# Log-sinh-cosh build research

Checked sources: CGLXZ arXiv:2210.14704v1, Definition 2.1 (sign and
normalization), Lemma 2.2 eq. (2.5), section 4 eq. (4.2) (a=log(phi) at z=1/4).
The actual paper, not a sibling's conclusion, was inspected. Full CGLXZ.html
SHA-256: 0e2770c516de8640792ccfa67738f561021bd0157e4af4419697f510d96c834a.
DLMF 4.36.1-2 was inspected through the web reader: the two product expansions
justify the derived kernel coefficients; computation.md gives the full derivation.

Conventions checked: H_(m,n)(x)=-integral A^(m-1)C^(n-1), with natural logs,
A=log(2sinh(t/2)), C=log(2cosh(t/2)); positive real x, ordered positive integer
m,n, no factorial or pi normalization, no magnitude. Source inner absolute
values can be omitted for positive t. General domain is larger than selection.
At a=log(phi) retain m>=1,n>=2,m+n<=10; at 2a retain n=2,1<=m<=8.
The finite prefix p=1..8 in (2.5) gives exactly these 44 rows. Bounds are
noncanonical. Details and every free-parameter role: argument-selection.md.

Live catalogue was fetched once and saved; already_here/asked and three
sample /api/lookup queries are saved in discovery.json. No duplicate family
or exact title was found. Existing tags integral, moments, special functions,
special values, number theory exist; polylogarithms does not. No new tag used.
Plausible companions' live metadata was refreshed immediately before claim.
T699 is trigonometric; T353 stores positive beta parameters rather than
mixed parameter derivatives at b=0; T176 stores rational polylog arguments;
T713 is the pure log-sinh sibling. None is independent numerical evidence.
T194 (Shi) and T195 (Chi) also have different, non-logarithmic kernels, as checked in live metadata. T713 was already offered by its controller; this build does not change it.

Draft claim: T714, created by this run for the selected title. Initial history
message retains proposal #317 and contributes-to wanted #263, both issue URLs
checked live. No answers/closure claim is made for #263. No unrelated issues
are promoted to direct origins.

Derived, with explicit analytic justification in table prose: duplication from
A(2t)=A(t)+C(t); H_1,2 dilogarithm antiderivative; incomplete-beta generating
function under q=tanh^2(t/2), including its b=0 parameter domain. Numerical
checks are distinct from proofs of universal identities. Larger endpoint
selections are unassessed and not included; no unresolved convention remains.

Builder ball evidence: generator 614 bits, N=160, all 44 finite RealBall
returns; minimum accuracy 552 bits. Every serialized 100-digit interval
contains its full certified ball. Existing deterministic dry_run.py/check.py
pass with no exactness/prose findings. Independent numerical checker and final
stored checks are separate artifacts; none is an independent critic verdict.

Skill-manifest.json archives matching local/live bundle
 d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab.
All paths are relative to this run's work directory unless specified.
See artifact-hashes.json for source, generator, numerical and final snapshots.