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3345 bytes, as of the version from 2026-11-02 17:55 (current). Recorded here, not run.
# Argument selection for T682
The archived local and live numberdb-table skills agree. The controlling manifest is
`agents/runs/20261102T172655Z-build-skills/manifest.json`; both SHA-256 digests are
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
The run retains a copy as `skill-manifest.json`.
The quantity is the raw signed logarithmic moment
m_k(S_n) = integral_0^1 (log|S_n(exp(2*pi*i*t))|)^k dt,
where S_n(x)=1+x+...+x^(n-1), n>=2 and integer k>=2. The logarithm is natural,
and Haar measure of the circle is one. The order is (n,k), both type Z.
## n: discrete_index
The index counts summands of the standard geometric-sum polynomial, not a sampled
continuous coefficient. Lalín–Sinha (arXiv:1106.1304), Theorem 7 and section 5.1,
item 1, study precisely this family. Its higher measures distinguish products of
cyclotomic factors even though their ordinary logarithmic measures vanish.
The selected prefix is every integer 2<=n<=50, in increasing order. This gives
primes (overlap with the cyclotomic companion), prime powers, and composite lengths
with two and three distinct prime divisors (30 and 42). There is no reduction to
squarefree indices: power substitution on S_n does not remove all the lower
cyclotomic factors. Keeping a complete prefix avoids unexplained arithmetic gaps.
The endpoint 50 is a noncanonical practical cutoff, keeping the degree at most 49
and the number of root differences small enough for independent 100-digit checks.
It is not an assertion of individual fame or applications for every length.
## k: discrete_index
The integer k is the order of the single moment family, not a selector of different
quantities. We select k=2 and k=3: the first nonzero even and odd moments, because
m_1(S_n)=0. Both admit finite certified formulas (Lalín–Sinha, Theorem 6, Lemma 10,
and section 5.1). Including both covers magnitude and signed skewness information
without introducing standardized moments. No absolute third moment is included.
Higher orders belong to this same family, but would require another independently
validated evaluation method. Arithmetic breadth across lengths takes priority over
a long list of orders for S_2 alone.
## Interactions and coverage
The two moment orders are evaluated for each of the 49 consecutive lengths,
for 98 entries. This small product is the same two low-order invariants over one
discrete polynomial family, not a Cartesian mesh of independently tunable continuous
arguments. The rational root angles j/n, 1<=j<n, are derived from the polynomial;
there is no free angular grid. Both parameters are discrete_index; no intrinsic,
reference_sample, or mixed continuous selection is claimed. Bounds describe current
coverage and are kept out of the permanent parameter constraints. Enumeration is
increasing n, then k=2,3. The completeness note states the same selection and reasons.
The family issue #302 and live companion T681 were read. T681 has the same integral,
parameter order, and signed convention; it reduces cyclotomic orders by symmetries
that do not apply to geometric sums. Prime overlaps are mathematically intentional.
The independent critique must evaluate this justification and bind its own argument
sidecar to the live selection and all generator dependencies. This build does not
supply an independent acceptance.