{"id":"1bd3c23b-6843-4be0-ba53-2eba790ffb6e","arxiv_id":"2608.09913","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For single-layer translation-invariant CSS codes, the nonzero superselection layer and its degree determine the stable translation-invariant Clifford class; multi-layer nonsplit extensions show individual layers do not classify translation SET order.","lead":"This paper develops homological invariants, called superselection profiles, for translation-invariant quantum error-correcting codes and shows when these invariants determine the code's topological phase. Its main theorem says a single nontrivial invariant layer plus its degree fix the code up to Clifford circuits; a family of eight three-dimensional models shows this fails when multiple layers coexist.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 4 rests on an unproved Frobenius-extension identification of Ext modules; if it fails in degree or module structure, the claimed minimal-coarse-graining threshold collapses.","rationale":"The reader identified the Frobenius-extension step in Corollary 4 as the weakest assumption, and I agree: this is the point where the proof makes a nontrivial ring-theoretic assertion without proof. I independently re-checked the main chain behind Theorem 2: the matrix-level Schanuel lemma, the dualization argument, the projective-freeness over Z_pm[Z^D], and the upgrade from free stable R-equivalence to Clifford stable equivalence via Whitehead's lemma all appear internally consistent. The eight-model construction and the GSD computation in Sec. VIII are also coherent, and the distinction between split and nonsplit extensions is backed by explicit finite-size invariants. The finite-length-free-resolution restriction for m>1 is explicitly acknowledged and Appendix F shows it is essential, so it is a scope limit rather than a hidden flaw. The remaining concern is precisely the unproved identification of positive-degree Ext modules under restriction of scalars. Because Corollary 4 is a headline result giving the exact minimal coarse graining, this gap warrants a conditional verdict rather than full acceptance. The proposed test on a nontrivial layer with nontrivial translation action would settle whether the identification holds in the needed degree and module structure.","tokens_in":49411,"tokens_out":49939,"duration_ms":449226,"concrete_test":"For D=2, p=2, Lambda=2Z^2, take M=coker phi_X for the full-length regular Koszul model with f=((x1-1)^2, x2-1); its unique layer E has a nontrivial translation action. Compute Ext^1_A(M,A) and Ext^1_R(M,R) as A-modules, with A=Z_2[Lambda], using explicit free resolutions and a Groebner-basis or syzygy calculation. If they are not isomorphic as A-modules, or if Ext^2_A(M,A) is nonzero while Ext^2_R(M,R) vanishes, then the Frobenius-extension identification in Corollary 4 fails in the needed direction and the minimal-coarse-graining theorem is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central rigidity theorem (Theorem 2, Sec. VII B 1) is supported by the generalized Schanuel lemma and the matrix/Clifford dictionary, and I found no flaw in that chain. The load-bearing unproved step is in the proof of Corollary 4 (Sec. VII C 2): \"Because R is a finite free Frobenius extension of R_Lambda, it also identifies the positive-degree Ext modules over R_Lambda with those over R.\" This is not a formal consequence of being a Frobenius extension without an explicit comparison of the two derived functors. The standard adjunction gives Ext_{R_Lambda}(M,R_Lambda) isomorphic to Ext_R(M, Hom_{R_Lambda}(R,R_Lambda)); the proof additionally needs the Frobenius isomorphism Hom_{R_Lambda}(R,R_Lambda) ≅ R as R-modules, which is not supplied. The identification must be degree-preserving and must respect the R_Lambda-module structure, so that the restricted layer E is literally the j-th layer of the toric-code stack over R_Lambda. If restriction shifts the nonzero Ext degree or introduces extra layers, the \"if\" direction of the iff fails and the claimed exact minimal coarse graining at Lambda_E is unsupported. The converse direction has the same dependence: it assumes the translation action on E is preserved by the stable Clifford equivalence. The paper neither proves this identification nor cites a specific statement of it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a homological framework for translation-invariant CSS codes over Laurent polynomial rings, centered on the topological superselection profile S_sigma = tau_{>=1} R Hom_R(coker phi_sigma, R) and its cohomology layers E_sigma^ell = Ext^ell_R(coker phi_sigma, R). It computes these layers for regular Koszul-complex stabilizer models, relates finite-size logical spaces to Tor, and proves a rigidity theorem (Theorem 2) asserting that for single-layer profiles with a finite-resolution hypothesis the nonzero layer and its degree determine the stable translation-invariant Clifford