Scope, conventions, and principal conclusions

Hawking's particle-creation calculation determines a semiclassical emission law once the spacetime, state, field, and asymptotic time normalization have been specified Hawking1975,Unruh1976. It does not by itself determine the channel transferring a specified infalling quantum message into accessible outgoing correlations. Hayden–Preskill recovery introduces additional assumptions concerning unitary dynamics, mixing, prior radiation, and decoding access HaydenPreskill2007. These assumptions are not consequences of substituting an astrophysical mass into a finite radiation model.

This paper separates three mathematical objects:

1. the semiclassical outgoing field state;

2. a declared message-to-radiation channel;

3. an observer's collection, storage, and measurement channel.

All recovery statements refer to the complete composition of the second and third objects. An emission spectrum is not substituted for that composition.

The source normalization follows the fixed-background interface reported in Cheung's Em03-A CheungEm03A. The operational separations are informed by the delayed-query, recovery, extraction, and record interfaces in Psy04-A/B, En02-B–E, Em05-A/B, and En03-A CheungPsy04A,CheungPsy04B,CheungEn02B,CheungEn02C, CheungEn02D,CheungEn02E,CheungEm05A,CheungEm05B,CheungEn03A. The relevant results are proved below rather than transferred between those manuscripts' different models.

Except where SI units are displayed, we use

The time $u$ is retarded time normalized by the asymptotic Schwarzschild Killing field. Frequencies $\omega$ are positive angular frequencies conjugate to that time. For normal states and finite-dimensional channels, respectively, write

Diamond-norm statements are made on the explicitly regulated interfaces; no finite-dimensional tensor-product factorization of the exact gravitational theory is asserted.

The principal established conclusions are:

1. the fixed-background scalar source law, Theorem thm:source;

2. exact delayed-query scoring and observer monotonicity, Theorem thm:score;

3. a conditional semiclassical null and its complete-attempt correction, Theorem thm:null;

4. an explicit finite-energy, spectrum-constrained completion and a nonrecovering completion with the same late marginal, Theorem thm:completion;

5. exact access-dependent scores and quantum recovery error, Theorem thm:completionoptima;

6. regulator, model-error, and sampling certificates.