# Sources and conventions checked for T591

The run used the archived local and live skills named in
`agents/runs/20261028T210416Z-build-skills/manifest.json`. Their common SHA-256 is
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The attached `skill-manifest.json` reproduces that manifest.

Peters and Regts, arXiv:1810.01699v3, equations (1.1)-(1.2), was read and
archived. Its field fugacity is the first argument, and its cut-edge weight is
the second. Substituting zero field gives the exact F(b) used here.
Liu, Sinclair and Srivastava, arXiv:1807.06577v2, Introduction and equation (1),
was read and archived. Their beta is exp(-2K), the present b. Their Fisher-zero
definition refers to the interaction polynomial.

Wu, Izmailian and Guo, Physical Review E 86, 041149 (2012), abstract and first
page, was read. Its free rectangular geometry and aspect-ratio discussion
support the family choice, not a demand for each selected rectangle. The
listed erratum concerns finite-size claims not used here.

Sage's official complex_roots documentation states that the disjoint complex
intervals each contain one root, with squarefree decomposition and retained
multiplicities. The QQbar documentation states exact algebraic arithmetic and
certified interval conversion. Both documentation pages were readable through
the web research tool, although a separate urllib archival request returned
HTTP errors. The run's numerical environment used Sage 10.9; the online manuals
were for later Sage versions, so all methods used were also exercised locally.

Sokal, arXiv:math/0503607, Sections 2.2 and 2.5, establishes the Potts/Tutte
connection cited in Similar tables. In this normalization, expand the edge
factor b+(1-b) delta_equal and sum over two spin labels. For a connected graph,
F(b)=b^E Z_RC(2,(1-b)/b), with
Z_RC(q,v)=q v^(N-1) T_G(1+q/v,1+v).
Thus F(b)=2 b^(E-N+1) (1-b)^(N-1)
T_G((1+b)/(1-b),1/b), interpreted as a polynomial after cancellation.
This general derivation establishes the relation for every selected rectangle;
the table does not transcribe numerical values from T126.

Other displayed identities are either checked as exact polynomial identities
for all 15 rectangles or follow directly from the edge weights. In particular,
every spin assignment has bond sum E-2d(S), proving the physical normalization.
The high-temperature expansion follows by writing each edge factor as
((1+b)+(1-b) sigma_u sigma_v)/2 and summing spins; only even edge subsets survive.
Reversing one bipartite class takes d to E-d and proves reciprocity. Positive
coefficients and F(0)=2 prove absence of positive real zeros.

Live corpus checks found no existing Fisher-zero table, and lookup of two
2-by-2 zeros returned no matches. T154, T155 and T126 were read, and their
actual URL slugs were taken from search results. The Lee-Yang, specific-heat
and Binder companions in issue 232 were still proposals, so no nonexistent
links were added. The tags statistical mechanics, physics, zero and algebraic
exist. An Ising model tag is proposed for this table, T154, T473, T474, T476 and
the three planned companions; it was not invented as a live tag.