. Four explicit external examples (degrees two and three for 1A and 2A) from Bannai–Kojima–Miezaki, pp. 24 and 26, agree after the documented shifts by 744 and 104. The first nine positive identity coefficients agree with T79, and nine 13A coefficients agree with OEIS A034319. The checker verifies exact squarefreeness, multiplicity-one comments, conjugation, ordering, root sums and root products over the whole selection. These checks are rerun against the stored database document after filling.

Sources checked:
- Bannai, Kojima and Miezaki, On the zeros of Hecke-type Faber polynomials, Kyushu Journal of Mathematics 62 (2008), 15–61, DOI 10.2206/kyushujm.62.15; normalization pp. 16–17 and Definition 2.2, recurrence §2.2, examples pp. 24 and 26, real-root theorem §4. https://www.jstage.jst.go.jp/article/kyushujm/62/1/62_1_15/_pdf
- Matsusaka, Research in Number Theory 3 (2017), article 23, DOI 10.1007/s40993-017-0090-x, §1 eta formulas and prime genus-zero list. https://link.springer.com/article/10.1007/s40993-017-0090-x
- OEIS A034319, zero-constant 13A Fourier expansion. https://oeis.org/A034319

Companions T634 and T635 were read from the authenticated API and their actual slugs from their file pages. They are drafts at build time, not published tables. Existing tags were confirmed using the tag API. `moonshine` is absent; it is proposed for T79, T255, T634, T635 and T636, without modifying those tables. Initial incorrect hand-written tag/lookup routes were replaced with the package's actual API calls; successful evidence is in screen-finish.json and lookup-final.json.

Reproduce with `sage -python checks.py` beside generate.py, T635.json and T79.json. To check the database, set NUMBERDB_API_KEY and run `sage -python checks.py --stored`; draft reads require authorization. The source and checks do not require a repository checkout. No table has been offered, reviewed or published by this run.