{"id":"4278bf3b-d491-4e09-b710-2dc2f25d16ec","arxiv_id":"2610.28253","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For every condensable algebra in a twisted quantum double D(G, ω), the paper fixes the SymTFT layout of its proliferation transition and shows twin algebras with identical anyon content give distinct Landau-Ginzburg transitions due to opposite relative cubic signs.","lead":"Anyons in a 2+1d topological phase can sometimes be condensed collectively, driving a phase transition to another topological phase. This paper supplies the complete data for such transitions in quantum doubles of finite groups, and shows that the multiplication rule of the condensate, not just its particle content, fixes the effective cubic interactions, so twin condensates with identical particle content give different transitions.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Twin G32 transitions are exchanged by an outer automorphism plus field redefinition, so 'distinct' is presentation-dependent unless a fixed reference is stipulated.","rationale":"The reader's conditional verdict already captures the central scoping problem: the categorical dictionary appears sound, but the physical claim of distinct twin transitions is weaker than the abstract states. My stress-test isolates a specific mechanism for that weakness: the braided autoequivalence φ of D(G32) combined with a field redefinition makes the two G32 transitions equivalent under standard identifications of symmetric theories. This is not an internal inconsistency; the mathematics of the dictionary and the algebraic computation of c± appear internally consistent as far as I could check. It is a correctness-of-interpretation risk for the headline claim. The proposed check is concrete and would settle whether the G32 example exhibits a genuine invariant distinction or only a convention-dependent one. Because the paper's own text already flags the exchange and because the reader's conditional verdict already conditions the headline claim on this scoping, I recommend no change to the verdict.","tokens_in":52651,"tokens_out":14126,"duration_ms":295829,"concrete_test":"Independently compute the full SymTFT sandwiches P_{A1}(r) and P_{A2}(r) of (4.7) for the G32 example, including the physical-boundary factor N_A and the gauged G_A action, and check whether applying the outer automorphism φ of (4.27) together with z3 → −z3 maps the A1 sandwich to the A2 sandwich term by term: quadratic masses, cubic tensor, quartic threshold, and boundary factor. If it does, the two transitions are equivalent under a symmetry of the parent theory and the 'distinct transitions' claim holds only in a fixed reference presentation; if it does not, exhibit the invariant quantity that survives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing physical claim is that twin algebras yield distinct proliferation transitions (§4.3). The paper itself shows that the two G32 algebras are exchanged by the outer automorphism φ of (4.27) together with the field redefinition z3 → −z3, and that this maps c+ to c− and flips the quartic threshold term J (4.22–4.26). Under the standard equivalence relation for phases and transitions, which allows relabelings that are symmetries of the parent theory, this makes P_{A1} and P_{A2} the same transition presented in two different bases, not two distinct transitions. The defense that the two are distinct 'in a single parent theory, with a fixed reference G32 action' discards precisely the automorphisms that are symmetries of that parent; a fixed reference is a convention, not a gauge-invariant distinction. The opposite cubic signs and opposite quartic corrections are therefore differences relative to a chosen labeling of z3, and by themselves do not establish inequivalent transitions. This concern does not touch the algebra-to-SymTFT dictionary of Theorems B.9, B.19, and Proposition B.24, but it undercuts the flagship example meant to demonstrate the new physics claimed in the abstract.