{"id":"191b3efd-a50c-4939-b11e-fe83576f1eca","arxiv_id":"2607.28811","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Crystalline SPT phases with symmorphic symmetry should be classified by a fully equivariant Freed–Hopkins invertible-field-theory ansatz, equipped with a natural free-to-interacting map from equivariant K-theory.","lead":"This paper proposes a new way to classify symmetry-protected topological (SPT) phases in crystals by treating the crystal's spatial symmetry equivalently to internal symmetries at every step, rather than collapsing it to a simpler 'crystalline equivalence principle.' If right, it changes how physicists compute which interacting topological phases can exist in materials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The crystalline Kitaev conjecture (eq 4) is underdetermined: Borel extensions can satisfy it, so it does not force the non-Borel interacting classification; the central argument relies on an unproved choice of equivariant extension.","rationale":"The reader's weakest_assumption already targets eq (4), and I agree that this is the load-bearing premise. All of the paper's structural conclusions—the inadequacy of [2], the non-Borel character of the interacting classification, and the naturality of the equivariant FTI map—are consequences of assuming eq (4) with a particular non-Borel equivariant extension. But eq (4) is stated with 'some P-equivariant generalized cohomology theory'; no uniqueness or naturality property is specified. The Borel extension always exists, so eq (4) alone cannot rule out CEP. The free-fermion input fixes KO_P, but that is exactly the input that the interacting case lacks. The paper's discussion in §§2–3 is honest about this: it says it does not prove conclusively that the CEP fails and instead points toward naturality. That is a legitimate research proposal, but the central claim as stated is conditional. I do not see an internal inconsistency; the issue is underjustification. The suggested test—an explicit comparison in 1+1D with C2—would provide the missing computational anchor. If the Borel and genuine groups coincide in that case, the paper's main motivation collapses; if they differ and physical models match the genuine group, it would be a strong vindication. The secondary reliance on the unproved eq (20) FTI map reinforces the same conditionality, but eq (4) is the more fundamental gap. Therefore no change to the reader's CONDITIONAL verdict is needed.","tokens_in":6840,"tokens_out":15997,"duration_ms":173938,"concrete_test":"Test eq (4) in a minimal interacting example where the classification is independently known: take d=1, P=C2 acting on T^1 by reflection, and internal symmetry H=Z2 (or the corresponding H(s) in the fermionic case). Compute the genuine equivariant group Ω^{3}_{H(s),C2}(T^1) via equivariant Pontryagin–Thom and compare it with the Borel/CEP group H^3(B(H×C2);Z) (or the [2] prediction) and with the known lattice classification of 1+1D C2-symmetric SPTs. If the genuine and Borel groups agree, eq (4) does not force non-Borel behavior; if they differ, the lattice classification decides which extension is physical. This would settle whether the extrapolation from free fermions is valid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proposal rests on the crystalline Kitaev conjecture, eq (4): π0(S_d(H,G)) = D(H)^d_P(T^d). The paper treats this as a principle that forces a non-Borel equivariant extension. But eq (4) is not a conjecture with unique content: for any spectrum D(H), the Borel construction D(H)^d_P(X) := D(H)^d(X ×_P EP) is another P-equivariant extension with the same non-equivariant limit. For the group-cohomology spectrum this Borel choice reproduces the CEP (eqs 7–9); for free fermions the physically correct extension is known from first principles to be the genuine non-Borel KO_P. The free-fermion case therefore selects the extension from physics, not from eq (4). For the interacting spectrum Ω_{H(s),P}, no analogous first-principles computation is supplied; the jump in §3 from 'free fermions are non-Borel' to 'the interacting classification is non-Borel' is an extrapolation. The authors themselves state in §7 that they do not prove the CEP fails and that the absence of computations is the biggest drawback. Until one exhibits an interacting model selecting a non-Borel extension, or shows a Borel extension is incompatible with eq (4), the central claim is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a fully point-group-equivariant version of the Freed–Hopkins ansatz for symmorphic crystalline SPT phases. It argues that the crystalline equivalence principle (CEP) used in Freed–Hopkins's spatial-symmetry extension [2] is unnatural, because for free fermions the correct classification is non-Borel equivariant KO, and the natural Atiyah–Bott–Shapiro-based free-to-interacting map into the [2] spectrum lands in a twisted Borel group rather than KO_P. The paper defines a genuinely equivariant Freed–Hopkins spectrum Ω_{H(s),P}^{d+2} using the equivariant Madsen–Tillmann