{"id":"2bb32204-bd06-493f-a1df-ead299233cd1","arxiv_id":"2603.13475","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Skewed approximate erasure-correcting codes have proto-area entropy that increases monotonically with bulk entropy, controlled by tripartite non-local magic in the encoding Choi state.","lead":"Approximate quantum error-correcting codes with non-local magic produce a state-dependent proto-area entropy that rises with bulk entropy, unlike exact codes. This gives a concrete information-theoretic route to gravitational backreaction in holographic toy models.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader’s strongest claim matches the paper’s theorems exactly. The weakest assumption correctly isolates the only soft spot—the qualitative geometric identification—but that identification is not load-bearing for the theorems themselves. The appendices supply explicit expansions, Weingarten evaluations, and positivity proofs that close the technical loop. Because the concern does not threaten the stated results, the ACCEPT verdict with high confidence stands; no adjustment is warranted.","tokens_in":66768,"tokens_out":365,"duration_ms":5033,"concrete_test":"Independently re-derive the leading-order coefficient of ��corr� for the mixed-state case (Eq. 4.13 / A.47) from the relative-entropy expansion (A.10) and Haar Weingarten rules (E.3.2) without using the Pauli-block positivity argument of A.53; if the coefficient of ��(�) remains non-negative, the monotonicity claim is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorems establishing Haar-averaged monotonicity of proto-area with bulk entropy/entanglement (4.2–4.6) and its control by tripartite non-local magic (5.1–5.2) are self-contained under the stated assumptions (flat or general resource spectrum, small GUE/skewing perturbation, optimal recovery by coherent-information maximization). The interpretive gap the reader flags—whether proto-area equals a QES or backreacted area—is already acknowledged by the authors (Sec. 3, Discussion) and is not required for the mathematical claims. No internal inconsistency or hidden assumption that would invalidate the leading-order expansions or the magic identification was found.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper argues that exact subsystem complementary recovery forces a state-independent area term and therefore cannot capture gravitational backreaction, so approximate erasure correction is required. It defines an RT-like decomposition for approximate codes: bulk matter entropy is the entropy of the optimally recoverable state (maximizing coherent information), and the complementary proto-area is the residual S(A) − S(recoverable bulk). For skewed codes obtained by small nonlocal unitary deformations of exact erasure-correcting codes, Haar-averaged proto-area is shown to increase monotonically with bulk entropy (mixed logical states) or bulk entanglement (pure logical states) at leading order in the skewing parameter, for both flat and general resource spectra (Theorems 4.2–4.6). The strength of the response is identified with perturbative tripartite non-local magic in the Choi state of the encoding map, which vanishes for stabilizer codes (Theorems 5.1–5.2). Detailed appendices supply the relative-entropy expansions, Weingarten/Haar averages, and spectral monotonicity proofs.","tokens_in":66933,"tokens_out":627,"duration_ms":6664,"significance":"If the results hold, the work supplies a concrete, calculable information-theoretic mechanism that links approximate quantum error correction, non-local magic, and state-dependent geometry. The central technical contribution is a set of explicit leading-order expansions and positivity/monotonicity proofs (Appendices A–E) that turn a qualitative no-go (exact codes cannot backreact) into a quantitative statement for a broad class of skewed codes. The identification of the coupling with tripartite non-local magic is falsifiable within the same framework and cleanly explains why stabilizer and local-unitary deformations fail. The interpretive gap between proto-area and actual QES/backreacted area is already flagged by the authors and does not undermine the mathematical claims; the paper therefore advances the program of emergent gravity from quantum codes in a precise and checkable way.","major_comments":[],"minor_comments":[{"comment":"The manuscript repeatedly uses “proto-area” and “PA entropy” interchangeably; a single consistent abbreviation after the first definition would improve readability.","section":null},{"comment":"Several figures (especially the circuit diagrams in Secs. 3–4) are dense; adding a short legend that maps A′, A″, χ, etc., to the text notation would help.","section":null},{"comment":"Appendix G’s AdS3 example is useful but sits somewhat apart from the main theorems; a one-sentence pointer in the Discussion clarifying that it is only qualitative support would avoid over-reading.","section":null},{"comment":"A few typographical inconsistencies appear in the arXiv source (e.g., missing spaces around some equation references and occasional font switches in the appendices); a light copy-edit pass would clean these up.","section":null}],"recommendation":"accept","confidential_remarks":"The paper is a solid technical contribution that fits JHEP well. The interpretive caution already present in the Discussion is appropriate; no hidden circularity or load-bearing gap that would require major revision was found."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The core result is clean. Exact subsystem complementary recovery forces a state-independent area term; small non-local skewing of those codes produces a Haar-averaged proto-area that rises with bulk entropy (mixed) or bulk entanglement (pure) at leading order in the perturbation, and the coupling strength is set by tripartite non-local magic in the Choi state. That is new relative to Harlow, Pastawski et al., and the earlier no-go on stabilizer area operators.