{"id":"b5296738-c640-4636-a33d-114cb7d534dd","arxiv_id":"2608.02795","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Zesting preserves the computational complexity of Reshetikhin-Turaev invariants for links colored by simple objects, via a new polynomial-time computable invariant J_zeta.","lead":"A new construction shows that zesting, an algebraic operation on fusion categories, does not change the computational complexity of the associated Reshetikhin-Turaev link invariants. This gives a way to organize topological quantum field theories into complexity classes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.12's polynomial-time claim is only proved for zesting data valued in a fixed finite extension of Q that is not part of the input; 'given zesting data' is ambiguous and Cor 4.13 inherits this unstated restriction.","rationale":"The reader's weakest assumption matches my reading: the only insecure point in the central complexity claim is the unstated fixed-finite-extension/fixed-data hypothesis in Theorem 4.12. The factorization Theorem 2.7, the tangle functor J_ζ, and the rack-cocycle identification are internally coherent, and I found no structural error in the diagrammatic arguments. The proof's phrase 'whose values we may assume lie in a fixed finite extension K of Q' (Section 4.4.1) is doing essential work: if ζ is input rather than fixed, the running time depends on the size of ζ; if k is not a number field, exact computation is ill-posed. This does not invalidate the mathematics but makes the theorem statement narrower than it appears. The admitted typo in footnote 11 is a presentation issue and not load-bearing for the main theorem. Since the reader already imposed a CONDITIONAL verdict asking for this clarification, my stress-test does not move the verdict.","tokens_in":35037,"tokens_out":32864,"duration_ms":343942,"concrete_test":"Treat ζ as part of the input: encode ν, t, ϵ as elements of a number field K = Q(α) with degree d and coefficient bit-length b counted in the input size, fix a link diagram D, and run the Section 4.4.1 local-formula algorithm; check whether the total bit complexity is polynomial in d, b, and n(D)+c(D). Then check Cor 4.13's reduction when the oracle for F_C returns elements of a fixed extension K0 and J_ζ(L) lies in a different fixed extension K1: if the compositum K0K1 must be constructed and its degree enters the bound, the equivalence requires the fixed-extension hypothesis. If either check fails, the theorems need an explicit 'fixed ζ and number field' hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central reduction (Cor 4.13) computes F_{C^ζ}(L) from F_C(L) as J_ζ(L)·F_C(L), so the entire complexity equivalence hangs on J_ζ(L) being efficiently evaluable. Theorem 4.12's proof (Section 4.4.1) establishes this only under the assumption, stated in the proof but not in the theorem, that the values of ν, t, ϵ 'lie in a fixed finite extension K of Q which is pre-computed independently of D or L.' If ζ is instead part of the input with unbounded algebraic degree or coefficient bit-length, the running time is measured only in n(D)+c(D), and arithmetic in K is no longer constant-time; if k is not a number field, there is no canonical encoding of J_ζ(L) at all. Cor 4.13 states the equivalence for arbitrary ribbon fusion categories C, C^ζ over an algebraically closed field of characteristic 0, which includes non-computable fields. Thus, under the input-ζ reading, the stated polynomial-time Turing equivalence is not established. The fix is routine: state Theorem 4.12 and Cor 4.13 for fixed C and fixed ζ (or include ζ in the input size and prove the bound in that measure), and require k to be a number field. This is not a mathematical gap in the factorization itself, which appears sound; it is an unstated hypothesis that is load-bearing for the complexity conclusion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the effect of ribbon zesting on the computational complexity of Reshetikhin-Turaev link invariants. The authors construct a 2-functor J_ζ from a 2-category of A-shadow-colored tangles to the delooping of the pointed subcategory of invertible objects, giving a local, diagrammatic definition of the zesting correction invariant J_ζ(T). They prove a factorization theorem F_{C^ζ}(T) = F_C(T) ⋊ J_ζ(T) for homogeneous tangles, extend this to links, and identify J_ζ(L) (up to a sign that vanishes in the pseudo-unitary