# Argument selection: Generalized log-sinh integrals

Build assessment, not an independent critique verdict. Archived policy: ../20261111T011141Z-build-skills/manifest.json, bundle SHA-256 d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab.

Quantity and domain: Lsh_n^(k)(x) = -integral_0^x t^k log(2 sinh(t/2))^(n-1-k) dt. Natural logs, x>0, integers n>=1 and 0<=k<n, parameter order (n,k,x); x is R. There are no factorial normalizations, absolute values of logarithms, regularizations or complex branches. Positive real radicals and logs define phi=(1+sqrt(5))/2, a=log(phi), b=2 log(1+sqrt(2)). BS §5.2 equation (33) inspected; its absolute value inside the logarithm is immaterial for positive t.

n and k are discrete_index: total order and integration-variable moment order. Include the complete non-elementary triangle 2<=n<=10, 0<=k<=n-2 at 2a and b. This covers basic, first and higher moments before extending depth. Omit the zero-logarithm edge k=n-1, whose integral is the elementary -x^n/n. Ten is a practical, noncanonical prefix. At a retain only k=0, 2<=n<=9: these are the basic boundary terms of the mixed-family duplication expansion through logarithmic degree eight. Nine is a practical companion prefix, not a theorem cutoff.

x has category mixed, with exhaustive disjoint groups:
- intrinsic, values [2logphi]: x=2a is the unique positive zero of A(x)=log(2sinh(x/2)), since A'(x)=coth(x/2)/2>0. BS Example 11 selects the same endpoint for the alternating Apery relation.
- documented_application, values [logphi]: x=a is the half-endpoint in CGLXZ Lemma 2.2 at z=1/4. Definition 2.1 includes the basic log-sinh boundary H_(m,1)=Lsh_m^(0). The elementary identity A(2t)=A(t)+log(2cosh(t/2)) gives the duplication expansion. Only this basic boundary slice is included, not an unsupported triangle at a.
- reference_sample, values [2logsilver]: x=b=2asinh(1). This is bounded exploratory coverage; no source is claimed to publish every selected value or to supply an application for every moment.

reference_policy for 2logsilver:
  purpose: Recognition of logarithmic moments at the negative inverse-binomial convergence endpoint, including the higher moments beyond k=1.
  bounds: Only x=2asinh(1), every 2<=n<=10 and 0<=k<=n-2.
  coverage_rationale: The normalization u=-4sinh(x/2)^2 gives u=-4 here and u=-1 at 2a. DK equation (1.1) defines the inverse-binomial series with u unscaled (the hypergeometric variable is u/4). Its coefficient ratio tends to |u|/4. Thus b marks the negative radius-of-convergence endpoint and exp(-b)=(sqrt(2)-1)^2 is algebraic. It also tests integration across the sign change of A. This single structurally motivated endpoint broadens the reference beyond the golden endpoint without manufacturing a radical grid. The endpoint map is derived, not attributed as a statement about every higher moment in DK.
  noncanonical_acknowledged: true
  notation_independent: true
  sources_checked: true

Entire enumeration: 45 rows each at 2logphi and 2logsilver; 8 at logphi; total 98. Increasing n, then k, then exact endpoint. complete=no. Parameter descriptions give the natural domain, while the completeness note, Programs and generator describe this finite selection. Negative endpoints yield parity transforms and are not separate coverage. No quantiles, probabilities or statistical critical values occur.

Checked sources: BS arXiv:1103.4298v1 §5.2 eqs. (33)-(35), Example 11; CGLXZ arXiv:2210.14704v1 Definition 2.1 and Lemmas 2.1-2.2; DK hep-th/0303162v4 eqs. (1.1)-(1.3). Copies BS.html, CGLXZ.html and DK.html are in the build evidence. The source passages, not sibling values, support the conventions.