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# Argument selection for T593
The table holds one quantity: the dimensionless heat capacity per spin,
C/(mn k_B), at zero field and K_c=log(1+sqrt(2))/2 on the simple vertex
grid P_m square P_n. All bonds have equal positive coupling and both
boundaries are free. Side lengths count vertices, not edges.
## Every free parameter
Both m and n are `discrete_index` arguments. They select the actual finite
spin system and its geometry, not samples of a continuous function. The
family domain is 2 <= m <= n, with n unbounded. Interchanging the lengths
rotates the graph, so only the ordered representative is stored. The lower
bound excludes chains and isolated spins: the intended domain is rectangles
with at least one square face. The chain remains a useful independent control.
The finite selection includes every pair 2 <= m <= n <= 6, in lexicographic
order. This gives five squares and ten nonsquares, including the full set of
width-two ladders through length six and intermediate aspect ratios. Breadth
across these shapes is retained before extending only the square sequence.
The inputs are sizes of finite models that can be independently enumerated or
used as finite-system thermodynamic reference examples. No claim is made
that each particular rectangle is independently famous.
Six is a practical, noncanonical bound, not a distinguished physical length.
It limits the row-transfer space to 64 spin states and the zero-field cut
polynomial to degree 60. The independent cycle-space verification at the
largest size enumerates 2^25 even edge subsets, making a complete all-shape
check practical. The 15 values are a consequence of that coverage and cost,
not an entry-count target. All degrees of elongation possible under this
small-size bound are included. Wider or longer rectangles are meaningful
extensions after verification cost has been assessed.
## Fixed inputs and conventions
K_c is intrinsically selected by the bulk square-lattice transition. It is
not a varying parameter or a proxy for a finite rectangle's heat-capacity
maximum. Zero field is the spin-reversal-symmetric slice. Free boundaries
specify a documented finite-size model, not a claim that they are the only
meaningful boundary choice. J has been removed by using K and dividing C by
mn k_B. There is no free field, anisotropy, continuous temperature or reference
mesh. Neither `reference_sample` nor a mixed-category assessment is needed.
## Supporting sources checked
- Wu, Zheng, Izmailian and Guo, arXiv:1308.2040, Section II equations (1)-(5),
defines the partition sum and specific heat per spin including the squared
inverse-temperature factor. Section III studies rectangular square-lattice
systems at K_c with varying aspect ratios. These establish the quantity,
normalization and geometric family. Their larger-size fits are not used as
exact values for the small rectangles here.
- Peters and Regts, arXiv:1810.01699, equations (1.1)-(1.2), defines the
vertex-subset cut polynomial with field fugacity first and bond weight
second. Its zero-field specialization is the exact combinatorial object
used by the generator.
- NumberDB T154, square,coupling, supplies the same bulk critical coupling.
Live T591 and T592 establish the companion tables' vertex-grid, free-boundary
and parameter-order conventions. They do not justify the bound by precedent;
the independent coverage and verification-cost reasons are given here.
The default generator, Parameters and completeness note all describe this
same selection. generate.py is self-contained; no attached count data are
used as input. checks.py and counts.json are verification evidence.
Policy: runner archive agents/runs/20261028T232356Z-build-skills/manifest.json.
Both archived local and live skill files, and the source canonical file,
have SHA-256 28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.
This build rationale is not an independent critique or an acceptance sidecar.