case) with a shadow rack cocycle invariant. Using Vogel's algorithm to convert link diagrams to braid closures, they then claim polynomial-time computability of J_ζ(L) and, as a corollary, polynomial-time Turing equivalence of the link invariants for zesting-related categories. The framework is extended to define invariants of closed 3-manifolds with A-structure, and a similar complexity equivalence is claimed for A-modular categories.","tokens_in":35288,"tokens_out":6368,"duration_ms":54027,"significance":"The paper gives a clean, local formalism for the zesting correction invariant, complementing the earlier link-level results of [DKP25] with a tangle-level 2-functor construction. The identification of the zest invariant with a shadow rack cocycle invariant is a novel connection between zesting and rack cohomology that may be of independent interest. The complexity result, if properly qualified, is a useful step toward organizing TQFT invariants into complexity-theoretic hierarchies, and the paper explicitly relates it to existing work on quantum braid group representations under zesting. The proofs are largely detailed and the factorization argument is sound; Appendix B verifies the shadow rack cocycle condition directly from the zesting axioms.","major_comments":[{"comment":"The proof of Theorem 4.12 assumes that the zesting data ν, t, ϵ take values in a fixed finite extension K of Q that is pre-computed independently of the diagram D or link L, but this hypothesis is not stated in the theorem and is omitted from Corollary 4.13. As stated, Corollary 4.13 quantifies over ribbon fusion categories over an algebraically closed field of characteristic 0, which need not be number fields, and for which 'polynomial-time Turing equivalence' is not canonically defined; if the zesting data ζ is instead given as part of the input with unbounded algebraic degree or coefficient bit-length, the running time measured only in n(D)+c(D) is insufficient to account for the cost of arithmetic in K. Because the complexity equivalence is obtained by multiplying or dividing by J_ζ(L), the efficient evaluability of J_ζ is load-bearing. I recommend restating Theorem 4.12 and Corollaries 4.13 and 5.19 for a fixed category C and fixed zesting data valued in a fixed number field, or explicitly including the bit-length of the zesting data in the input size and proving the bound in that measure.","section":"Section 4.4.1, Theorem 4.12, Corollary 4.13"},{"comment":"Corollary 5.19 is asserted to follow obviously from the argument of Theorem 4.12 and therefore inherits the same unstated fixed-field/number-field restriction on the zesting data and the ground field. In addition, the definition of the A-manifold invariant Z_ζ(M,ω) in equation (38) uses J_ζ(e,L,ω) on a surgery presentation, so a precise complexity statement requires explicit conventions for how the surgery link, its linking matrix, and the A-coloring are encoded. The current statement, 'ZC(M,ω) and ZCζ(M,ω) are polynomial-time Turing equivalent,' is not well-posed unless these inputs and the field are specified. Please state the precise input conventions and the hypotheses under which the equivalence holds.","section":"Section 5.3.1, Corollary 5.19"}],"minor_comments":[{"comment":"The displayed equality 'J_ζ(loop) = J_ζ(loop) = ϵ(a)' is ambiguous because the two sides refer to the counterclockwise and clockwise oriented loops, yet both are written with the same symbol; please use distinct notation and clarify that both values equal ϵ(a) because ϵ(a) = ϵ(a)^{-1}.","section":"Section 3.3, Proposition 3.12"},{"comment":"The phrase 'we can perform the multiplications in the formula for J(B) for free' is informal; since K is fixed, the intended meaning is 'at constant cost per field operation,' which should be stated explicitly.","section":"Section 4.4.1"},{"comment":"The passage from the morphism-valued zesting conditions in Definition 2.4 to the scalar-valued equations (28)–(31) is somewhat compressed; a brief remark on how the braiding of the pointed subcategory is evaluated to obtain the β-term would help the reader verify the equations.","section":"Section 4.2, equations (28)–(31)"},{"comment":"The cancellation of the ν terms and all but one of the t terms is stated without showing the intermediate algebra; adding a line or two of computation would make the example self-contained and easier to check.","section":"Example 4.2"},{"comment":"The