# T497 build handoff Draft: https://numberdb.org/T497 Title: Minkowski-Siegel masses of the genera of unimodular lattices Family: numberdb-data issue #206. The draft was created and filled in this run. It is unpublished and has not been offered for review. The separate critique and repair stages remain the controller's responsibility. ## Scope and convention 37 exact rational masses: type I (odd), dimensions 1 through 32; type II (even), dimensions 0, 8, 16, 24, 32. The quantity is the sum of reciprocal orders of full integral isometry groups over the isometry classes of a positive definite integral unimodular genus. Parity indexes the genus, not a second quantity. Dimension zero has mass 1. Orientation-reversing maps are included. No decimal approximations or digit count are used. The range follows the classic reference tables of Conway and Sloane (1982), including E8, the two dimension-16 even classes, the 24 Niemeier classes, and the dimension-32 transition to large masses. Measurements at dimensions 32, 48, 64 and 96 gave longest entries of 116, 298, 602 and 1604 characters, respectively. The chosen 37-entry table occupies about 3.4 KB including comments. The parameter definitions describe the whole infinite domain. The family did not settle the rank-zero inclusion, parity parameterization, or upper dimension. These follow the classical mass literature and its published reference range. The classical-lattice `family` parameter is not used because a genus is not an individual named lattice. ## Source discrepancy resolved The 1982 Theorem 2 and Table 2 give 1 at odd rank one. This conflicts with the full-isometry definition: the only class is Z, and its full group is {+1,-1}, so its mass is 1/2. King (2003), Table 6, Finch (2005), page 3, and Sage's general routine give 1/2. The draft has a comment explaining the exception, and the generator handles it directly. Every other entry matches the exact fractions in the original tables. Primary PDF pages 4, 6, 7 and 8 are preserved as PNG evidence, alongside the downloaded PDFs. ## Checks actually run - The required dry_run.py, using agents/sage.sh, computed all 37 entries and passed exactness, named-ring, prose, and size checks. - SageMath 10.9's general genus-mass routine agreed exactly for all 37 entries. Controls for ranks 1 and 2 and the E8 genus ran before the full comparison. - 36 exact fractions were independently transcribed from the 1982 Tables 2 and 4 and compared; the rank-one exception was checked from its definition. - Independent class sums passed for odd ranks 1 through 11 and even ranks 0, 8, 16 and 24. The rank-24 check used all 24 orders read from T496. - The odd/even identity passed at dimensions 8, 16, 24 and 32. - Doubling the E8 mass and omitting the Leech class both failed the controls. - Generator.verify(sample=None) on the filled draft reported 37/37 matched, zero differing, missing or extra entries. - The general genus-mass, published-value, class-sum and parity-identity checks all passed again on the values read back from T497. The Niemeier orders were read live from T496 again for this pass. The generator, checks, helper, published-value transcription, Niemeier order snapshot and evidence were attached to T497. source_values.json is a check fixture; the generator computes its values independently. ## Audit and critique items The site audit reports exactly one finding, and `clean` is false: the symbolic parameter `parity` has labels I and II, each with several entries. This is a structural heuristic rather than a numerical failure. These labels specify which unimodular genus is the argument of the same mass function. The definition, convention comment and number-header all say so. The parameter was retained, without disguising its values or splitting the requested mathematical family. The separate critique should assess this rationale. No other audit findings remain. T495 and T496 are currently drafts, accessible to the program account. T497 uses their stable T-number HREFs, including T495#E,8 read from its Numbers, because the public search cannot return their slugs. The audit accepts these references between drafts. Their publication status should be considered when the controller later offers this table; no published table was edited. The proposed `lattices` tag does not exist. Existing `discrete geometry` and `number theory` tags are used. A future `lattices` tag would fit T147, T150, T402, T495, T496 and this table; no tag was invented in this run. ## Artifacts - generate.py, checks.py, check.py, source_values.json, niemeier_orders.json - numbers.json and stored-document.json - dry-run.log, preflight-checks.log, readback.log, audit.json - screening.json and successful lookup-*.json/tag-*.json responses - conway-sloane.pdf, king.pdf, finch.pdf and source-page-*.png - evidence.txt with source hashes; dependency-versions.json - lessons.md with the rank-one exception and Sage initialization findings No publication, offer-for-review, review, deployment, push, commit, or pipeline source edit was performed.