# Argument selection for T653

This is the builder's justification, not the independent critique decision.
The canonical instructions used are archived in
`agents/runs/20261101T152208Z-build-skills/manifest.json`. Both archived copies
were read and have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The family conventions are recorded in numberdb-data issue 287 and in the
run's `family-287.md`. The authoritative enumeration is `generate.py`.

## Quantity and domain

The one stored quantity is the full finite polynomial
H_G(t) = sum_j dim_Fp(I^j/I^(j+1)) t^j for I the augmentation ideal in F_p[G],
with I^0 = F_p[G]. The formal variable t is not a sampled argument.
Coefficients are integer dimensions, not elements of F_p.
The sole declared parameter G is a nontrivial finite p-group. Within the
selected matrix families, p, n, and r are independent discrete structural
indices embedded in the group identity, and each is assessed here. There is
no free real input or probability level. The proposed companion's fixed
one-third threshold is not an argument or value of this polynomial table;
it comes from the three-fold zero-product application and belongs to the
separate scalar table.

## Upper unitriangular branch: 13 groups

U_n(F_p) consists of upper triangular n by n matrices with diagonal entries 1.
The canonical identity is UT-n-p. Matrix size n changes nilpotency class n-1
and the counts n-j of weight-j matrix coordinates. The prime p changes the
coefficient characteristic and the allowed exponents in a Jennings basis.
These are discrete group invariants, not continuous arguments renamed as
indices. The appropriate review category for G is `discrete_index`.

| Prime p | Matrix sizes n | Exact identities |
|---|---|---|
| 2 | 3, 4, 5, 6 | UT-3-2, UT-4-2, UT-5-2, UT-6-2 |
| 3 | 3, 4, 5, 6 | UT-3-3, UT-4-3, UT-5-3, UT-6-3 |
| 5 | 3, 4, 5 | UT-3-5, UT-4-5, UT-5-5 |
| 7 | 3, 4 | UT-3-7, UT-4-7 |

This is exactly p prime at most 7, 3 <= n <= 6, with
(p-1)n(n^2-1)/6 <= 80. Starting at n=3 focuses this pilot on nonabelian
matrix groups; n=2 is the cyclic prime-order degeneration. The four small
characteristics include 2 and cases with p smaller than, equal to, or greater
than the nilpotency class. This tests the characteristic-sensitive grading
without extending to a large catalogue of matrix sizes.

## Heisenberg branch: 12 further groups

Heis_r(F_p) consists of triples (a,b,c), a,b in F_p^r, c in F_p, with product
(a+a', b+b', c+c'+a dot b'). Its canonical identity is Heis-r-p.
The rank r varies the number 2r of weight-one generators while keeping class
two and one weight-two central coordinate. Thus this branch supplies breadth
not obtained merely by increasing unitriangular matrix size. Its role is
also `discrete_index`, with the same prime-characteristic interpretation.

| Prime p | Ranks r | Exact identities |
|---|---|---|
| 2 | 2, 3, 4 | Heis-2-2, Heis-3-2, Heis-4-2 |
| 3 | 2, 3, 4 | Heis-2-3, Heis-3-3, Heis-4-3 |
| 5 | 2, 3, 4 | Heis-2-5, Heis-3-5, Heis-4-5 |
| 7 | 2, 3, 4 | Heis-2-7, Heis-3-7, Heis-4-7 |

The omitted rank-one rows are already represented by UT-3-p, since the triple
law is precisely multiplication in U_3(F_p). In particular the p=2 law fixes
the dihedral group of order 8 rather than the quaternion group. No ambiguous
extraspecial-group naming convention is used. Petrov writes H_(n-1), so our
rank r equals his n-1.

## Bounds, breadth, and evidence

Both branches are studied together in the group-algebra approach of Petrov,
Section 7, and the unitriangular Jennings degrees are explicitly stated by
Blasiak, Church, Cohn, Grochow and Umans, Example 3.15. Bartholdi and
Grigorchuk, Section 2, equation (3), fixes the dimension-subgroup convention
and Hilbert-series product. These sources justify the families, definitions,
and application context, not an individually published use for every row.

The prime bound 7, matrix-size bound 6, Heisenberg-rank bound 4, and polynomial
degree ceiling 80 are noncanonical reference choices. They are independent of
decimal spelling and give a bounded first reference for identifying Loewy-layer
polynomials in small characteristics. They are not claimed to be intrinsic
endpoints. The dry run measures a longest expanded entry of 1007 characters
at UT-5-5 and an annotated block of about 10.6 KiB. This supports readability
without pruning either structural branch to a few named exceptional groups.
The two branches are explicit slices, not a Cartesian product of unrelated
simple inputs. Dihedral, quaternion, other extraspecial groups, arbitrary
SmallGroups catalogues, and non-prime coefficient fields are deferred to a
future coverage decision. No quantile or statistical critical-value family
is involved.

The enumeration is increasing p; for each p it lists increasing n followed
by increasing r. There are 25 distinct group identities. The completeness
note and default generator state these exact bounds. No selected identity
comes from rounding, numerical root finding, or a display label.

## Checked sources

- Bartholdi and Grigorchuk, *Lie Methods in Growth of Groups and Groups of
  Finite Width*, Section 2, especially equation (3):
  https://arxiv.org/html/math/0002010#S2
- Blasiak, Church, Cohn, Grochow and Umans, *Which groups are amenable to
  proving exponent two for matrix multiplication?*, Definition 3.5 and
  Example 3.15: https://arxiv.org/html/1712.02302#S3.SS2
- Petrov, *Combinatorial results implied by many zero divisors in a group
  ring*, Section 7: https://arxiv.org/html/1606.03256#S7

The independent critic must inspect both branches, verify all embedded
indices, and bind its own decision to the live selection and attachment
hashes. This build supplies no acceptance sidecar and takes no review action.