shadow rack 2-cocycle condition is written with fractions, which makes the exponents hard to read; presenting it as a product of weights with explicit powers would improve readability.","section":"Definition 4.5, equation (32)"}],"recommendation":"major_revision","confidential_remarks":"The paper's factorization content appears mathematically sound, and the tangle-level 2-functor formalism is a genuine contribution beyond the earlier link-level results. The main issue is the unstated fixed-number-field hypothesis in the complexity theorem, which is load-bearing for the paper's central claim. The fix is routine, so major revision rather than rejection seems appropriate. The connection to shadow rack cocycle invariants is a nice byproduct that will broaden the paper's appeal. I would encourage the authors to also check that Corollary 5.19 is stated with precise input encoding assumptions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is real: zesting preserves the polynomial-time Turing complexity of Reshetikhin–Turaev invariants for links colored by simple objects. The new piece is the tangle-level invariant J_ζ, built as a 2-functor from shadow-colored tangles into the delooping of the pointed subcategory. That gives a local, diagrammatic computation, a polynomial-time algorithm, and the connection to shadow rack cocycle invariants. The rack cocycle identification is a nice byproduct and may have legs beyond this paper. The extension to A-manifold invariants is a sensible bonus, and the relative-complexity framing is stated honestly.\n\nI read the main proofs as sound. The factorization theorem for tangles is proven by generators and relations, with invariance reduced to the ribbon property of C^ζ from Delaney–Galindo–Plavnik–Rowell–Zhang. The rack cocycle verification in Appendix B is direct. The reliance on earlier work — especially [Del+20] and [DKP25] — is substantial, but the new local formalism and the complexity reduction are not simply restatements; the circularity burden is low. The citation pattern is honest.\n\nThe soft spot is exactly the one the stress-test flags. Theorem 4.12 says \"given ribbon zesting data ζ and a diagram D\" the invariant is computable in polynomial time. But the proof assumes the values of ν, t, ϵ lie in a fixed finite extension K of Q precomputed independently of the diagram. That assumption is load-bearing: computing J_ζ by the local formula requires arithmetic in K as constant-time operations. If ζ is part of the input with unbounded degree or coefficient size, the stated complexity bound does not follow. Corollary 4.13 inherits the problem, since as written it ranges over arbitrary ribbon fusion categories over algebraically closed fields of characteristic 0, including non-computable fields. This is not a gap in the factorization math; it is an unstated hypothesis in the complexity statement. The fix is routine: state the theorem for fixed C and fixed ζ, or include ζ in the input size and prove the bound in that measure, and require the ground field to be a number field. The footnote typo about t(i,i)^{-1} should be corrected but is minor.\n\nWho benefits: people working on complexity of TQFT invariants, topological quantum computation, and zesting as a structural operation. The paper deserves a serious referee; it is not a desk-reject. I would send it out, with the expectation that the authors clarify the complexity assumptions and fix the typo.","headline":"A solid, genuinely useful paper proving that zesting preserves the polynomial-time complexity of simply colored Reshetikhin–Turaev link invariants, provided the zesting data is fixed and number-field valued; the unstated complexity hypothesis needs to be made explicit.","tokens_in":35872,"tokens_out":1420,"would_cite":true,"duration_ms":15538,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K16","18M20","57K12"],"pacs":[],"model":"deepseek-v4-flash","headline":"When two ribbon categories are related by zesting, their Reshetikhin-Turaev link invariants are polynomial-time equivalent.","keywords":["zesting","Reshetikhin-Turaev invariants","ribbon fusion categories","computational complexity","Turing equivalence","shadow rack cocycle invariants","homotopy quantum field theory","link invariants"],"falsifier":"One concrete observation that would settle the claim: exhibit two ribbon fusion categories over a number field related by zesting and a family of simply colored