class. The paper also proves a realizability criterion (Proposition 4), derives an exact minimal coarse-graining result to toric-code stacks (Corollary 4), and constructs eight qubit degree-(1,2) toric-code extensions that share identical layers and translation actions but have different size-dependent ground-state degeneracies, thereby demonstrating that inter-layer gluing carries information invisible to individual layers.","tokens_in":49603,"tokens_out":9872,"duration_ms":90805,"significance":"If correct, the paper gives a substantial unification of toric-code hierarchies and bivariate-bicycle codes within one homological framework, and its rigidity theorem provides a sharp classification of single-layer translation-invariant CSS codes at the stable Clifford level. The core computations, including Fact 1 (Ext layers of regular Koszul models), Facts 3 and 4 (Tor formulation), Fact 10 (common layers of the extensions), and Fact 11 (size-dependent ground-state degeneracies of the eight models), are internally consistent and were checked in detail. The paper is unusually honest about its scope: Appendix F explicitly demonstrates that the finite-resolution hypothesis is necessary for m>1, and the limitation to Clifford circuits is stated repeatedly. The third-dimensional multi-degree extension result is a valuable counterexample to the naive expectation that individual superselection layers classify translation SET order.","major_comments":[{"comment":"The central proof step asserts that restriction of scalars to a finite-index translation subgroup Lambda preserves topological exactness and 'identifies the positive-degree Ext modules over R_Lambda with those over R'. This is not a formal consequence of R being a finite free Frobenius extension of R_Lambda. The Frobenius property gives an isomorphism Hom_{R_Lambda}(R, R_Lambda) ≅ R as bimodules, but one must prove that for every M admitting a finite free resolution P_bullet, the complex Hom_{R_Lambda}(P_bullet|_{Lambda}, R_Lambda) computes the same Ext groups in the same degrees and with the same R_Lambda-module structure as Ext_R(M,R) restricted to Lambda. If restriction shifted the nonzero Ext degree or introduced extra layers, the 'if' direction of the iff would fail and the claimed exact minimal coarse graining at Lambda_E would be unsupported. The converse direction likewise assumes without proof that stable translation-invariant Clifford equivalence preserves the layer E together with its translation action. Because Corollary 5 and the headline minimal-coarse-graining threshold inherit this step, a complete proof using the Frobenius-extension adjunction, or a precise citation to a stated theorem, is required.","section":"§VII C 2, proof of Corollary 4"},{"comment":"The rigidity theorem is conditional on coker phi_X admitting a finite-length free resolution, and the paper correctly notes that this is automatic for prime qudits and essential for m>1 (Appendix F). However, as stated, Corollary 3 is formulated only for prime qudits, so the m>1 statements in Section VII C 1 are a source of potential over-reading: for prime-power qudits the paper does not prove that identical single-layer profiles imply stable Clifford equivalence unless the finite-resolution hypothesis is verified. I recommend that the abstract and Section I state this scope limitation more prominently, so that the headline 'nonzero layer and degree determine the stable translation-invariant Clifford class' is not interpreted as applying unconditionally beyond prime qudits.","section":"§VII B 1, Theorem 2 and Corollary 3"},{"comment":"The stable-equivalence framework is sound, but the proof of Proposition 1 invokes Whitehead's lemma without a citation or a brief justification, and Proposition 2's forward implication is asserted while its converse is deferred to Appendix C. Since the rigidity theorem depends on the dictionary between matrix equivalence, module stable isomorphism, and Clifford equivalence, these two results deserve short self-contained proofs or explicit references in the main text rather than a sentence referring to standard K-theory facts.","section":"§VII A 2–3; Proposition 1 and Proposition 2"}],"minor_comments":[{"comment":"The phrase 'local order parameters of a classical code' in the 0-th superselection layer is not defined elsewhere; a brief physical definition would improve readability.","section":"§IV C, bullet list after Eq. (81)"},{"comment":"The asterisk in the table row for E_sigma^ell is attached to a footnote-like sentence inside the table entry; it should be formatted as a standard table footnote to avoid ambiguity.","section":"Table I"},{"comment":"The cyclic ordering of the basis of K_2 in three and four dimensions is natural, but a one-sentence explanation of why this ordering is chosen would help readers verify the matrix signs in the Koszul differentials.","section":"§III C, Eqs. (46b) and (53b)"},{"comment":"The identity pd_R(M) = max{ell : Ext^ell_R(M,R) != 0} in Eq. (131) is used without proof; it is standard for finite-projective-dimension modules over Gorenstein rings, but a citation or one-line justification would make the argument self-contained.","section":"§VII B 1, Proposition 3 proof"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Fresh eyes: this is a substantial theory paper. The genuinely new results are Theorem 2 and the eight-model degree-(1,2) extension family, and both hold up under hand-checking. The Koszul unification and the Ext-layer packaging are partly repackaging of known material, but the rigidity theorem and the multi-layer counterexample are real advances.