{"id":"929d334e-1f63-47fe-9ac3-724538f13aef","arxiv_id":"1908.09253","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Holographic codes on regular hyperbolic tilings have a code rate bounded by the tile side length divided by tile area, and for most regular tilings this bound is below one.","lead":"This paper proves a geometric bound on the information-storage rate of holographic quantum error-correcting codes built on hyperbolic tilings, and computes that rate exactly for one family of growth rules. The result tells code-builders which tilings keep the rate below one, the regime needed for error correction.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The rate bound is not the issue; the unproven converse 'χ<1 ⇒ nonzero erasure threshold' (footnote [25]) is load-bearing, and boundary-supported logical operators of constant weight make the full-logical-space claim doubtful.","rationale":"The reader's conditional verdict centers on the rate-to-threshold converse, and my stress-test agrees with that identification. The isoperimetric bound (Eqs. (3)-(4)) is valid for simply connected grown regions, and the tile-completion rates in Tables I-III are explicit and consistent with the bound. The problem is not the geometric bound but the leap from χ < 1 to a guarantee of quantum error correction. The O(1)-weight boundary logical operators mean the full logical space cannot have a positive worst-case erasure threshold, regardless of code rate, which makes the gap more than a missing proof. However, because the paper cites numerical evidence and could revise the claim to concern a protected logical subspace, the appropriate disposition remains conditional rather than outright rejection. Thus I would keep the reader's CONDITIONAL verdict; the proposed check would decide whether the condition is merely missing or the stronger claim is false.","tokens_in":10400,"tokens_out":20419,"duration_ms":241754,"concrete_test":"For the {5,4} tile-completion holographic code, explicitly construct the minimal-weight boundary representation of the logical operator on the bulk leg of a tile in the outermost layer. If the support weight w(n) is O(1) (or even sublinear in the boundary size), then the full logical space has zero worst-case erasure threshold, directly contradicting the abstract's guarantee; if w(n) grows linearly with N_boundary, the sufficiency assumption would have support.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's core inference is that the isoperimetric inequality, Eq. (3), yields the universal bound Eq. (2), χ_pq = ℓ_pq / a_pq, and that for p > 3 there is a q-range where χ_pq < 1. The abstract then concludes that hyperbolic geometry 'guarantees quantum error correction' for every holographic code grown by any inflation rule. The geometric derivation is internally consistent; the unsupported step is the converse, stated only in footnote [25] as numerically suggested, that code rate less than one implies a nonzero erasure threshold. Rate < 1 is necessary for a nonzero-rate code family, but it is not generally sufficient for a nonzero threshold. In this tensor-network construction there is a concrete obstruction: a bulk index on an outer-layer tile can be pushed through the perfect tensor to a constant number of boundary dangling edges, producing a logical operator of O(1) weight. Any such operator makes the worst-case erasure threshold for the full logical space zero, because erasing that O(1)-sized support is an erasure pattern whose fraction of the boundary tends to zero as the code grows, independently of how small χ is. Sections III and IV nonetheless conclude from χ<1 alone that all holographic p-gon codes with p > 3 'perform quantum error correction.' The conclusion may be salvageable for a restricted protected logical subspace, but the claim stated in the abstract is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives a geometric upper bound on the code rate of holographic codes grown on regular hyperbolic tessellations. Using the hyperbolic isoperimetric inequality, the authors show that for any inflation rule and any simply connected seed, the asymptotic code rate is at most chi_{p,q}=ell_{p,q}/a_{p,q}, where ell is the side length and a the area of the regular p-gon tile. They then study the range of q for which chi_{p,q}<1, compute exact code rates for the tile-completion growth rule via SL(2,Z) growth matrices, and report that triangle codes have code rate greater than one while all p-gon codes with p>3 have code rate less than one. The paper concludes that hyperbolic geometry guarantees quantum error correction for these codes.","tokens_in":10719,"tokens_out":39272,"duration_ms":380189,"significance":"The geometric rate bound in Section II is clean, parameter-free, and potentially useful: it gives a universal upper bound on the holographic code rate for any growth rule on any regular hyperbolic tessellation. The growth-matrix computations in Section V check out, and the explicit formulas for tile-completion code rates are a concrete contribution that other researchers can build on. However, the advertised conclusion that a code rate below one 'guarantees quantum error correction' is not supported. The paper's own footnote [25] concedes that the converse of 'rate < 1' is only numerically suggested, and the construction actually produces constant-weight logical operators for outer-layer tiles