{"id":"d7cc1c11-b671-4f36-be4f-c9a195b0501c","arxiv_id":"2508.02661","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Fermionic superconductors with exact fermion parity and average crystalline symmetry are classified across all 2D wallpaper groups and 3D point groups, yielding many intrinsically average-symmetry-protected phases.","lead":"This paper classifies topological superconducting phases that stay robust when crystalline symmetries are only preserved on average, for example under disorder or decoherence. It provides a full catalog for all 2D wallpaper and 3D point group symmetries, including phases that exist only because of disorder or decoherence.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The completeness of the classification rests on the conjectured crystalline equivalence principle of Sec. III E, which underpins the spectral-sequence cross-check and the stacking relations for D3d, D2d, and Td; if that conjecture fails, those results are unsupported.","rationale":"The reader's weakest assumption identifies the conjectured crystalline equivalence principle in Sec. III E as the key unsupported step. I agree: this is the single most load-bearing concern because the paper claims completeness of a classification and validates it with a spectral-sequence computation whose applicability to average crystalline symmetries is explicitly left as a conjecture. The importance is amplified by the paper's own admission that the stacking relations for D3d, D2d, and Td are unresolved in the block-state construction and are taken from the spectral sequence alone. A failure of the conjecture would not invalidate the worked block-state examples, but it would remove the independent confirmation and leave a substantial part of Tables III–VI (especially the blue intrinsic entries for those three point groups) without support. The concern is not that the conjecture is false; it is that the paper's strongest claim is conditioned on it. Since the reader already rendered CONDITIONAL for precisely this reason, my assessment leaves the verdict unchanged. I have not identified a stronger objection than the reader's: the internal mechanics of the block-state construction for pmm, p2, and C2v are detailed and coherent, and the paper is transparent about the three unresolved stacking cases. The concrete test I propose would turn the conjecture into a checkable statement in the simplest cases where orientation-reversing symmetries and nontrivial omega2 both appear.","tokens_in":65307,"tokens_out":3720,"duration_ms":38689,"concrete_test":"Prove or disprove the crystalline equivalence principle for average SPTs in a minimal nontrivial case. Concretely: take the 2D pmm group analyzed in Sec. II C and the 3D C1h group in Appendix B.4. For each, (i) compute the onsite ASPT classification by the AHSS of Sec. III E with the modified omega2 and antiunitary treatment; (ii) construct an explicit map from the block-state data in Tables III and V (E1D, E0D, G and E2D, E1D, E0D, G) to the E2 page and differentials of the AHSS, including the effects of the h0/h1 truncations; and (iii) verify that the map is an isomorphism of abelian groups that preserves stacking, i.e., that the block-state extension for pmm (Z8_2) and the stacking relation for C1h (Z8 reduction) are reproduced by the spectral sequence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is a complete classification of average crystalline SPTs for all 2D wallpaper groups and 32 3D point groups under decoherence and disorder, summarized in Tables III–VI. The two methods used are the generalized block-state construction and a generalized spectral-sequence computation. The block-state construction is physically motivated, but the spectral-sequence method is connected to the crystalline problem only through an explicit conjecture. Sec. III E states: 'We conjecture that there is a crystalline equivalent principle similar to the case of clean SPTs [70],' after replacing orientation-reversing crystalline symmetries by antiunitary onsite symmetries and modifying omega2 (spinless ACSPTs corresponding to spin-1/2 onsite ASPTs and vice versa). No derivation or independent test of this conjecture is provided. This matters for two reasons. First, the spectral-sequence calculation is presented as an independent consistency check of the block-state results; if the equivalence principle fails, the agreement between the two methods is not evidence for either. Second, the paper's Conclusion explicitly says that stacking relations for D3d, D2d, and Td are unresolved in the block-state construction and 'we currently rely on the generalized spectral-sequence results alone for their stacking relations.' Since these three point groups are among the cases with intrinsic ACSPTs highlighted as a main result, their classifications are load-bearing for the paper's headline claims. Additionally, the modification of the AHSS by setting h0(Zf2)=0 for decoherence and h0=h1=0 for disorder, while ignoring obstructions in those layers, is an input to the spectral-sequence computation that is motivated by results for onsite ASPTs (Ref. [51]) but whose extension to the crystalline equivalence is exactly what is being conjectured.