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# Argument selection for T662
The only free parameter is the integer m, the order of each cyclic factor in
H_m=(C_m)^3. The family is m>=5. The table stores the unique exponent beta_m
in (2,3) determined by the exact complex character spectrum of H_m wr S_2
and the explicit triple-product subsets of Cohn, Kleinberg, Szegedy and Umans
(CKSU), Section 2, Lemma 2.1. It does not range independently over q, n,
an exponent argument, a coupling, or a choice of root. In the family's
notation q=m^3 and n=2. The parameter m is therefore a discrete_index, with
a documented construction application.
Selection: every integer 5<=m<=40, in increasing order, for 36 entries.
The lower bound starts where this construction beats cubic cost. The range
includes its initial nontrivial cases, the source's coarse optimum at m=17,
and a substantial continuation on the other side. All intermediate sizes
belong to the same published construction, whether individually famous or
not. The cutoff 40 is a noncanonical practical reference boundary, independent
of numeral notation. Extending it costs little, but adds depth without a new
construction mechanism. The existing companion T661 already gives breadth
across symmetric, signed-permutation and other abelian-base wreath products;
those are character costs, not additional beta arguments to be absorbed here.
No reference mesh or Cartesian product is present. Every selected member has
the same argument role, so no mixed or reference_sample classification is
needed. A later critique must make its own assessment.
The coarse largest-degree estimate and the root from the full spectrum are
different quantities. The proposal explicitly selects the latter. Only beta_m
is stored. The coarse expression appears as a comparison formula. The source
calls the cyclic order n; this table calls it m so n continues to mean wreath
permutation degree in T661. All powers are positive real powers; every
irreducible character, including every linear character, is counted.
Checked sources:
- CKSU, Group-theoretic Algorithms for Matrix Multiplication (2005), Theorem
1.8, Section 2, Lemma 2.1:
https://arxiv.org/html/math/0511460v1#S2 . This specifies the subsets,
their cardinality 2m(m-1), their triple-product property, and the coarse
comparison bound with its reported minimum at 17.
- Gao and Jing, Irreducible Characters of the Generalized Symmetric Group,
Journal of Lie Theory 35 (2025), Theorem 4.1, equation (4.4), page 605:
https://jolt.centre-mersenne.org/item/10.5802/jolt.1399.pdf . For permutation
degree two the colored-partition degrees give 2q characters of degree one
and binomial(q,2) of degree two. For any abelian base of order q the same
counting follows by swapping pairs of its q linear characters.
- Live T661 was read through the authenticated NumberDB API, including its
conventions, complete selection, formulas and numbers. Its cubic and 9/4
moments at q=m^3,n=2 are checked against our spectrum for every selected m.
The table's enumeration, parameter domain and completeness note agree with
this selection. The numerical checks and measurement are attached separately.
No assertion of current-record status or algebraicity of beta_m is made.
Policy provenance: agents/runs/20261101T231203Z-build-skills/manifest.json,
also attached as skill-manifest.json. Both archived versions were read and
matched the canonical snapshot, SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.