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4139 bytes, as of the version from 2026-10-26 21:30 (current). Recorded here, not run.
# Numerical evidence for T528
Run: 20261026T210447Z-build. The source instructions are recorded at
agents/runs/20261026T210447Z-build-skills/manifest.json, copied verbatim to
skill-manifest.json. The canonical local skill, both archived versions and the
live skill fetched during this run are byte-identical: SHA-256
b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18.
## Convention and scope
Janson, arXiv:0704.2289v3, equation (20): C(n,n+k) ~ rho_k n^(n+(3k-1)/2),
where C counts simple connected graphs on a fixed labelled vertex set. Store
only rho_k for k=0,...,11. Tree value rho_{-1}=1 is a formula/control. The
argument-selection.md attachment explains every parameter and the finite bound.
## Computation and independent checks before filling
- The generator uses the K recurrence (7) over QQ. The relation
sigma_k=2^(k+1) K_(k+1) and equation (21) give the constants. Gamma factors
are simplified exactly before any approximation; odd k stays rational.
- Independent coefficients e_r=(6r)!/[2^(5r) 3^(2r) (3r)! (2r)!] and
c_r=e_r-(1/r) sum_(j=1)^(r-1) j c_j e_(r-j) use equations (62)–(65).
They are the formal logarithm of the graph generating series, computed
without calling the generator recurrence. Equation (34) converts c_k
independently to rho_k. All 12 rational factors agree exactly (k=0 has
the separately known unicyclic value sqrt(2*pi)/4).
- The first four constants agree with Table 1, page 91. The paper's printed
pages 87 and 92 were visually inspected to verify the powers and factorials.
- The sigma recurrence (23) holds for every required coefficient. The formula
in the Programs section agrees with the independent computation at all 12
indices, using direct Arb Gamma evaluation at 640 bits.
- Direct enumeration of every labelled simple graph on 2 through 5 vertices
gives tree counts 1,3,16,125, verifying Cayley's known formula before using
this enumeration as a control. Unicyclic counts are 1,15,222 at n=3,4,5,
agreeing with (n-1)!/2 * sum_(j=0)^(n-3) n^j/j!. These are controls for the
graph convention, not a high-precision estimate of the asymptotic limit.
- Every even-k value uses Arb at 512 bits. Measured accuracy is at least
509 bits (over 153 decimal digits); the maximum observed ball radius is
8.9460872e-155. The 100-significant-digit written decimal's exact rational
endpoints contain both the original ball and the independent 640-bit ball.
Comparisons first require finite, narrow balls. They test containment, not
merely midpoint agreement or overlap of unrestricted intervals.
- Exactness and named-ring checks pass. The prescribed dry_run.py passes.
The actual 12 written values have maximum length 106 characters and an
entries-block estimate of 1.13 KiB. Six rows are exact rational fractions;
six are 100-significant-digit decimal enclosures.
Reproduce with `sage -python checks.py` beside generate.py and check.py.
With an authorized NUMBERDB_API_KEY it additionally reads all entries from
T528, checks them against the independent graph generating series, runs
verify(sample=None), and checks rho_(n-1)=E[A^n]/n! against all 12 entries
read from the companion Brownian-excursion draft T523. This last comparison
is a cross-table consistency check, not a substitute for the independent
coefficient calculation. The authenticated checks are performed after filling;
the run artifacts contain their actual result.
Environment: SageMath 10.9, numberdb 0.1.10. Scripts ran through agents/sage.sh.
The PDF reader is disposable research tooling; its version and the source
PDF SHA-256 are recorded in research-versions.json in the run artifacts.
No independent critique acceptance is asserted here. No offer, review or
publication action is part of this build.
## Read-back outcome
Authenticated checks completed successfully: VerifyReport T528 reports 12/12
matched, zero differing, zero missing and zero extra. All 12 stored entries
also pass the independent graph-series comparison; all 12 stored excursion
moments in T523 pass the factorial relation. The exact output is attached as
checks-stored-final.json.