spectrum of Galatius–Szűcs, posits π0(S_d(H,G)) = Ω_{H(s),P}^{d+2}(T^d) (eq 18), and constructs an equivariant FTI map (eq 22). The authors explicitly acknowledge in §7 that they do not prove that the CEP fails and that they provide no computations of the proposed groups.","tokens_in":7131,"tokens_out":11648,"duration_ms":114786,"significance":"If the proposed framework were established, it would be a significant conceptual correction: interacting crystalline SPT phases would be classified by genuinely equivariant, non-Borel cohomology theories, and the Freed–Hopkins spatial-symmetry ansatz would have a naturality defect. The paper is transparent about its limitations, builds on published mathematical work (equivariant cobordism [21], Anderson duality [23]), and has no fitted parameters or circular predictions. Its main value is in articulating a concrete naturality criterion and a research programme for comparing CEP-based and fully equivariant classifications. However, the central conclusions are currently not proven: eq (4) is underdetermined, eq (20) is asserted rather than demonstrated, and eq (18) is a conjecture. The paper is therefore more a position/proposal paper than a proof of the non-CEP interacting classification.","major_comments":[{"comment":"The crystalline Kitaev conjecture as stated does not force a non-Borel P-equivariant extension. For any spectrum D(H), the Borel construction D(H)^d_P(X) := D(H)^d(X ×_P EP) also satisfies eq (4), and the group-cohomology case (eqs 7–9) illustrates exactly this. Free fermions select KO_P by first-principles physics, not by eq (4). The jump from 'free fermions are non-Borel' to 'the interacting spectrum should be non-Borel' is an extrapolation. To make this load-bearing, the authors need either an interacting model that selects a non-Borel extension or a theorem showing no Borel extension can satisfy eq (4) together with other naturality conditions. As written, eq (18) is a conjecture whose evidence is naturality, not derivation.","section":"§3, Eq. (4)"},{"comment":"The equivariant FTI map (22) depends on an equivariant ABS map φ_P : MT H(s)_P → Σ^s KO_P that is asserted without proof. The cited reference [22] concerns higher coherences for equivariant K-theory and does not obviously construct a map from the equivariant Madsen–Tillmann spectrum with H(s)-tangential structure. Since eq (22) is advertised as a main result in the abstract ('we show there is a natural equivariant FTI map'), the construction of φ_P, as well as the P-equivariant Anderson self-duality eq (21), must be supplied in detail or explicitly marked as a conjecture.","section":"§6, Eq. (20)"},{"comment":"The 'wrong FTI map' argument is not conclusive, and the text concedes this: 'we have not shown conclusively that such a map from the ansatz in [2] does not exist.' The argument shows that the specific ABS-based map (13) targets a twisted Borel group, not that every natural FTI map to E^{hP}(T^d) must do so. As a result, the title and abstract overstate the case: the CEP-based ansatz is not shown to be wrong, only that one natural construction fails. Either prove uniqueness/naturality of the FTI map or soften the claims accordingly.","section":"§4, Eqs. (13)–(15)"},{"comment":"The identification π0(S_d(H,G)) = Ω_{H(s),P}^{d+2}(T^d) is posited, not derived. The passage from the equivariant cobordism category of [21] to a spectrum MT H(s)_P that classifies reflection-positive invertible field theories with P-action is not demonstrated. In particular, the role of Anderson duality in imposing reflection positivity in the presence of a nontrivial P-action is asserted rather than proved. Without a proof or at least a detailed spectral construction, eq (18) remains a conjecture; this is also acknowledged in §7 as the biggest drawback.","section":"§5, Eq. (18)"}],"minor_comments":[{"comment":"The displayed exponents appear inconsistent: eq (14) uses S^V while eq (15) uses S^{-V}, and Σ^{s-2} in eq (14) becomes Σ^{-2} in eq (15). Please check the Atiyah-duality and suspension calculations; as written, the chain is not transparent.","section":"§4, Eqs. (14)–(15)"},{"comment":"Minor typo: 'classifies interacting SPTS' should be 'SPTs'.","section":"Introduction"},{"comment":"References [3] and [13] appear to be the same paper ('Failure of the crystalline equivalence principle for weak free fermions') listed twice with slightly different page data; consolidate.","section":"References"},{"comment":"The notation 'hP' is defined as homotopy orbits, but the text later says 'homotopy fixed points becomes a Borel construction.' Please use consistent terminology (fixed points vs. orbits) to avoid confusion.","section":"§4–§5"},{"comment":"Eq (21) states I_Z(KO_P) ≃ Σ^4 KO_P, while §4 uses I_Z(KO) ≃ Σ^{-4} KO. These are equivalent under 8-periodicity of KO, but the relation should be stated explicitly to avoid apparent contradiction.","section":"§6"}],"recommendation":"major_revision","confidential_remarks":"This is a thought-provoking proposal that fits the journal's scope. The authors are honest about the main gaps, particularly the absence of computations and of a proof that the CEP fails. I do not recommend rejection because the central idea is defensible and the missing piece—a detailed construction of eq (20) or a clear reframing as a conjecture—may be fillable. However, the abstract and title currently overclaim relative to the proof content, and the underdetermination of eq (4) is a serious logical gap that needs to be addressed head-on."