{"id":"fc13b223-ff99-446a-92d2-407623234f0e","arxiv_id":"1908.06237","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Heegaard Floer invariants of closed 3-manifolds are shown to be natural with respect to diffeomorphisms over the integers, up to an overall sign.","lead":"This paper extends a foundational consistency result for Heegaard Floer homology, a powerful invariant of 3-dimensional spaces, to integer coefficients up to sign. It proves the invariant can be defined naturally for all choices of data, provided maps differing by a sign are treated as the same.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Simple handleswap invariance (Theorem 7.2, axiom 4) depends on Proposition 8.2, whose proof is omitted as 'nearly identical' to 8.1; if the beta-side triangle count is 0 rather than ±1, the composition g∘f∘e is 0, not Id, and Theorem 1.3 is unsupported.","rationale":"The reader's weakest assumption correctly identifies the omission of Proposition 8.2 as the load-bearing gap. I re-read the structure: Theorem 1.3 hangs on the four axioms; axioms 1–2 are imported from JTZ12 (legitimate), axiom 3 is Theorem 7.1 (proved in the text, though with one 'we omit the technical details' forwarded to the proved Lemma 8.13), and axiom 4 is Theorem 7.2, which splits into the α-side (Proposition 8.1, proved in detail) and the β-side (Proposition 8.2, stated without proof). Because the target category is P(Kom(Z[U]-Mod)), a common sign in the composition is harmless, so the decisive question is whether the β-side triangle count is ±1 (not 0) and independent of the matched divisor. The manuscript neither proves this nor reduces it to 8.1 by a formal symmetry; the 'nearly identical' claim is an assertion of support, not support. I also checked whether another gap is more load-bearing: the orientation-transport path-independence in Section 6.5 defers to the distinguished-rectangle and handleswap commutativity (which is exactly what Sections 7–8 must prove), but that deferral lands on the same axiom 4, so it does not displace Proposition 8.2 as the critical point. Credit where due: the paper is careful about sign handling, makes the role of projectivization explicit (Section 4, Remark 4.4), proves the α-side count in detail, and flags the β-side omission honestly. The gap is plausibly repairable by an α↔β symmetry check or a transcription of the 8.1 proof, but until one exists the handleswap axiom — and hence the strong Heegaard invariant property — is unsupported. This leaves the reader's CONDITIONAL verdict unchanged.","tokens_in":55894,"tokens_out":14549,"duration_ms":137871,"concrete_test":"Verify the β-analogs of Lemmas 8.18–8.21 needed in Proposition 8.2: (i) ∂ = 0 on ĈF(Σ0, β0, β′0) with the Lemma 6.2 orientation; (ii) #M_{(b,Θ′,c)}(d) = ±1 independent of generic matched divisors d; (iii) the index-2 strip count #M_{(b,b)}(c) = ±1 in the Step-3 orientation convention and the product formula. Equivalent fast check: exhibit a symmetry of Figures 9 and 10 interchanging α and β, preserving the basepoint divisor condition, and check it identifies the coherent orientation systems, so 8.2 follows from 8.1 with sign absorbed in ±. A count of 0 on the β-side kills the handleswap relation g∘f∘e = Id and with it Theorems 1.3 and 1.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 7 verifies axioms 1–3 of a strong Heegaard invariant and defers axiom 4 (simple handleswap invariance) to Theorem 7.2. In the proof of Theorem 7.2, the α-side computation uses Proposition 8.1, which is proved in detail, while the β-side computation uses Proposition 8.2, stated and then explicitly dropped: 'Since a nearly identical proof can be used to establish Proposition 8.2, we omit the proof of that result.' This is load-bearing, not cosmetic. Proposition 8.2 asserts that the β-side triangle map splits through the model diagram T′0 as F_{T′#T′0}((x×b)⊗(y×Θ′)) = ±F_{T′}(x⊗y)×c. The proof of Theorem 7.2 concludes g∗∘Φf∘Φe = ±Id by multiplying signs from Propositions 8.1, 8.2, and Theorem 7.1; in P(Kom(Z[U]-Mod)) a common sign is absorbed, so sign ambiguities are benign — but only if each factor is an isomorphism. The β-side analog of Lemma 8.21 must give #M_{(b,Θ′,c)}(d) = ±1 for every generic matched divisor d ∈ Sym^k(∆)\\Diag, independent of d; if the signed count is 0 for some divisor (an orientation cancellation on the β-diagram), the map Φf is 0, the composition is 0 ≠ Id, and axiom 4 fails. Nothing in the manuscript rules this out: the claimed 'nearly identical' proof is not a formal α↔β symmetry argument, and the model diagrams T0 and T′0 (Figures 9 and 10) place the basepoint and the special curve pair differently relative to the F/R regions, so the