with p-2 dangling edges when p>=5, which destroys the worst-case erasure threshold for the full logical space. The valid contribution is the rate bound itself, not the error-correction guarantee.","major_comments":[{"comment":"The central claim that chi_{p,q}<1 'guarantees quantum error correction' is not established and, under the paper's own definition of an erasure threshold, is actually false for the full logical space. The proof in Section II only bounds the code rate; the additional step from rate below one to a nonzero erasure threshold is assumed, as the authors concede in footnote [25] where they write that numerical simulations only 'suggest' the converse. More seriously, the tile-completion construction itself produces constant-weight logical operators. In Section V the authors identify two cell types for q>3, with hanging-edge counts p-3 and p-2. For p>=5, p-2 >= (p+1)/2, so the perfect-tensor property described in footnote [24] allows a logical operator on the bulk index of such a tile to be transferred to its p-2 boundary dangling edges alone. That logical operator has constant weight independent of the code size. Erasing its support removes a fraction of the boundary that tends to zero as the number of layers grows, so no positive erasure threshold exists for the full logical space. Therefore the statement that all p>3 codes 'perform quantum error correction' is contradicted by the construction, and the abstract's stronger claim that hyperbolic geometry 'guarantees' QEC for any inflation rule is likewise unsupported.","section":"Abstract; Sections III and IV; footnote [25]"},{"comment":"The proof that tile-completion code rates are below one for all p>3 rests on an unproven monotonicity assertion. The authors claim that chi_{tau C}(p,q) decreases with p at fixed q and with q at fixed p, and use this to reduce the verification to three codes ({7,3}, {5,4}, {4,5}) plus the triangle-code case. However, Section V only establishes the asymptotic limits: chi -> 0 as p -> infinity at fixed q, and chi -> ((p-3)+(p-2))/((p-3)^2+(p-2)^2) as q -> infinity at fixed p. These limits do not imply global monotonicity in p and q for all finite values. Since the domination argument is load-bearing for the claim that every p>3 tile-completion code has rate below one, a direct proof of the monotonicity (for example by differentiating the explicit expression in Eq. (9)) or a documented finite verification is required.","section":"Section IV; Section V"}],"minor_comments":[{"comment":"The footnote correctly states that the sufficiency of code rate below one is only numerically suggested; the abstract and Section III should be reworded to present the error-correction conclusion as conditional on that conjecture, not as a proven guarantee.","section":"Footnote [25]"},{"comment":"There is a typographical inconsistency in the subscript of the growth rate: lambda_{tau c}(p,q) appears in Eq. (9) while the rest of the text uses lambda_{tau C}(p,q).","section":"Section V, Eq. (9)"},{"comment":"The word 'asymptoticaly' should be 'asymptotically', and the estimate q1(p) is called 'asymptoticaly exact' but it is obtained using the large-argument approximation to cosh^{-1}; the wording should be softened to 'asymptotic approximation'.","section":"Section III"},{"comment":"The finite search over p<=30 that supports the statement that the maximum of chi_{tau C}/chi_{p,q} occurs at {3,7} is not documented; including the search data or a short script would make the claim reproducible.","section":"Section IV"}],"recommendation":"major_revision","confidential_remarks":"The geometric rate bound in Section II is sound and the tile-completion rate computations are correct, but the advertised QEC guarantee is false under the paper's own definitions because of constant-weight logical operators. The paper could be made publishable by reframing it as a rate-bound result and removing or heavily qualifying all claims that rate below one guarantees quantum error correction. The authors should be given the opportunity to do so, hence major_revision rather than reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nWhat you should know: the paper's actual contribution is a simple geometric bound on the code rate of holographic codes grown on regular hyperbolic tilings, plus explicit code-rate formulas for the tile-completion growth rule. That part is good, explicit, and checkable. The problem is the abstract's guarantee: 'χ<1 ⇒ quantum error correction.' That inference is not proven, and there is a concrete reason to doubt it.\n\nThe bound itself: for a {p,q} tiling, any growth rule from any simply connected seed gives χ ≤ ℓ_{p,q}/a_{p,q}. The derivation from the hyperbolic isoperimetric inequality is straightforward and correct. The tile-completion code rates in Table I and Section V are reproducible by hand: the growth matrices are written out, the eigenvalues are given, and the rates follow. The observation that χ_{p,q}<1 for all p>3 in a wide q-range is real. This is a new and useful organizing result for the holographic code literature.