\n\nWhat it does well: the computations in Fact 1 and Fact 10 are clean; the matrix-level Schanuel lemma is a nice bridge from module theory to stable Clifford equivalence. The eight models sharing identical layers yet differing in size-dependent GSD is a sharp demonstration that layer-wise Ext data is incomplete in multi-layer settings. Appendix F is honest about the m>1 limitation and gives a concrete obstruction, which I read as a scope note, not a flaw.\n\nThe main soft spot is the proof of Corollary 4. The sentence 'Because R is a finite free Frobenius extension of R_Lambda, it also identifies the positive-degree Ext modules over R_Lambda with those over R' is load-bearing and unproved. The standard adjunction gives a comparison involving Hom_{R_Lambda}(R,R_Lambda); you need a degree-preserving, R_Lambda-linear Frobenius isomorphism as an R-module. I think it is true for finite-index group rings over Z_p, but it is not a formality and the paper neither proves it nor cites a specific statement. Without that identification, the 'if' direction of Corollary 4 is unsupported. This is the one gap I would insist on closing.\n\nMinor: the eight-model count fixes coordinate directions. Under lattice rotations the weight-1 and weight-2 vectors are equivalent, so if rotations are part of the symmetry data, the number of inequivalent SET orders is smaller. The paper does not address this. It is a clarification, not a deep problem. Also, the paper's equivalence relation preserves the translation action on layers; the converse direction of Corollary 4 assumes that, which is fine but should be stated explicitly.\n\nVerdict: deserves a serious referee. The central rigidity theorem and the counterexample are likely correct, and the missing Frobenius-extension justification is an addressable gap. I would send it to review, with a request to prove or properly cite that step and to comment on the rotation action on the eight models.","headline":"A substantial, mostly correct rigidity theorem for single-layer CSS codes, with a sharp multi-layer counterexample; the main gap is the unproved Frobenius-extension step in Corollary 4.","tokens_in":50219,"tokens_out":6316,"would_cite":true,"duration_ms":55212,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P70","13D02","18G10"],"pacs":["03.67.Lx"],"model":"deepseek-v4-flash","headline":"For single-layer topological CSS codes, the nonzero superselection layer and its degree determine the stable translation-invariant Clifford class; multi-layer extensions hide extra phase data in inter-layer gluing.","keywords":["translation-invariant CSS codes","topological superselection profile","Koszul complexes","Ext and Tor invariants","toric code hierarchy","symmetry-enriched topological order","stable Clifford equivalence","ground-state degeneracy"],"falsifier":"A concrete test: find two prime-qudit translation-invariant topological CSS codes with single-layer $X$-profiles supported in the same degree $j$ and isomorphic $j$-th superselection layers whose size-dependent logical dimensions differ on some torus; Theorem 2 predicts they must be stably translation-invariant Clifford equivalent, so unequal logical dimensions would refute the rigidity claim. Alternatively, a computation showing a sub-threshold translation subgroup (one that acts nontrivially on the layer $E$) nevertheless decoupling the code to a toric-code stack would refute Corollary 4.","tokens_in":49105,"feed_emoji":"🌀","tokens_out":14189,"duration_ms":118883,"temperature":0.7,"pith_summary":"This paper introduces a homological invariant for translation-invariant CSS stabilizer codes, the topological superselection profile, whose layers organize pointlike and extended excitations together with the lattice-translation action; the framework is developed on Koszul-complex stabilizer models, which unify the toric-code hierarchy and bivariate-bicycle codes in one algebraic picture. Its main result is a rigidity theorem: when a code's profile has exactly one nonzero layer and the reduced Pauli module admits a finite-length free resolution, that layer together with its degree determines the whole code up to translation-invariant Clifford circuits and ancillas; the resolution condition is automatic for prime qudits. A corollary gives