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5034 bytes, as of the version from 2026-10-28 09:23 (current). Recorded here, not run.
# Argument selection for T576, Zonal polynomials
This build follows family issue 228 and the matched archived skill versions in
`agents/runs/20261028T084155Z-build-skills/manifest.json`. Both the canonical
local skill and the archived live skill have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The manifest SHA-256 is
`468f5191d13df1eaecac71d659cc26682a40378c5982ce84143ed004d1933cc7`.
This is the builder's justification. Independent critique remains required.
The quantity is the real zonal polynomial in C normalization, not a numerical
evaluation, quantile, critical value, or sample of a continuous parameter.
The relation to integral Jack J at alpha=2 is Koev and Edelman, equation (3.1),
with the hook product in (3.2). The trace-power normalization is DLMF (35.4.6).
All polynomials are expanded exactly in QQ[x1,...,xn].
## Every free parameter
* `n` is a `discrete_index`: the alphabet size, equivalently the dimension of
the real symmetric matrix whose eigenvalues are x1,...,xn. DLMF 35.4(i)
and Jiu and Koutschan, Sections 1 and 2, define the family for arbitrary
positive dimension. Dimensions 2 through 5 give several small matrix sizes
and increasing numbers of parts, before extending one alphabet deeply.
The maximum dimension is a noncanonical readability choice, below the
six-variable search limit. One variable gives C_(d)(x1)=x1^d and is stated
in the boundary formula instead of stored as another monomial listing.
* `lambda` is `arithmetic_structure`: a partition encodes the symmetric-function
shape. Its length cannot exceed n. Every partition of each included degree
is retained, in descending lexicographic order, including row, column and
interior shapes whenever they are admitted. The degree d is its weight,
not a separately chosen parameter. Jiu and Koutschan, Definition 2.11,
Theorem 3.3 and Remark 3.4, support this partition-indexed structure.
We do not claim a separate application for every partition.
The symbols x1,...,xn are polynomial indeterminates, not arguments sampled
on a numerical mesh. The Jack parameter has the fixed value alpha=2 mandated
by the real zonal family; it is neither a free parameter nor a grid of couplings.
There is therefore no exploratory reference_sample group or hidden Cartesian
product of continuous inputs. These are bounded discrete polynomial families.
## Breadth, extent and measurements
The proposal's 87 entries (n=2,3,4,5, each degree 2 through 6) were all measured
before choosing the final extent. The longest expanded entry was 4,597
characters at n=5, lambda=(5,1). Complete degree slices were shortened as
follows, without dropping difficult individual shapes:
| n | included degrees | entries | longest retained polynomial |
|---|---|---:|---:|
| 2 | 2 through 6 | 14 | 95 characters |
| 3 | 2 through 6 | 21 | 494 characters |
| 4 | 2 through 5 | 16 | 949 characters |
| 5 | 2 through 4 | 10 | 1,157 characters |
The next omitted slices reach 1,705 characters at (n,d)=(4,6) and 2,254
at (5,5). The final 61 entries need about 18 KB in the mechanical dry run.
The largest retained polynomial is C_(4) in five variables: 70 monomials,
1,157 characters, with short rational coefficients. It remains inspectable
as a complete homogeneous quartic expansion. The issue's original rectangle
would be substantially harder to read even though it fits the byte limit.
The choices of dimension and degree bounds are practical and noncanonical,
independent of numeral base. Degree 6 is retained for the smaller alphabets
because the complete slices remain compact and include several interior
shapes. Breadth across four alphabet sizes comes before any greater degree.
The zero and first degrees are stated as C_empty=1 and C_(1)=sum x_i in the
boundary formula. No other simple entries in a retained slice are discarded.
The parameter domains remain all positive n and partitions of length at most n.
The completeness note, default enumeration and Programs example use precisely
these conventions. Enumeration is n increasing, then weight increasing, then
partition parts in descending lexicographic order, without zero padding.
## Checked sources
* NIST DLMF 35.4: https://dlmf.nist.gov/35.4 (definitions (35.4.2)-(35.4.3),
eigenvalue interpretation, trace-power normalization (35.4.6)).
* L. Jiu and C. Koutschan, Calculation and Properties of Zonal Polynomials,
https://arxiv.org/html/2001.11599v2 (Sections 1-3 and Example 3.5).
* P. Koev and A. Edelman, The Efficient Evaluation of the Hypergeometric
Function of a Matrix Argument, https://arxiv.org/html/math/0505344
(equations (3.1)-(3.2), real zonal specialization alpha=2).
* SageMath Jack symmetric functions documentation:
https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/jack.html
(Sage's zonal basis is monic P, so the C normalization requires scaling).
The run's source archive and skill-evidence.json record provenance. No
independent acceptance sidecar is written by this build actor.