back to table · edit · history · where entries came from · files · download
5529 bytes, as of the version from 2026-10-28 22:35 (current). Recorded here, not run.
# Argument selection for T592
This is the build author's justification, not the independent critique verdict.
The critique stage must evaluate it independently and create its own fingerprint
sidecar. The governing skill is recorded in
`agents/runs/20261028T215936Z-build-skills/manifest.json`: archived local and live
copies both have SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
Both were read, and their bytes were verified against the manifest.
## The single quantity and conventions
The quantity is a distinct zero in field fugacity z of
P(m,n;z,b_c) = sum over S of z^|S| b_c^d(S), with b_c = sqrt(2)-1.
The graph is P_m square P_n; path sizes count vertices. Both boundaries are
free, edges appear once, spins are +/-1, and there are no periodic, fixed-spin,
ghost, diagonal, or parallel bonds. The physical normalization is
Z(K,H) = exp(KE+HN) P(z,b), z=exp(-2H), b=exp(-2K),
N=mn, E=2mn-m-n, K=J/(k_B T), H=h/(k_B T), J>0.
Peters and Regts, equations (1.1)-(1.2), explicitly gives this cut-polynomial
normalization. The positive square root chooses the real embedding.
The critical slice is the infinite square lattice's bulk coupling
K_c=log(1+sqrt(2))/2, not a finite rectangle's singularity or peak location.
T154's square,coupling entry and Codello, Table 3 and equation (9), support
this choice. The field fugacity is the dependent root coordinate; it is not
another independently sampled input. This is one named quantity, not a bundle
of temperature zeros, moments, and response functions.
## Every free parameter
- **m: discrete_index.** The number of vertices on the shorter side specifies
the finite graph. The domain is integers 2 <= m <= n. The finite selection
uses 2 <= m <= 6. It includes the ladder family, intermediate widths, and
square geometries. Width one would be a one-dimensional path, outside the
agreed two-dimensional rectangle family. No continuous coupling is hidden
behind an integer label.
- **n: discrete_index.** The longer side also specifies the graph. The domain
is integers n >= m. For each selected m the build includes every m <= n <= 6.
Omitting rotations removes graph-isomorphic duplicates. These are all 15
rectangles, including five squares and ten nonsquare rectangles. Wu,
Izmailian and Guo, opening paragraphs and the abstract, establishes free
rectangles and varying aspect ratio as a finite-size Ising setting. It does
not specifically prescribe our small-size cutoff or claim applications for
each of these 15 shapes.
- **j: intrinsic.** This is the index, starting at 1, of every distinct root of
the exact polynomial for each selected graph. Roots are sorted by exact
(real part, imaginary part), with multiplicities stated in entry comments.
Both conjugates and all real roots are included. It is not a choice of only
the leading zero or a subset selected by decimal appearance. Peters-Regts,
Introduction, defines the field-zero object and states the Lee-Yang circle
theorem. Sage's algebraic-number and complex-root documentation supports
certified root identities and exact comparisons.
## Bounds, breadth, and tradeoffs
The cutoff six is practical and noncanonical. It bounds the transfer state
space by 64 row masks, the full field polynomial degree by 36, and the reduced
real root problem by degree 18. It covers all small widths and aspect ratios
within that bound before taking a long sequence of squares or just the first
root. The resulting 245 roots have maximum serialized length 214 characters;
the entries block is about 67 KiB by the shared check.py estimate. This permits
full root sets while keeping the table readable and easily reproducible.
Larger rectangles are legitimate extensions after their cost and conditioning
are assessed. The cutoff is not inferred from a target row count.
These are discrete model indices plus an intrinsic root index. There is no
exploratory continuous reference group, rational coupling mesh, angular mesh,
or multidimensional product of unrelated simple arguments. Thus neither
reference_sample nor mixed is appropriate. No quantile family is involved.
The completeness note and the generator's default limit=6 use precisely the
same set, ordered lexicographically by m,n and then j.
## Checked sources
- H. Peters and G. Regts, *Location of zeros for the partition function of the
Ising model on bounded degree graphs*, https://arxiv.org/abs/1810.01699,
p. 1, equations (1.1)-(1.2) and the following Lee-Yang statement.
- X. Wu, N. Izmailian and W. Guo, *Finite-size behavior of the critical Ising
model on a rectangle with free boundaries*, https://arxiv.org/abs/1207.4540,
abstract and Introduction. The final paper has an erratum; no fitted
thermodynamic coefficients from either version are used here.
- A. Codello, *Exact Curie temperature for the Ising model on Archimedean and
Laves lattices*, https://arxiv.org/abs/1008.4720, Table 3 and equation (9).
- SageMath, *Algebraic numbers*, opening implementation description:
https://doc.sagemath.org/html/en/reference/number_fields/sage/rings/qqbar.html.
- SageMath, *Isolate Complex Roots of Polynomials*, algorithm description:
https://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/complex_roots.html.
The archived source bytes and hashes are listed in sources.json. The family
issue #232 was read before screening and construction. Its conventions were
retained; the new selection was reassessed against the archived current skill.