the exact minimal coarse graining to a stack of toric codes: a translation subgroup works precisely when it acts trivially on the nonzero layer. The paper also constructs multi-layer extensions and shows that eight three-dimensional qubit toric-code extensions have identical individual layers yet different size-dependent ground-state degeneracies, so the layers alone underdetermine translation-symmetry-enriched topological order.","feed_headline":"A superselection layer plus its degree pins a stabilizer code's phase","feed_subtitle":"It also fixes the exact coarse graining to toric-code stacks and exposes inter-layer data invisible to individual layers.","key_machinery":"The central object is the topological superselection profile, a positive-degree derived-dual object built from the reduced Pauli modules; its cohomology layers are Ext modules that encode fusion and translation action. The arguments run on three mechanisms: the Koszul complex $K_\\bullet(f)$ of a regular sequence, whose exactness makes the associated stabilizer model topological and whose structure module $S=R/(f)$ controls all its superselection layers; a matrix-level generalized Schanuel lemma, which turns stable $R$-isomorphism of cokernels into stable equivalence of stabilizer matrices, together with Whitehead's lemma, which upgrades stable $R$-equivalence to stable Clifford equivalence after adding ancillas; and the natural symmetry of Tor, which expresses finite-size logical modules as $\\operatorname{Tor}_j^R(S,R_L)$ and ties size-dependent ground-state degeneracy to the translation action. A final construction couples Koszul models in different degrees by an off-diagonal matrix block, producing the split and nonsplit extensions that exhibit inter-layer gluing.","core_discovery":"On the paper's own terms, the central discovery is that the topological superselection data of a translation-invariant topological CSS code is largely contained in one derived object $\\mathcal{S}_\\sigma=\\tau_{\\ge1}\\mathbf{R}\\operatorname{Hom}_R(\\overline{\\operatorname{coker}\\varphi_\\sigma},R)$ and its cohomology layers $E^\\ell_\\sigma\\cong_R\\operatorname{Ext}_R^\\ell(\\overline{\\operatorname{coker}\\varphi_\\sigma},R)$. For codes whose $X$-sector profile is single-layer in degree $j$ and whose reduced module has a finite-length free resolution, Theorem 2 states that $E=\\operatorname{Ext}_R^j(\\overline{\\operatorname{coker}\\varphi_X},R)$, together with $j$, determines the stable translation-invariant Clifford class: $\\varphi_X\\sim_{R,\\mathrm{Cl},\\mathrm{st}}\\partial_j^\\dagger$ for any partial resolution of $E$. For prime qudits this yields an unconditional phase-identification criterion, and when $E$ is finite the translation kernel of $E$ is exactly the minimal coarse-graining subgroup needed to reach a toric-code stack. The paper's eight degree-$(1,2)$ extensions of the three-dimensional qubit toric code show that the single-layer hypothesis is essential: all eight have identical layers $E^\\ell_\\sigma$ with trivial translation action, but their ground-state degeneracies depend differently on the torus size, so they realize distinct translation SET orders that individual layers cannot detect.","pith_inferences":["A direct extension would classify multi-layer profiles by finite Postnikov stages, with inter-layer gluing encoded by Maurer-Cartan equations; this would recover the single-layer rigidity theorem as the base case.","The eight-model degeneracy tables suggest a numerical protocol: compute logical dimensions on tori of varying parity for a fixed coupling vector and check that the predicted parity classes appear, giving an experimental distinguisher for inter-layer gluing.","Over $\\mathbb{Z}_{p^m}$, replacing the free-resolution hypothesis by a projective-dimension condition, or moving to general finite-depth local unitaries, may extend rigidity to the cases where vanishing layers are Clifford nontrivial, since the paper's appendix shows the obstruction is real.","The criterion that retained translations are exactly those acting trivially on the layer suggests an algorithm for minimal cellulation: compute the mobility lattice $\\Lambda_E$ from a presentation, then verify toric-code-stack equivalence by testing periodicity of logical dimensions along $\\Lambda_E$."],"forward_implications":["Every full-length regular Koszul-complex model with structure module $S$ is stably Clifford equivalent, after coarse graining by the mobility lattice $\\Lambda_S$, to $\\operatorname{rank}_{\\mathbb{Z}_{p^m}}S$ copies of the degree-$j$ toric code.","Over prime qudits, any two translation-invariant topological CSS codes with single-layer $X$-sector profiles supported in the same degree $j$ and with isomorphic $j$-th layers are connected by a finite-depth translation-invariant Clifford circuit up to ancillas.","When the nonzero layer $E$ is finite, a translation subgroup can decouple the code to the toric-code-stack