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper develops a complete dictionary, for every condensable algebra A=A(H,N,gamma,epsilon) in the twisted quantum double D(G,omega), between the algebra classification data and the data needed to construct the corresponding proliferation transition in the SymTFT framework of [10]. The main mathematical results are: the generated subcategory is <A>=S(K_A,L_A,B_A) (Theorem B.9); the Mueger center is E_A congr Rep(G_A,z_A) with G_A constructed from R_A,P_A and z_A=(B_A circ Delta,1) (Theorem B.19); type-(i) algebras are exactly those equivalent, in another presentation, to purely electric algebras Fun(G'/H') (Theorem B.17); and, in the Tannakian case, the boundary MTCs are M_A congr (Vec^{omega_M}_{K_A/R_A})^{K_A/I_A} and N_A congr (Vec^{omega_N}_{L_A/R_A})^{L_A/I_A} (Proposition B.24). Section 4 then proposes that for type-(i) algebras the multiplication structure of A fixes the cubic terms of the scalar LG theory, and applies this to twin algebras in G_32, obtaining opposite relative cubic signs and opposite quartic threshold corrections. The paper also treats super-Tannakian Mueger centers, with the minimal example D(Z_2 x A_4), and discusses partial LG descriptions for type-(ii) algebras.","tokens_in":54538,"tokens_out":1553,"duration_ms":53197,"significance":"If the dictionary is correct, this is a substantial and useful contribution: it is, to my knowledge, the first systematic translation of the full condensable-algebra classification of twisted quantum doubles into SymTFT transition data, including the super-Tannakian case. The proofs in Appendix B are real proofs rather than sketches; Theorems B.9, B.17, B.19 and Proposition B.24 give explicit, checkable formulas, and the examples in Section 3 provide a valuable test bed. The proposed relation between algebra multiplication and the cubic tensor is a genuinely new and falsifiable prediction within the LG ansatz. The central limitation is explicitly acknowledged at b77, namely that the categorical construction alone does not establish a direct transition, its order, or a universal critical theory. The remaining weakness is that the flagship physical claim, that twin algebras give distinct proliferation transitions, is presented more strongly than the authors' own equations justify.","major_comments":[{"comment":"The claim that the G_32 twins produce distinct proliferation transitions is not established as stated. The authors themselves show that the field redefinition z_3 -> -z_3 combined with the outer automorphism phi of (4.27) maps c_+ to c_- and J to -J, and that this is a braided autoequivalence of the parent D(G_32) exchanging the two algebras. Under the standard equivalence relation for phase transitions, which identifies theories related by relabelings that are symmetries/automorphisms of the parent, this makes P_{A1} and P_{A2} the same transition in two different presentations. The defense that the transitions are distinct \"in a single parent theory, with a fixed reference G_32 action\" amounts to fixing a convention for which field carries which representation; it does not supply a physical observable that distinguishes the twins. This issue is load-bearing because it is the abstract's central new claim (b6). The authors should either define and justify an equivalence relation on transitions that excludes such relabelings, or weaken the claim to \"twin algebras give inequivalent effective cubic tensors in a fixed presentation.\"","section":"Section 4.3, Eqs. (4.22)-(4.27)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The identification of the cubic tensor with the algebra multiplication coefficients is conditional on the LG ansatz: one scalar multiplet per anyon endpoint, an ordered phase whose vacua are X=G_A/H_A, and the exclusion of SPTs (b295). The paper states this honestly at b77, but the abstract's phrasing \"properties of the transition can then be inferred directly from the algebra classification data\" risks overstating the scope. Since the same condition also underlies the twin example, the twin-sign prediction should be presented as a prediction of this effective LG model rather than as a theorem about the true proliferation transition. I do not flag this as an error, but the wording should be adjusted for proportionality.","section":"Sections 4.1-4.2 and b295/b77","authors_acknowledge":"The categorical construction alone does not establish a direct transition, its order, or a universal critical theory.","authors_acknowledge_p":0.84},{"comment":"The super-Tannakian SymTFT construction is presented as a candidate bulk via D(Sigma E_A), with the paper noting at b130 that the original SymTFT construction in [10] assumes a Tannakian center, and Eq. (2.45) is labelled