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4191 bytes, as of the version from 2026-10-30 10:01 (current). Recorded here, not run.
# Sources and convention checks for T622
Read on 2026-10-30. The family proposal was read through `queue.py show 248`;
its conventions are not substituted for checking the sources.
1. Enrico Toffoli, *Rho invariants for manifolds with boundary and low-dimensional
topology*, University of Regensburg dissertation (2019).
https://epub.uni-regensburg.de/40688/1/thesis.pdf
(also https://d-nb.info/1194646867/34).
Checked Section 3.3.4, pp.84–88, especially equation (3.7), Theorem 3.3.27,
and the definition of s_{x,y}. The theorem states eta=-4*s_{0,u/p} for the
round metric. Appendix A.1, pp.135–138, fixes the quotient action and the
marked generator; coordinate exchange sends the character exponent to its
inverse-residue multiple. Appendix A.2 defines the sawtooth at integers and
records classical reciprocity. The printed p>=0 in Theorem 3.3.27 includes
p=0 although the sum divides by p; this table uses only p>=2, as already
assumed in the surrounding discussion. No convention needs guessing.
2. Enrico Toffoli, *The Atiyah–Patodi–Singer rho invariant and signatures of
links*, Proceedings of the Edinburgh Mathematical Society 65(2) (2022),
404–440. DOI: https://doi.org/10.1017/S0013091522000153
https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/atiyahpatodisinger-rho-invariant-and-signatures-of-links/4BC45AB3FD65076115A82564AAAB23E0
Checked Section 2.2 for even-degree complex forms and unreduced spectral eta;
Section 2.4, Proposition 2.10 and equation (2.4), for the weighted lattice
triangle. Its k labels chi_{kq}; the checker converts k=u*q^(-1) modulo p.
The trivial character is handled separately in this check. In particular,
the q=1 formula 2*u*(p-u)/p-1 applies only to nontrivial u, despite the
statement of Corollary 2.11 not explicitly excluding the trivial character.
The defining difference and Proposition 2.10 give rho_0=0.
3. Murat Sağlam, *Holomorphic curves in the symplectizations of lens spaces:
An elementary approach*, Münster Journal of Mathematics 15 (2022), 389–440.
https://www.uni-muenster.de/FB10/mjm/vol_15/mjm_vol_15_12.pdf
DOI: https://doi.org/10.17879/21089689454
Checked Theorem 1.3 and the following paragraph on p.392: keeping orientation
reduces the equivalence to inversion, without the extra negation allowed
for unoriented lens spaces. This supports the q enumeration.
## Cotangent identity used by the independent numerical checker
Let f(a)=((a/p)) on Z/p. Direct summation of a geometric series and its derivative
at a nontrivial pth root gives
sum_a f(a)*exp(2*pi*i*j*a/p) = -(i/2)*cot(pi*j/p) for j != 0,
and the zero Fourier coefficient vanishes. Fourier inversion in
sum_a f(a)*f(q*a+u) consequently gives
eta_u = -(1/p)*sum_{j=1}^{p-1} cot(pi*j/p)*cot(pi*q*j/p)*cos(2*pi*u*j/p).
Pair j with p-j to obtain the real cosine form. This derivation independently
connects the exact finite sum to the ball expression; the weighted lattice
check with a reciprocity baseline supplies a separate exact computation.
## Live context and screening
T599 was read through `numberdb.table('T599')`; its address was independently
read from the search result: `Ray_Singer_torsion_of_lens_spaces`. The new table
uses the same quotient, parameter order and complex line labels. T599 removes
orientation because positive torsion ignores it; T622 retains it. The rho
proposal has no live companion at this screening, so no identifier is invented.
`already_here`, `already_asked`, text searches, four exact-value `api/lookup`
queries and `/tags` are archived in screening.json. The matches are unrelated
families, including Dedekind eta values rather than spectral eta invariants.
Existing tags analysis and number theory are used. The absent tag topology is
proposed for T622, the planned rho companion, and T599, without editing T599.
Both archived skills matched the manifest in
agents/runs/20261030T094236Z-build-skills/manifest.json. Their SHA-256 is
28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.
A copy with the absolute manifest path is attached as skill-manifest.json.