normal form if and only if it acts trivially on $E$; hence $\\Lambda_E$ is the exact minimal coarse-graining scale.","In two dimensions every nonvanishing $X$-sector profile is single-layer in degree one, so the first superselection layer, including anyon permutation data, fixes the stable translation-invariant Clifford class; in one dimension every such code is stably Clifford trivial.","The eight qubit degree-$(1,2)$ toric-code extensions have identical individual layers but eight different size-dependent ground-state degeneracy functions, so they are pairwise distinct translation SET orders with the original translation symmetry fixed, although uniform $2\\times2\\times2$ blocking brings all eight to the same decoupled toric-code stack."],"supporting_citations":[{"why":"Supplies the Ext-based charge-module hierarchy of superselection sectors and the grade bound that the paper restates in CSS terms.","marker":"[13]"},{"why":"Establishes the Laurent-polynomial module formalism for commuting Pauli stabilizers on which the whole CSS setup is built.","marker":"[1]"},{"why":"Introduces the mobility sublattice and the logical–anyon correspondence used for translation enrichment and finite-size logical modules.","marker":"[7]"},{"why":"Provides the bivariate-bicycle code family whose two-polynomial checks are length-two Koszul complexes, one of the two families unified here.","marker":"[9]"},{"why":"Supplies the standard facts on Koszul complexes, regular sequences, grade, and Ext used throughout the computations.","marker":"[34]"},{"why":"Provides the derived-functor foundations for the superselection profile and for retaining inter-layer gluing data.","marker":"[35]"},{"why":"Supplies Whitehead's lemma, used to prove that stable R-equivalence coincides with stable Clifford R-equivalence under ancilla stabilization.","marker":"[51]"},{"why":"Provides the Laurent-polynomial freeness result that lets the rigidity proof treat finitely generated projective modules as free.","marker":"[54]"}],"fun_headline_variants":["A layer and its degree fix a topological CSS code's phase","Superselection data: one layer suffices to pin the SET order","Inter-layer gluing reveals stabilizer codes layers miss","Single-layer superselection profile determines Clifford phase","Nonsplit extensions hide phase data in inter-layer gluing"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing premise is that passing to a finite-index translation subgroup preserves the code's topological exactness and keeps the higher-level excitation modules identified with the original ones; if this ring-theoretic identification fails, the claimed exact minimum for coarse graining is unsupported.","fun_headline_variants_meta":{"raw":{"variants":["A layer and its degree fix a topological CSS code's phase","Superselection data: one layer suffices to pin the SET order","Inter-layer gluing reveals stabilizer codes layers miss","Single-layer superselection profile determines Clifford phase","Nonsplit extensions hide phase data in inter-layer gluing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000303,"raw_usage":{"total_tokens":1902,"prompt_tokens":1265,"completion_tokens":637,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":881,"completion_tokens_details":{"reasoning_tokens":554}},"tokens_in":881,"tokens_out":637,"duration_ms":6236,"temperature":1.0,"reasoning_tokens":554,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:27:40.949681+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test: find two prime-qudit translation-invariant topological CSS codes with single-layer $X$-profiles supported in the same degree $j$ and isomorphic $j$-th superselection layers whose size-dependent logical dimensions differ on some torus; Theorem 2 predicts they must be stably translation-invariant Clifford equivalent, so unequal logical dimensions would refute the rigidity claim. Alternatively, a computation showing a sub-threshold translation subgroup (one that acts nontrivially on the layer $E$) nevertheless decoupling the code to a toric-code stack would refute Corollary 4.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the Laurent-polynomial module formalism for commuting Pauli stabilizers on which the whole CSS setup is built."},{"cited_title":"This hypothesis is needed only when rigidity is invoked to compare an arbitrary code with that representative","cited_arxiv_id":null,"evidence_quote":"Introduces the mobility sublattice and the logical–anyon correspondence used for translation enrichment and finite-size logical modules."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the bivariate-bicycle code family whose two-polynomial checks are length-two Koszul complexes, one of the two families unified here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Whitehead's lemma, used to prove that stable R-equivalence coincides with stable Clifford R-equivalence under ancilla stabilization."}],"review_version":2}