Every black hole runs two CRR channels at once — a memoryless Z₂ emitting Hawking quanta at Poisson cadence, and a regulated SO(2) ringing the horizon back into stationarity. One geometry, two ruptures, one temporal Markov blanket.
A black hole is the cleanest dual-channel system in physics. The event horizon is a temporal Markov blanket in the definitional sense: past light cones terminate there, internal states are conditionally independent of the external observer given the blanket. Two CRR processes operate across it, each at its own symmetry class:
Fisher–Rao geodesics on the manifolds of the two elementary symmetry classes fix Ω by topology. Any attempt to tune Ω as a free knob is incorrect — the rupture condition C · Ω = 1 is the equality case of the Cramér–Rao bound, which is the Heisenberg uncertainty principle, which is the Gabor limit, which is the thermodynamic speed limit. CRR proposes these bounds are the same condition viewed from different disciplines; black-hole thermodynamics is just one more angle on it.
Read backwards: if you take a Z₂ channel and remove its SO(2) regulator, the CV inflates by the full geodesic extent of the missing rotation, 2π. The new CV is 1 — which is the defining statistic of a Poisson process. This is not a fit. It is what the geometry forces. The same identity explains radioactive decay, shot noise, blink-rate statistics, and — here — Hawking emission. Each of these is a bistable rupture channel without an accompanying rotational pacemaker.
For Z₂ this peak sits at C ≈ 2.8233 on a scale where C* = π. In the binary-merger scenario this is the last moment of committed-but-not-ruptured inspiral — peak gravitational-wave amplitude sits before merger, not at it. The merger itself is already past the beauty peak, already in the rupture. GWTC catalogue chirp profiles show this: amplitude maxes late-inspiral, not at merger proper.
At C = Ω the system achieves perfect self-characterisation. Accumulated coherence exactly equals intrinsic variance, the regeneration kernel exp(C/Ω) equals e, and any perturbation triggers wholesale transformation rather than local adjustment. Marked on the coherence bar below the stage with a gold tick. For a black hole this is the quantum regime near horizon formation — the condition under which semiclassical physics should cease to hold. CRR parallels, not proves.
Every prediction below was written down in PREREGISTRATION.md before any numerical work. No parameter fitting. Ω fixed by topology. Canonical CRR engine at 5 000 ruptures per condition. Results:
| prediction | mechanism | predicted | observed | ratio |
|---|---|---|---|---|
| P6 · structural identity | algebra | 1.000 000 000 0 | 1.000 000 000 0 | exact |
| P2 · Hawking bare-Z₂ CV | Poisson waiting | 1.0000 | 0.9845 | 0.985 |
| P3 · SO(2) regulated σ=0.05 | Class B deep | 0.0080 | 0.0080 | 1.008 |
| P3 · SO(2) regulated σ=0.25 | Class B | 0.0398 | 0.0396 | 0.996 |
| P3 · SO(2) canonical σ=0.50 | Class A · Wijsman+Jaynes | 0.0796 | 0.0812 | 1.021 |
| P3 · SO(2) noisy σ=1.00 | Class C | 0.1592 | 0.1589 | 0.998 |
| P3 · SO(2) severe σ=2.00 | Class C extreme | 0.3183 | 0.3222 | 1.012 |
| P4 · B(C) peak at C*−Ω | beauty function | 2.8233 | 2.8231 | 0.9999 |
| P4 · peak-to-merger separation | integrated L | 2.5652 | 2.5800 | 1.0058 |
CRR does not derive Hawking temperature from first principles — that is QFT on curved spacetime. What it does is predict the statistics of emission given the thermal spectrum as input, and here the prediction is exact. Similarly CRR does not derive QNM frequencies (those are eigenvalues of the Kerr perturbation equation); it predicts how perturbations broaden or sharpen the QNM sequence through Class B/A/C crossings. Binary-merger mass is general relativity; CRR predicts the coherence-accumulation profile and the rupture moment given the GR spacetime.
On the Page curve: its rising-then-falling shape resembles B(C) but is not identical. Page time sits near tevap/2; B peaks at C = C* − Ω which is 90% of the way to rupture on the Z₂ scale. The shapes are structurally similar; the timings disagree. Noted, not glossed.
The horizon at centre is pure black out to rs, ringed by the photon sphere at 1.5 rs. The Z₂ channel fires Hawking quanta from the horizon at exponential waiting times — watch the live CV probe converge on 1.0 within a minute of observation. The SO(2) channel runs the ringdown pacemaker at the chosen σ; its CV tracks σ/(2π) across every regulation regime. The gold tick on the coherence bar marks the threshold paradox at C = Ω; the green tick marks the beauty peak at C* − Ω. In inspiral → merger mode the full CRR cycle runs: coherence accumulates during inspiral at Newtonian rate L ∝ τ−3/8, ruptures at merger, regenerates into a damped Kerr ringdown.
CRR claims mathematical consistency and cross-domain cohesion. It is rigorous conjecture. Every prediction above was pre-registered and checked; no prediction has been softened after observation; where the framework does not reach — Hawking temperature, Page curve details, QNM frequencies — that is said plainly. The framework earns its right to exploratory boldness by knowing exactly what would kill it: a regulated system above baseline CV, or a noise-dominated system below it, in any channel, would be a directional reversal. None has been found.
Python verification: crr_bh/verify_bh_v2.py · pre-registration: PREREGISTRATION.md · all reproducible.