link diagrams whose sizes grow polynomially but for which the local formula for $J_\\zeta$ requires super-polynomially many arithmetic operations, or for which $F_{\\mathcal{C}^\\zeta}$ is provably not polynomial-time Turing reducible to $F_{\\mathcal{C}}$; the theorem states that no such pair exists.","tokens_in":34789,"feed_emoji":"🪢","tokens_out":11952,"duration_ms":92803,"temperature":0.7,"pith_summary":"This paper proves that the zesting construction, which modifies a ribbon fusion category by cohomological data to produce a new category, does not change how hard it is to compute Reshetikhin-Turaev invariants of links colored by simple objects. If two categories are related by zesting, their link invariants are polynomial-time Turing equivalent: computing one from the other costs only polynomial overhead. The reason is a new tangle invariant, the zest invariant, which factors the zested invariant into the original invariant times a correction depending only on the zesting data, and which can be evaluated quickly from a link diagram. The result matters because exact quantum link invariants range across a wide complexity landscape, and a structural relation that guarantees equal complexity lets researchers transfer hardness or easiness between topological quantum field theories and organize topological phases by computational class.","feed_headline":"Zesting preserves the computational complexity of link invariants","feed_subtitle":"Categories related by zesting have Reshetikhin-Turaev link invariants that are polynomial-time equivalent.","key_machinery":"The central object is the zest invariant $J_\\zeta$, defined as a 2-functor from the 2-category of $A$-shadow-colored, oriented, framed tangles to the delooping of the pointed subcategory of invertible objects. A shadow coloring labels each region of a tangle diagram by an element of the universal grading group $A$, with prescribed rules at strands and crossings. The invariant is computed locally: crossings contribute scalars $\\phi_i(a,b)=\\nu(i,b,a)^{-1}t(a,b)\\nu(i,a,b)$, cups and caps contribute factors built from the zesting data $(\\lambda,\\nu,t,\\epsilon)$, and closed loops contribute $\\epsilon(a)$. This local calculus is what lets the paper evaluate $J_\\zeta(L)$ in polynomial time: convert the link diagram to a braid closure with a polynomial-time algorithm, read off the shadow coloring, multiply the crossing factors, and finish with the cup and cap corrections. The factorization identity relating $J_\\zeta$ to the Reshetikhin-Turaev functor is the load-bearing mechanism that turns a local invariant into a statement about complexity.","core_discovery":"The paper's central claim is Corollary 4.13: if $\\mathcal{C}^\\zeta$ is a ribbon zesting of $\\mathcal{C}$, then for any framed, oriented link $L$ colored by simple objects, the Reshetikhin-Turaev invariant $F_{\\mathcal{C}^\\zeta}(L)$ is polynomial-time Turing equivalent to $F_{\\mathcal{C}}(L)$. Equivalently, zesting preserves the computational complexity of simply colored link invariants. The proof works by defining a local invariant $J_\\zeta$ of $A$-shadow-colored tangles, showing that $F_{\\mathcal{C}^\\zeta}(T)$ is obtained from $F_{\\mathcal{C}}(T)$ by interleaving it with $J_\\zeta(T)$ through the warp product, and then giving a polynomial-time algorithm that evaluates $J_\\zeta$ on links by converting a diagram to a braid closure and multiplying Boltzmann weights at crossings. In the pseudo-unitary setting $J_\\zeta(L)$ coincides with a shadow rack cocycle invariant, so the correction factor is a discrete state-sum invariant rather than a generic quantum invariant.","pith_inferences":["The paper leaves open whether zesting preserves the complexity of $3$-manifold invariants without $A$-structure; because the surgery color mixes grading degrees, the same factorization does not directly apply, and the paper's own Remark 5.20 suggests the relative complexity there is genuinely subtler.","The shadow rack cocycle identification suggests that the correction factor, and hence the complexity-equivalence class, depends only on a cohomology class of the zesting data rather than on the full category, a structural reduction the paper does not pursue.","The framework suggests a testable extension: apply the local formula to families of finite-group gauge theories where the zesting data arise from group cohomology, to see whether the known hardness or easiness