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# Argument selection for T661
This is the builder's justification, not an independent critique or an acceptance
sidecar. The controller must obtain a separate critique of the attached sources,
actual entries and this selection.
The governing instructions are archived in
`agents/runs/20261101T212519Z-build-skills/manifest.json`. Both local and live
copies were read and checked byte for byte. Each has SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The family proposal is numberdb-data issue 292, read before this build and
preserved in the run artifacts. Its proposed range was reassessed under these
instructions. No quantile or statistical critical-value family is involved.
## Quantity and conventions
For integers q>=1 and n>=2, W_(q,n)=C_q wr S_n, with S_n permuting the n
coordinates. D_(q,n)(s) sums chi(1)^s over every inequivalent irreducible complex
character, with multiplicity of equal degrees and including every linear
character. Powers are positive real powers. The domain of s is rational s>=2.
There is no division by group order, number of characters or a Plancherel weight.
For every abelian H of order q, H wr S_n has this same degree multiset, because
all q irreducible characters of H have degree one. This is an equality of
spectra, not an assertion that the groups are isomorphic.
The parameters q, n, s are respectively the base order, permutation degree,
and exponent of one named power sum. The exponent is an argument of that sum,
not a selector of unrelated quantities. The construction exponents in the
family's second proposal are outside this table.
## Selection by parameter and by slice
The generator emits these disjoint pairs in the order listed, with s=9/4 then
s=3 at every pair:
| q | n | Number of pairs | Role |
|---|---|---:|---|
| 1 | 3 through 20 | 18 | Distinguished symmetric-group specialization |
| 2 | 2 through 12 | 11 | Distinguished signed-permutation specialization |
| 3 | 2 through 10 | 9 | Bounded exploratory generalized-symmetric-group reference |
| m^3, for every integer m from 2 through 40 | 2 | 39 | Published CKSU construction's abelian base orders |
Thus there are 77 group pairs and 154 values. The generator does not form a
Cartesian product of all base orders and all permutation degrees. In particular,
large cube base orders occur only at n=2.
**q is mixed.** Values 1 and 2 select established discrete group specializations:
S_n and the hyperoctahedral group, respectively. Gao and Jing, Introduction,
p. 594, explicitly identify both. Values m^3 arise from H_m=(C_m)^3 in CKSU
Section 2. Our m is their n; their factor-swapping group has permutation
degree two. The abelian order, rather than the cyclic factor order, is therefore
q=m^3. Every m=2,...,40 is included, retaining the small unsuccessful algorithmic
cases m=2,3,4 as part of the same group family and covering the source's coarse
optimum near m=17 and larger orders. The cutoff 40 is a practical reference
extent; it is not a mathematical stopping point or a claim of optimality.
The remaining value q=3 is explicitly a **reference_sample**. Its purpose is
recognition of low-order generalized-symmetric-group character costs beyond
the real reflection families and reference checks of the degree formula with
three colors. It is the smallest cyclic base beyond q=2, a simple structural
comparison that does not depend on numeral notation. Its chosen extent is
2<=n<=10. The choice of this comparison base and its cutoff is noncanonical;
no individual application of every q=3 row is claimed. Gao and Jing establish
the family, normalization and degree formula, rather than endorsing this
particular reference range. For a critique's reference_policy, purpose, bounds
and coverage_rationale must reflect precisely this paragraph, with explicit
noncanonical, notation-independent and source-check attestations.
**n is a discrete_index**, conditionally on q. It is the number of permuted
factors and hence the rank/size parameter of each group family. The q=1 slice
starts at n=3 since S_2 is abelian and has only a trivial constant moment. The
q=2 and q=3 slices start at the first nonabelian case n=2. For q=m^3, n=2 is
fixed by the published construction, not chosen from a mesh. The initial
ranges of 18, 11 and 9 groups give depth in each of the three structural slices;
there is no reason to make their endpoint sizes equal. Counting multipartitions
gets more expensive as the number of colors increases, so the multicolor
slices stop sooner. These cutoffs are practical choices, not domain restrictions.
**s is documented_application.** The cubic value is the sum-of-cubes cost in
CKSU Section 2 and Theorem 1.8, obtained from the matrix block dimensions of
the complex group algebra. The comparison exponent 9/4 is selected by Amanov,
van den Berg and Zuiddam, Theorem 1.1, which states an upper bound for matrix
multiplication. It is not asserted to equal the unknown exponent and does not
assert that any row alone supplies a fast multiplication algorithm. These two
arguments are not samples of an interpolation grid. The square moment is an
exact group-order check and a formula, rather than another slice of stored
small counts. There is no need to add all intervening rational exponents.
## Breadth, bounds and representation
Breadth was reviewed before depth. The selection includes the symmetric,
signed-permutation, generalized-symmetric reference, and application-derived
large-base families. A single q=1 sequence would omit the nontrivial base
behavior; restricting to the m=17 historical example would omit the varying
character cost of the construction family. Adding every q to a rectangle would
lack a corresponding purpose. Nonabelian bases require additional data and a
separate family justification and are deliberately outside this proposal.
All cubic sums are exact integers. The other 77 values use finite certified
Arb balls, with exact integer degrees and positive fourth roots. The delivered
text is measured by checks.py at 100 significant digits. This range remains
compact; numerical-evidence.json records its actual longest entry, block size,
accuracy and the group-order and independent spectrum comparisons. Readability
and computation both support keeping the proposed extents without imposing a
minimum number of entries or pruning the exploratory family to famous groups.
The completeness note, parameters, generator enumeration and stored identities
must agree with these bounds. All selection logic is in generate.py. checks.py
contains the independently constructed spectra and validation, and check.py is
the unmodified campaign checking utility. The attached flatten.py is the repository's unmodified parser for nested and flat stored entries. No imported data controls enumeration.
## Sources actually checked
* Gao and Jing, *Irreducible Characters of the Generalized Symmetric Group*,
Journal of Lie Theory 35 (2025), 593–616, Introduction p. 594 and Theorem 4.1,
equation (4.4), p. 605. https://doi.org/10.5802/jolt.1399
* Cohn and Umans, *A Group-theoretic Approach to Fast Matrix Multiplication*,
Sections 1.3, 4 and 7.2. https://arxiv.org/html/math/0307321v1
* Cohn, Kleinberg, Szegedy and Umans, *Group-theoretic Algorithms for Matrix
Multiplication*, Theorem 1.8 and Section 2, especially the definition
H=Cyc_n^3 and the factor-swapping construction.
https://arxiv.org/html/math/0511460v1
* Amanov, van den Berg and Zuiddam, *Multiplying matrices via polynomials:
exposition and barriers*, Theorem 1.1.
https://arxiv.org/html/2610.30054v1#S1
Primary sources are archived with hashes in sources-manifest.json in the run
artifacts. The q=3 reference selection is our choice, not a literature claim.