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3910 bytes, as of the version from 2026-10-28 21:43 (current). Recorded here, not run.
# Argument selection for T591
This assessment follows the two byte-identical archived skills in
`agents/runs/20261028T210416Z-build-skills/manifest.json`. Both the local and
live copies have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The manifest is also attached as `skill-manifest.json`.
The quantity is the zero set of the zero-field cut polynomial for the simple
rectangular graph P_m square P_n. There is one quantity, expressed throughout
in the bond variable b=exp(-2J/(k_B T)). A path P_m has m vertices. All bonds
have the same coupling; each occurs once. Both directions have free boundaries.
No field, anisotropy, continuous coupling slice, or probability is sampled.
* **m: discrete_index.** The shorter side length specifies an actual finite
lattice. The domain is 2 <= m <= n; excluding m=1 removes path graphs,
whose only Fisher zero is b=-1. The first build takes m=2,...,6.
* **n: discrete_index.** The longer side length determines the size and aspect
ratio. The domain is n>=m. The first build includes every m<=n<=6, once per
rotational equivalence class. This gives five squares, five ladders (including
the 2 by 2 square), and six other nonsquare shapes. Thus there are 15 shapes,
with overlap between the square and ladder descriptions, not 16.
* **j: intrinsic.** The index starts at 1 and runs over all distinct roots,
ordered lexicographically by exact real and imaginary parts. Both conjugates
and all real roots are included. Multiplicity belongs in the comment; it
does not duplicate a row. The generator recomputes algebraic root identities
from exact integer coefficients, rather than using decimal approximations
as parameter identities.
Six is a practical, noncanonical cutoff, with at most 64 row states and degree
60. It is independent of numeral notation. Covering every small shape gives
breadth in geometry before increasing the depth of the square sequence alone.
The complete root sets retain the entire finite-volume zero pattern, rather
than just a leading root or a modulus. This is a bounded part of a discrete
model family, not a reference mesh in continuous coupling space and not a
claim that all 15 geometries have individually documented applications.
Sources checked:
1. H. Peters and G. Regts, *Location of zeros for the partition function of the
Ising model on bounded degree graphs*, arXiv:1810.01699v3, equations
(1.1)-(1.2), establishes the cut-polynomial normalization and conversion
from field and bond Boltzmann weights. https://arxiv.org/pdf/1810.01699
2. J. Liu, A. Sinclair and P. Srivastava, *Fisher zeros and correlation decay in
the Ising model*, arXiv:1807.06577v2, Introduction, equation (1), defines
Fisher zeros in precisely this interaction variable, called beta there.
https://arxiv.org/html/1807.06577
3. X. Wu, N. Izmailian and W. Guo, *Finite-size behavior of the critical Ising
model on a rectangle with free boundaries*, Phys. Rev. E 86, 041149 (2012),
abstract and opening discussion, supports free rectangles and aspect ratio
as a studied finite-size family. It studies thermodynamic observables, not
the specific 15 zero sets selected here. Its erratum does not affect the
geometry or the model definition used here.
https://doi.org/10.1103/PhysRevE.86.041149
4. Sage documentation, `complex_roots`, and the algebraic-number implementation,
specify certified isolated roots, multiplicities, and exact comparisons.
https://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/complex_roots.html
https://doc.sagemath.org/html/en/reference/number_fields/sage/rings/qqbar.html
The build's completeness note and default enumeration use this same selection.
Independent argument-selection approval is reserved for the separate critique
stage; this file is the builder's justification, not an approval sidecar.