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious research program, not a completed result. The best concrete output is in §4: they show that the natural would-be FTI map for Freed–Hopkins's spatial-symmetry extension [2] has domain a twisted Borel group, not the actual crystalline free-fermion group KO_P^{d+s-2}(T^d). That is a well-defined, checkable problem with the existing ansatz, and it gives the paper its strongest argument.\n\nThe construction of Ω_{H(s),P} via the Galatius–Szűcs equivariant cobordism spectrum is a natural non-Borel candidate, and the paper is honest that it is a proposal, not a proof. The authors explicitly concede they have not shown the CEP fails in the interacting case and that the absence of computations is the biggest drawback. They are right on both points.\n\nThe soft spot, as the stress-test note says, is eq (4). The crystalline Kitaev conjecture is underdetermined: for any non-equivariant spectrum D(H), the Borel extension D(H)^d_P(X)=D(H)^d(X ×_P EP) is another equivariant extension with the same non-equivariant limit, and for group cohomology it reproduces the CEP. Free fermions select the non-Borel extension because we can compute KO_P from physics, not because eq (4) forces it. No analogous first-principles input is supplied for the interacting spectrum, so the jump to full equivariance is an extrapolation. Likewise, the equivariant ABS map of eq (20) is cited, not proved, and it is load-bearing for the naturality of the FTI map.\n\nNone of this is disqualifying. The wrong-FTI-map observation is a genuine hit, and the proposed spectrum is the right object to study. The paper deserves a serious referee, but a referee should ask for either a written proof of the equivariant ABS extension or at least one explicit computation that distinguishes Ω_{H(s),P} from the Borel alternative. I would bring it to a reading group working on SPT classification and would cite the wrong-FTI point in my own writing.","headline":"A serious proposal for a fully equivariant Freed–Hopkins ansatz, with a real hit against the Borel-style extension, but the central claim remains conditional on an unproved generalization and missing computations.","tokens_in":7665,"tokens_out":3413,"would_cite":true,"duration_ms":36067,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N91","55P91","19L47","81T45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Crystalline SPT phases are classified by non-Borel equivariant invertible field theories, and the natural free-to-interacting map points to this conclusion.","keywords":["symmetry-protected topological phases","crystalline symmetries","free-to-interacting map","equivariant homotopy theory","Kitaev conjecture","Borel equivariant cohomology","invertible field theories","equivariant K-theory"],"falsifier":"A concrete computation would settle the matter: for d=2 and a point group P like a single reflection, compute the group Ω_{H(s),P}^{4}(T^2) and the image of FTI_P. If this group equals the Borel-type classification, or if FTI_P factors through a Borel theory, the paper's central claim collapses. Alternatively, finding an explicit element in the Borel ansatz that is spurious (not in the image of the actual free-fermion group) and showing it does not appear in the equivariant classification would confirm the distinction.","tokens_in":6692,"feed_emoji":"⚛️","tokens_out":4273,"duration_ms":43582,"temperature":0.7,"pith_summary":"The paper argues that for materials with symmorphic crystalline symmetries, the correct classification of interacting symmetry-protected topological (SPT) phases is a fully equivariant version of the standard invertible-field-theory ansatz, not the Borel-type extension that treats spatial symmetries as internal. It motivates this through a crystalline version of Kitaev's conjecture, which forces the spectrum to be equivariant with respect to the point-group action on the unit-cell torus. The paper then shows that the Borel-style extension has a natural free-to-interacting map that lands in the wrong free-fermion group, a twisted Borel version rather than genuine equivariant K-theory. It constructs an alternative equivariant ansatz and a natural equivariant free-to-interacting map, concluding that full equivariance should replace the crystalline equivalence principle.","feed_headline":"Crystalline SPT phases need full equivariance, not Borel symmetry","feed_subtitle":"The natural free-to-interacting map lands in non-Borel equivariant K-theory, overturning the Borel-style ansatz.","key_machinery":"The central object is the equivariant Freed–Hopkins