schematic. This is acceptable as an exploratory extension, but it means the super-Tannakian examples in Section 3.4 should be read as a proposed framework rather than as a derivation of the transition. The main dictionary claims (Theorems B.9 and B.19) do not depend on this, so this is a scope issue rather than a technical defect.","section":"Section 2.6 and Corollary 2.1","authors_acknowledge":"Note the SymTFT construction in [10] assumesE A to be Tannakian.","authors_acknowledge_p":0.87}],"minor_comments":[{"comment":"There is a typo: \"determime\" should be \"determine\".","section":"Section 4, first paragraph","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"\"It would be interesting so see\" should be \"to see\".","section":"Acknowledgments","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The notation D(Sigma E_A) is used without an explicit definition of the functor D; a one-sentence definition or a pointer to [48] would help readers unfamiliar with the dependence of the Walker-Wang defect category on its input.","section":"Section 2, text near Eq. (2.45)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sentence \"As the formulation maybe rather abstract we will not discuss examples without and with anomaly\" appears garbled; presumably it should read \"with and without anomaly\".","section":"Section 3.4 opening","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The tables are dense and very useful, but Table III in particular would benefit from a note in the caption explaining how the entries labeled by epsilon_x, gamma_16, gamma_17, and B_+- are normalized, since several entries depend on cocycle representatives rather than cohomology classes.","section":"General","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The categorical dictionary in Appendix B appears sound and is the strongest part of the paper; if the authors reframe the twin claim along the lines suggested (\"inequivalent effective LG descriptions in a fixed presentation\" rather than \"distinct transitions\"), the paper would be close to acceptable. The current abstract overstates the physical conclusion, and this is the main obstacle to acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"The genuinely new content is the complete dictionary from condensable-algebra data (H, N, γ, ε) in D(G, ω) to the generated subcategory, Müger center, bulk group G_A, fermion parity z_A, and boundary MTCs M_A, N_A. The appendix carries real proofs of the central statements — Theorem B.9 for ⟨A⟩ = S(K_A, L_A, B_A), Proposition B.13 for the Müger center, Theorem B.17 for the type-(i) characterization, Theorem B.19 for E_A ≅ Rep(G_A, z_A), and Proposition B.24 for M_A, N_A. I checked the internal logic at the level I can verify and did not find an inconsistency. The five super-Tannakian algebras in D(Z₂ × A₄) are worked out concretely and the resulting T/A identifications are backed by explicit computation, which is a real asset. The observation that the algebra multiplication fixes the cubic tensor in the LG description is also solid, and the threshold-sign difference in the G32 example follows from the stated EFT if one accepts the LG ansatz.\n\nThe main soft spot is the headline claim that twins give distinct proliferation transitions. The paper itself shows that the outer automorphism φ of (4.27) together with the field redefinition z₃ → −z₃ exchanges the two G32 algebras, maps c₊ to c₋, and flips the quartic threshold term. Under the standard equivalence relation for phases and transitions, which allows relabelings that are symmetries of the parent theory, that makes them the same transition presented in two bases, not two distinct transitions. The paper's defense — that they are distinct in a single parent theory with a fixed reference G32 action — is a convention, not a gauge-invariant distinction. This does not touch the algebra-to-SymTFT dictionary, but it undercuts the flagship physical claim. The authors should either soften the claim or identify a reference-invariant observable that genuinely distinguishes the two transitions.\n\nThe other caveats are minor and mostly acknowledged: the super-Tannakian SymTFT sandwich is verified in examples rather than proven generally, the ν_M = ν_N = 0 assumption is deferred, and the Table V data is marked 'proposed' and needs a fixed GAP version and clearer provenance from the same-author preprint [35]. The paper also states plainly that the categorical construction alone does not establish a direct transition, its order, or a universal critical theory, so readers should not expect a proof of an actual critical point.