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5198 bytes, as of the version from 2026-10-28 09:51 (current). Recorded here, not run.
# Argument selection for Schur Q-polynomials (T577)
Policy read: `agents/runs/20261028T093728Z-build-skills/manifest.json`.
Both archived local and live skills have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The canonical source snapshot has that same hash. `skill-evidence.json`
records the manifest and the verified digests. This is the builder's rationale,
not the separate critic's acceptance or argument-review sidecar.
## Mathematical object and parameter roles
The stored quantity is the fully expanded Schur Q-polynomial
Q_lambda(x1,...,xn), in the usual Q normalization, with integer coefficients.
The generator constructs q_r from product_i (1+x_i*z)/(1-x_i*z), uses
Q_(a,b)=q_a*q_b+2*sum_{j=1}^b (-1)^j q_(a+j)*q_(b-j) for a>b>=0,
and applies the Pfaffian after appending one zero when the strict partition
has odd length. No variables or parameters select a second quantity.
* `n` is a **discrete_index**, the positive alphabet size. It controls the
number of variables and the allowed tableau labels. The stored alphabets
are n=2,3,4,5. These give explicit expansions across several small alphabets
before increasing the degree of a single alphabet. This is not a continuous
argument relabelled as an integer.
* `lambda` is an **arithmetic_structure**, a strict partition of the degree
with length at most n. Its parts specify a shifted Young diagram and the
shape index of the Schur Q-function. For every included degree and alphabet
we include every admissible shape. There is no redundant degree parameter,
and no choice of shapes based on the appearance of their coefficients.
* The x variables are formal polynomial indeterminates, not numerical
arguments. There are no selected rational, angular, coupling or probability
inputs, and no exploratory reference_sample group. Q has no additional
formal deformation parameter. The t=-1 Hall-Littlewood relationship fixes
the named object rather than introducing a free parameter.
## Extent, breadth and readability
Final enumeration, in increasing n and degree, then descending lexicographic
partition order:
| n | included degrees | strict shapes / entries |
|---|---|---:|
| 2 | 2 through 6 | 11 |
| 3 | 2 through 6 | 12 |
| 4 | 2 through 6 | 12 |
| 5 | 2 through 4 | 5 |
Total: 40. Degree 6 includes the first three-row strict shape (3,2,1).
The selections include one-row shapes and every admissible multirow shape,
not merely named staircases. In the degree range there cannot be strict
four-row shapes, whose least weight is 4+3+2+1=10. Increasing the alphabet
still changes every polynomial and permits more marked-tableau weights.
The issue's proposed n=2..5, degree=2..6 gives 47 entries, a 34.52 KiB
measured entries estimate, and a longest value of 3542 characters at
n=5, lambda=(5,1). We revised only the five-variable bound, preserving the
complete degree slices for all other alphabets. The five-variable degree-5
slice reaches 1937 characters; degree 4 reaches 957. For n=4, degree 6
reaches 1299 characters, while degree 7 reaches 1977. The final maximum of
1299 is long but still inspectable, and keeps the three-row shape in two
alphabets. These are noncanonical choices based on readability, independent
of any numeral base, not a storage limit or entry-count target. The final
block estimate and full size evidence are in numerical-evidence.md and the
run's measure-selection.log.
We do not extend n=2 or 3 to higher degrees merely because those expansions
are short. Complete low-degree slices across four alphabets provide a
coherent initial reference. The parameters' domain remains all positive n
and all strict partitions with length at most n, allowing later extensions.
Degree zero, degree one, and the one-variable alphabet are omitted from
stored entries because their general boundary formulas suffice:
Q_empty=1, Q_(1)=2*sum x_i, and Q_(d)(x1)=2*x1^d for d>=1.
No other simple members inside the chosen slices are removed.
## Sources actually checked
* H. Rosengren, *Schur Q-polynomials, multiple hypergeometric series and
enumeration of marked shifted tableaux*, arXiv:math/0603086v3,
Section 2.1, equations (2.1)-(2.3), pp. 3-4.
https://arxiv.org/pdf/math/0603086
This specifies strict shapes, finite alphabets, the Q normalization,
marked shifted-tableau weights, and the symmetrization formula.
* S. Okada, *Pfaffian Formulas and Schur Q-Function Identities*,
arXiv:1706.01029, Section 1 and Theorem 4.1, Lemma 4.2.
https://arxiv.org/html/1706.01029v1
This fixes the one-row generating function, Pfaffian and stability.
* SageMath reference, *Hall-Littlewood polynomials*, P() and Q() examples.
https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/hall_littlewood.html
This independently identifies the t=-1 Schur Q basis and its relationship
to the subalgebra generated by odd power sums.
These sources establish the family and conventions. They do not prescribe
our finite cutoffs, and no claim of a particular application for every small
shape or alphabet is needed. The separate critique must independently check
this selection and the generator's actual enumeration and dependencies.