of coloring invariants is stable under the complexity-equivalence relation.","As an immediate corollary of the Turing equivalence, any hardness proof for the Reshetikhin-Turaev invariant of one category automatically transfers to every zesting of that category, which could be used to certify hardness of invariants in less-studied categories."],"forward_implications":["If $\\mathcal{C}$ and $\\mathcal{C}^\\zeta$ are related by zesting, any algorithm that computes $F_{\\mathcal{C}}$ on simply colored links yields an algorithm for $F_{\\mathcal{C}^\\zeta}$ with only polynomial overhead, and conversely.","The zest invariant $J_\\zeta(L)$ itself is polynomial-time computable from a link diagram, so the multiplicative correction never introduces super-polynomial work under the paper's assumptions.","For pseudo-unitary categories, zesting-related Reshetikhin-Turaev invariants differ by a shadow rack cocycle invariant, so the complexity of those rack state-sum invariants is bounded by the complexity of the parent theory.","For $A$-manifolds built from $A$-modular categories, the analogous $A$-manifold invariants are also polynomial-time Turing equivalent, extending the result beyond links to a class of $3$-manifold invariants.","Zesting equivalence classes give candidate equivalence classes for complexity-theoretic hierarchies of $(2+1)$-dimensional topological quantum field theories and topological phases."],"supporting_citations":[{"why":"Supplies the original Reshetikhin-Turaev ribbon functor from tangles to the category whose invariants are being compared.","marker":"[RT90]"},{"why":"Defines braided zesting and proves the zested category is ribbon, providing the algebraic relations the paper rewrites in scalar form.","marker":"[Del+20]"},{"why":"Records the earlier factorization of zested link invariants by a factor $J_\\zeta$, which this paper extends from links to tangles.","marker":"[DKP25]"},{"why":"Gives the polynomial-time conversion of link diagrams to braid closures used to evaluate $J_\\zeta$ efficiently.","marker":"[Vog90]"},{"why":"Provides the polynomial-time arithmetic in fixed finite extensions of $\\mathbb{Q}$ that supports the Turing-equivalence conclusion.","marker":"[Len92]"},{"why":"Defines shadow rack cocycle invariants and Boltzmann weights, which are identified with the zest invariant in the pseudo-unitary case.","marker":"[Car+03]"},{"why":"Supplies homotopy quantum field theory and $A$-manifold surgery invariants, the setting for the paper's extension result.","marker":"[Tur10]"}],"fun_headline_variants":["Zesting keeps link invariants equally complex","Zesting: RT invariant complexity unchanged","Zesting doesn't change link invariant complexity","Zesting preserves complexity of RT invariants","Zesting: same complexity for RT link invariants"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The polynomial-time equivalence assumes the zesting data are fixed algebraic numbers in a finite extension of $\\mathbb{Q}$ known before the link diagram is given; if the data arrive as part of the input with unbounded size, or the ground field is not a number field, the proof's complexity bound no longer applies.","fun_headline_variants_meta":{"raw":{"variants":["Zesting keeps link invariants equally complex","Zesting: RT invariant complexity unchanged","Zesting doesn't change link invariant complexity","Zesting preserves complexity of RT invariants","Zesting: same complexity for RT link invariants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001267,"raw_usage":{"total_tokens":5234,"prompt_tokens":1044,"completion_tokens":4190,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":660,"completion_tokens_details":{"reasoning_tokens":4123}},"tokens_in":660,"tokens_out":4190,"duration_ms":27023,"temperature":1.0,"reasoning_tokens":4123,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:00:26.709898+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"One concrete observation that would settle the claim: exhibit two ribbon fusion categories over a number field related by zesting and a family of simply colored link diagrams whose sizes grow polynomially but for which the local formula for $J_\\zeta$ requires super-polynomially many arithmetic operations, or for which $F_{\\mathcal{C}^\\zeta}$ is provably not polynomial-time Turing reducible to $F_{\\mathcal{C}}$; the theorem states that no such pair exists.","supporting_citations":[],"review_version":2}