\n\nWho this is for: anyone working on anyon condensation, SymTFT realizations of Dijkgraaf–Witten theory, or the algebra classification program. The dictionary is the kind of thing that will be cited as a reference tool. It deserves a serious referee: send it out, but ask for the twin-claim scoping and the reproducibility details before acceptance.","headline":"The algebra-to-SymTFT dictionary is new and mostly well proven; the twin-algebra 'distinct transitions' claim needs explicit scoping because the two G32 transitions are exchanged by a symmetry of the parent theory.","tokens_in":53706,"tokens_out":1822,"would_cite":true,"duration_ms":15341,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["18M20","18M15","81T45"],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"Every anyon condensation in a group quantum double is mapped to complete transition data, and twin condensations with identical anyon content are shown to give distinct phase transitions.","keywords":["proliferation transitions","anyon condensation","condensable algebras","quantum doubles of groups","Symmetry Topological Field Theory","gauged Landau-Ginzburg transitions","twin algebras","super-Tannakian categories"],"falsifier":"Compute the effective potential of the two twin condensations in $D(G_{32})$ from a microscopic model rather than from the algebra: the paper predicts relative cubic couplings $g_2/g_1=\\pm2$ and $g_3/g_0=\\pm2$ and, after integrating out the heavy $\\sigma$ and $z_2$ fields, opposite quartic threshold corrections $\\Delta\\lambda^\\pm_J=\\mp w^2/(8M_2^2)$; a realization with the correct anyon content that yields identical sign structures for the twins would refute the claim that multiplication controls the transition. For the categorical statements, a direct computation in $D(\\mathbb{Z}_2\\times A_4)$ of the twist eigenvalues of the transparent lines should reproduce $E_{\\mathcal{A}}\\cong\\mathrm{Rep}(\\mathbb{Z}_6,z)$, and the existence of any condensable algebra whose generated subcategory differs from $S(K_{\\mathcal{A}},L_{\\mathcal{A}},B_{\\mathcal{A}})$, or of a super-Tannakian center of type (i), would refute Theorems B.9 and B.17.","tokens_in":53979,"feed_emoji":"🌀","tokens_out":24025,"duration_ms":143862,"temperature":0.7,"pith_summary":"Anyon condensation is a topological operation that relates two 2+1d topological orders, but deciding whether a given condensation can actually be realized as a phase transition is a dynamical question. This paper answers it for the entire family of quantum double topological orders $D(G,\\omega)$ for finite groups $G$ with possible twist $\\omega$: every condensable algebra $\\mathcal{A}=\\mathcal{A}(H,N,\\gamma,\\epsilon)$ is mapped, purely from its classification data, to the full data of its proliferation transition — the subcategory it generates, its Müger center (the condensate-transparent lines), the symmetry group, and the two boundary theories of the SymTFT, the one-higher-dimensional topological field theory whose sandwich construction encodes the transition. For the broad class of type (i) condensations the transition is a gauged Landau-Ginzburg theory, and the central new step is that the cubic interactions of the scalar fields are not free: they are the multiplication coefficients of the algebra $\\mathcal{A}$ itself. Because twin algebras can share the same anyon content yet differ in their multiplication, twins give distinct proliferation transitions, with concrete signatures — opposite relative cubic signs and opposite quartic threshold corrections in the $G_{32}$ example. The paper states plainly that the categorical construction alone does not fix a transition's order or its universal critical theory; the claim is that within the gauged-scalar ansatz, the algebra's multiplication fixes the interaction structure any such transition must carry.","feed_headline":"Multiplication decides: twin condensations are distinct transitions","feed_subtitle":"In group quantum doubles, the algebra's product table fixes the cubic couplings of the transition.","key_machinery":"The argument runs through four linked objects. First, the classification input: a