orientation-coherence steps (Lemmas 8.13, 8.19, 8.20, and Step 3 of 8.21) do not transfer automatically. Moreover, even the α-side Step 3 of Lemma 8.21 is a sketch ('As in the proof of stabilization invariance in [Lip06]'). The same gap feeds back into Section 6.5, where path-independence of the induced orientation systems is deferred to 'verifying the commutativity ... in a simple handleswap' — i.e., to the very result needing Proposition 8.2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a projectivized (up-to-sign) naturality statement for the Heegaard Floer invariants of closed, connected, oriented, based 3-manifolds over Z[U]. It introduces the projectivized homotopy category P(Kom(Z[U]-Mod)) and the category Trans(P(Kom(Z[U]-Mod))) of transitive systems, and states that the morphisms \\hat{CF}, CF^-, CF^+, CF^\\infty : Gman -> Trans(P(Kom(Z[U]-Mod))) are strong Heegaard invariants (Theorem 1.3). From this it derives functors HF^\\circ : Man_* -> Trans(P(Z[U]-Mod)) whose values are isomorphic to the Ozsvath-Szabo modules (Theorem 1.1). The proof follows the structure of JTZ12, concentrates on the new integral simple-handleswap computation, and includes an application to involutive Heegaard Floer homology over Z. The paper is honest about the fact that the beta-side triangle count (Proposition 8.2) is not proved, and that Lemma 8.21's sign count is only sketched.","tokens_in":56358,"tokens_out":8514,"duration_ms":83817,"significance":"If the central claim is correct, the paper fills a recognized gap: it provides naturality over Z up to sign for all Heegaard Floer variants, on the chain level, and establishes groundwork for integral cobordism naturality and mixed invariants over Z/±. The projectivized transitive-system formalism is a sensible and carefully described framework for the result, and the verification of axioms 1–3 in Section 7 is substantive. The alpha-side triangle count in Proposition 8.1 is proved in detail, and the dependence on coherent orientation systems is discussed explicitly. However, the main new ingredient—simple handleswap invariance—is not fully supported as written: Proposition 8.2 is stated without proof, and the crucial nonzero sign count in Lemma 8.21 rests on a sketch. Since the target category is projectivized, sign ambiguities are benign only if the relevant counts are ±1 rather than 0; exactly this is left unproved for the beta-side.","major_comments":[{"comment":"Proposition 8.2 is load-bearing for Theorem 7.2 but its proof is omitted, with the sentence 'Since a nearly identical proof can be used to establish Proposition 8.2, we omit the proof of that result.' In the proof of Theorem 7.2, the beta-side computation of Φf uses Proposition 8.2 to replace F_{T'#T'_0}((x×b)⊗(y×Θ')) by ±F_{T'}(x⊗y)×c. If the beta-side count were 0 for some divisor rather than ±1, the composition g*∘Φf∘Φe would be 0, not Id, and Theorem 7.2 would fail. The 'nearly identical' claim is not a formal α↔β symmetry: the model diagrams T0 and T'0 (Figures 9 and 10) differ in the placement of the basepoint and in the arrangement of the special curve pair relative to the F/R regions, so the orientation-coherence arguments used on the alpha side—Lemmas 8.13, 8.19, 8.20, and Step 3 of Lemma 8.21—do not automatically transfer. The proof of Proposition 8.2, or at least a precise symmetry reduction to Proposition 8.1, must be supplied.","section":"Section 8, Proposition 8.2"},{"comment":"Lemma 8.21 is the quantitative heart of Proposition 8.1, but its proof is only a sketch. Step 2 asserts independence of the signed count from the divisor d using Lemma 8.13, and Step 3 finds one divisor with the desired count via a stabilization argument involving the twice-stabilized bigon and an appeal to [OS04b, Lemma 8.7] to extend the orientation system. The text does not give the sign bookkeeping for the extension, nor does it prove that the resulting coherent orientation system yields the same sign for all generic d and for the beta-side configuration needed in Proposition 8.2. Since Lemma 8.21 must produce #M_{(Θ,a,b)}(d) = ±1 for every generic d, and since a zero count would destroy the handleswap relation, this lemma needs a complete proof rather than a sketch. In particular, the orientation system on the matched moduli spaces and the sign in Step 3 must be made explicit and shown to be compatible with the path-independence argument in Step 2.","section":"Section 8.3, Lemma 8.21"},{"comment":"In Section 6.5, after defining CF^-(H,s) for a fixed isotopy diagram H, the paper asserts that the coherent orientation system induced on a diagram H' is independent of the path γ chosen from H to H', citing [JTZ12, Proof of Theorem 2.38 and Remark 