Nonlocal Interior Completion

If a semiclassical interior is an independent local subsystem and Hawking quanta carry no state-dependent correlations, information cannot reach the exterior in time for unitary evaporation. “Nonlocal completion” names several ways to reject that premise, not one theory. Direct atmosphere couplings, holographic encoding, non-isometric effective maps, topology change, and baby- universe sectors differ in their observables and causal claims. Each must state where locality fails, by how much, and what replaces it.

Helpful background. Semiclassical Gravity supplies stress-energy and backreaction constraints; Quantum Information supplies channels and error norms; Quantum Fields in Curved Spacetime supplies local effective field theory; Black-Hole Information supplies the transfer-rate target; Quantum Gravity supplies relational observables; and Causality and Global Structure supplies causal separation and global topology.

Locality must be stated as an observable claim

Local quantum field theory assigns commuting algebras to spacelike-separated regions, subject to gauge constraints, and propagates influences inside causal cones. Gravity already complicates exact localization: a gauge-invariant excitation needs dressing to relational or asymptotic data. That structural nonlocality does not automatically transfer a black hole’s microscopic state into outgoing radiation. A recovery mechanism needs a quantitative channel.

Let $\mathcal H_B$ contain the remaining black-hole degrees of freedom and $\mathcal H_{\mathrm{atm}}$ the atmosphere modes. A schematic transfer model adds

$$H_{\text{int}}(t) =\sum_{\alpha}\lambda_{\alpha}(t) \mathcal O_{\alpha}^{B}\mathcal O_{\alpha}^{\mathrm{atm}}.$$

The equation is only a parametrization until the operators, constraints, spatial support, state dependence, and coefficients follow from dynamics. One must calculate both the channel capacity from $B$ to the atmosphere and the induced stress-energy, commutators, scattering phases, and detector response. A coupling can be “soft” in energy or wavelength while remaining strong enough in information capacity to invalidate a local-factorization assumption.

Atmosphere-scale transfer

Giddings proposes parametrized interactions that transfer information to relatively soft modes in a region of order the horizon scale rather than at a Planck-thick surface. The construction explores rates compatible with unitary decay and asks whether infalling observers can avoid violent high-energy excitations Giddings 2013, §§ 2–5. It is a model class, not an established mechanism. Its functions and operators are chosen to meet consistency targets; they are not derived from a microscopic Hamiltonian.

After the Page regime, a finite evaporating model typically needs an order-one amount of quantum information to leave per characteristic emission interval. That target cannot be checked by energy flux alone. The relevant comparison is between a reduced exterior channel $\mathcal N$ and the Hawking reference channel $\mathcal N_0$, together with an energy budget. One may bound

$$\lVert\mathcal N-\mathcal N_0\rVert_{\diamond}$$

on a declared state family. A small full-domain bound limits output distinguishability; converting it to a capacity bound also requires the output dimension or an energy constraint. A proposal seeking large capacity with small simple-observable effects must therefore explain the restricted norm, complex encoding, or state dependence that separates those claims.

An adverse test couples the same atmosphere operator to a simple probe. Order-one anomalous scattering is an observable effect, but need not be energetically violent if all transferred frequencies are soft. Quantifying infalling stress and detector excitation is necessary to decide that claim. At the opposite limit, if the complete accessible observable algebra has exactly preparation-independent statistics, it carries no distinguishable message. Equality only of selected low-point correlators still allows information in higher correlations. The viable window is model dependent.

Encoding is different from propagation

Holographic error correction offers a second meaning of nonlocality. An interior operator may be encoded in a boundary algebra or radiation-plus-black- hole code rather than stored in an independent tensor factor. Reconstruction then changes the representation of the same logical information; it need not describe a signal crossing the horizon. The controlled code-subspace statements are developed in Quantum Error Correction and Complementary Recovery.

Non-isometric effective encodings sharpen this possibility. Akers and collaborators show in random-model settings how an effective interior map can approximately preserve a subexponential family of states while failing on sufficiently complex or outside-domain operations Akers et al. 2022, §§ 2–8. The fundamental dynamics in those models remains unitary; the non-isometry belongs to the effective map. This can explain why simple observers see a smooth interior without supplying a unique four-dimensional microscopic theory.

