# Build handoff: T696 Built **Odd period polynomials of rational newforms**, https://numberdb.org/T696, from family issue #309. The table is populated and remains an unpublished, in-progress draft. It has not been offered, reviewed or published. The controller must arrange the separate independent critique and any repair before offering it. No independent argument-review sidecar has been authored by this build actor. ## Instruction authority Read and byte-checked both archived versions in `agents/runs/20261109T182554Z-build-skills/manifest.json`. The local and live skills match at SHA-256 `886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`. The manifest itself has SHA-256 `c879ba3e40020535135932a0ac7d93e894d04b267cf416aa702c3c96617ec363`. `skill-evidence.json`, `manifest.json`, and both numerical evidence files identify this archive. No remembered or later skill version was substituted. ## What was built There are 18 exact primitive integer polynomials, with positive leading coefficient, proportional to the odd-in-X part of the identity-coset period integral. The original variable X and all forced factors are retained. The minus superscript denotes odd powers, not the functional-equation sign. Exact rational modular symbols, not numerical rational reconstruction, produce the coefficients. The longest value is 138 characters. The enumeration is the family proposal's two meaningful discrete slices: all rational forms at level one in nonempty even weights through 26, and all rational weight-twelve newforms at exact levels 2 through 10. Six forms come from level one and twelve from higher level. All odd parts are nonzero. The full spaces have 33 embedded forms; rational filtering preserves their lexicographic indices and gaps. Level two has no weight-twelve newforms; level seven and level-one weight 24 have no rational forms. Argument-selection.md justifies N, k and i separately and their dependencies, the noncanonical finite bounds, the coefficient-field restriction, and breadth before depth. This is a `discrete_index` selection, not exploratory continuous sampling. The family settled the convention; this run confirmed that Sage sign -1 selects the required moments, that all eighteen odd parts are nonzero, and that their actual degree is k-3. The reciprocal action still uses degree k-2. Source-notes.md records the derivation and the documentation typos resolved by coefficient identities and the independent Delta control. ## Checks actually run * Cheap screening: already_here, already_asked, authenticated paginated catalogue, polynomial API lookup, and live tag queries. No duplicate found. * Family: read live T694 and T695 and their attached reproducible sources; read the relevant critical-value, Fourier, Hecke and norm companions. Canonical URLs came from live search results or actual long-URL links. * Dry run: all 18 values exact; no float coefficients or prose findings. * Roster: independent full/new dimension formulas agree in all 21 spaces; a separate PARI decomposition agrees through Sturm bounds on all 33 forms. All 33 indices agree exactly with T694. The eighteen rational identities agree with T695. Sage and PARI agree on 100 Fourier coefficients per rational form. An integer Delta eta-product controls those coefficients. * Exact relations on all entries: primitivity, positive leading coefficient, odd parity, degree, and the Fricke reciprocal relation. Every level-one entry also satisfies the order-three relation and vanishes at +/-1. * Nonzero controls: the published Delta odd polynomial and three implemented exponential integrals at t=2,T=3 for s=1,2,3, compared with exp(-6) times 1/2, 7/4 and 25/4, respectively. * Independent analytic comparisons on every coefficient ratio: PARI at 160 and 220 working digits, and explicit split Mellin sums at 150/450 and 190/600 working-digit/Fourier-term settings. Largest higher-setting errors are below 6.6e-229 and 1.7e-198, respectively. The analytic errors are not certified; they corroborate the exact modular-symbol construction. * The displayed Sage program was executed and its Delta answer checked. * After filling, verify(sample=None) reports 18/18 matched, zero differing, missing or extra. The complete exact and independent analytic checks were rerun on values read back from the API, with all 18 passing. All prose sections read back identically. Every initial attachment was fetched back and its SHA-256 checked. * The filled-draft audit returns `clean: true`, `findings: []`. This API audit does not perform the database command's optional outward-link crawl. ## Attachments and critique inputs The draft carries generate.py, checks.py, check.py, dry_run.py, argument-selection.md, roster.json, T694-roster.json, source-notes.md, numerical-evidence.json, skill-evidence.json and manifest.json. Additional read-back evidence, its runner, this handoff and the clean audit are attached. The generators use named Sage imports and initialize numberdb.sage first. Their first docstring line identifies T696. The database is authoritative; local files under this run's build-work directory are working artifacts. The live fingerprint command was run, with no acceptance verdict: `python3 agents/argument_review.py fingerprint T696 --generator generate.py --generator checks.py --generator check.py --generator dry_run.py --generator argument-selection.md --generator roster.json --generator T694-roster.json` Selection SHA-256: `524a5f3a408128a33f19f3492cb2835d326c0b5e39148d4f03844d6436e28640`. Exact source hashes are in selection-fingerprint.json. The independent critic must perform its own numerical and selection assessment, obtain current live hashes, and use its own NUMBERDB_RUN_ID in the required sidecar. Existing tags modular form, number theory and polynomial are used. The tag **period polynomial** is absent; it is proposed jointly for T694, T695 and T696. This build does not create the tag or edit companions. T694 and T695 are explicitly identified as drafts in the links. Later family review can backfill their links to this newly built odd-part table. No published table was edited, no extra table was built, no batch-exhausted marker was created, and no queue completion or request closure was claimed. In particular #262 remains a broader partially addressed request. No code was deployed or committed. The initial submission helper stopped before writing because Entries is not a top-level package export; it was corrected to use the package's internal writer, and the successful run filled the numbers once. Both attempt logs are preserved.