2.39] and a verification in the five types of distinguished rectangle and in a simple handleswap. The simple handleswap verification is precisely the content that is deferred to Section 8 and Theorem 7.2. As written, this makes the definition of CF^-(H') for vertices of Gman appear to presuppose the theorem it is meant to help prove. The author should either move the definition of CF^-(H') to after the handleswap proof, or state explicitly that the orientation-level independence is a weaker statement proved independently of the map-level handleswap invariance.","section":"Section 6.5"}],"minor_comments":[{"comment":"The notation for T'_0 is inconsistent: the line 'β'_0={β_1,β_2} and β'_0={β'_1,β'_2}' should read 'β_0={β_1,β_2} and β'_0={β'_1,β'_2}'.","section":"Section 8, before Proposition 8.2"},{"comment":"There are several typos, including 'orienation' in Theorem 7.1 and 'stablization' in Section 6.6; these should be corrected.","section":"Throughout"},{"comment":"The proof of Theorem 1.7 is only a sketch and relies on a choice among 'at least one' of the two maps in {±ι'}; the text should clarify in what sense the resulting unordered pair is independent of that choice, since the application to involutive Heegaard Floer homology is a stated motivation.","section":"Theorem 1.7"}],"recommendation":"major_revision","confidential_remarks":"The central technical risk is the omitted proof of Proposition 8.2 and the sketched sign count in Lemma 8.21. If the author can supply complete proofs for these, the paper is likely acceptable; as it stands, the main theorem is not fully supported. I would not recommend rejection on grounds of correctness, but the load-bearing nature of the omission makes major revision appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper fills a real and known gap: extending JTZ12 naturality from F2 to Z, up to sign, for all standard Heegaard Floer variants. The move to transitive systems in the projectivized homotopy category is a sensible way to phrase the result, and the categorical framework in Sections 4–5 is clearly laid out. The paper also proves most of the structural axioms in Sections 6–7, and Theorem 7.1 is given a real proof. Section 8 contains a detailed proof of Proposition 8.1, and Lemma 8.21's step 2 is handled carefully. This is not a paper that handwaves the whole project; it does serious work and is honest about what it imports from JTZ12.\n\nThe soft spot is exactly where the reader's stress-test lands. Proposition 8.2 is load-bearing for the handleswap axiom: the proof of Theorem 7.2 multiplies signs from Propositions 8.1 and 8.2 to conclude g∘f∘e = ±Id. But Proposition 8.2 is stated and then dropped with the note that the proof is nearly identical to Proposition 8.1. The beta-side diagram T0′ is not symmetric to the alpha-side diagram T0: the basepoint and the special curve pair sit differently relative to the F/R regions, so the orientation-coherence arguments do not automatically transfer. If the beta-side triangle count were 0 for some matched divisor, the composition would be 0, not Id, and Theorem 1.3 would fail. The manuscript does not rule that out. This is not a cosmetic omission.\n\nThere is a second, smaller gap on the alpha side: Step 3 of Lemma 8.21, which fixes the nonzero signed count, is a sketch that defers to stabilization invariance in Lip06. And Section 6.5 explicitly defers path-independence of the induced orientation systems to verification in a simple handleswap, i.e. to the very theorem that depends on Proposition 8.2. So the missing proof has a downstream effect on the coherence of the orientation systems themselves.\n\nI want to be fair: I see no circularity, no parameter fitting, and no invented entities. The overall strategy mirrors JTZ12 and is plausible. The gaps are specific and likely fixable, but they are real. This paper deserves a serious referee and would likely be accepted after a major revision that supplies the proof of Proposition 8.2 and fleshes out the stabilization argument in Lemma 8.21. I would send it to review and, as a referee, make acceptance conditional on those additions.","headline":"A credible candidate for integral projectivized naturality in Heegaard Floer homology, but the load-bearing beta-side triangle count is explicitly unproved, so it deserves refereeing rather than acceptance as is.","tokens_in":56945,"tokens_out":1695,"would_cite":false,"duration_ms":21750,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57R58"],"pacs":[],"model":"deepseek-v4-flash","headline":"Heegaard Floer invariants are natural over Z up to sign","keywords":["Heegaard Floer homology","naturality","projectivization","transitive systems","strong