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a generalized real-space block-state construction for average crystalline symmetry-protected topological phases (ACSPTs) in fermionic systems with exact fermion parity and crystalline symmetries that are preserved only on average under disorder or decoherence. The authors classify ACSPTs for all 17 2D wallpaper groups and all 32 3D point groups, for both spinless and spin-1/2 fermions, reporting results in Tables I–VI. The classification is obtained by decorating coarse-grained cells with onsite ASPTs, imposing generalized obstruction-free conditions and bubble equivalence relations, and is cross-checked against a generalized spectral-sequence computation. The paper highlights numerous intrinsic ACSPTs that have no clean-system analogue. It concludes with a discussion of stacking relations, noting that for D3d, D2d, and Td those relations are currently taken from the spectral-sequence computation alone.","tokens_in":65569,"tokens_out":4428,"duration_ms":55294,"significance":"If the classification is correct, this is a substantial advance: it provides the first systematic enumeration of average crystalline topological superconductors across all wallpaper groups and 3D point groups, and it significantly enlarges the known landscape of intrinsic ASPTs. The worked examples in Sec. II (pmm, p2, and C2v) are carefully presented, with explicit Majorana-mode counting, K-matrix edge analyses, and anomaly-indicator computations, and the appendix delivers a per-group construction that is a valuable reference resource. The paper also makes a useful conceptual point that the obstruction-free conditions are relaxed under decoherence/disorder, enabling phases that are strictly forbidden in clean systems. The absence of fitted parameters and the transparent physical reasoning are additional strengths. However, the completeness claim and the independent-check claim both rest on a conjecture that is not proved, which limits the certainty of the full classification.","major_comments":[{"comment":"The crystalline equivalence principle is explicitly conjectural: the text states, \"We conjecture that there is a crystalline equivalent principle similar to the case of clean SPTs [70],\" and then uses the onsite ASPT classification with antiunitary symmetries and modified omega2 as the basis for the spectral-sequence computation around Eq. (52). This conjecture is load-bearing because the spectral-sequence results are presented as an independent consistency check of the block-state classification. Without a proof or at least a nontrivial independent test, agreement between the two methods is evidence only that both computations embody the same assumption. The manuscript should either prove the equivalence in the average-symmetry setting, supply a concrete derivation or explicit counterexample-free argument, or clearly label the spectral-sequence-based results as conditional on this conjecture.","section":"Sec. III E"},{"comment":"The Conclusion states that stacking relations for D3d, D2d, and Td are unresolved within the block-state construction and that \"we currently rely on the generalized spectral-sequence results alone for their stacking relations.\" Since these three point groups include intrinsic ACSPTs highlighted among the main results, their entries in Tables V and VI (for example, D3d spinless decohered G = Z2 × Z4 and Td spinless decohered G = Z2 × Z4) depend entirely on the conjectural spectral-sequence bridge. This is a genuine gap in the completeness claim: either derive these stacking relations by block-state arguments, or explicitly mark the affected rows as conditional and temper the completeness statement.","section":"Sec. V"},{"comment":"For the decohered spin-1/2 case, the paper argues that certain obstruction functions are trivialized by decoherence because they live in the bosonic layer, and it derives several intrinsic decorations (e.g., the Zf4 ASPT on 1D blocks). The logic is plausible, but the argument relies on a layer-by-layer truncation of the general fermionic obstruction data from Ref. [81]. The manuscript does not provide a fully explicit account of why the remaining higher-layer obstructions and differentials do not generate additional constraints in the crystalline block-state setting. Since the spin-1/2 classification is a central and highly nontrivial part of the tables, a more detailed derivation of the obstruction-free conditions in this case would materially strengthen the paper.","section":"Sec. III C 1 and Sec. III C 2"}],"minor_comments":[{"comment":"In the p3 section, the text labels the block list as \"BBlocks and onsite symmetries,\" which appears to be a typo for \"Blocks and onsite symmetries.\"","section":"Appendix A 13"},{"comment":"The disordered classifications appear twice: in Tables I–II in the Introduction and in Tables IV and VI in the Results. The duplication is confusing, especially because the captions describe the data differently; the authors should either remove one set or explicitly state that the tables are identical and explain why both are needed.","section":"Tables I, II, IV, and VI"},{"comment":"The description of replacing orientation-reversing crystalline symmetries by antiunitary onsite symmetries and modifying omega2 is very brief. A concrete example showing how this replacement works for one wallpaper group or point group would make the spectral-sequence computation much easier for readers to verify.","section":"Sec. III E"},{"comment":"The notation E1D0, E0D0, G0, and their spin-1/2 counterparts is used in Tables III and V without being defined in the main text; a definition should be given near its first appearance.","section":"Sec. IV"},{"comment":"The subsection heading \"Fermionic APSTs and intrinsic ASPTs\" contains a typo (APSTs should be ASPTs).","section":"Sec. II A"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically rich and the worked examples are of high quality, but the completeness claim is weakened by the unproved crystalline equivalence conjecture and by the admitted reliance on the spectral sequence for the stacking relations of D3d, D2d, and Td. I see no issue with novelty or fit; the requested revision is conceptual rather than presentational. I would be satisfied if the authors either prove or substantially test the equivalence principle, or explicitly condition the affected results on it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, what's genuinely new: the block-state construction extended to average crystalline symmetries, applied to all 17 wallpaper groups and all 32 point groups, for both decohered and disordered fermions, spinless and spin-1/2. The worked examples (pmm, p2, C2v) are careful, with Majorana-mode counting, K-matrix edge theories, and anomaly indicators. The classification tables are large and physically plausible, and the identification of intrinsic ACSPTs is a real result. The paper is transparent about its methods, which I respect.