QEC Evidence, Current Disputes, and Status

As of 10 August 2026, holographic QEC has several firm but non-equivalent layers: exact finite-dimensional code theorems; leading semiclassical JLMS and entanglement-wedge recovery; quantitative approximate and alpha-bit results in specified models; large-$N$ gravitational-algebra constructions; and non-isometric proposals. A new finite-$N$ objection to a shared region-independent logical algebra is material and unresolved. The durable conclusion is therefore conditional recovery with declared algebra, dressing, code, norm, and limit—not a consensus exact finite-$N$ code for gravity.

Required background. Approximate Finite-N Recovery, Alpha-Bits, and Error Bounds supplies the quantitative claims. Claim–Evidence Records, Replication, and Retraction Handling supplies source-parity rules.

Helpful background. Non-Isometric Encoding Proposals gives the alternative encoding class. Scrambling Evidence and Claim-Status Matrix and Relativistic Communication Protocols: Assumptions and Status provide comparison discipline.

Claim matrix at the 2026 cutoff

Claim class	What is established	Missing or disputed ingredient
Exact toy codes	Exact erasure recovery and redundant logical representatives for specified finite encodings	Gravity, continuum algebras, dressing, backreaction, finite $N$
JLMS and wedge recovery	Leading semiclassical relative-entropy relation and conditional recovery on a fixed code	Uniform finite-$N$ norm, nonperturbative error, unique dressing/algebra
Operator-algebra centers	Exact block structure in finite codes; central area-like term under exact complementary recovery	Universal state-dependent gravitational area operator
Approximate QEC	Recovery–leakage bounds for named channels and state families	Model-independent gravitational channel and energy-uniform error
Alpha-bits	Subspace-dependent recovery up to entropy-scaled code sizes in black-hole models	One decoder or full-algebra recovery on the complete microstate space
Large-$N$ gravitational algebras	Explicit Type II constructions in specified large-$N$ limits and ensembles	Generic finite-$N$ wedge algebra and commuting order of limits
Non-isometric codes	Solvable models, kernel/state-dependence relations, computational protection scenarios	General causal, linear, state-independent dictionary and UV completion
Shared finite-$N$ logical algebra	Standard QEC interpretation supplies it in idealized codes	A 2026 locality/dressing argument disputes it for ordinary finite-$N$ CFT supergravity sectors

The exact toy-code and semiclassical rows support the usefulness of QEC. They do not decide the last row.

The present finite-N dispute

The conventional argument begins from entanglement-wedge relative-entropy matching and operator-algebra recovery Dong, Harlow, and Wall 2016. Approximate-code work in 2026 shows how state-dependent geometry can coexist with controlled, non-exact recovery Cao et al. 2026, and Witten derives a quantitative hierarchy between recovery and area-function corrections in that framework Witten 2026.

Terashima instead argues that if one insists on a single region-independent, code-preserving finite-$N$ logical operator with representatives in several boundary regions, boundary locality forces it into the commutant of complementary code-preserving local algebras. Smeared stress tensors and gravitational dressing are then claimed to make that common noncentral algebra trivial for ordinary supergravity fields Terashima 2026.

These conclusions cannot be compared by slogans. A decisive comparison must use the same:

1. finite-$N$ code projector and energy window;
2. physical dressed operator and asymptotic anchor;
3. region algebras and code-preserving condition;
4. center and superselection convention;
5. operator or channel error norm; and
6. order of the $N\to\infty$, regulator, and code-size limits.

Until that comparison exists, “finite-$N$ holographic QEC is established” and “holographic QEC is impossible” are both too broad.

First application

Evaluate one scalar excitation using a six-column record: code sector, $A$- and $\bar A$-anchored dressings, candidate common algebra, complementary leakage, recovery error, and $N$ scaling. Apply the standard recovery construction and then the commutant test to the same matrices $P O_A P$ and $P O_B P$. If the representatives agree only at $N=\infty$ but differ by an $O(1/N)$ stress-tensor-detectable term, record leading semiclassical recovery and a failed exact shared-algebra claim separately.

For an approximate magic-enriched code, also compute the proto-area response and recovery fidelity. Agreement there is evidence for that model; it does not answer the CFT dressing objection unless the models’ algebras are matched.

Adversarial control

Require source parity: place the original claim, an independent analysis, and the material counterclaim under identical hypotheses. Do not cite the exact HaPPY theorem as direct evidence for finite-$N$ gravitational dressing, and do not cite one 2026 preprint as a theorem excluding every QEC interpretation. Attempt to construct the allegedly shared operator explicitly and test its commutators with complementary stress-tensor smearings.

The surviving statement must be no stronger than the common domain of the compared sources.

Evidence ceiling, update triggers, and handoff

This page’s cutoff is 10 August 2026. Update it if the finite-$N$ commutant argument receives a published proof or counterexample, if a common dressed algebra with an energy-constrained error bound is constructed, if the approximate area-function framework is embedded in a holographic CFT, or if a non-isometric proposal supplies a causal state-independent observable algebra.

The evidence ceiling supports exact toy recovery, leading semiclassical wedge recovery, and model-dependent approximate statements. It does not settle a nontrivial shared finite-$N$ logical algebra. Live adjudication belongs in the Research dossier; the next stable conceptual handoff is Holographic Complexity Proposals and Diagnostics where computational protection becomes a separately defined proposal rather than an error-correction theorem.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Cao, C., Cheng, G., Karthikeyan, K., Li, C., and Preskill, J. (2026), “State-Dependent Geometries from Magic-Enriched Quantum Codes,” preprint, revised June 2026. arXiv:2603.13475.
• Dong, X., Harlow, D., and Wall, A. C. (2016), “Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality,” Physical Review Letters 117, 021601. DOI; arXiv:1601.05416.
• Hayden, P., and Penington, G. (2019), “Learning the Alpha-Bits of Black Holes,” Journal of High Energy Physics 2019(12), 007. DOI; arXiv:1807.06041.
• Terashima, S. (2026), “Entanglement Wedge Reconstruction without Holographic Quantum Error Correction,” preprint. arXiv:2607.08684.
• Witten, E. (2026), “A Note on Corrections to Entanglement Wedge Reconstruction,” preprint, revised June 2026. arXiv:2606.18639.