The Page Time Is 64.6%, Not 50%. Here's a Simulator to Show You Why.

The Hawking radiation calculation isn’t wrong. It’s an approximation that misses the correlations. That’s the whole paradox in one sentence, and it took fifty years of theoretical physics to unpack what it means.

If you throw a book into a black hole, quantum mechanics says the information can’t be destroyed — unitarity is non-negotiable, it’s built into the mathematical structure of the theory. But Hawking’s 1974 calculation shows black holes radiate thermally, and thermal radiation carries no information. When the black hole finishes evaporating, the book is gone. The two best theories in physics give directly contradictory answers on this point, and neither one is obviously wrong.

The resolution — as best as anyone currently understands it — is that the information does come back out, encoded in correlations across all the Hawking radiation ever emitted, in a form so scrambled that decoding it would require a quantum computer with access to the entire radiation field and the exact microscopic state of the black hole at every moment. The information is present in principle. Recovering it is computationally equivalent to cracking a cipher with a key you don’t have.

The simulator makes this concrete using a Feistel cipher as the black hole model. A Feistel network is a good analog for several reasons: it scrambles input bits in a way that’s exactly invertible (mirroring unitarity), errors in the scrambled bits amplify nonlinearly on inversion (mirroring how Hawking noise damages recovered information), and the round structure lets you tune scrambling depth the same way you’d tune how deep information sinks before it starts coming back.

The Page Curve tab runs a Monte Carlo over collection fractions — for each fraction of radiation bits collected, it replaces the rest with random noise and measures fidelity after inversion. The flat-then-rising curve that comes out is the computational Page curve. The transition happens at 64.6% collection, not 50% — that’s the exact Page time, derived by solving for when the radiation entropy equals the remaining black hole entropy. A lot of sources round this to “halfway through evaporation.” It isn’t halfway.

The Entropy tab plots all three curves: Bekenstein-Hawking entropy falling monotonically, Hawking’s radiation entropy rising monotonically (the information-loss scenario), and the Page curve following Hawking until Page time then falling back to zero (what unitarity requires). The gap between the second and third curves is a measure of the information paradox. The 2019 island formula results from Penington and Almheiri et al. derived the Page curve from first principles using replica wormholes — the problem is now considered solved in the sense that we have the right answer, even if the physical mechanism isn’t fully understood.

The Wormhole tab computes real numbers: Schwarzschild radius, Hawking temperature, scrambling time, and evaporation time for any mass you enter, from the actual formulas at double-precision float accuracy. A 0.001 solar-mass black hole has a scrambling time under a microsecond and would evaporate in about $2 \times 10^58$ years. The Hayden-Preskill panel shows exactly when Bob can recover Alice’s qubit depending on where in the evaporation she throws it — the recovery delay drops sharply at the Page time transition.

The unsolved parts are documented in the companion paper: the firewall paradox, the physical interpretation of replica wormholes, and what the actual decoding procedure looks like from first principles. The Page curve being correct is widely accepted. Why it’s correct is still an open question.

Full paper →