condensable algebra $\\mathcal{A}(H,N,\\gamma,\\epsilon)$ records the unbroken subgroup $H$, the condensed fluxes $N\\trianglelefteq H$, the flux-operator multiplication phases $\\gamma$, and the charges $\\epsilon$ that surviving gauge transformations assign to the condensed fluxes. Second, the generated subcategory $\\langle\\mathcal{A}\\rangle = S(K_{\\mathcal{A}},L_{\\mathcal{A}},B_{\\mathcal{A}})$: the standard classification of fusion subcategories of $D(G,\\omega)$ packages all anyons generated by $\\mathcal{A}$, with $K_{\\mathcal{A}}$ the normal closure of the condensed fluxes, $L_{\\mathcal{A}}$ the subgroup of core centralizers acting on flux sectors by scalars, and $B_{\\mathcal{A}}$ the resulting bicharacter. Third, the Müger center $E_{\\mathcal{A}}=\\mathcal{Z}_2(\\langle\\mathcal{A}\\rangle)\\cong\\mathrm{Rep}(G_{\\mathcal{A}},z_{\\mathcal{A}})$, the transparent lines that become the bulk gauge symmetry of the SymTFT; the group $G_{\\mathcal{A}}$ is a semidirect extension of the Pontryagin dual of the transparent fluxes by $G/K_{\\mathcal{A}}L_{\\mathcal{A}}$, and the central involution $z_{\\mathcal{A}}$ decides which lines are fermionic (the super-Tannakian case). Fourth, the symmetry-boundary Landau-Ginzburg theory for type (i): one real scalar multiplet per nontrivial anyon in $\\mathcal{A}$, ordered vacua indexed by the set $X=G_{\\mathcal{A}}/H_{\\mathcal{A}}$, and a cubic tensor $C_{IJK}=Q(m(e_I,e_J),e_K)$ defined from the multiplication $m$ of the algebra in normalized coset coordinates — the step that converts algebra structure into interaction coefficients.","core_discovery":"The paper's central claim is a complete dictionary from condensable-algebra data to proliferation-transition data in twisted quantum doubles $D(G,\\omega)$. For each algebra $\\mathcal{A}(H,N,\\gamma,\\epsilon)$ the paper proves that the generated fusion subcategory is $\\langle\\mathcal{A}\\rangle = S(K_{\\mathcal{A}}, L_{\\mathcal{A}}, B_{\\mathcal{A}})$ with the flux subgroup $K_{\\mathcal{A}}$, the scalar-action subgroup $L_{\\mathcal{A}}$, and the bicharacter $B_{\\mathcal{A}}$ all given by explicit group-theoretic formulas; that the Müger center of transparent lines is $\\mathrm{Rep}(G_{\\mathcal{A}}, z_{\\mathcal{A}})$ with $G_{\\mathcal{A}} = \\widehat{R}_{\\mathcal{A}} \\rtimes (G/P_{\\mathcal{A}})$ and parity element $z_{\\mathcal{A}} = (B_{\\mathcal{A}}\\circ\\Delta, 1)$; and that in the bosonic case the two SymTFT boundary theories are $M_{\\mathcal{A}}\\cong(\\mathrm{Vec}^{\\omega_M}_{K_{\\mathcal{A}}/R_{\\mathcal{A}}})^{K_{\\mathcal{A}}/I_{\\mathcal{A}}}$ and $N_{\\mathcal{A}}\\cong(\\mathrm{Vec}^{\\omega_N}_{L_{\\mathcal{A}}/R_{\\mathcal{A}}})^{L_{\\mathcal{A}}/I_{\\mathcal{A}}}$. From this dictionary the paper derives a structural classification of the transitions: an algebra is of type (i), meaning its transition is a gauged Landau-Ginzburg (LG*) theory, if and only if it is purely electric, $\\mathcal{A}\\cong\\mathrm{Fun}(G'/H')$, in some equivalent presentation of $D(G,\\omega)$, so these transitions are always Higgsings. The key step that carries the physical conclusion is that the cubic tensor of the symmetry-boundary scalar theory equals the multiplication structure of $\\mathcal{A}$, which is why twin algebras — identical anyon decompositions, different multiplication — yield inequivalent transitions, demonstrated in $D(G_{32})$ by opposite relative cubic signs and opposite quartic threshold corrections.","pith_inferences":["My inference: the dictionary turns proliferation-transition design into a finite group computation, so the natural next step — which the paper does not take — is to build the predicted transitions in lattice realizations (perturbed quantum double or string-net models) for a sequence of groups, reading off all transition data in advance from $(H,N,\\gamma,\\epsilon)$.","My inference: the twin prediction is a sharp falsifiable target for any microscopic model of $D(G_{32})$, because the two condensations share field content, symmetries, and even the reduced topological order, leaving the sign structure of the effective potential as the only possible difference.","My inference: the paper leaves open whether the twin distinction survives at the critical point; the multiplication-fixed cubic tensor is derived for the gapped Higgs description, and whether it controls the leading relevant