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# Source and convention evidence for T599
Read using the archived skill manifest
`agents/runs/20261029T032928Z-build-skills/manifest.json`; both versions have
SHA-256 `28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The two primary articles were read through their PDF text and downloaded
for this run. Original files, hashes and extracted text remain in the run's
artifact directory; `sources.json` records their URLs and SHA-256 digests.
* https://sigma-journal.com/2022/091/sigma22-091.pdf
A. Kitaoka, SIGMA 18 (2022), 091. Pages 2–3 identify the action exponents,
inverse residues and complex characters. Section 4 defines the de Rham
torsion with one half of the alternating degree-weighted zeta derivatives.
Equation (4.5) gives the scale change, which vanishes for an acyclic bundle.
Proposition 5.1 on page 15 states the positive product used here.
Theorem 1.1(2), equation (1.5), and Corollary 1.3 also supply a route through
Hurwitz zeta derivatives in dimension three. Applying
zeta'(0,a) = log Gamma(a) - log(2*pi)/2 gives the independent check
(2*pi)^2/[Gamma(a)Gamma(1-a)Gamma(b)Gamma(1-b)], with a=u/p and
b=(u*q^{-1} mod p)/p. The gamma check is not the rigorous enclosure;
Arb sine arithmetic is the enclosure calculation.
* https://www.uni-muenster.de/FB10/mjm/vol_15/mjm_vol_15_12.pdf
M. Sağlam, Münster J. Math. 15 (2022), 389–440. Theorem 1.3, page 392,
gives q' = +/- q^(+/-1) modulo p for unoriented diffeomorphism.
Additional algebraic checks used here are elementary deductions: evaluating
(x^p-1)/(x-1) at x=1 gives product_{u=1}^{p-1}(1-zeta_p^u)=p.
Multiplication by q^{-1} permutes nonzero residues, so the full character
product of the torsions is p^2. Complex conjugation and coordinate exchange
supply the stated symmetry identities. Exact reduction of T^2 modulo Phi_p
proves rationality or nonrationality without inspecting approximate digits.
The complex line, positive product, inverse residue and parameter order were
all specified by the family proposal and confirmed from the primary article.
No source disagreement or unresolved convention was found.
Live corpus checks found no analytic/Reidemeister lens-space torsion table.
The numeric lookup samples matched unrelated algebraic and spectral contexts.
The existing `algebraic` and `analysis` tags were confirmed. `topology` does
not exist; it would serve this table, T141 (hyperbolic knot volumes), and T313
(Chern–Simons invariants), among others, and is proposed for human consideration.
No tag was created. The root-of-unity link was read from T60's search result
and its document, rather than inferred from its title.