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2535 bytes, as of the version from 2026-11-11 08:15 (current). Recorded here, not run.
# Certified polynomial/log-moment evaluation
Source: DLMF 4.36.1 and 4.36.2 (sinh and cosh infinite products), inspected
2026-11-11. Taking logarithms and expanding log(1+u) gives, for |t|<pi,
B(t)=log(2sinh(t/2)/t)=sum_{j>=1} b_j t^(2j),
D(t)=log(cosh(t/2))=sum_{j>=1} d_j t^(2j), where
b_j=(-1)^(j+1) zeta(2j)/(j(2pi)^(2j)) and d_j=(4^j-1)b_j.
These coefficients are derived from the products, not copied table values.
Set t=xy, r=m-1, s=n-1, 0<y<=1. Truncate both expansions after N terms.
P(y^2)=log(x)+sum b_j x^(2j)y^(2j),
Q(y^2)=log(2)+sum d_j x^(2j)y^(2j).
The approximants to A(xy),C(xy) are log(y)+P(y^2), Q(y^2).
For q_B=(x/(2pi))^2 and q_C=(x/pi)^2,
e_B=2 q_B^(N+1)/((N+1)(1-q_B)),
e_C=2 q_C^(N+1)/((N+1)(1-q_C))
bound the omitted kernels uniformly on y in [0,1], since zeta(2j)<2.
For the selected endpoints the generator proves in ball arithmetic
2q_B/(1-q_B)<1 and log(2)+2q_C/(1-q_C)<1.
Consequently |B|,|B_N|<1, |C|,|C_N|<1 and
|A|,|A_N| <= c-log(y), c=|log(x)|+1.
The exact finite polynomial integral is
-x sum_{h=0}^r binomial(r,h)(-1)^h h!
sum_j [y^(2j)](P^(r-h) Q^s)/(2j+1)^(h+1).
This uses integral_0^1 y^(2j)log(y)^h dy=(-1)^h h!/(2j+1)^(h+1).
The difference-of-powers identity bounds the integrand error by
r e_B(c-log(y))^(r-1) + s e_C(c-log(y))^r.
Writing J_v=sum_{h=0}^v binomial(v,h)c^(v-h)h!, its integrated absolute
error is at most x(r e_B J_(r-1)+s e_C J_r); the first term is zero if r=0.
The generator adds this entire bound to the result radius. This also applies
to n=1 controls and m=n=1, though these are not extra stored rows.
Exact endpoint radicals and logarithms are reconstructed in Arb balls and all
coefficient, polynomial and moment rounding is enclosed. No rounded endpoint
or sibling decimal is used. At 100 requested digits use 614 working bits and
N=160; larger requests increase both. RealBall native return types, finiteness
and retained accuracy are asserted. Output decimal intervals must contain
the complete balls. Actual measured precision is recorded in numerical evidence.
Independent checks use mpmath transformed quadrature t=x exp(-v), a method
with no polynomial expansion, and direct harmonic-weighted central-binomial
series from CGLXZ (2.5), with independent low-order dilogarithm and elementary
controls. They are numerical corroboration, not the rigorous tail proof above.
No claim of independent scientific acceptance is made by the builder.
Instructions: skill-manifest.json, bundle SHA-256
d6661c2f21613524889fa3a57d3c95f8330344ca62e5a856211c08a282b7dbab.