\n\nThe soft spot, as you flagged: Sec. III E conjectures a crystalline equivalence principle that maps the crystalline ACSPT classification to an onsite ASPT classification with antiunitary symmetries and modified omega2. The spectral-sequence computation uses that conjecture. So the 'independent consistency check' is not independent — it's checking block states against a conjectural mathematical model. The paper says so itself, which is honest, but it weakens the check. More concretely, the Conclusion admits that the stacking relations for D3d, D2d, and Td are unresolved by the block-state construction and are taken from the spectral-sequence results alone. Those three point groups are among the headline intrinsic-ACSPT cases. If the equivalence conjecture fails, those stacking relations are unsupported. This is a real gap, but it is localized: the block-state construction is a separate physical computation, and the agreement with the spectral sequence across many groups is evidence that the conjecture is probably right. I would not call the main classification wrong; I would call it conditional on that conjecture.\n\nMinor points: they reference a public spectral-sequence package, but don't ship the full computation inputs, so reproducibility is partial. The citation pattern is fine — they build on Refs. [51] and [77] and cite them properly.\n\nVerdict: this deserves a serious referee. I'd send it out, and ask the authors to either prove the equivalence conjecture in a follow-up or at least soften the 'independent check' language and make the D3d/D2d/Td stacking caveat prominent in the main text. The tables are the kind of thing the community will use, so they need to be right.","headline":"First systematic classification of average crystalline topological superconductors across 2D wallpaper and 3D point groups, built from a physically motivated real-space construction; the spectral-sequence cross-check rests on an unproved equivalence conjecture, leaving three stacking relations conditional.","tokens_in":66177,"tokens_out":2416,"would_cite":true,"duration_ms":25080,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A complete classification of average crystalline topological superconductors: all 17 two-dimensional wallpaper groups and all 32 three-dimensional point groups are classified, under both decoherence and disorder, and many phases have no…","keywords":["average symmetry-protected topological phases","crystalline topological superconductors","real-space block-state construction","intrinsic ACSPT","decoherence","disorder","spectral sequence","fermion parity"],"falsifier":"For the decohered spinless pmm model, the paper predicts that a single Z2 fSPT decoration on a 1D block is obstruction-free, with a gappable edge in the doubled space. An exact-diagonalization or tensor-network calculation that finds a protected degeneracy or nonlocal correlation on that edge, even after all symmetric gapping terms are included, would falsify the relaxed obstruction-free condition and with it the classification.","tokens_in":65111,"feed_emoji":"","tokens_out":5865,"duration_ms":63064,"temperature":0.7,"pith_summary":"Topological superconductors are normally thought to need exact crystalline symmetry to protect their boundary states. This paper claims that when disorder or decoherence breaks a crystalline symmetry locally while preserving it on average, the phase can remain stable as long as exact fermion parity survives. It extends the real-space block-state construction—decorating symmetry-fixed cells with lower-dimensional SPT states—to this average setting, and derives obstruction-free and bubble-equivalence rules for the resulting decorations. The result is a full classification for all 2D wallpaper groups and all 32 3D point groups, for spinless and spin-1/2 fermions under both decoherence and disorder, with many intrinsic average phases that have no clean-system analog.","feed_headline":"Disorder-protected superconductors get a full classification","feed_subtitle":"All 2D wallpaper groups and 3D point groups are classified under decoherence or disorder, exposing phases with no clean-system analog.","key_machinery":"The generalized real-space block-state construction partitions the crystal into cells fixed by crystalline little groups, decorates a p-dimensional cell with a p-dimensional ASPT protected by the cell's average onsite symmetry plus exact fermion parity, and then imposes two checks: the folded intersection of decorated edge modes must be a trivial ASPT, and decorations that differ by nucleating and shrinking closed SPT bubbles are identified. In the average setting the key modification is that bosonic anomalies are ignored, so previously forbidden decorations become obstruction-free. The construction carries the classification because every candidate phase is built from these decorated cells, and the bubble equivalence is what reduces naive decoration counts to final groups. A generalized spectral sequence, with h0(Z2^f)=0 for decoherence and h0=h1=0 for disorder, serves as an independent check on the same data.","core_discovery":"The paper's central