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6422 bytes, as of the version from 2026-10-28 08:03 (current). Recorded here, not run.
# Build handoff: T575, Jack polynomials
Draft: https://numberdb.org/T575 (slug Jack_polynomials).
Family: numberdb-data issue #228, read first with `python3 agents/queue.py show 228`.
The draft was filled by this run. It has not been published, reviewed or offered.
The controller must run a separate critique and any required repair before offering.
This build has not written an independent acceptance sidecar.
## Instructions and evidence
The archived skill manifest is
`/home/ubuntu/numberdb-campaign-20261026-hundred/repo-balanced-coverage/agents/runs/20261028T074202Z-build-skills/manifest.json`.
Both archived skill versions were read and matched byte-for-byte. Each has SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
`skill-evidence.json` and `numerical-evidence.json` record this manifest and the
individual hashes. `fingerprint.json` binds the final prose, selection and live
attached sources. The critic must obtain a fresh fingerprint for its own verdict.
## What was built
34 exact integral Jack polynomials J_lambda^(alpha), fully expanded in
Z[x1,...,xn,a], with a explicitly denoting alpha. The entry parameters are
n and lambda; alpha remains a formal variable, never a numerical sample.
The normalization is J=c_lambda P, where P is monic in m_lambda and orthogonal
for the alpha-deformed power-sum scalar product, and
c_lambda=product_s(alpha*arm(s)+leg(s)+1).
The complete slices are:
| Alphabet size n | Weights | Number of entries |
|---|---|---:|
| 2 | 2 through 5 | 10 |
| 3 | 2 through 5 | 14 |
| 4 | 2 through 4 | 10 |
Every partition within each slice and the length restriction is included.
Enumeration is increasing n, increasing weight, descending lexicographic shape.
The mathematical domains in Parameters remain extensible.
The queued proposal's n=4, degree-5 slice produced a 3341-character expansion.
It was omitted as a complete slice. The retained maximum is 1412 characters
at (n,lambda)=(4,4), and the entries block is 13,480 bytes in measured YAML.
The modest length over the usual 1100--1300 guideline retains every degree-4
shape, including the first four-row partition. This choice needs independent
critique; it is explicitly justified in argument-selection.md and completeness.
The spelling `alpha` requested in the family is unsupported by the site's
polynomial parser. `a` is used as an explicitly declared alias, preserving the
formal parameter, normalization and alphabet names. This is the only other
change to the family's conventions.
## Corpus and sources
`already_here` returned only unrelated lattice tables matching "jack".
`already_asked` returned the family and generic polynomial requests, no existing
Jack table. Direct symbolic lookups with `a` returned no Jack entries. The
column 2*x1*x2 matched unrelated Gegenbauer and Zernike values, which does not
duplicate this family. The same lookups using `alpha` exposed the unsupported
variable spelling. Records are in screening.json and lookup-screening.json.
Existing polynomial, algebra, combinatorics and bases-for-symmetric-polynomials
tags were checked. Neighbor documents and returned slugs were read; all HREFs
resolved in the final audit.
Primary conventions were checked against Nakajima, Section 2(i), Theorem 2.3
and equation (2.4), and Stanley (1989), Theorem 1.1 and Propositions 1.2 and 2.2.
Sage's J/P convention was checked in its official manual. Jiang and Wang,
equations (1.1)--(1.3), support the independent differential check.
The Nakajima HTML and Jiang--Wang PDF are archived with source hashes.
Stanley and Sage documentation were available through the web tool; direct
file downloads were refused, recorded in the run evidence. This did not prevent
checking the relevant definitions and formulas.
## Verification performed
- The shared dry-run checker exited 0: exact coefficients and prose checks
passed. Its 1412-character readability warning is retained and explained.
- Every one of the 34 generated polynomials agreed exactly with Sage's Jack J.
- A separate differential-operator calculation passed symbolically for all
34 values. The eigenvalues are distinct over Q(a) in every selected block.
- All 34 alpha=1 specializations matched hook-product times a Schur polynomial
computed by an alternant determinant, independently of the generator.
All 26 overlapping shapes also matched the stored T122 values.
- Symmetry, alphabet homogeneity, dominance support and leading hook coefficient
passed for all 34. One-row formulas passed for 11, column formulas for 6,
zero specialization for 11 full-length shapes, and alphabet stability for 19.
- Empty, degree-one and one-variable boundary formulas were checked separately.
- Nonzero known controls and a deliberately corrupted polynomial were checked
before trusting the numerical methods. No tolerances or floating point occur.
- The table's displayed Sage program was executed and matched its selected row.
- `verify(sample=None)` on the authenticated draft reported 34/34 matched,
zero differing, missing or extra. Every independent check was repeated on
values read back from the website. The later prose-only edit preserved
those entries exactly.
- The first audit asked for a shorter definition. The storage-spelling sentence
was moved to Comments; the final audit is clean with no findings.
Generator and checks are attached to the website: generate.py, checks.py,
check.py, argument-selection.md, and the T122.json corpus snapshot used by the
checker. The generator is self-contained apart from Sage and numberdb;
checks.py additionally imports check.py and optionally reads T122.json.
SageMath 10.9 and numberdb 0.1.10 were used.
## Handoff and limitations
The authoritative draft is populated and has passed this build's checks.
Independent argument-selection review remains pending. In particular, review
the complete four-variable degree-4 slice and the declared a=alpha spelling.
The build did not mark the proposal published or close a request issue.
No batch-exhausted sentinel was created. No source snapshot files, skills,
pipeline scripts or controller state were modified; no Git operation or
publication action was performed.
The retained run artifacts include raw check logs, measurement output,
source downloads, draft documents, audit responses and a lesson proposal on
variable names and initializing Sage's symbolic ring before Symmetrica.