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# Central binomial sums: argument selection
Quantity: U_p(v) = sum_{n>=0} binomial(2n,n) v^n/(2n+1)^p.
The natural domain is integer p>=1 and rational -1/4<=v<=1/4;
all these series converge absolutely. Order is (p,v). The n=0 term is 1,
negative v contributes (-1)^n, and no pi or factorial scales the stored value.
All logarithms in representations are natural; inverse sine and inverse
hyperbolic sine use their real principal branches.
Selection: every p=1,...,9 at v=-1/4,-1/16,1/16,1/4, in that order,
36 entries. Both parameters have been assessed separately.
* p: discrete_index. This is the denominator power and one plus the logarithmic
moment order, not a grid of a continuous variable. The prefix includes the
elementary boundary, both parities, and successive moments at all four
argument families. Nine is a noncanonical finite computation cutoff. It also
supplies U_(p+1) through the planned harmonic companion's order eight under
H_(2n+1)=H_(2n)+1/(2n+1). This connection does not merge quantities or assume
that the companion has been built. Higher orders remain in the natural family.
* v: mixed, with exhaustive disjoint groups:
* [-1/16,1/16]: documented_application. CGLXZ equations (1.1)-(1.2)
select these signed arguments in Apery-like series. Lemma 2.1, equations
(2.2)-(2.3), gives the unweighted terms needed after the harmonic-index
shift. In that lemma v=+/-z^2 and its denominator order is p+1, so our
order p is the lemma's p+1. At z=1/4 the trigonometric/hyperbolic endpoint
is pi/6 or asinh(1/2)=log((1+sqrt(5))/2).
* [-1/4,1/4]: intrinsic. These are the two real endpoints of the disk of
absolute convergence. Their endpoint sums exist for every p>=1. In
Lemma 2.1 these are z=1/2, giving pi/2 and asinh(1). Both signs are
retained; alternating signs are not replaced by magnitudes.
Breadth before depth: the full application pair and both convergence boundaries
are covered at the same prefix of orders. The known elementary order is not the
only slice. No arbitrary denominator mesh, decimal grid, unexplained radical
set or exploratory reference sample is introduced. The formula U_p(0)=1 records
the elementary zero argument without filling the table with repeated ones.
The p=0 generating function is a computational identity outside this table's
natural domain. There are no quantile or statistical critical-value entries.
Sources actually inspected: Charlton, Gangl, Lai, Xu and Zhao,
On two conjectures of Sun concerning Apery-like series, arXiv:2210.14704v1,
Introduction (1.1)-(1.2), Lemma 2.1 (2.1)-(2.3), and Lemma 2.2's defining
harmonic sums. The source proves the two identities initially called conjectures.
https://arxiv.org/html/2210.14704v1 . Euler's beta integral, DLMF 5.12.1,
independently fixes the positive-endpoint normalization.
The current catalogue and candidate metadata were checked on 2026-11-11.
T698 is inverse-binomial with denominator n^s; T697 and T713 are logarithmic
moments. The transformations require factorial rescalings or linear
combinations, so those families do not replace this quantity. T713 is currently
an offered private draft, so its cross-link must remain a handoff plan until
it is publicly available. No published table is changed.
Skill evidence: skill-manifest.json, archived local/live bundle SHA-256
d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab.
This is the builder's rationale, not an independent argument-selection verdict.