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No Small Observer Can Verify a Black Hole Firewall

Michael Timothy Bennett

Published Apr 7, 2026• 0 citations
Publisher

Abstract

Alice stays outside a black hole; Bob falls in. Alice never sees the interior. Bob crosses the horizon but can never signal back. Physics holds subjectively for both, yet their realities objectively contradict. This illustrating a conflict between quantum and classical mechanics. AMPS is a thought experiment about this. It asks one observer to distil a purifier of a late Hawking mode \( B \) from early radiation $R$, then compare it with the interior partner \( A \) inside a single causal patch. The hidden cost is not just computation but control. If the correct decoder depends on the black hole microstate, the observer must carry a physical selector for the decoder family. I model that selector as a finite control register bounded by the covariant entropy bound on the patch. A covering argument shows that any fixed, state-independent decoder works on at most \( 2^{\alpha_* S_{\mathrm{BH}}+c_\varepsilon} \) microstates, so the observer needs at least \( (1-\alpha_*)S_{\mathrm{BH}}-c_\varepsilon \) control bits---requiring a patch with area fraction \( \rho^2 \ge 1-\alpha_* - c_\varepsilon/S_{\mathrm{BH}} \). A time-sharing objection fails because single-observer verification requires co-instantiation, not mere sequential occurrence; serialisation creates no free room. The conclusion: a subhorizon observer generally cannot certify the AMPS contradiction. In quantum gravity, control information is physical, and single-observer certification is a capacity problem. If the contradiction can never be measured, is it real?

DOI
10.20944/preprints202603.0227.v2

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A bstract When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time β/ 2 π log S BH into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-time Hawking modes, despite the absence of a firewall. By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions. A crucial role is played by the existence of tiny, non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time, interior operators can only be reconstructed from the Hawking radiation if the initial state of the black hole is known. As the black hole continues to evaporate, reconstructions become possible that simultaneously work for a large class of initial states. Using similar techniques, we generalise Hayden-Preskill to show how the amount of Hawking radiation required to reconstruct a large diary, thrown into the black hole, depends on both the energy and the entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-independent interior reconstruction exists for any code space of microstates with entropy strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state dependence to avoid the AMPSS typical-state firewall paradox.

