Interior Consistency Checklist

Reproducing one thermal two-point function is necessary but not sufficient for a black-hole interior. A candidate must define its state space and operator domain, preserve linear quantum mechanics, realize a consistent algebra and dynamics, match the full semiclassical observables it claims, coexist with radiation purification, and control causal, energetic, and nonperturbative errors. The checks are independent: passing one does not compensate for failing another.

Required background. Quantum Information supplies channels, code subspaces, recovery, Born probabilities, and error norms used to formulate the tests.

Helpful background. Semiclassical Gravity supplies horizon and backreaction scales; Quantum Fields in Curved Spacetime supplies local algebra, Hadamard, stress-tensor, and detector tests; Black-Hole Information supplies the Page-curve and AMPS constraints.

State space and operator domain

First declare the microscopic Hilbert space or observable algebra and the class of states for which an interior is claimed. A code subspace $\mathcal C$ should specify its energy window, excitation number, conserved sectors, and whether it includes old holes entangled with arbitrary references. An operator $A_{\mathrm{inside}}$ is not defined merely by naming its action on one vector; its domain must contain the superpositions and apparatus states used in the proposed experiments.

If different reference states define different code subspaces $\mathcal C_i$, give transition maps on every overlap. For $\lvert\psi\rangle\in\mathcal C_i\cap\mathcal C_j$, the two representations must predict the same physical correlators to a stated error. Disjoint domains alone do not settle the issue: if their coherent superpositions are physical, the proposal must define its predictions on those states too. Excluding such superpositions needs a justified restriction of the claimed regime or a physical superselection rule.

The first diagnostic is therefore a domain table: state class, allowed algebra, excluded excitations, approximation parameter, and failure boundary. Without it, no claim of smoothness or locality has a quantifiable range.

Linearity and algebra

For every fixed domain, test linearity on arbitrary allowed superpositions,

$$A_{\mathrm{inside}}(\alpha\lvert\psi\rangle+\beta\lvert\phi\rangle) =\alpha A_{\mathrm{inside}}\lvert\psi\rangle +\beta A_{\mathrm{inside}}\lvert\phi\rangle.$$

Then test adjoints, products, associativity, and the commutator algebra. A proposed interior annihilation operator should reproduce its canonical commutator within the code error, have the correct commutator with the Hamiltonian, and approximately commute with exterior operators declared spacelike separated. Gauge charges and gravitational dressing must be included rather than discarded as inconvenient zero modes.

Born probabilities must depend on the density operator and a physical apparatus, not on an unrecorded ensemble decomposition. A state-dependent rule that assigns different outcomes to the same density matrix fails unless an accessible reference register distinguishes the preparations. Harlow identifies this measurement-theory burden for mirror proposals Harlow 2014, §§ 3–5, while Marolf and Polchinski give an adverse generic-state test Marolf and Polchinski 2016, § 2 and Appendix B.

Correlators, stress energy, and dynamics

The observable checklist should go beyond one Wightman function:

Check	Quantity	Failure it detects
Vacuum spectrum	Two-point function across frequencies and angular modes	Wrong thermal weights or missing entanglement
Operator ordering	Time-ordered, commutator, and out-of-time-order correlators	An assignment that matches one ordering but not the algebra
Ultraviolet regularity	Hadamard short-distance form	Hidden divergent infalling stress energy
Local response	Renormalized $T_{\hat\mu\hat\nu}$ and detector rate	Firewall or frame-dependent drama
Interactions	Three- and four-point functions with stated coupling order	Free-field reconstruction that fails under perturbations
Evolution	Heisenberg equation and Hamiltonian commutator	Interior operators inconsistent at different times

The state, regulator, smearing, and error must be common across rows. A two-point match at leading large $N$ does not establish associativity or locality at the first order where information recovery appears. Conversely, demanding exact local commutation in gravity may be too strong; the appropriate requirement is a controlled error below the physical effect being inferred.

Information and observer consistency

An interior construction must be combined with the radiation algebra in one formulation. If $E$, $B$, and the proposed $A$ are independent factors, the AMPS entropy inequality applies Almheiri et al. 2013, § 2. If $A$ is encoded in $E$ or another exterior algebra, the construction must give the encoding map, recovery error, and commutators that replace independence.

No-cloning should be checked on reference-entangled arbitrary inputs, not only orthogonal basis records. Observer complementarity should state whether it changes the global operator algebra or only prevents an experiment. A causal obstruction cannot by itself repair inconsistent Born probabilities on a common state.

Finally, distinguish existence from execution. A formal reconstruction can require exponentially complex operations, more radiation than is available, or energy that causes order-one backreaction. Causal time, circuit size, detector resolution, collection efficiency, and stress energy are part of the consistency claim whenever an observer is invoked.

