Manuscript Hawk01-A@1 · Hawk01-A.tex · theorem · 11979653

Theorem: Conserving completions with identical late radiation

An explicit Hawking-mode completion and its nonidentifiability › Paired occupation sectors and the early resource · lines 1550–1643 · proof included or linked

Hawk01-A@1#thm:completion, Hawk01-A@1#eq:transferchannel, Hawk01-A@1#eq:completionmarginals, Hawk01-A@1#eq:nullcompletion

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The map $V_{1}$ is an isometry on the prepared subspace. Every term has total energy $M$ , and the residual mass obeys the cutoff (eq:energycut). With input (eq:earlykey) and with $B K$ inaccessible, its accessible channel is

N_{1} (X) = \frac{1}{4} r \sum ∣ r ⟩ ⟨ r ∣_{E} \otimes [n, k \sum p_{n, k} J_{n, k} P_{r} X P_{r}^{†} J_{n, k}^{†} + Tr (X) 0 \leq n \leq L n even \sum w_{n} ∣ n, n ⟩ ⟨ n, n ∣] .

For every operator $X$ ,

Tr_{E} N_{1} (X) = Tr (X) σ_{L} (q), Tr_{R} N_{1} (X) = Tr (X) \frac{I _{E}}{4} .

There is a second energy-conserving isometry $V_{0}$ , on a sufficiently large but otherwise identical declared residual sector space, whose accessible channel is

N_{0} (X) = Tr (X) \frac{I _{E}}{4} \otimes σ_{L} (q) .

The two completions have the same entire late-radiation state, the same early marginal, and the same emitted-energy distribution for every input. Their assisted delayed-query scores differ whenever $p_{L} (q) > 0$ .

Proof. Each paired map $J_{n, k} P_{r}$ and each residual map $I_{n}$ is an isometry on the message. Orthogonality of the residual labels and (eq:weightscomplete) give $V_{1}^{†} V_{1} = I$ . Distinct key inputs remain orthogonal. Every term carries radiation energy $nω$ and residual energy $M - nω$ , so energy is conserved. The radiation is neutral and s-wave, and the stipulated residual labels conserve the declared charge and angular momentum.

Apply $V_{1}$ to the fixed $E K$ resource and trace the orthogonal residual sectors and $K$ . This gives (eq:transferchannel). The Pauli twirl satisfies

\frac{1}{4} r \sum P_{r} X P_{r}^{†} = \frac{Tr X}{2} I_{A} .

Since $p_{n, k} /2 = d_{n}$ , the twirled paired sectors and unpaired terms assemble exactly into $σ_{L} (q)$ . Tracing $R$ gives the uniform early marginal.

For the second completion, choose mutually orthogonal residual two-dimensional embeddings $I_{n, j}^{(0)}$ of energy $M - nω$ , and set

V_{0} (∣ ψ ⟩_{A} ∣ r ⟩_{K} ∣ M ⟩) = ∣ r ⟩_{K} n = 0 \sum^{L} j = 0 \sum^{n} d_{n} I_{n, j}^{(0)} ∣ ψ ⟩_{B} \otimes ∣ n, j ⟩_{R} .

Normalization $\sum_{n} (n + 1) d_{n} = 1$ proves that this is an isometry. Each term conserves energy and the declared charges. Tracing its residual and $K$ gives (eq:nullcompletion).

Both isometries may be embedded in a common enlarged residual Hilbert space. Within each finite total-energy sector, an isometry on a prepared subspace extends to a unitary after completing orthonormal bases of that sector. This extension establishes finite-interface unitary physicality, not a microscopic black-hole Hamiltonian.

Finally, the score difference follows from Theorem thm:completionoptima below and the message-independent channel (eq:nullcompletion).

Displayed formulas in this passage

Hawk01-A@1#eq:transferchannel · line 1557
$N_{1} (X) = \frac{1}{4} r \sum ∣ r ⟩ ⟨ r ∣_{E} \otimes [n, k \sum p_{n, k} J_{n, k} P_{r} X P_{r}^{†} J_{n, k}^{†} + Tr (X) 0 \leq n \leq L n even \sum w_{n} ∣ n, n ⟩ ⟨ n, n ∣] .$
Hawk01-A@1#eq:completionmarginals · line 1573
$Tr_{E} N_{1} (X) = Tr (X) σ_{L} (q), Tr_{R} N_{1} (X) = Tr (X) \frac{I _{E}}{4} .$
Hawk01-A@1#eq:nullcompletion · line 1583
$N_{0} (X) = Tr (X) \frac{I _{E}}{4} \otimes σ_{L} (q) .$
unlabeled · line 1609
$\frac{1}{4} r \sum P_{r} X P_{r}^{†} = \frac{Tr X}{2} I_{A} .$
unlabeled · line 1620
$V_{0} (∣ ψ ⟩_{A} ∣ r ⟩_{K} ∣ M ⟩) = ∣ r ⟩_{K} n = 0 \sum^{L} j = 0 \sum^{n} d_{n} I_{n, j}^{(0)} ∣ ψ ⟩_{B} \otimes ∣ n, j ⟩_{R} .$

Theorem: Conserving completions with identical late radiation

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