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# Argument selection for T578: integral Macdonald polynomials
This build read numberdb-data family issue #228 before selecting arguments.
The policy is the coordinated archived canonical skill, recorded in
`../20261028T101159Z-build-skills/manifest.json`. Both `skill-local.md` and
`skill-live.md` have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The same manifest is attached as `skill-manifest.json` for a portable record.
## Quantity and conventions
One named object is stored: the symmetric integral Macdonald polynomial
J_lambda(x1,...,xn;q,t)=c_lambda(q,t) P_lambda(x;q,t), specialized from the
symmetric-function ring. P is monic in the monomial basis and orthogonal for
the standard q,t deformation of the Hall product. The normalizing factor is
product_(i,j in lambda)(1-q^(lambda_i-j)t^(lambda'_j-i+1)). The coordinates
are one-based row and column indices. All coefficients in the fully expanded
ring ZZ[x1,...,xn,q,t] are integers. q and t, in that order, are formal
polynomial variables; they are not continuous inputs chosen for evaluation.
The n alphabet variables are also indeterminates. No rational or decimal
mesh, root-indexed coupling, or statistical quantity is present.
## Each free parameter
- `n`, type Z, is the number of alphabet variables, a **discrete_index**.
Alphabet size is an intrinsic index of finite-variable symmetric
polynomials. We retain n=2,3,4 to cover the first nontrivial alphabets before
extending one alphabet to much higher degree. Four alphabet variables plus
q,t reach the six-variable representation limit. The lower exclusion n=1
is explicit: J_(d)(x1;q,t)=x1^d product_(j=0)^(d-1)(1-q^j t).
- `lambda`, type Symbolic, is a partition, an **arithmetic_structure**.
Its positive nonincreasing parts specify the symmetric-function shape,
not a numeric sampling rule. We include every admissible partition in each
complete degree slice, with length at most n. Rows, columns and interior
shapes are retained wherever they fit. Identities are comma-separated
positive parts without zero padding. Weight is derived, not an extra free
parameter. Degrees 0 and 1 have boundary formulas in the table; individual
simple shapes within a retained degree are not removed.
These categories describe the mathematical roles. The finite cutoffs are
noncanonical editorial choices for a compact reference. They are independent
of numeral base, not claims of an application for each individual partition.
Sources establish the objects and conventions, not the preferred cutoffs.
## Measurement and breadth review
The proposal suggested n=2,3, degrees 2 through 4, giving 16 entries. We
computed that selection and also measured n=4 before deciding the extent.
The maximum expanded lengths by complete slice were:
| n | degree 2 | degree 3 | degree 4 |
|---|---:|---:|---:|
| 2 | 147 | 642 | 1906 |
| 3 | 327 | 1916 | 7185 |
| 4 | 577 | 4198 | 18882 |
Final selection:
| n | retained degrees | entries |
|---|---|---:|
| 2 | 2 through 4 | 7 |
| 3 | 2 through 3 | 5 |
| 4 | 2 | 2 |
The 14 entries occupy about 8.5 KiB in the check helper's serialization.
The two largest retained entries, J_(4) in two variables and J_(3) in three,
have 113 and 124 terms and 1906 and 1916 characters. This exceeds the
helper's advisory 1300-character warning. We retain these two complete
slices deliberately: degree 4 gives the first (2,2) shape in two variables,
and degree 3 in three variables supplies the first three-part column beside
the (2,1) interior shape. The four-variable quadratic slice adds alphabet
breadth at just 577 characters. Dropping entire slices at an automatic
1300-character threshold would sacrifice these distinctions. This is a
readability judgment for independent critique, not a size-limit exception.
The omitted three-variable degree-4 slice reaches 7185 characters; the
four-variable cubic slice reaches 4198. Those are a substantial readability
jump, so they remain measured evidence rather than stored entries.
Enumeration is n increasingly, then degree increasingly, then partitions in
descending lexicographic order. The generator default, exact stored identities,
and completeness note must agree. A checker using Sage's separate Partitions
enumerator confirms all 14 identities and their order. No difficulty-dependent
holes or famous-only choices occur.
## Checked supporting sources
1. Mark Haiman, *Macdonald polynomials and geometry*, Section 2, pp. 4–9,
especially the q,t inner product preceding (2.8), triangularity (2.9),
the difference operator calculation on p. 8, and integral form (2.16):
https://math.berkeley.edu/~mhaiman/ftp/nfact/msri.pdf
2. SageMath reference manual, *Macdonald polynomials*, J() and P():
https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/sf/macdonald.html
J() explicitly names the integral form and its independent determinantal
implementation. Its examples fix the P-to-J factors for (2) and (1,1).
3. Live corpus T121 and T122 fix partition identities and alphabet names;
T261 supplies the Hall–Littlewood P normalization. All 14 q=0
specializations were checked against T261, with the required scalar factor;
all 14 q=t specializations were checked against T122 and alternants.
This is the builder's justification. It is not an independent acceptance.
The controller must obtain a separate critique and selection fingerprint,
including generate.py, checks.py, check.py and this document. The critic must
supply its own reviewer_run and -arguments.json sidecar. This build neither
writes that sidecar nor offers or publishes the draft.