Error budget and independent failure tests

Collect every approximation in one budget: large-entropy corrections, code-subspace reconstruction error, gravitational commutators, finite regulator, late-time corrections, and backreaction. State a norm for each. If an entropy contradiction has gap $\Delta$, combined factorization and continuity errors must be compared with $\Delta$, not merely called small.

Three adverse tests are especially useful. Enlarge the code subspace until two reference-state domains overlap; apply the operator to a coherent superposition rather than an ensemble; and perturb the state with a simple exterior unitary whose infalling effect is known. A construction that survives only by redefining the observable after each test has not supplied one quantum measurement rule.

Papadodimas and Raju provide a positive benchmark by constructing mirror commutators and correlators on a small algebra Papadodimas and Raju 2014, §§ 2–5. The critical analyses above supply independent tests of how far that domain can be extended. Both sides are needed for a meaningful assessment.

Results and open questions as of 2026-08-19

No accepted construction has demonstrated every item for generic old, evaporating four-dimensional black holes. It remains unresolved whether one state-independent algebra, compatible code-subspace maps, or a more radical modification is required. Competing positions emphasize different success criteria.

Evidence or proposal	Result	Scope and limitation
AMPS common-state test Almheiri et al. 2013, § 2	Any independent $EBA$ interior must satisfy the entropy inequalities	Identifies inconsistency but does not construct a replacement algebra
Mirror construction Papadodimas and Raju 2014, §§ 2–5	Reproduces interior commutators and correlators on a reference-state small algebra	Domain size, overlaps, measurement interpretation, and generic evaporation remain qualified
Measurement-theory critique Harlow 2014, §§ 3–5	Exposes ambiguous physical interpretations and unitary-apparatus problems	Does not negate controlled linear reconstruction on every restricted code subspace
Born-rule adverse test Marolf and Polchinski 2016, § 2 and Appendix B	Finds large violations for state dependence strong enough to smooth generic typical states	Conditional on generic-state smoothness and the stated state-dependent interpretation
Semiclassical EFT diagnostics	Hadamard, stress-tensor, detector, and higher-correlator tests constrain drama	Require a regulator and do not by themselves define microscopic factorization

The permissive position gives primary weight to correct correlators on a small semiclassical algebra. The conservative code position additionally requires overlap and quantum-channel control. The state-independent position demands one global operator, while the firewall position changes the state instead of the algebra. Their disagreement depends on state class and required accuracy, so the evidence does not support a universal verdict.

The conclusions could change after a construction publishes uniform error bounds for overlapping code subspaces, or after a microscopic evaporation model passes entropy, algebra, correlator, and backreaction tests together, or after a decisive counterexample proves that one item cannot coexist with the others under stated gravitational assumptions.

A minimal worked assessment

Suppose a candidate matches the vacuum two-point function on one reference state but gives no action on superpositions of two nearby reference states. It passes the first correlator row and fails the domain, linearity, and overlap tests. One cannot infer its commutators, Born probabilities, stress tensor, or compatibility with late radiation. The correct conclusion is “insufficiently defined on the claimed state class,” not “smooth interior established” and not “firewall proved.”

Generic operator and channel theory belongs to Quantum Information, local stress diagnostics to Quantum Fields in Curved Spacetime, and explicit holographic constructions elsewhere.

Exercise

A candidate has proved that its code, factorization, and continuity corrections shift the same entropy inequality by at most $10^{-3}$, $4\times10^{-3}$, and $2\times10^{-3}$ nats, respectively. An AMPS-style entropy gap is $0.02$ nats. Can the combined worst-case corrections hide the contradiction?

Solution

The triangle inequality bounds the total shift by $0.007$ nats, leaving a gap of at least $0.013$ nats. This worst-case sum remains valid even when the errors are correlated; statistical independence would be needed for a smaller quadrature estimate, not for the sum. Raw operator-norm, trace-distance, and entropy errors could not be added this way without first proving their contributions to the same inequality. Those conversions are part of the problem’s hypothesis.

References

• Almheiri, Ahmed, Donald Marolf, Joseph Polchinski, and James Sully. 2013. “Black Holes: Complementarity or Firewalls?” Journal of High Energy Physics 2013 (02): 062. Open text.
• Harlow, Daniel. 2014. “Aspects of the Papadodimas–Raju Proposal for the Black Hole Interior.” Journal of High Energy Physics 2014 (11): 055. Open text.
• Marolf, Donald, and Joseph Polchinski. 2016. “Violations of the Born Rule in Cool State-Dependent Horizons.” Journal of High Energy Physics 2016 (01): 008. Open text.
• Papadodimas, Kyriakos, and Suvrat Raju. 2014. “State-Dependent Bulk-Boundary Maps and Black Hole Complementarity.” Physical Review D 89: 086010. Open text.