# Build evidence for T568

Policy: `agents/runs/20261028T013533Z-build-skills/manifest.json`, attached as
`skill-manifest.json`. Local and live archived canonical skills were read and
both hash to `e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The source snapshot's canonical skill also has that hash.

The defining normalization was written in definition-first.md before
computation. DLMF 10.41.9 supplies the exact recurrence for U_k and its
conversion to V_k; DLMF 10.41.5–10.41.8 fixes the expansions and branches.

The generator computes in QQ[p], never with floating-point coefficients.
All 13 full polynomials for 0 <= k <= 12 passed agents/table-build/dry_run.py
and the shared exactness and names_its_rings checks. Maximum length is 702
characters at k=12; the shared measure reports a 3.78515625 KiB entries block.
Exact polynomials have no precision limitation or decimal error claim; the
campaign's 100-digit aim is inapplicable to these rational coefficients.

Independent construction: start with DLMF 10.32.9, write the integral over
the real line and differentiate with respect to the full Bessel argument.
The saddle of z*cosh(t)-t is t0=asinh(1/z). After t=t0+sqrt(p/nu)*s,
remove the Gaussian factor and the leading K' prefactor. The remaining
amplitude is cosh(sqrt(p)*epsilon*s)+p*sinh(sqrt(p)*epsilon*s), with
epsilon=nu^(-1/2). Expand the nonquadratic exponential formally through
epsilon^24 and apply exact Gaussian moments. The coefficient of epsilon^(2k)
is (-1)^k V_k(p). This algorithm uses neither the U recurrence nor the V
conversion. checks.py derives and implements the expansion. Every coefficient
of all 13 polynomials agrees exactly, not merely at selected values of p.

Controls run first: Gaussian moments of orders 0,1,2,4,6 are 1,0,1,3,15;
expectations of 1, s^2 and p*s^4 are 1,1,3p. A deliberately altered
coefficient at k=12 is rejected by agrees_with. Odd epsilon terms vanish.
The initial value and all three explicit polynomials in DLMF 10.41.11 agree.
Degree 3k, parity (-1)^k, lowest exponent k and the zero at p=0 for k>0
were checked throughout. Additional read-back checks compare the stored
V values with the stored companion T567 and rerun the independent construction.
Their actual outputs are kept in postfill-checks.log and readback.json in
this run's artifacts; the handoff reports their result.

The expansion checks establish the exact coefficients. They do not give a
100-digit approximation or a truncation-error bound for Bessel function
values at any chosen z and nu. No such assertion appears in the table.

Sources checked through the browsing tool: https://dlmf.nist.gov/10.41#ii
(equations 5–11) and https://dlmf.nist.gov/10.32#E9. Direct urllib downloads
returned HTTP errors, so no downloaded HTML is represented as source evidence.
The standard integral and recurrence are documented in checks.py and in the
table's references. This build is not the independent campaign critique.

Final read-back outcome: verify(sample=None) reports 13/13 matched, zero
differing, missing or extra entries. All 13 read-back polynomials also pass
the independent integral expansion; all 13 endpoint identities and all 12
derivative-conversion identities pass against T567. The displayed Programs
snippet reproduces all 13 polynomials. Runtime: Python 3.12.3, Sage 10.9,
numberdb distribution 0.1.10. No unresolved numerical discrepancies.