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3921 bytes, as of the version from 2026-10-26 18:05 (current). Recorded here, not run.
# Certified computation of T522
All stored values are Arb ball enclosures narrowed to 100 significant decimal digits. No approximate floating result is wrapped as a certified ball. The definition is the expectation for independent uniform angles and unit steps in two dimensions.
## Three-step seeds and recurrence
Equations (5) and (7) of Borwein–Straub–Wan's October 2010 preprint give
A = 3*2^(1/3)*Gamma(1/3)^6/(16*pi^4) = W_3(-1),
B = A + 27*2^(2/3)*Gamma(2/3)^6/(4*pi^4) = W_3(1).
The recurrence is
(s+4)^2 W_3(s+4) - 2(5s^2+30s+46) W_3(s+2)
+ 9(s+2)^2 W_3(s) = 0.
## Four-step seeds and recurrence
Let K(m) use the elliptic parameter. Equations (40) and (58) of that paper become
J_p = integral_0^1 k^p K(1-k^2)^2 dk,
W_4(-1) = 4 J_0/pi^3,
W_4(1) = 16 (J_0 - 3 J_2)/pi^3.
The paper's K'(k) is the complete integral at complementary modulus, not a derivative. Squaring the modulus gives parameter 1-k^2. The proved recurrence (Borwein–Nuyens–Straub–Wan, Theorem 4 and Example 1) is
(s+4)^3 W_4(s+4) - 4(s+3)(5s^2+30s+48) W_4(s+2)
+ 64(s+2)^3 W_4(s) = 0.
## Treatment of the improper endpoint
DLMF 19.12.1 and 19.12.3 give, for 0<k<1,
K(1-k^2) = sum_{j>=0} c_j k^(2j) (L-d_j),
c_j = ((1/2)_j/j!)^2,
d_j = 2(H_{2j}-H_j), L=log(4/k).
Here 0<c_j<=1 and 0<=d_j<log(4). In particular all summands are positive for 0<k<=1/2. If S is the sum for j<N and R=K-S, then
0 <= R <= L*k^(2N)/(1-k^2),
0 <= S <= K <= L/(1-k^2),
0 <= K^2-S^2 <= (32/9) L^2 k^(2N).
Thus the remainder in J_p over (0,1/2) is bounded by (32/9) I(2N+p), where
I(q) = 2^(-q-1)/(q+1) * (log(8)^2 + 2*log(8)/(q+1) + 2/(q+1)^2).
The generator integrates S^2 exactly as a finite polynomial in k and log(4/k), with all coefficients, logarithms and rational arithmetic enclosed in balls. It adds a zero-centered ball with radius equal to an upper bound on this remainder. The two endpoint values themselves have measure zero and require no sampling.
On [1/2,1], Sage's ComplexBallField.integral provides certified adaptive quadrature. The integrand is analytic in Re(k)>0 because 1-k^2 avoids the elliptic K branch cut [1,infinity). The callback explicitly refuses analytic evaluation on any ball whose real part is not strictly positive. Both returned components must be finite and the imaginary component must contain zero.
Working precision is numberdb.bits(working_digits, losing=48); N=bits//2+12. Default working_digits=150. Repeating at 200 digits checks numerical stability in addition to the enclosure proof. Recurrences propagate the seed enclosures. Every resulting ball must be positive, finite, and sufficiently narrow; the NumberDB writer additionally enforces its requested 100 digits.
## Independent computation
The checks use the proved representations in Borwein–Straub–Wan–Zudilin, Section 7, equations (75) and (76), Lemmas 1 and 3:
W_3(s) = 3^(s+3/2)/(pi*2^s*Gamma(s/2+1)^2)
* integral_0^infinity t^(s+1) I_0(t) K_0(t)^2 dt,
W_4(s) = 4^(s+2)/(pi^2*Gamma(s/2+1)^2)
* integral_0^infinity t^(s+1) I_0(t) K_0(t)^3 dt.
These numerical quadratures neither use the elliptic seeds nor generate their values by recurrence. They are independent agreement checks, not a second enclosure proof. Exact normalization and second-moment controls W_n(0)=1 and W_n(2)=n must pass first. The elliptic quadrature also has a polynomial integral control. Checks must be repeated against the stored decimal intervals after filling.
## Instructions
Run manifest: ../20261026T172724Z-build-skills/manifest.json. Both archived instructions and the live version fetched during this run have SHA-256 b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18. Numerical results and exact dependency versions are recorded in numerical-evidence.json and the run logs. This method note is not an independent critique.