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# Argument selection for T606
This is the builder's justification, not the independent critique required by
agents/special-values/campaign-gate.md. The sole free parameter p is a positive
rational moment exponent in the probability-average real Rademacher inequality.
The mathematical function is defined for all real p > 0. No vector length,
coefficient vector, sign probability, or alternate quantity is a free parameter.
The enumeration uses reduced fraction identities in increasing numerical order.
The archived skill manifest is
agents/runs/20261029T093155Z-build-skills/manifest.json. Both archived versions
have SHA-256 28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.
The exact digest record is also attached as skill-evidence.json.
## Mixed selection, with exhaustive disjoint groups
1. Intrinsic anchors: p = 1 and 2. These are the first absolute moment and the
second moment that defines the coefficient normalization. Both upper constants
are exactly 1. The entire plateau 0 < p <= 2 is specified in the formulas;
repeating its value at a dense selection of other exponents adds no coverage.
2. Arithmetic structure: integer p = 3,...,32. At even p = 2m the power
B_p^p = (2m-1)!! counts pairings, while at odd p = 2m+1 the Gaussian absolute
moment is 2^(m+1/2) m! / sqrt(pi). These two simple moment families provide
useful initial higher moments. The bound 32 covers the first sixteen even
orders and interleaved odd orders. It is a practical reference cutoff, not
a distinguished endpoint; the constant remains compact beyond it. The
rational parameter type is retained, and these are not called discrete_index.
3. Exploratory reference sample: every reduced fraction 2 < a/b < 4 with
2 <= b <= 6. These cover fractional moments on both sides of p=3, including
the theorem's historically delicate interval (2,3), and the first adjacent
interval. The fractions give simple exact moment exponents; no application
is claimed for every fraction. The denominator bound is noncanonical, keeps
the selection small, and includes halves, thirds, quarters, fifths and sixths
without favoring decimal spelling or pretending to supply interpolation data.
4. Exploratory reference sample: p = 9/2,11/2,13/2,15/2,17/2,19/2. This single
slice extends fractional moments beyond 4 through moderate orders below 10.
Half-integers are a simple nonintegral reference sequence. The choice and
cutoff are noncanonical; they are not evidence of a documented application.
No higher-denominator rectangle is extended to large p.
## Coverage review and reference policy
The selection has 60 entries, a consequence of these families rather than a
size target. It covers the unit plateau, both integer parities, the nonintegral
transition interval, and a modest fractional tail before increasing integer
depth. It is not restricted to exceptional named constants. There is no Cartesian
product and no decimal mesh. The purpose of both reference groups is to make
sharp upper constants at simple normalized fractional moment exponents
identifiable. Their exact bounds are the groups just stated. The coverage
tradeoff is breadth in the first two nonconstant intervals and a single sparse
higher-order slice, with the more extensive integer family justified by exact
moment identities. Neither fraction group is called intrinsic or an application.
For a future independent assessment, p should be classified as mixed. Each
reference_sample group requires its own reference_policy with purpose, bounds,
coverage_rationale, noncanonical_acknowledged=true,
notation_independent=true, and sources_checked=true. Those flags must be the
critic's findings, not adopted from this builder document.
## Sources and mathematics checked
Mordhorst, Colloquium Mathematicum 147 (2017), pp. 203-204, Theorem 1.1,
defines real coefficients, independent unbiased signs, probability expectation,
and the sharp upper constant as max(1,G(p)). Section 2 (pp. 204-205) gives the Gaussian lower bound from normalized
equal-coefficient sums. The limit also follows from the central limit theorem
and uniform integrability, using an even moment of higher order. The Gaussian density
integral and integration by parts give M_(p+2)=(p+1) M_p; M_0=1 and
M_1=sqrt(2/pi) yield the even and odd formulas. These relationships support
the integer family; they do not select the exploratory fractions or their bounds.
The paper is checked for the family and conventions, not for use of every input.
See sources.md for URLs and download evidence.
## Completeness wording
it holds p=1,2, all integer exponents 3<=p<=32, every reduced noninteger
fraction 2<a/b<4 with b<=6, and the half-integers 9/2 through 19/2; integer
moments form the factorial and pairing families, while the noncanonical
fraction bounds provide compact reference coverage near the branch transition
and a single fractional slice at moderate orders; the remaining unit plateau
0<p<=2 is specified by the unit-branch formula
## Exact identities in each group
intrinsic: 1, 2
arithmetic_structure: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32
reference_sample: 13/6, 11/5, 9/4, 7/3, 12/5, 5/2, 13/5, 8/3, 11/4, 14/5, 17/6, 19/6, 16/5, 13/4, 10/3, 17/5, 7/2, 18/5, 11/3, 15/4, 19/5, 23/6
reference_sample: 9/2, 11/2, 13/2, 15/2, 17/2, 19/2