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4824 bytes, as of the version from 2026-10-28 06:01 (current). Recorded here, not run.
# Sources and mathematical checks for T573
The instruction evidence is
`agents/runs/20261028T054431Z-build-skills/manifest.json`;
its matching local/live SHA-256 is
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
- D. N. Page, *Average entropy of a subsystem*, Physical Review Letters 71
(1993), 1291–1294. DOI: 10.1103/PhysRevLett.71.1291.
https://arxiv.org/abs/gr-qc/9305007v2
Equations (1)-(5) define partial traces, entropy, and the Haar average;
equation (9) gives the eigenvalue density; equation (11) states the finite
conjecture subsequently proved. The paper explicitly distinguishes mean
entropy from entropy of the mean state.
- S. Sen, *Average entropy of a quantum subsystem*, Physical Review Letters
77 (1996), 1–3. DOI: 10.1103/PhysRevLett.77.1.
https://arxiv.org/abs/hep-th/9601132
Equation (9) proves the finite Page formula. Equations (2)-(3) give the
monic generalized Laguerre coefficients and their norm. The unnumbered
integral following (4) gives a separate route for numerical validation.
- K. Życzkowski and H.-J. Sommers, *Induced measures in the space of mixed
quantum states*, Journal of Physics A 34 (2001), 7111–7125.
DOI: 10.1088/0305-4470/34/35/335.
https://arxiv.org/abs/quant-ph/0012101v3
Equations (3.5)-(3.8) fix the complex induced density, normalization and
Gaussian construction. At K=N the measure equals normalized complex
Hilbert–Schmidt volume measure.
All three PDFs were downloaded, hashed in source-downloads.json in the run
workspace, and read. The Sánchez-Ruiz (1995) proof appeared in search results,
but both full-text download routes failed; it is not relied upon here.
## Independent exact calculation
Set alpha = K-N. For each degree j = 0,...,N-1, expand the monic polynomial
p_j(x) with coefficient of x^i equal to
(-1)^(j-i) binomial(j,i) (j+alpha)!/(i+alpha)!.
Its squared norm for x^alpha exp(-x) is j!(j+alpha)!.
Sum the squared normalized polynomials against this weight to obtain the
Wishart one-point density R(x). Direct coefficient integration first verifies
integral R(x) dx = N and integral x R(x) dx = NK.
The entropy is psi(NK+1) - integral x log(x) R(x) dx/(NK).
For integer r >= 0, integral x^r exp(-x) log(x) dx = r!(H_r-gamma).
The Euler-constant terms cancel by the trace control, leaving exact rational
arithmetic. This calculation does not use the generator's harmonic-difference
formula. It agrees with every one of the 200 dimension pairs.
The Sen preprint has inconsistent superscripts in two unnumbered integral
expressions: p^(m-n) appears despite alpha=n-m in the weight and preceding
orthogonality definition. The check uses alpha=K-N from the density and from
Sen equations (2)-(3). This is determined by the orthogonality requirement,
not selected by agreement with a desired answer. The normalization and first
moment controls are checked for every polynomial and every pair. The polynomial coefficients were read from equation (2) and the PDF page was
visually inspected to distinguish genuine superscript inconsistencies from
text-extraction artifacts.
A separate two-qubit control integrates the probability density
3(2*x-1)^2 on [0,1]. Its integral is 1; integrating the binary spectral entropy
using integral -x^r log(x) dx = 1/(r+1)^2 gives 1/3 nat. The 2-by-3 control is
9/20 nat.
## Independent bit enclosures
With t=1/3, log(2) = 2 sum_(j>=0) t^(2j+1)/(2j+1).
After J=150 terms the remainder is positive and at most
2*t^(2J+1)/((2J+1)*(1-t^2)), less than 10^(-145).
Exact rational arithmetic thus bounds the logarithm without calling Arb log.
Dividing each independently obtained positive rational entropy by the two
endpoints gives an independent enclosure for its value in bits.
All these enclosures and all generating Arb balls lie inside the serialized
100-significant-digit intervals. Every tested ball is finite and has relative
width less than 10^(-117), before any comparison is accepted.
## Corpus and tags
Fresh already_here/already_asked, Page/entropy/entanglement/quantum searches,
and lookups of 1/3, 9/20 and their bit values found no existing Page family.
The rational 1/3 has unrelated hits, which do not duplicate this family.
The table documents for T571 and T572 are readable to the bot, but their
public pages return 404: they are still drafts. Stable T-number HREF targets
are therefore used, without guessing slugs. They need coordinated publication
before T573's public links can serve readers.
The existing physics, probability theory and special values tags are used.
The live two-page tag list lacks quantum information. Propose that tag for
T571 (Hilbert–Schmidt volumes), T572 (Bures volumes), T573 (Page entropy), and
the queued induced-state trace moments. This build changes no other table
and creates no singleton tag.