# Argument selection for T594

This build follows the archived canonical skill whose local and live SHA-256
are both `28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
The authoritative instruction manifest is
`agents/runs/20261028T234955Z-build-skills/manifest.json`; the corresponding
run evidence is attached as `skill-evidence.json`.

The named quantity is the Binder cumulant
U = 1 - <M^4>/(3<M^2>^2), with total magnetization M. Selke,
cond-mat/0701515, Section 2, equation (4), explicitly supplies this convention.
Equations (1) and (2) give the spin normalization, isotropic nearest-neighbor
Hamiltonian, and bulk critical temperature. Sections 1 and 3 discuss dependence
on boundary conditions and shape; the free-boundary study in Section 3 is
primarily for squares and circles. We do not claim that paper tabulates each
of the fifteen small free rectangles chosen here. These are finite-size values,
not thermodynamic limits or critical-value thresholds in statistical testing.

## Every free parameter

- `m`, type Z, category `discrete_index`: number of vertices in the shorter
  path in P_m square P_n, m >= 2. It changes the actual lattice and is not an
  integer encoding of a sampled temperature or field. Its practical range in
  this build is 2 through 6.
- `n`, type Z, category `discrete_index`: number of vertices in the longer
  path, n >= m. Each n changes the finite lattice. For every selected m, this
  build includes all n from m through 6. The m <= n convention removes
  rotations, not distinct graph shapes.

The joint selection is all fifteen pairs 2 <= m <= n <= 6, in lexicographic
order. It covers five squares, the five two-row ladders (including the smallest
square), and the intermediate rectangles. There are ten nonsquare shapes in
all. This breadth gives comparisons across aspect ratio and boundary fractions
before spending the budget on deeper square-only sequences. It is an initial
complete size-bounded discrete family, not a Cartesian product of unrelated
simple real arguments and not a requirement that each shape be famous.

Six is a practical, noncanonical cutoff. The largest graph has 36 sites and
60 bonds, and the widest row transfer has 64 states. Both exact spin-count
and independent edge-subgraph computations finish for all fifteen shapes in
under a second after Sage initialization on this builder. The written values
have 100 significant digits and at most 102 characters each. Larger sizes are
possible extensions, but first covering this small family gives reproducible
finite-lattice benchmarks without implying thermodynamic extrapolation. The
cutoff is not chosen to reach a target number of rows or the storage ceiling.

## Fixed choices and sources

Free boundaries in both directions, uniform ferromagnetic J > 0, spins +/-1,
and one bond per nearest-neighbor pair define the graph model. They follow the
family's conventions and are compatible with Selke's free-boundary model.
Zero field is the spin-reversal symmetric slice. Kc = log(1+sqrt(2))/2 is the
bulk square-lattice critical coupling, not a fitted finite-size transition.
No free continuous coupling, anisotropy, field, probability or root index is
hidden in the enumeration. There are no exploratory reference groups requiring
a `reference_sample` or `mixed` classification.

Peters and Regts, arXiv:1810.01699, equations (1.1)-(1.2), define the cut
polynomial in field fugacity and bond weight. Here z=exp(-2H), b=exp(-2K),
and z=1, b=sqrt(2)-1 at the selected slice. T154's `square,coupling` entry
independently identifies this coupling. The siblings T591, T592 and T593
fix identical vertices, boundaries and parameter order. The generator
recomputes all coefficients; it does not consume sibling rounded values.

Checked primary sources:
- https://arxiv.org/pdf/cond-mat/0701515 (Section 2, equations (1), (2), (4);
  Sections 1 and 3 for geometry and boundary dependence).
- https://arxiv.org/pdf/1810.01699 (equations (1.1)-(1.2)).
- https://numberdb.org/Critical_couplings_of_the_Ising_model_on_lattices
  (square-lattice coupling entry, key `square,coupling`).

This file is the builder's justification. Independent argument-selection
assessment, with the live selection and source fingerprints, belongs to the
separate critique stage. No acceptance sidecar is supplied by this build.