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Alice stays outside a black hole; Bob falls in. Alice never sees the interior. Bob crosses the horizon but can never signal back. Physics holds subjectively for both, yet their realities objectively contradict. This illustrating a conflict between quantum and classical mechanics. AMPS is a thought experiment about this. It asks one observer to distil a purifier of a late Hawking mode ( B ) from early radiation RRR, then compare it with the interior partner ( A ) inside a single causal patch. The hidden cost is not just computation but control. If the correct decoder depends on the black hole microstate, the observer must carry a physical selector for the decoder family. I model that selector as a finite control register bounded by the covariant entropy bound on the patch. A covering argument shows that any fixed, state-independent decoder works on at most ( 2^{\alpha_* S_{\mathrm{BH}}+c_\varepsilon} ) microstates, so the observer needs at least ( (1-\alpha_)S_{\mathrm{BH}}-c_\varepsilon ) control bits---requiring a patch with area fraction ( \rho^2 \ge 1-\alpha_ - c_\varepsilon/S_{\mathrm{BH}} ). A time-sharing objection fails because single-observer verification requires co-instantiation, not mere sequential occurrence; serialisation creates no free room. The conclusion: a subhorizon observer generally cannot certify the AMPS contradiction. In quantum gravity, control information is physical, and single-observer certification is a capacity problem. If the contradiction can never be measured, is it real?
A bstract We reconsider the black hole firewall puzzle, emphasizing that quantum error- correction, computational complexity, and pseudorandomness are crucial concepts for understanding the black hole interior. We assume that the Hawking radiation emitted by an old black hole is pseudorandom, meaning that it cannot be distinguished from a perfectly thermal state by any efficient quantum computation acting on the radiation alone. We then infer the existence of a subspace of the radiation system which we interpret as an encoding of the black hole interior. This encoded interior is entangled with the late outgoing Hawking quanta emitted by the old black hole, and is inaccessible to computationally bounded observers who are outside the black hole. Specifically, efficient operations acting on the radiation, those with quantum computational complexity polynomial in the entropy of the remaining black hole, commute with a complete set of logical operators acting on the encoded interior, up to corrections which are exponentially small in the entropy. Thus, under our pseudorandomness assumption, the black hole interior is well protected from exterior observers as long as the remaining black hole is macroscopic. On the other hand, if the radiation is not pseudorandom, an exterior observer may be able to create a firewall by applying a polynomial-time quantum computation to the radiation.
The monogamy of quantum entanglement, applied by Almheiri-Marolf-Polchinski-Sully (AMPS) to black holes, obstructs a smooth horizon vacuum after the Page time. We transcribe this argument to Hawking-like phonon radiation from a sonic horizon in the Unruh acoustic metric. An exact purity identity shows that post-Page-time unitarity forces the entanglement between an outgoing phonon and its interior partner to vanish, selecting a non-Hadamard (Boulware-like) phonon state, which we define as an acoustic firewall. Its renormalized stress tensor differs from the smooth state by a constant, negative near-horizon flux, and the thermal-atmosphere energy density it removes, measured by a static calorimeter, grows as (δr)−2(δr)^{-2}(δr)−2 in the radial coordinate toward the horizon (singular in the free-fall frame), cut off at the healing length. The construction is kinematic and does not resolve the information paradox; it yields one concrete, falsifiable prediction: a differential phonon-calorimetry signal R(δr)=∣ΔE∣/E(0)→(ℓκ/δr)2\mathcal{R}(δr)=|Δ\mathcal{E}|/\mathcal{E}^{(0)}\to(\ell_κ/δr)^{2}R(δr)=∣ΔE∣/E(0)→(ℓκ​/δr)2, present only after the analogue Page time in a Bose-Einstein condensate.
The black hole information paradox presents a fundamental tension between the global unitarity of quantum mechanics and the apparent thermal, mixed-state nature of Hawking radiation. We propose a resolution grounded strictly in the Operational Coherence Bound (OCB), a covariant entropy constraint asserting that while formal (global) information may be unbounded, the operationally accessible and distinguishable information available to any single observer must be finite. By defining operational entropy via vacuum-subtracted relative entropy to ensure finiteness in continuum quantum field theory, we prove the Operational Unitarity Theorem: in a black hole spacetime satisfying the OCB, no single observer can operationally distinguish between unitary and non-unitary evolution of the global state. This framework resolves the Hawking information loss claim, the Page curve verification problem, and the AMPS firewall paradox by demonstrating that the causal geometry of black holes dynamically enforces an observer-indexed cap on cumulative information access. Global unitarity is preserved, but its operational verification is strictly forbidden by the geometric constraints of the observer’s causal diamond.This paper was originally integrated into the Unified Framework for Yang-Mills Mass Gap, QCD Confinement, and Black Hole Information paper uploaded in March 2026 https://doi.org/10.5281/zenodo.18829529 Version 3 updates a duplication error I made in the last version.
When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time β/2πlog⁡SBHβ/2π\log S_{BH}β/2πlogSBH​ into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-time Hawking modes, despite the absence of a firewall. By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions. A crucial role is played by the existence of tiny, non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time, interior operators can only be reconstructed from the Hawking radiation if the initial state of the black hole is known. As the black hole continues to evaporate, reconstructions become possible that simultaneously work for a large class of initial states. Using similar techniques, we generalise Hayden-Preskill to show how the amount of Hawking radiation required to reconstruct a large diary, thrown into the black hole, depends on both the energy and the entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-independent interior reconstruction exists for any code space of microstates with entropy strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state dependence to avoid the AMPSS typical-state firewall paradox.
Black holes offer an exciting area to explore the nature of quantum gravity. The classic work on Hawking radiation indicates that black holes should decay via quantum effects, but our ideas about how this might work at a technical level are incomplete. Recently Almheiri-Marolf-Polchinski-Sully (AMPS) have noted an apparent paradox in reconciling fundamental properties of quantum mechanics with standard beliefs about black holes. One way to resolve the paradox is to postulate the existence of a ``firewall'' inside the black hole horizon which prevents objects from falling smoothly toward the singularity. A fundamental limitation on the behavior of quantum entanglement known as ``monogamy'' plays a key role in the AMPS argument. Our goal is to study and apply many-body entanglement theory to consider the entanglement among different parts of Hawking radiation and black holes. Using the multipartite entanglement measure called negativity, we identify an example which could change the AMPS accounting of quantum entanglement and perhaps eliminate the need for a firewall. Specifically, we constructed a toy model for black hole decay which has different entanglement behavior than that assumed by AMPS. We discuss the additional steps that would be needed to bring lessons from our toy model to our understanding of realistic black holes.
A bstract When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time β/ 2 π log S BH into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decreasing entanglement entropy of the black hole is exactly consistent with the semiclassical bulk entanglement of the late-time Hawking modes, despite the absence of a firewall. By studying the entanglement wedge of highly mixed states, we can understand the state dependence of the interior reconstructions. A crucial role is played by the existence of tiny, non-perturbative errors in entanglement wedge reconstruction. Directly after the Page time, interior operators can only be reconstructed from the Hawking radiation if the initial state of the black hole is known. As the black hole continues to evaporate, reconstructions become possible that simultaneously work for a large class of initial states. Using similar techniques, we generalise Hayden-Preskill to show how the amount of Hawking radiation required to reconstruct a large diary, thrown into the black hole, depends on both the energy and the entropy of the diary. Finally we argue that, before the evaporation begins, a single, state-independent interior reconstruction exists for any code space of microstates with entropy strictly less than the Bekenstein-Hawking entropy, and show that this is sufficient state dependence to avoid the AMPSS typical-state firewall paradox.