style spectrum Ω_{H(s),P}^{d+2}(X) = [X ∧ MT H(s)_P, Σ^{d+2} IZ]^P, where MT H(s)_P is the equivariant Madsen–Tillmann spectrum for finite point groups. The identity that carries the argument is the equivariant Atiyah–Bott–Shapiro map φ_P: MT H(s)_P → Σ^s KO_P, combined with Anderson self-duality IZ(KO_P) ≃ Σ^4 KO_P, to assemble the equivariant free-to-interacting map FTI_P: KO_P^{d+s-2}(T^d) → Ω_{H(s),P}^{d+2}(T^d). The crystalline Kitaev conjecture π0(S_d(H,G)) = D(H)^d_P(T^d) is the principle that forces non-Borel equivariance.","core_discovery":"The discovery is that the free-to-interacting map from free-fermion phases to interacting invertible field theories, when extended to crystalline symmetries, must be equivariant with respect to the point group P: the domain is KO_P^{d+s-2}(T^d), and a natural map to the interacting classification requires a spectrum Ω_{H(s),P}^{d+2} built from the equivariant Madsen–Tillmann spectrum, not the homotopy-orbit (Borel) construction. This leads to the proposal π0(S_d(H(s),G)) = Ω_{H(s),P}^{d+2}(T^d), which does not satisfy the crystalline equivalence principle. The paper claims this is mathematically and physically more natural than the earlier Borel extension, even though explicit computations o","pith_inferences":["A concrete next step is to compute Ω_{H(s),P}^{d+2}(T^d) for a simple case (e.g., d=2, P a reflection or a cyclic group) and compare with the Borel ansatz; any discrepancy would confirm the paper's central distinction.","The failure of the crystalline equivalence principle for weak free fermions, previously observed, may be a symptom of a general rule: spatial symmetries always require full equivariance, and the Borel approximation only works when the point group acts trivially enough to be invisible.","If full equivariance is adopted, existing classifications based on the Borel-style ansatz may need revisiting for all space groups, not just the symmorphic ones considered here.","The existence of a natural equivariant FTI map suggests that the Atiyah–Bott–Shapiro map should be regarded as an equivariant construction, which could have implications for the topology of Dirac operators on orbifolds."],"forward_implications":["If the proposal is correct, interacting crystalline SPT classifications must be computed with P-equivariant cohomology, and the crystalline equivalence principle fails for interacting phases, not just free fermions.","The kernel of FTI_P identifies which symmorphic crystalline free-fermion phases are killed by interactions, and the cokernel identifies interaction-enabled crystalline phases.","The Borel-style ansatz admits spurious free phases—elements in its domain that are not genuine free-fermion phases—whose images would falsely appear in the interacting classification.","The framework motivates developing computational tools for equivariant homotopy theory to make the new groups explicitly computable.","For non-symmorphic or mixed internal/spatial symmetries, the equivariant spectrum would require twisting, pointing to a broader de-Borelianization program."],"fun_headline_variants":["Equivariant map fixes crystalline SPT classification","Crystalline SPTs: Full equivariance beats Borel ansatz","Freed-Hopkins ansatz needs full equivariance for crystals","Why crystalline SPT phases demand equivariant K-theory","Equivariant FTI map overturns Borel symmetry in SPTs"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole argument rests on the crystalline Kitaev conjecture (eq 4)—that interacting SPT phases with crystalline symmetries are classified by a P-equivariant generalized cohomology theory—which is assumed rather than derived; if this conjecture fails, the preference for non-Borel equivariance loses its foundation.","fun_headline_variants_meta":{"raw":{"variants":["Equivariant map fixes crystalline SPT classification","Crystalline SPTs: Full equivariance beats Borel ansatz","Freed-Hopkins ansatz needs full equivariance for crystals","Why crystalline SPT phases demand equivariant K-theory","Equivariant FTI map overturns Borel symmetry in SPTs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000637,"raw_usage":{"total_tokens":2749,"prompt_tokens":694,"completion_tokens":2055,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":1967}},"tokens_in":438,"tokens_out":2055,"duration_ms":14870,"temperature":1.0,"reasoning_tokens":1967,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T00:18:43.189903+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete computation would settle the matter: for d=2 and a point group P like a single reflection, compute the group Ω_{H(s),P}^{4}(T^2) and the image of FTI_P. If this group equals the Borel-type classification, or if FTI_P factors through a Borel theory, the paper's central claim collapses. Alternatively, finding an explicit element in the Borel ansatz that is spurious (not in the image of the actual free-fermion group) and showing it does not appear in the equivariant classification would confirm the distinction.","supporting_citations":[],"review_version":1}