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# Argument selection for T605
The only free parameter is the positive rational moment exponent p. The
underlying constant is defined for every real p > 0. Coefficient-vector length
is universally quantified; no length, coefficient vector or distribution is
selected as an additional table parameter. All signs are independent and
uniform on {-1,1}; all coefficients are real; expectation has total mass one.
The selection is mixed, with disjoint groups and increasing reduced-fraction
identities. The generator's selection_groups() is the authoritative enumeration.
1. Intrinsic: p = 1, 2. The first absolute moment is the classical sharp
first-moment case; p = 2 is the second-moment identity and the beginning of
the identically-one branch. These are structural endpoints of the usual
L1-to-L2 comparison, not generic integer sampling.
2. Documented application: p = 2m/(m+1), 2 <= m <= 20. Proposition 3.1 of
Bayart, Pellegrino and Seoane-Sepulveda, arXiv:1310.2834v2, section 3,
explicitly uses the real Khintchine constants at these exponents in
multilinear Bohnenblust-Hille bounds. Their A_{R,p} is our 1/A_p; section 3
defines the normalization. The integer m indexes that application family,
but the table's p remains type Q and this group is not discrete_index.
The finite cutoff 20 is a practical reference choice. It includes the early
application cases and both active nonconstant branches (m = 13 through 20
are on the Gaussian branch) without chasing the accumulation at p = 2.
3. Exploratory reference: every reduced p = a/b with 0 < a < 2b, b <= 6,
excluding groups 1 and 2. Purpose: identify lower constants encountered
at simple subquadratic moment exponents, including quasi-norm exponents
below 1 that the application family misses. The denominator cutoff is
noncanonical and notation-independent; it is not evidence of a published
application at every fraction. It covers several small denominators and
the small-p branch without a dense decimal mesh or an arbitrary extension
of the application sequence. Bounds yield 1/6 <= p <= 11/6 before
overlaps are removed. Numerator bounds follow from the nonconstant regime,
rather than an independent Cartesian product.
Breadth was assessed before depth: first and second moments, subunit moments,
the interval from 1 to 2, and both active formulas are represented. Larger
application indices mostly approach the already represented endpoint. A denser
rational reference mesh was not needed to cover another regime. No p > 2 is
stored because its constant is always exactly 1; the formula states the entire
constant branch. The transition exponent is defined implicitly by an equality
of Gamma values and is not a rational parameter supplied by this proposal.
There is no claim that a rounded approximation to that exponent is an exact
rational argument. No quantiles, percentiles or critical values are involved.
Sources checked: Mordhorst, Theorem 1.1, for the real-sign convention and
complete sharp-constant formula; Bayart et al., section 3, Proposition 3.1 and
Corollary 3.3, for the application family and reciprocal convention. These
sources support the function and relationships, not a claim that every
exploratory rational has a documented use.
Completeness statement: it holds p = 1, 2, the application exponents
p = 2m/(m+1) for 2 <= m <= 20, and every reduced fraction 0 < a/b < 2 with
b <= 6, with overlaps removed. The application cutoff covers early cases and
both nonconstant branches; the noncanonical fraction bound supplies compact,
notation-independent exploratory coverage, especially for p < 1. Values for
p > 2 are identically 1 and are described by the formula.
The independent critique must assess each group and bind generate.py and this
argument-selection.md attachment (plus any additional source it relies on)
using agents/argument_review.py fingerprint. This is the builder's justification,
not independent approval and not a critique sidecar.
Instruction provenance: ../20261029T085909Z-build-skills/manifest.json in the
run artifacts; the exact manifest is also attached as skill-manifest.json.
Both archived skill files have SHA-256
28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.