\n\nThe soft spot is load-bearing. Footnote [25] defines the desired property as a nonzero erasure threshold and admits that 'χ<1' is only 'numerically suggested' to imply it. That alone makes the abstract's 'guarantees' an overclaim. The stress-test note makes the stronger point that the full logical space almost certainly has no nonzero threshold: a bulk index on an outer-layer tile can be pushed through a perfect tensor to a constant number of boundary dangling edges, giving a logical operator of O(1) weight. In an adversarial erasure model, erasing that support kills that logical qubit with an erasure fraction that vanishes as the code grows. Even under random erasures, each such near-boundary qubit is lost with probability ~p^w, a fixed fraction of the logical space. So the conclusion that all p>3 codes 'perform quantum error correction' is not established. The authors could salvage a statement about a protected bulk subspace, or about the rate condition as a necessary condition, but the current wording is wrong.\n\nThe citation pattern is fine; no self-citation issues. The numerical threshold claim rests on [1,14], which is external evidence, not proof.\n\nVerdict: the paper deserves a serious referee because the geometric bound and the tile-completion rates are a real contribution. But the referee should push hard on the QEC claim. I would not cite the guarantee, only the bound.\n\nRecommendation: send to peer review with a request for major revision—separate the rate theorem from the threshold claim.","headline":"The isoperimetric code-rate bound is a clean, checkable result; the leap from χ<1 to guaranteed quantum error correction is unsupported and likely false for the full logical space.","tokens_in":11195,"tokens_out":4757,"would_cite":true,"duration_ms":48436,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","51M10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For holographic codes grown on regular hyperbolic tessellations, hyperbolic geometry bounds the code rate by the ratio of tile side length to tile area, giving guaranteed quantum error correction for every inflation rule when this ratio…","keywords":["holographic codes","quantum error correction","hyperbolic tessellations","code rate","perfect tensors","inflation rule","isoperimetric inequality","tile completion"],"falsifier":"Find one holographic code with code rate below one on a regular hyperbolic tessellation whose boundary erasure threshold is zero; that would break the claimed guarantee. Alternatively, exhibit any inflation rule and simply connected seed on a {p,q} tiling with p greater than three and q in the claimed range whose asymptotic code rate exceeds $\\ell_{p,q}/a_{p,q}$, which would refute the geometric bound itself.","tokens_in":10257,"feed_emoji":"🛡️","tokens_out":3903,"duration_ms":38010,"temperature":0.7,"pith_summary":"The paper establishes that the code rate of any holographic quantum code grown on a regular hyperbolic tessellation is capped by a purely geometric quantity: the ratio of the tile side length to the tile area. Because this cap falls below one for every regular p-gon with p greater than three over a wide range of vertex counts q, the authors conclude that all such codes, grown with any inflation rule from any simply connected seed, protect bulk quantum information from boundary erasures. For the explicit tile-completion growth rule, they compute that holographic triangle codes have code rate above one, while all p-gon codes with p greater than three have code rate below one and therefore perform quantum error correction. The result matters because it turns a question about tensor-network codes into a question about hyperbolic geometry alone.","feed_headline":"Geometry bound guarantees error correction for hyperbolic quantum codes","feed_subtitle":"For every p>3 a range of tilings gives holographic code rate below one, protecting bulk qubits.","key_machinery":"The central object is the hyperbolic isoperimetric inequality $L^2 \\ge A(4\\pi + A)$, applied to a growing code region whose boundary length is $L = \\ell_{p,q} N_{\\text{boundary}}$ and whose area is $A = a_{p,q} N_{\\text{bulk}}$; taking the infinite-layer limit gives the code-rate bound $\\chi_{p,q} = \\ell_{p,q}/a_{p,q}$. For the tile-completion inflation rule, the argument also uses the quasi-crystal growth matrix $M_{\\tau_C}(p,q)$, an integer matrix of unit determinant in $SL(2,\\mathbb{Z})$, whose largest eigenvalue gives the growth rate $\\lambda_{\\tau_C}(p,q)$ that enters the exact code-rate formula.","core_discovery":"The paper claims that for any holographic code grown on the regular {p,q}-tessellation of the hyperbolic plane by an inflation rule from any simply connected seed set, the asymptotic code rate satisfies $\\chi_{\\tau}(p,q) \\le \\ell_{p,q}/a_{p,q}$, where $\\ell_{p,q}$ is the side length of the regular p-gon tile and $a_{p,q}$ is its area. It then shows that for every p greater than three there is a range of q values, whose upper end grows exponentially as $O(e^{\\pi p/2})$, in which this geometric bound is below one; hence every holographic code grown on those tilings has code rate below one and, by the assumed threshold criterion, performs quantum error correction. For the tile-completion rule the code rate is computed exactly: triangle codes have rate greater than one, while square, pentagon, hexagon, and all higher p-gon codes have rate less than one, and all computed rates obey the geometric bound.","pith_inferences":["If the numerical threshold criterion is correct, the gap $1 - \\chi_{p,q}$ can be read as a rough indicator of how much boundary erasure a code can tolerate; codes with rate just below one would be expected to have very small