Count the bits on a horizon. Watch the defect settle.
Black-hole information is bounded by area. On a discrete recognition lattice that bound becomes a
capacity per face, and the interesting number is the defect d = 4 − κ: how far the capacity
falls short of four bits per pixel. This script extends the Lean-checked GF(2) cut-marginal computations
(verified by enumeration at N=2,3 in SharedCutMarginal.lean) to arbitrary lattice sizes by exact
linear algebra: strips, open patches, and the torus, which is the horizon topology class. Every number is an
exact fraction. No floating point anywhere in the computation.
Strips, patches, torus
1×N open strips, n×n open patches, and n×n tori under two closure conventions (GLOBAL and LOCAL XOR). For each host it reports the total dimension, subregion marginals, capacity per face, the defect, and a gluing-law check across every split of the N=12 host.
Does the defect go transcendental?
The live bet is that the defect calibrates 1/π in G. The preregistered kill condition: a clean integer asymptote with an integer seam ("structureless rational"), unless the seam itself carries the structure.
GLOBAL vs LOCAL
Both closure conventions run side by side on every host. Per-pixel traced marginals give κ = 4 (GLOBAL) or κ = 3 (LOCAL) exactly, on every size. Joint marginals fall toward 1, or toward a perimeter law. The fork is the instrument check: an effect that appears in only one convention is a convention, not physics.
The measured defect, both conventions
Each square is one exact computation. The strip defects climb toward integer limits: the finite-N values fit the closed forms 2(N−2)/(N−1) for GLOBAL and (3N−5)/(N−1) for LOCAL, whose limits are exactly 2 and 3. That presses directly on the live bet, exactly as the kill condition anticipates, unless the integer seam is where the structure lives.
Strip defect table
| N | GLOBAL per face | GLOBAL defect | LOCAL per face | LOCAL defect |
|---|---|---|---|---|
| 2 | 4 | 0 | 3 | 1 |
| 3 | 3 | 1 | 2 | 2 |
| 4 | 8/3 | 4/3 | 5/3 | 7/3 |
| 6 | 12/5 | 8/5 | 7/5 | 13/5 |
| 8 | 16/7 | 12/7 | 9/7 | 19/7 |
| 12 | 24/11 | 20/11 | 13/11 | 31/11 |
Gluing law on the N = 12 host: D(m+n) − D(m) − D(n) = 2 on all sixteen tested splits, both conventions. The seam is an exact integer.
Run it yourself
One file, Python standard library only (math, fractions). Finishes in under a second.
$ python3 bekenstein_spectrometer.py BEKENSTEIN PHASE-0 SPECTROMETER (exact GF(2) linear algebra) reference constants: 1/pi = 0.31831 2pi = 6.28319 phi^5/pi = 3.53011 STRIP 1xN ... closure = GLOBAL asymptotic defect/face -> 1.8182 [nearest 2 (|err| = 0.1818)] gluing law check: D(m+n) - D(m) - D(n) = 2 (every split)
MEASURED instrument. The Lean theorems cover N = 2, 3 by enumeration; larger N here are exact-arithmetic extensions of the same definitions. The Bekenstein 1/4 coefficient itself remains CONDITIONAL in the holography program; this page proves nothing about it. What the instrument shows so far cuts against the "defect calibrates 1/π" bet: the finite-N tables are consistent with closed forms whose limits are the integers 2 and 3, and the seam is exactly 2. Under the preregistered reading that is pressure toward the kill condition, reported as found. The open escape is the seam-carries-structure branch.