\n\nWhat they do well is the formal work. They define matter entropy via coherent-information-maximizing recovery and proto-area as the residual, then prove the monotonicity statements (Theorems 4.2–4.6) for flat and non-flat resource spectra by explicit second-order expansions, Weingarten calculus, and spectral-function positivity. The appendices are thorough; the magic identification (5.1–5.2) is a direct calculation once the Pauli blocks are isolated. Circularity is low: the definitions are introduced and then related by computation, not assumed.\n\nThe soft spot is interpretive and already flagged by the authors. Proto-area is not shown to be a quantum extremal surface or a backreacted area; the optimal-recovery condition need not coincide with QES extremization, and the link to linearized Einstein constraints remains qualitative. That does not touch the theorems as stated. Minor residual couplings at finite logical dimension are left open, but they are higher-order in the large-dimension limit they emphasize.\n\nThis is for people working on holographic codes, approximate QEC, and information-theoretic gravity. The math is solid enough that a serious referee should see it. I would bring it to reading group, cite the monotonicity and magic results, and accept it for peer review.","headline":"Solid math on approximate codes: proto-area responds monotonically to bulk entropy under small non-local skewing, controlled by tripartite non-local magic; geometric identification stays qualitative as the authors say.","tokens_in":67522,"tokens_out":454,"would_cite":true,"duration_ms":7811,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Approximate quantum codes with non-local magic make geometry respond to bulk matter.","keywords":["quantum error correction","holographic codes","proto-area entropy","non-local magic","approximate recovery","quantum extremal surfaces","gravitational backreaction","Ryu-Takayanagi"],"falsifier":"Construct or sample a small non-local deformation of an exact code, compute the averaged proto-area versus bulk entropy at leading order, and check whether the slope is proportional to the measured tripartite non-local stabilizer Rényi entropy of the Choi state; a vanishing or negative slope for generic magic-rich deformations would refute the claim.","tokens_in":67660,"feed_emoji":"🔮","tokens_out":609,"duration_ms":9376,"temperature":0.7,"pith_summary":"Exact holographic quantum codes treat geometry as a fixed background: the entropic area term never depends on the logical bulk state, so matter cannot backreact. This paper argues that the rigidity is built into exact complementary recovery and that genuine matter-geometry coupling requires approximate recovery. It defines a Ryu-Takayanagi-like split for approximate codes, taking the bulk (matter) entropy to be that of the optimally recoverable state and the residual proto-area entropy to be the leftover geometric piece. For a broad family of codes obtained by small non-local deformations of exact codes, the averaged proto-area rises monotonically with bulk entropy or bulk entanglement, matching the qualitative response of quantum extremal surfaces. The size of the response is controlled by tripartite non-local magic in the encoding map's Choi state, a resource that vanishes for stabilizer codes. The result supplies a concrete information-theoretic mechanism by which geometry can become state-dependent once exact error correction is relaxed.","feed_headline":"Magic-enriched codes make geometry respond to matter","feed_subtitle":"Approximate recovery plus non-local magic turns the area term state-dependent","key_machinery":"Proto-area entropy: the difference between boundary entropy and the entropy of the bulk state recovered by maximizing coherent information. Its leading response to bulk entropy is proportional to a tripartite non-local magic measure of the encoding map's Choi state.","core_discovery":"In skewed approximate subsystem erasure-correcting codes, the Haar-averaged proto-area entropy increases monotonically with bulk entropy (mixed case) or bulk entanglement (pure case) at leading order in the deformation, and the coupling strength is set by tripartite non-local magic in the Choi state of the encoding isometry.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Skewed codes make proto-area rise with bulk entropy via non-local magic","Approximate recovery turns area state-dependent through tripartite magic","Non-local magic couples bulk entropy to geometry in erasure codes","Magic-enriched codes yield state-dependent Ryu-Takayanagi proto-area","Tripartite magic sets matter-geometry link in approximate holographic codes"],"cache_read_input_tokens":128,"weakest_assumption_plain":"That the residual after optimal state recovery is the right geometric quantity to compare with quantum extremal surfaces or gravitational backreaction, even though the paper notes the correspondence is still only qualitative.","fun_headline_variants_meta":{"raw":{"variants":["Skewed codes make proto-area rise with bulk entropy via non-local magic","Approximate recovery turns area state-dependent through tripartite magic","Non-local magic couples bulk entropy to geometry in erasure codes","Magic-enriched codes yield state-dependent Ryu-Takayanagi proto-area","Tripartite magic sets matter-geometry link in approximate holographic codes"]},"model":"grok-4.5","effort":"low","cost_usd":0.003666,"raw_usage":{"total_tokens":1115,"prompt_tokens":708,"num_sources_used":0,"completion_tokens":96,"cost_in_usd_ticks":36660000,"prompt_tokens_details":{"text_tokens":708,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":311,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":708,"tokens_out":96,"duration_ms":3187,"temperature":1.0,"reasoning_tokens":311,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T21:51:04.849012+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct or sample a small non-local deformation of an exact code, compute the averaged proto-area versus bulk entropy at leading order, and check whether the slope is proportional to the measured tripartite non-local stabilizer Rényi entropy of the Choi state; a vanishing or negative slope for generic magic-rich deformations would refute the claim.","supporting_citations":[],"review_version":1}