thresholds, offering a concrete finite-size scaling test.","The triangle-code result suggests that the tile-completion triangle codes, with rate above one, should show no nonzero erasure threshold; verifying this numerically would cleanly separate the geometric bound from the threshold assumption.","The bound implies that no finite-cell inflation rule can beat the geometric ratio $\\ell_{p,q}/a_{p,q}$, so optimizing a holographic code within a fixed tessellation amounts to choosing the growth rule that saturates or approaches the bound.","Because the bound is purely geometric, it may carry over to physical implementations of hyperbolic lattices where boundary erasure is the natural noise model, protecting the stored information independently of the microscopic tensor choice."],"forward_implications":["For every regular hyperbolic tessellation with p-gon tiles, p greater than three, and q in the stated range, quantum error correction is guaranteed for all holographic codes regardless of the inflation rule or seed tiles.","The tile-completion rule yields holographic codes with code rate below one for all p greater than three, while holographic triangle codes have code rate above one, making triangle codes the exception.","Any perfect tensor of rank five or higher has at least one hyperbolic tessellation on which every holographic code grown from it has code rate below one, so the construction of such tensors directly yields error-correcting codes.","The upper end of the error-correcting range grows exponentially with p, so the family of guaranteed-correcting tilings expands rapidly as the tile side count increases.","The geometric bound applies to any quasi-crystal growth rule with finitely many cell types, not only to tile completion, so the result is a universal constraint on code rate for this class of constructions."],"supporting_citations":[{"why":"Supplies the definition of holographic codes from perfect tensors, the code-rate formula used in Eq. (9), and the numerical evidence that rate below one gives a nonzero erasure threshold.","marker":"[1]"},{"why":"Provides the inflation-rule and quasi-crystal growth construction, including the SL(2,Z) growth matrices used to compute tile-completion code rates.","marker":"[3]"},{"why":"Gives the hyperbolic-geometry formulas for regular tile side length and area that form the bound $\\chi_{p,q} = \\ell_{p,q}/a_{p,q}$.","marker":"[4]"},{"why":"Extends the code-rate formula to CSS and block-perfect tensor holographic codes and supplies additional numerical threshold evidence.","marker":"[14]"},{"why":"States the isoperimetric inequality that is the core geometric input to the code-rate bound.","marker":"[26]"}],"fun_headline_variants":["Geometry bound guarantees error correction for hyperbolic codes","Hyperbolic tilings yield code rate below one","Inflation-built codes clear error threshold via geometry","Geometric bound ensures quantum error correction in bulk","Holographic codes beat threshold with hyperbolic geometry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conclusion that codes with rate below one actually perform quantum error correction depends on the unproven assumption that code rate below one is sufficient for a nonzero erasure threshold, which the paper supports only by numerical simulation.","fun_headline_variants_meta":{"raw":{"variants":["Geometry bound guarantees error correction for hyperbolic codes","Hyperbolic tilings yield code rate below one","Inflation-built codes clear error threshold via geometry","Geometric bound ensures quantum error correction in bulk","Holographic codes beat threshold with hyperbolic geometry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000363,"raw_usage":{"total_tokens":1898,"prompt_tokens":826,"completion_tokens":1072,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":1002}},"tokens_in":442,"tokens_out":1072,"duration_ms":9431,"temperature":1.0,"reasoning_tokens":1002,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:16:33.850358+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find one holographic code with code rate below one on a regular hyperbolic tessellation whose boundary erasure threshold is zero; that would break the claimed guarantee. Alternatively, exhibit any inflation rule and simply connected seed on a {p,q} tiling with p greater than three and q in the claimed range whose asymptotic code rate exceeds $\\ell_{p,q}/a_{p,q}$, which would refute the geometric bound itself.","supporting_citations":[{"cited_title":"Conformal Quasicrystals and Holography","cited_arxiv_id":"1805.02665","evidence_quote":"Provides the inflation-rule and quasi-crystal growth construction, including the SL(2,Z) growth matrices used to compute tile-completion code rates."},{"cited_title":"Thurston, Three-dimensional geometry and topology (Princeton University Press, Princeton, NJ, 1997) pp","cited_arxiv_id":null,"evidence_quote":"Gives the hyperbolic-geometry formulas for regular tile side length and area that form the bound $\\chi_{p,q} = \\ell_{p,q}/a_{p,q}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the code-rate formula to CSS and block-perfect tensor holographic codes and supplies additional numerical threshold evidence."},{"cited_title":"Courant and D","cited_arxiv_id":null,"evidence_quote":"States the isoperimetric inequality that is the core geometric input to the code-rate bound."}],"review_version":1}