Heegaard invariants","handleswap invariance","integral coefficients","involutive Heegaard Floer homology"],"falsifier":"On the model diagram $T_0$, choose a divisor $d\\in \\mathrm{Sym}^k(\\Delta)$ away from the fat diagonal and compute the signed count $\\#\\mathcal{M}_{\\Theta,a,b}(d)$ with the coherent orientation system constructed in Section 8.3. If the result is $0$ for any such $d$, the projective handleswap identity $g\\circ f\\circ e=\\mathrm{Id}$ fails, disproving Theorem 7.2. Independently, writing out the omitted proof of Proposition 8.2 and checking that $F_{T'\\#T'_0}((x\\times b)\\otimes(y\\times \\Theta'))=\\pm F_{T'}(x\\otimes y)\\times c$ for all $x,y$ would settle the remaining sign question.","tokens_in":55705,"feed_emoji":"🔁","tokens_out":11973,"duration_ms":103288,"temperature":0.7,"pith_summary":"This paper proves that the Heegaard Floer invariants of closed, based 3-manifolds are natural with respect to diffeomorphisms even with integer coefficients, provided one identifies every map with its negative. Earlier naturality results worked over $\\mathbb{F}_2$, where a sign is invisible; this paper removes that restriction by passing to a projectivized category in which $f$ and $-f$ are declared equal. Concretely, Theorem 1.1 produces functors from the category of based 3-manifolds to transitive systems in $P(\\mathbb{Z}[U]\\text{-Mod})$, whose values are isomorphic to the original Heegaard Floer modules, and isotopic diffeomorphisms act as the identity. The point is that all geometric and analytic choices can be made consistently up to an overall sign, which is exactly the ambiguity left over when working over $\\mathbb{Z}$ instead of $\\mathbb{F}_2$.","feed_headline":"Heegaard Floer invariants are natural over Z up to sign","feed_subtitle":"A projective quotient absorbs the sign ambiguity, making based 3-manifold invariants functorial to transitive systems.","key_machinery":"The projectivization $P(\\mathcal{C})$ of an additive category $\\mathcal{C}$ is the quotient that identifies every morphism $f$ with $-f$; the paper works with $P(\\mathbb{Z}[U]\\text{-Mod})$ and with the projectivized homotopy category of $\\mathbb{Z}[U]$-module chain complexes. A transitive system is a directed-indexed family of objects and isomorphisms satisfying $f_{i,k}=f_{j,k}\\circ f_{i,j}$, and $\\mathrm{Trans}(\\mathcal{C})$ is the resulting category. On the geometric side, the graph $G_{\\mathrm{man}}$ has vertices given by pointed isotopy Heegaard diagrams and edges for strong $\\alpha$-equivalences, strong $\\beta$-equivalences, stabilizations, and diffeomorphisms. A strong Heegaard invariant is a weak one that also satisfies functoriality, commutativity for distinguished rectangles, continuity for isotopies, and simple handleswap invariance; the transitivity theorem for strong Heegaard invariants then turns such data into a genuine invariant. The load-bearing new check is the simple handleswap triangle count on the model diagram $T_0$, carried out through matched moduli spaces and coherent orientation systems, with the key identity being the signed count $\\#\\mathcal{M}_{\\Theta,a,b}(d)=\\pm 1$ independent of the divisor $d$.","core_discovery":"The central assertion is that the four chain-complex invariants $\\widehat{CF}$, $CF^-$, $CF^+$, and $CF^\\infty$ are strong Heegaard invariants valued in transitive systems in the projectivized homotopy category $P(\\mathrm{Kom}(\\mathbb{Z}[U]\\text{-Mod}))$. On homology this yields strong invariants into $P(\\mathbb{Z}[U]\\text{-Mod})$, and the abstract machinery of strong Heegaard invariants converts them into functors $\\mathrm{Man}_* \\to \\mathrm{Trans}(P(\\mathbb{Z}[U]\\text{-Mod}))$ whose values on a based 3-manifold agree with the original modules up to isomorphism, with isotopic diffeomorphisms acting trivially. The main new work is the verification of the last required axiom, simple handleswap invariance, by counting holomorphic triangles on a fixed genus-two model diagram; the decisive signed count is $\\pm 1$, which is all one needs after projectivization.","pith_inferences":["The same projectivization device should apply to any Floer invariant whose only obstruction is handleswap-type monodromy and whose orientation systems are canonical up to sign, not just to Heegaard Floer homology.","The unproved mirror-side triangle count (Proposition 8.2) is the most likely place for a hidden sign inconsistency; an independent proof would directly test the construction of the integral involutive invariant.","If the transitive systems here are homotopy coherent, a homotopy colimit could replace the transitive-system packaging by one projectivized chain complex per 3-manifold, simplifying