# Certified values for T698 Instruction provenance: `skill-manifest.json`, archived at `agents/runs/20261109T224707Z-build-skills/manifest.json`. Both canonical skills have SHA-256 `886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`. ## Interior series Put a_j=u^j/(j^s binomial(2j,j)). For s>=2 and 0<|u|<4, |a_(j+1)/a_j| = |u| j^s / (2(2j+1)(j+1)^(s-1)) <= |u|/4 = r < 1. After adding a_N, the entire remaining absolute tail is at most |a_N| r/(1-r). This also bounds the alternating series, without assuming cancellation. The recurrence b_1=u/2, b_(j+1)=b_j*u*(j+1)/(2(2j+1)) computes b_j=u^j/binomial(2j,j), and then a_j=b_j/j^s. All operations are Arb ball operations; no Python-int division enters. The loop ends when the computed upper-bounding ball for the tail is strictly below 10^(-digits-60), and adds its upper endpoint as an error radius. Rational arguments enter exactly through QQ. ## Positive boundary Let p=s-2 and theta=2 arcsin(sqrt(u)/2). DK equation (1.2), with its log-sine definition (1.4), gives the equivalent integral for 0=1} zeta(2m) x^(2m)/(m*4^m), F_s(4) = 2^p*pi^2/p! integral_0^1 x [A-log(x)+H(x)]^p dx. The generator multiplies H as a polynomial in x^2, truncating after degree N. Truncation preserves all lower coefficients. For each retained coefficient of H^j, it integrates the logarithmic factor analytically: J_d(m) = integral_0^1 x^(2m+1) (A-log(x))^d dx, J_0(m)=1/(2m+2), J_d(m)=(A^d+d J_(d-1)(m))/(2m+2). The singular endpoint is integrated, not cut off numerically. This endpoint method adapts the Euler-product construction inspected in the T697 attachment; its archived source hash is recorded in generate.py. T698 contains a complete implementation, so it does not depend on future changes to that draft. For the omitted terms, let B(z)=sum_{m>=1} zeta(2m) z^m/m. Its coefficients are positive, and B(1/2)<2 log(2)=C since zeta(2m)<2. If c_(j,m) is the coefficient of B(z)^j, then sum_{m>N} c_(j,m)/4^m <= 2^(-N-1) C^j. This bounds the absolute tail of H(x)^j uniformly for 0<=x<=1. Multiplication by the other factor and integration uses M_d = sum_{l=0}^d binomial(d,l) |A|^(d-l) l!/2^(l+1) >= integral_0^1 x |A-log(x)|^d dx. Thus an absolute error bound for F_s(4) is E = 2^p*pi^2/p! sum_{j=1}^p binomial(p,j) 2^(-N-1) C^j M_(p-j). The bound is evaluated in Arb and its upper endpoint is added to the answer. The p=0 integral is elementary and has no series remainder. Arb also encloses pi, the logarithm, zeta coefficients, polynomial products, and all arithmetic. The code requires finite output and at least bits(digits+20) relative accuracy. At 100 requested digits it uses 525 working bits and N=532. The evidence records the measured worst-case accuracy, not just the nominal working precision. ## Independent validation and output rounding The defining F_2 has the elementary forms 2 asin(sqrt(u)/2)^2 for 0<=u<=4, and -2 asinh(sqrt(-u)/2)^2 for -4<=u<0. Repeated integration of the absolutely convergent defining series gives, for s>=3, F_s(u) = 2/(s-3)! integral_0^infinity y^(2s-5) F_2(u*exp(-y^2)) dy. The independent checker evaluates this integral at 150 and 200 decimal digits. It does not use the endpoint polynomial or central-binomial recurrence. Before comparison it checks the nonzero known integrals obtained by replacing F_2(u*exp(-y^2)) by exp(-y^2); the result is 1 at s=3 and s=10. A deliberately wrong answer is rejected. This quadrature does not supply the generator's error radii and is not described as certified. Every entry is regenerated at 140 requested digits. Its entire higher-precision Arb enclosure must fit inside the interval represented by the stored 100-significant-digit decimal (one unit in the last place on each side). The high-precision independent quadrature value must fit there too. A deliberate ten-unit perturbation is rejected. The checker tests ten closed-form identities and forty relations to live companion entries. After filling it repeats these checks on all read-back decimal intervals and runs Generator.verify(sample=None). The evidence states exactly which checks ran before and after filling.