back to table · edit · history · where entries came from · files · download
5162 bytes, as of the version from 2026-10-28 05:42 (current). Recorded here, not run.
# Numerical evidence for T572
This run uses the matching instructions recorded in
`agents/runs/20261028T051113Z-build-skills/manifest.json`, copied unchanged as
`skill-manifest.json`. The local and live SHA-256 are both
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The table is a private draft; this is build evidence, not an independent
critique, review, or publication.
## Convention and sources
Use Sommers–Życzkowski, arXiv:quant-ph/0304041v3 (17 February 2005),
(3.5), (4.19), (5.15), (5.16), and correction note [54]. The downloaded v3
PDF numbers the general formula (5.15); the proposal's (88) is not the label
in this copy. Rank r means actual rank, d=N-r is nullity, and
D=r-1+beta*r*(2*N-r-1)/2. The Bures metric is
one half of the sum of |d rho_ij|^2/(lambda_i+lambda_j) over positive
lambda_i+lambda_j, so the qubit hemisphere has radius 1/2.
Each gamma factor is at an integer or half-integer. The generator reduces
these factors to exact rationals times powers of sqrt(pi), combines the
powers, and obtains q*pi^k with q a positive rational and k a positive
integer. Therefore none of these entries is rational; no decimal disguises
an exactly rational member. Arb evaluates only this final expression at
397 bits (100 requested decimal digits plus a 64-bit guard).
## Checks and independence
All tests in `checks.py` pass before filling. All 270 serialized decimal
intervals contain both their 397-bit generating balls and a separate direct
gamma evaluation at 560 bits. The least accurate generating ball retains
389 relative bits (more than 117 decimal digits). Comparison balls are
required to be finite, positive and narrow; interval containment uses exact
rationals, including exponents down to e-227. A deliberately wrong decimal
is rejected by the same containment routine.
The gamma evaluation is a separate implementation of the same volume formula;
it is a transcription and precision check, not independent mathematical
confirmation. Independent checks are:
* All 135 complex cases: integrate the Bures eigenvalue weight by exact
Pfaffian elimination, independently of the product formula. Following
Sarkar–Kumar, arXiv:1901.09587v2, (11), (15), (21)–(23) and (37), put
a=d+1/2 and i,j=0,...,r-1. The skew moments are
H_ij=Gamma(a+i)*Gamma(a+j)*(j-i)/(2*a+i+j), augmented by
Gamma(a+i) when r is odd. The unrestricted integral is r!*Pf(H).
Divide by Gamma(r*(2*N-r)/2) to impose trace one. The Bures orbit factor
is the product of pi^(k-1)/Gamma(k) for k=d+1,...,N, with diagonal factor
2^(1-r) and permutation divisor r!. Gamma factors are removed using an
independently written recurrence. The rational Pfaffian is computed by
elimination, not its closed product. All resulting exact q,k pairs agree
with the generator, and their 560-bit values fit every complex entry.
* All 30 rank-two cases: the eigenvalue integral over lambda in (0,1) is
the integral of [lambda*(1-lambda)]^((beta*d-1)/2)*|2*lambda-1|^beta.
Substitution t=2*lambda-1 and then u=t^2 gives
2^(-beta*d)*B((beta+1)/2,(beta*d+1)/2). Multiply by one quarter of the
orbit volume, the product for k=N-1,N of
pi^((k-1)*beta/2)*Gamma(beta/2)/Gamma(k*beta/2). One factor one half is
from the diagonal metric, and the other from the eigenvalue permutation.
This derivation uses the metric's spectral coordinates, not (5.15).
* All 30 pure-state volumes: real projective space has volume
pi^(N/2)/Gamma(N/2), half the unit (N-1)-sphere; complex projective
space has volume pi^(N-1)/(N-1)!, also the unit 2(N-1)-ball volume.
* All 30 full-rank volumes equal the volume of a D-dimensional hemisphere
of radius 1/2. This is an equality of total volumes for general N; no
isometry to a hemisphere is claimed except for N=2.
* The complex boundary/interior ratio is
2*N*Gamma(N^2/2)/(sqrt(pi)*Gamma((N^2-1)/2)). The ratio is checked
against the serialized intervals for every N=2,...,16.
The checker first verifies Pfaffians of known 2-by-2 and 4-by-4 matrices,
including a case requiring pivoting and a negative answer, several exact
gamma recurrence values, and the rebit and qubit controls pi/2 and pi^2/8.
The independent tests cover every complex row. For real intermediate ranks
3<=r<N, the claim rests on the sourced theorem, checked by the separate
gamma implementation; no independent high-dimensional integration is claimed.
`dry_run.py` reports 270 entries, no exactness or prose issues, and a longest
raw ball representation of 140 characters. Serialized entries have maximum
length 107 characters. `numerical-evidence.json` records measured counts and
sizes; `runtime-versions.json` records the software. A later stored-check
attachment records the same identities on the values read from the database,
and `verify(sample=None)` compares every stored value with the generator.
One checker portability issue was resolved before values were sent: Sage
10.9 calls its relative-accuracy method `accuracy()`, not the python-flint
method `rel_accuracy_bits()`. The failed call produced no numeric result;
the documented Sage method and the complete rerun establish the final result.