# Source reconciliation for T603

Sources read in this run:

* Cunha and Freitas, arXiv:2404.12114v1, Section 3.1, Theorem 3.2,
  Appendix A equations (A.3)–(A.6) and Appendix B Table 2.
* Dowker, arXiv:hep-th/9312080v1, Section 2, especially equations (2), (3),
  (8), and the unnumbered d=3 difference on printed page 7.

The shared definition is unambiguous: unit radius, minimally coupled scalar
Laplacian, Dirichlet at the equator, eigenvalues k(k+d-1), multiplicities
binomial(d+k-2,d-1), k>=1. The actual convergence half-plane is Re(s)>d/2,
by multiplicity growth; restrictive half-planes printed during the shifted
calculation are not used to redefine it. No novelty claim is taken from the
2024 paper, since Dowker already treats hemispheres.

The 1993 page-7 difference has an apparent arithmetic misprint. At d=3,
write x=k+1. The multiplicity is (x^2-x)/2, and the Barnes residue at 2 is
-1/2. The two Barnes derivatives in equation (8) are respectively
(ζ'(-2)+ζ'(-1))/2 and (ζ'(-2)-3ζ'(-1)+2ζ'(0))/2.
The anomaly contributes +1/2. Hence

    Z_D'(0) = ζ'(-2) - ζ'(-1) + ζ'(0) + 1/2.

Together with Z_S3'(0)=2ζ'(-2)+2ζ'(0)+log(2), this implies

    Z_N'(0)-Z_D'(0) = 2ζ'(-1)+log(2)-1.

The printed difference instead adds +1/2, differing by exactly 3/2.
The independent shifted-spectral continuation implemented here gives the
constant -1 and excludes +1/2 by disjoint enclosures. It agrees with
Cunha–Freitas Theorem 3.2. This was not resolved by selecting whichever
numeric answer looked plausible. `relations.py` reproduces the comparisons.

Both primary and independent methods also agree with every one of the first
20 hemisphere values in the source's Table 2 after rounding to six decimal
places. Those source decimals are a check, not inputs to the generator.
`source-table2.json` records the extracted hemisphere columns.

The original PDFs/HTML are retained in the run work directory, with source
hashes in numerical-evidence.json. The archived policy manifest is
`agents/runs/20261029T070407Z-build-skills/manifest.json`; both skill versions
have SHA-256 `28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.