deformation of the critical theory is a separate renormalization-group question.","My inference: the super-Tannakian examples point to a family of transitions with an emergent fermionic parity on the symmetry boundary even though both endpoint topological orders are bosonic; characterizing that boundary sector is a natural next problem, and the five algebras in $D(\\mathbb{Z}_2\\times A_4)$ are concrete test beds."],"forward_implications":["For any finite group $G$ and twist $\\omega$, the proliferation-transition data of every condensable algebra becomes a direct group-theoretic computation: the generated subcategory $\\langle\\mathcal{A}\\rangle=S(K_{\\mathcal{A}},L_{\\mathcal{A}},B_{\\mathcal{A}})$, the Müger center, the symmetry group $G_{\\mathcal{A}}$, and the boundary theories $M_{\\mathcal{A}}$, $N_{\\mathcal{A}}$ all follow from $(H,N","Type (i) transitions are always Higgsings: the condensate $\\mathcal{A}\\cong\\mathrm{Fun}(G'/H')$ breaks the symmetry group to a stabilizer subgroup, which is why this entire class admits gauged Landau-Ginzburg descriptions.","The multiplication of the condensate is visible in the dynamics: it fixes the relative cubic couplings of the gauged scalar theory, so twin condensations with identical anyon content are described by inequivalent transition theories.","For the $G_{32}$ twins the paper derives concrete signatures — opposite relative cubic couplings ($g_2/g_1=\\pm2$, $g_3/g_0=\\pm2$) and opposite quartic threshold corrections $\\Delta\\lambda^\\pm_J=\\mp w^2/(8M_2^2)$ after integrating out heavy fields.","Transparent fermions in the Müger center occur only for type (ii) algebras and do not force the reduced phase to be a spin theory: in the $D(\\mathbb{Z}_2\\times A_4)$ examples the odd lines are transparent only inside the generated subcategory, and the reduced orders are ordinary $D(\\mathbb{Z}_2)$ and $D(\\mathbb{Z}_2^2)$.","key_machinery_placeholder"],"supporting_citations":[{"why":"Supplies the SymTFT sandwich construction for proliferation transitions and the type (i)/(ii)/(iii) taxonomy that the paper applies to every condensable algebra of $D(G,\\omega)$.","marker":"[10]"},{"why":"Provides the full classification of condensable algebras $\\mathcal{A}(H,N,\\gamma,\\epsilon)$ in $D(G,\\omega)$ and the twin-algebra examples, including the $G_{32}$ twins used for the threshold-sign prediction.","marker":"[35]"},{"why":"Classifies fusion subcategories of $D(G,\\omega)$ as triples $(K,L,B)$; the generated-subcategory computation (Theorem B.9) and the centralizer and intersection formulas rest on this classification.","marker":"[44]"},{"why":"Establishes the compatibility conditions on $(H,N,\\gamma,\\epsilon)$ and the reduced-order formula $\\mathcal{T}/\\mathcal{A}=D(H/N,\\alpha)$ that the dictionary uses as input and endpoint.","marker":"[34]"},{"why":"The theorem on symmetric tensor categories and super fiber functors that upgrades the Müger center to $\\mathrm{Rep}(G_{\\mathcal{A}},z_{\\mathcal{A}})$ in Theorem B.19, including the super-Tannakian case.","marker":"[31]"},{"why":"Supplies equivariantization and de-equivariantization, and the Lagrangian-subcategory result used to prove the type-(i) characterization (Theorem B.17) and to construct $M_{\\mathcal{A}}$ and $N_{\\mathcal{A}}$.","marker":"[45]"},{"why":"Supplies fermionic equivariantization and the minimal-modular-extension test used for the super-Tannakian Müger centers in section 2.6.","marker":"[43]"}],"fun_headline_variants":["Multiplication structure decides proliferation transition in quantum doubles","Twin condensable algebras yield distinct anyon transitions","Same anyon content, different transition: it's the algebra product","Proliferation transitions in quantum doubles turn on algebra multiplication","Twin algebras split: multiplication fixes the proliferation transition"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The load-bearing premise is the paper's gauged-scalar ansatz for the transition — one scalar multiplet per anyon in the condensate, a symmetric trivially gapped phase, ordered vacua $X=G_{\\mathcal{A}}/H_{\\mathcal{A}}$, and no symmetry-protected-topological phases in the scalar theory — together with the assumption of vanishing crossed-extension obstructions ($\\nu_M=\\nu_N=0$) for the SymTFT sandwich; if a real transition carries extra degrees of freedom, interaction channels, or non-zero