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5093 bytes, as of the version from 2026-11-01 16:17 (current). Recorded here, not run.
# Argument selection for T654
This build follows the archived matching local and live skills in
`agents/runs/20261101T154704Z-build-skills/manifest.json`.
Both SHA-256 hashes are
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The attached `skill-manifest.json` preserves that manifest. This is the
builder's rationale, not an independent critique acceptance.
The only declared parameter is the finite p-group G. It selects an actual
algebraic object, not a continuous input hidden behind a symbolic identity.
Its selection category is `discrete_index`. Both structural branches require
assessment, including their embedded characteristic and size choices.
## Object and normalization
Work over the prime field F_p. Define H_G from successive powers of the
augmentation ideal, including I^0=F_p[G], and d_G=degree H_G. The single stored
quantity is min_{0<t<=1} H_G(t)t^(-d_G/3). It is not normalized by |G|.
Natural logarithms are used in its entropy interpretation. The optimizer t is
an internally determined unique root, never a free input or a second column.
The threshold d_G/3 is fixed because the product of three filtration terms
of degree greater than d_G/3 vanishes. It is not a sampled tail probability,
quantile, or adjustable coupling. The direct-power interpretation and the
optimization formula are derived in `derivation.md`; Petrov's Sections 6–7
supply the group-algebra and Chernoff setting, rather than the name C_G.
## Two discrete branches
* Upper unitriangular groups U_n(F_p): n is the matrix size, n>=2, and p is
prime. Increasing n changes the nilpotency class and the multiplicities of
the augmentation weights. Select primes p<=7, 3<=n<=6 and
d_G=(p-1)n(n^2-1)/6<=80. This selects 13 nonabelian groups. The n=2 abelian
boundary is outside this initial nonabelian pilot, not excluded from the
definition of C_G. This is the same pilot as the Hilbert-series companion.
* Heisenberg groups Heis_r(F_p): r>=1 counts the paired vector coordinates
in (a,b,c), with product (a+a',b+b',c+c'+a.b'). Select the same four
primes and 2<=r<=4, giving 12 further groups. This varies generator count
while keeping nilpotency class two. Heis_1(F_p) is U_3(F_p), already included
for each prime; retaining one identity avoids duplicate descriptions of
the same group. The explicit law also fixes the characteristic-two case.
The characteristic p is an arithmetic index of a group and its prime
coefficient field. The four smallest primes include characteristic two,
class equal to characteristic, and both sides of the class/characteristic
comparison. These finite prime, size, rank and degree bounds are noncanonical
reference choices. They are independent of decimal spelling, not a claim of
an application for each individual group. Their purpose is to identify
weighted counting and growth rates for small explicit nonabelian algebras.
Breadth was reviewed before depth: one branch changes class, the other changes
generator count at fixed class, and all four characteristics occur in both.
The degree ceiling is inherited from the readable expanded-polynomial range
of T653 (longest polynomial 1007 characters). Scalar optimization could go
farther, but the initial paired coverage makes each constant traceable to its
full grading. Increasing only one rank deeply or multiplying unrelated
parameters into a grid would lose that balance. No finer continuous mesh is
being proposed. Other p-groups and prime-power coefficient fields remain
outside this pilot. No sound existing entry was pruned.
## Exhaustive membership and ordering
Rows are ordered by increasing p, then U_n by n, then Heis_r by r:
* p=2: UT-3-2, UT-4-2, UT-5-2, UT-6-2, Heis-2-2, Heis-3-2, Heis-4-2.
* p=3: UT-3-3, UT-4-3, UT-5-3, UT-6-3, Heis-2-3, Heis-3-3, Heis-4-3.
* p=5: UT-3-5, UT-4-5, UT-5-5, Heis-2-5, Heis-3-5, Heis-4-5.
* p=7: UT-3-7, UT-4-7, Heis-2-7, Heis-3-7, Heis-4-7.
`generate.py:selected_groups`, the completeness note and the stored identities
must describe precisely these 25 groups. No quantity-selecting parameter exists.
## Checked sources
* Bartholdi–Grigorchuk, https://arxiv.org/html/math/0002010#S2,
Section 2, equation (3): Jennings Hilbert product and dimension filtration.
* Blasiak–Church–Cohn–Grochow–Umans,
https://arxiv.org/html/1712.02302#S3.SS2, Example 3.15:
unitriangular p-degrees n-j. This source also explains the matrix
multiplication context, which supplies no algorithm merely from C_G.
* Petrov, https://arxiv.org/html/1606.03256#S6 and #S7:
weighted monomial counting, zero products and the explicit matrix groups.
Petrov's H_(n-1) is our Heis_r with r=n-1.
* T653, fetched through the NumberDB API during this run:
https://numberdb.org/Augmentation_Hilbert_series_of_finite_p-groups.
Its 25 entries, explicit multiplication law, indexing, and integer-valued
coefficients fix the companion convention. Its snapshot is attached.
These sources establish the mathematics and the families. The finite bounds
are this reference selection, not endpoints prescribed by those papers.