Topology and baby-universe sectors

Topology change introduces yet another structure. If amplitudes connect the parent universe to baby-universe degrees of freedom, tracing the latter gives a mixed parent state. In gravitational path integrals, summing topology can also look like averaging over couplings or Hamiltonians. Marolf and Maxfield show how a baby-universe Hilbert space and fixed alpha sectors can organize factorization and conditional amplitudes Marolf and Maxfield 2020, §§ 2–5.

These alternatives must not be conflated. Information permanently escaping to an inaccessible baby universe does not restore a unitary S-matrix for the original asymptotic system. Conditioning on a fixed alpha sector may restore factorization in a controlled model, but it raises the question of which sector describes a single theory and how its state is selected. The ensemble and replica issues are treated in Factorization, Ensembles, and Gravitational Path Integrals.

Results and open questions through 2026-08-19

This discussion reflects results available through 2026-08-19. One position permits horizon-scale, low-energy departures from local effective field theory if they transfer the required information without unacceptable stress. A conservative position allows only the nonlocal encoding already forced by gravitational constraints and holography, demanding no new acausal propagation. A third position assigns the missing correlations to topology or baby-universe sectors. These competing positions differ in formulation, subsystem choice, and observable regime.

Evidence or framework	Result	Scope or limitation
Parametrized atmosphere interactions in Giddings 2013, §§ 2–5	Soft, horizon-scale couplings can be designed to carry information and expose phenomenological checks.	Couplings are ansätze; microscopic origin, exact capacity, and backreaction remain model dependent.
Non-isometric random encodings in Akers et al. 2022, §§ 2–8	Simple state families can be approximately preserved while complex operations expose non-isometry.	Random-model and code-domain assumptions; not a derived astrophysical interior Hamiltonian.
Baby-universe Hilbert space in Marolf and Maxfield 2020, §§ 2–5	Fixed sectors can clarify conditional factorization and apparent ensemble behavior.	Controlled gravitational models; sector selection and single-theory interpretation remain unresolved.
Exact local Hawking channel as an adverse limit	Vanishing state-dependent exterior coupling produces no recoverable interior signal.	Demonstrates the transfer obstruction under strict factorization, not that every gravitational algebra factorizes this way.

The table contains constructive frameworks and a no-transfer limit, but no single row derives unitary evaporation, a smooth generic interior, and ordinary causal observables in a complete four-dimensional theory.

A discriminating test suite

For any proposal, name the complete asymptotic algebra and compute the induced channel on at least two nonorthogonal inputs. Report transfer rate, energy flux, stress fluctuations, and commutators of dressed probe observables. Vary the state family and complexity cutoff. If topology changes, distinguish a trace, an ensemble average, and conditioning on a sector. If the map is effective, state the norm and code domain in which it approximates a unitary map.

The central unresolved question is whether enough information can be moved or re-encoded while all accessible low-complexity observables retain controlled semiclassical behavior. The conclusions could change after a microscopic Hamiltonian derives an atmosphere coupling with verified capacity and backreaction, or a single-theory calculation resolves the baby-universe and factorization choice, or an operational adverse test detects—or decisively bounds—the required nonlocal commutator in the claimed regime.

Exercises

A model takes its Hawking reference channel to output a fixed radiation state, independent of the stored message. It claims that its exterior channel equals that channel on every input state, yet it also claims that an arbitrary qubit stored in the black hole can be recovered from that exterior output. Can both statements hold for the same complete channel and state domain?

Solution

No. If the two channels agree on every input density operator in the complete domain, linearity makes every exterior measurement probability identical, so the output contains no distinguishable dependence on the stored qubit beyond whatever the Hawking channel already carried. Recovery requires the channels to differ on some states or correlations. The model can evade the contradiction only by restricting the comparison to a smaller observable or state class, using a different subsystem algebra, or abandoning the assumed linear-channel description; each choice must be stated and tested.

References

• Akers, Chris, Netta Engelhardt, Daniel Harlow, Geoff Penington, and Shreya Vardhan. 2022. “The Black Hole Interior from Non-Isometric Codes and Complexity.” arXiv:2207.06536. arXiv.
• Giddings, Steven B. 2013. “Nonviolent Nonlocality.” Physical Review D 88, 064023. DOI. arXiv:1211.7070.
• Marolf, Donald, and Henry Maxfield. 2020. “Transcending the Ensemble: Baby Universes, Spacetime Wormholes, and the Order and Disorder of Black Hole Information.” Journal of High Energy Physics 08, 044. DOI. arXiv:2002.08950.