obstructions, the strong conclusions (distinct twin transitions with opposite threshold signs) are not established by the paper's construction.","fun_headline_variants_meta":{"raw":{"variants":["Multiplication structure decides proliferation transition in quantum doubles","Twin condensable algebras yield distinct anyon transitions","Same anyon content, different transition: it's the algebra product","Proliferation transitions in quantum doubles turn on algebra multiplication","Twin algebras split: multiplication fixes the proliferation transition"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"cost":0.0001835652,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0001835652,"upstream_inference_prompt_cost":4.14232e-05,"upstream_inference_completions_cost":0.000142142},"total_tokens":2073,"prompt_tokens":1292,"completion_tokens":781,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":1046,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":716}},"tokens_in":1292,"tokens_out":781,"cost_source":"openrouter_usage","duration_ms":7850,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":716,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-21T07:29:45.193077+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Compute the effective potential of the two twin condensations in $D(G_{32})$ from a microscopic model rather than from the algebra: the paper predicts relative cubic couplings $g_2/g_1=\\pm2$ and $g_3/g_0=\\pm2$ and, after integrating out the heavy $\\sigma$ and $z_2$ fields, opposite quartic threshold corrections $\\Delta\\lambda^\\pm_J=\\mp w^2/(8M_2^2)$; a realization with the correct anyon content that yields identical sign structures for the twins would refute the claim that multiplication controls the transition. For the categorical statements, a direct computation in $D(\\mathbb{Z}_2\\times A_4)$ of the twist eigenvalues of the transparent lines should reproduce $E_{\\mathcal{A}}\\cong\\mathrm{Rep}(\\mathbb{Z}_6,z)$, and the existence of any condensable algebra whose generated subcategory differs from $S(K_{\\mathcal{A}},L_{\\mathcal{A}},B_{\\mathcal{A}})$, or of a super-Tannakian center of type (i), would refute Theorems B.9 and B.17.","supporting_citations":[{"cited_title":"Proliferation Transitions for Non-Abelian Anyons","cited_arxiv_id":"2608.12303","evidence_quote":"Supplies the SymTFT sandwich construction for proliferation transitions and the type (i)/(ii)/(iii) taxonomy that the paper applies to every condensable algebra of $D(G,\\omega)$."},{"cited_title":"Twin Algebras: Condensable Algebras beyond Anyons","cited_arxiv_id":"2605.31602","evidence_quote":"Provides the full classification of condensable algebras $\\mathcal{A}(H,N,\\gamma,\\epsilon)$ in $D(G,\\omega)$ and the twin-algebra examples, including the $G_{32}$ twins used for the threshold-sign prediction."},{"cited_title":"Fusion subcategories of representation categories of twisted quantum doubles of finite groups","cited_arxiv_id":"0810.0032","evidence_quote":"Classifies fusion subcategories of $D(G,\\omega)$ as triples $(K,L,B)$; the generated-subcategory computation (Theorem B.9) and the centralizer and intersection formulas rest on this classification."},{"cited_title":"On lagrangian algebras in group-theoretical braided fusion categories,","cited_arxiv_id":null,"evidence_quote":"Establishes the compatibility conditions on $(H,N,\\gamma,\\epsilon)$ and the reduced-order formula $\\mathcal{T}/\\mathcal{A}=D(H/N,\\alpha)$ that the dictionary uses as input and endpoint."},{"cited_title":"Cat´ egories tensorielles,","cited_arxiv_id":null,"evidence_quote":"The theorem on symmetric tensor categories and super fiber functors that upgrades the Müger center to $\\mathrm{Rep}(G_{\\mathcal{A}},z_{\\mathcal{A}})$ in Theorem B.19, including the super-Tannakian case."},{"cited_title":"On braided fusion categories. I,","cited_arxiv_id":null,"evidence_quote":"Supplies equivariantization and de-equivariantization, and the Lagrangian-subcategory result used to prove the type-(i) characterization (Theorem B.17) and to construct $M_{\\mathcal{A}}$ and $N_{\\mathcal{A}}$."},{"cited_title":"Categorical fermionic actions and minimal modular extensions,","cited_arxiv_id":null,"evidence_quote":"Supplies fermionic equivariantization and the minimal-modular-extension test used for the super-Tannakian Müger centers in section 2.6."}],"review_version":1}