claim is that fermionic phases protected by exact fermion parity and average crystalline symmetry are classified by generalized block-state data: each cell decorated by an onsite ASPT built from the cell's little-group symmetry, subject to a relaxed obstruction-free condition and identified by bubble equivalence. The obstruction-free condition is relaxed because bosonic Berry-phase obstructions vanish under decoherence and charge-localization obstructions vanish under disorder, and this relaxation is the mechanism that creates intrinsic ACSPTs. The full classification, summarized in Tables I-VI, shows that many crystalline topological superconductors survive realistic imperfections and that a substantial number of the resulting phases are intrinsic, existing only because the symmetry is average.","pith_inferences":["The paper leaves U(1) charge conservation and time-reversal symmetry out; if the same coarse-graining logic carries over, an analogous classification for systems with those symmetries is a natural next step.","The three point groups D3d, D2d, and Td have stacking relations taken from the spectral sequence alone; a direct block-state derivation for those groups would either confirm the conjectured equivalence or expose a discrepancy.","The predicted intrinsic phases could in principle be probed in small disordered or decohered lattice models with exact diagonalization or tensor-network methods, checking whether the doubled-space edge is indeed gappable.","The boundary LSM constraints for intrinsic ACSPTs may be qualitatively different from those for extrinsic ones; the paper leaves this open, so a classification of these boundary anomalies is a testable extension."],"forward_implications":["If the classification is correct, a wide class of crystalline topological superconductors remains well defined in the presence of disorder or decoherence, not just in idealized clean crystals.","The intrinsic ACSPT phases should evolve into intrinsic gapless phases as disorder or decoherence is removed, giving a new route to quantum critical states.","The boundary of each average SPT should impose an average-symmetry Lieb-Schultz-Mattis constraint, so the classification implies a family of disorder-robust LSM constraints.","Because the block-state construction does not depend on a free-fermion limit, the classification should hold for strongly interacting systems as well as noninteracting ones."],"supporting_citations":[{"why":"Supplies the general theory of onsite ASPTs, including the modified Kunneth formula and the notion of intrinsic ASPT that this paper generalizes to crystalline symmetries.","marker":"[51]"},{"why":"States the crystalline equivalence principle for clean SPTs that this paper conjectures to extend to average crystalline SPTs, underpinning the spectral-sequence cross-check.","marker":"[70]"},{"why":"Provides the cell decomposition and coarse-graining justification for the real-space block-state construction used here.","marker":"[74]"},{"why":"Establishes the clean crystalline topological superconductor classification, with Majorana bubble rules, that serves as the baseline for identifying intrinsic phases.","marker":"[77]"},{"why":"Gives the obstruction functions and higher spectral-sequence differentials for fermionic SPTs that the paper adapts to average symmetries.","marker":"[81]"},{"why":"Provides the Z2 anomaly indicator used to check whether folded edge theories become trivial under decoherence or averaged symmetry.","marker":"[79]"}],"fun_headline_variants":["Average symmetries unlock full classification of disordered superconductors","New framework classifies superconductors that withstand disorder","Intrinsic topological superconductors emerge under average symmetry","Real-space construction reveals disorder-robust superconductor phases","Classifying average crystalline topological superconductors under disorder"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the conjectured crystalline equivalence principle: that a crystal with only average crystalline symmetry can be faithfully replaced, for classification purposes, by an onsite system with the same group structure, orientation-reversing operations treated as antiunitary and the fermion-parity extension adjusted; if that conjecture fails, the spectral-sequence computation is not an independent validation and the stacking relations for D3d, D2d, and Td rest on an unverified assumption.","fun_headline_variants_meta":{"raw":{"variants":["Average symmetries unlock full classification of disordered superconductors","New framework classifies superconductors that withstand disorder","Intrinsic topological superconductors emerge under average symmetry","Real-space construction reveals disorder-robust superconductor phases","Classifying average crystalline topological superconductors under disorder"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000212,"raw_usage":{"total_tokens":1403,"prompt_tokens":912,"completion_tokens":491,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":528,"completion_tokens_details":{"reasoning_tokens":416}},"tokens_in":528,"tokens_out":491,"duration_ms":5760,"temperature":1.0,"reasoning_tokens":416,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:36:28.493283+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the decohered spinless pmm model, the paper predicts that a single Z2 fSPT decoration on a 1D block is obstruction-free, with a gappable edge in the doubled space. An exact-diagonalization or tensor-network calculation that finds a protected degeneracy or nonlocal correlation on that edge, even after all symmetric gapping terms are included, would falsify the relaxed obstruction-free condition and with it the classification.","supporting_citations":[],"review_version":2}