# Sources and conventions for T680

Read against the archived canonical instructions in
`agents/runs/20261102T145807Z-build-skills/manifest.json`.
Local and live SHA-256 both:
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.

The build read numberdb-data#301 with `python3 agents/queue.py show 301`.
It also read live documents T679, T674, T675, T25, T178, T348, T665 and T323,
and the source/check attachments of T679. Its period values are comparison
inputs, not copied canonical-coordinate values. The basic ordinary Taylor
recurrence in numerics.py adapts T679's checks.py; the logarithmic transport
and conifold Frobenius matching are new. The independent check uses mpmath's hypergeometric parameter derivative on
smooth fibers and a gamma-ratio asymptotic tail at the conifold, and does not
use that recurrence.

Checked primary sources on 2026-11-02:

* https://arxiv.org/html/1102.5375v1, Krattenthaler and Rivoal, Section 1,
  equations (1.1)–(1.4): cyclotomic scaling C_N, factorial-normalized
  hypergeometric coefficients, harmonic coefficients, and canonical coordinate.
  Their variable x is z/C_A here. Their mirror map is the compositional inverse;
  this table follows the common alternative terminology only in its title and
  keywords, and explicitly defines the forward canonical coordinate.
  Sections 3–4 give analytic continuation and the conifold singular expansion.
* https://arxiv.org/html/1705.01663v4, Long, Tu, Yui and Zudilin, Table 1 and
  Section 5: the fourteen parameter multisets and their geometric context.
* https://arxiv.org/html/2109.00027v1, Roberts and Rodriguez Villegas,
  Section 10, "Wild, tame, and good primes", and Section 11: dependence of bad
  primes on the rational fiber and the corresponding Frobenius polynomials.
* https://arxiv.org/html/2203.09426v1, Bönisch, Klemm, Scheidegger and Zagier,
  Section 2.1, Table 1, and equations (2.4)–(2.5): normalized Frobenius solutions
  versus an integral symplectic period basis. Their conifold variable is scaled;
  ours always has conifold z=1. No unspecified geometric period is substituted.

The corpus screening used `already_here` and `already_asked`, text queries for
mirror/canonical coordinate/Calabi/hypergeometric, and api/lookup at
0.5814620307094641, 0.5797235865463786, 0.5788162943265502. The number lookups
returned no results. No duplicate named family was found. Existing tag spellings
are number theory, special functions and special values. Calabi-Yau was absent
from both tag pages; it is proposed for T674, T675, T679 and T680 in the handoff,
and no existing table is edited or retagged.

All companion slugs were read from search results or the authenticated table's
file page. T679 is a private companion draft during this build; links from this
draft to it remain useful for family review. Both drafts remain unpublished.

Elliptic control sources: DLMF 20.9(i) gives the nome in terms of complete
elliptic integrals, and DLMF 19.7 gives the connection formulas used for the
negative and upper-bank controls. DLMF uses the modulus; T25 and this run use
the parameter m=k^2. The rank-two Q is sixteen times the elliptic nome.