future computations.","A concrete testable extension is to compute the pair $\\{CFI^\\pm\\}$ for a lens space or small Seifert-fibered space and compare it with the known $\\mathbb{F}_2$ involutive invariant, since torsion differences would show where the sign ambiguity matters."],"forward_implications":["Each flavor of Heegaard Floer homology becomes a well-defined functor on based 3-manifolds with values in $\\mathrm{Trans}(P(\\mathbb{Z}[U]\\text{-Mod}))$, matching the classical invariants up to sign.","Isotopic diffeomorphisms act as the identity, so mapping class group actions descend to the projectivized category without extra choices.","The chain complexes themselves fit into transitive systems in the projectivized homotopy category, so different Heegaard diagrams for the same manifold are related by homotopy equivalences that are coherent up to sign.","An integral version of involutive Heegaard Floer homology is obtained: the unordered pair of cone complexes $CFI^\\pm$ is a diffeomorphism invariant of the based 3-manifold.","The projectivized framework is positioned to upgrade naturality of cobordism maps and the mixed invariants of 4-manifolds from $\\mathbb{F}_2$ coefficients to $\\mathbb{Z}/\\pm$ coefficients."],"supporting_citations":[{"why":"Supplies the graph $G_{\\mathrm{man}}$, the notions of weak and strong Heegaard invariants, distinguished rectangles, simple handleswaps, and the transitivity theorem that the paper imports wholesale.","marker":"[JTZ12]"},{"why":"Defines the Heegaard Floer chain complexes and establishes that they are weak invariants over $\\mathbb{Z}$ and $\\mathbb{F}_2$; these are the objects whose projectivized naturality is proved.","marker":"[OS04b]"},{"why":"Constructs triangle maps, continuation maps, stabilization maps, and the change-of-complex-structure transitive systems used to define the weak invariants.","marker":"[OS06]"},{"why":"Provides the cylindrical formulation, orientability of moduli spaces, coherent orientation systems, and gluing results for holomorphic triangles on which Section 8 relies.","marker":"[Lip06]"},{"why":"Constructs canonical coherent orientation systems; the paper states that its results hold in particular for these systems.","marker":"[OS04a]"},{"why":"Defines involutive Heegaard Floer homology over $\\mathbb{F}_2$, whose cone construction is adapted to produce the integral invariant of Theorem 1.7.","marker":"[HM17]"},{"why":"Identified simple handleswap moves as candidate loops with monodromy, motivating the axiom whose projective verification is the main new result.","marker":"[Sar15]"},{"why":"Supplies the Maslov index formula for Whitney n-gons used to compute indices of holomorphic triangles in the triangle-count proof.","marker":"[Sar11]"}],"fun_headline_variants":["Heegaard Floer natural now holds up to sign only","Projective quotient makes Heegaard Floer functorial","Sign ambiguity resolved: Heegaard Floer is projective-natural","Handleswap count ±1 proves projective naturality"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on two triangle counts on a small model diagram: one is stated without proof, and the other must come out exactly plus or minus one for every divisor; if either count ever came out zero, the handleswap relation would fail and the whole projectivized naturality would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Heegaard Floer natural now holds up to sign only","Projective quotient makes Heegaard Floer functorial","Sign ambiguity resolved: Heegaard Floer is projective-natural","Handleswap count ±1 proves projective naturality"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000462,"raw_usage":{"total_tokens":2360,"prompt_tokens":1042,"completion_tokens":1318,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":658,"completion_tokens_details":{"reasoning_tokens":1248}},"tokens_in":658,"tokens_out":1318,"duration_ms":10356,"temperature":1.0,"reasoning_tokens":1248,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:51:50.216482+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the model diagram $T_0$, choose a divisor $d\\in \\mathrm{Sym}^k(\\Delta)$ away from the fat diagonal and compute the signed count $\\#\\mathcal{M}_{\\Theta,a,b}(d)$ with the coherent orientation system constructed in Section 8.3. If the result is $0$ for any such $d$, the projective handleswap identity $g\\circ f\\circ e=\\mathrm{Id}$ fails, disproving Theorem 7.2. Independently, writing out the omitted proof of Proposition 8.2 and checking that $F_{T'\\#T'_0}((x\\times b)\\otimes(y\\times \\Theta'))=\\pm F_{T'}(x\\otimes y)\\times c$ for all $x,y$ would settle the remaining sign question.","supporting_citations":[],"review_version":1}