{"id":"2de3d2c4-0469-4e86-9e81-7c38f4008cd2","arxiv_id":"2608.00256","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Real Heegaard Floer homology becomes a natural functor on based real 3-manifolds, with an equivariant mapping class group action and a new involutive variant.","lead":"Real Heegaard Floer homology — an invariant of three-manifolds equipped with an involution — is shown to be natural: it upgrades from an isomorphism-class invariant to a functor, and the equivariant mapping class group acts on it. The proof introduces a new singularity, a new Heegaard move, and a new 'simple trade' loop, and yields a new involutive real Heegaard Floer invariant.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 2.25's simple trade pentagon has incompatible surface genera: the stated arrow h: H4→H5 cannot exist, so Axiom (5) is ill-posed as written.","rationale":"The reader's weakest assumption was the simple trade invariance computation in Section 8. My concern is different but related: the simple trade loop itself is not well-defined as stated in Definition 2.25. The genus mismatch is checkable directly from the manuscript and, unless it is a typo, it invalidates the statement of Axiom (5) and the resolution in Section 6.6. This does not discredit the overall strategy—the intended definition is almost certainly H5=Σ#Σ0#Σ1#Σ2—but it reinforces the conditional verdict: the text must be corrected and the trade computation independently checked. I therefore keep the reader's CONDITIONAL verdict unchanged rather than escalating, since the issue is concrete but likely fixable and does not by itself show the mathematics is wrong.","tokens_in":65121,"tokens_out":9236,"duration_ms":83633,"concrete_test":"Analytical check: compute the genus of each vertex in Definition 2.25 from the stated connected sums. If g(H4)≠g(H5), the arrow h is impossible, confirming the definition needs correction; the likely fix is H5=Σ#Σ0#Σ1#Σ2. Then verify that the five arrows e,f,g,h,i close as a pentagon in G_R with the corrected H5, and check that the Section 8 verification of Axiom (5) explicitly covers this corrected pentagon. If Section 8 is unavailable, mark Axiom (5) as unverified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 2.25(1) assigns H1=(Σ#Σ0), H2=(Σ#Σ0#Σ1), H3=H4=(Σ#Σ0#Σ1#Σ2), and H5=(Σ#Σ0), with each Σ_i a genus-1 surface. Item (2) declares h: H4→H5 to be an equivariant diffeomorphism. But g(H4)=g(Σ)+3 while g(H5)=g(Σ)+1, so no equivariant diffeomorphism of surfaces exists between them. For the pentagon to close—two {1}-stabilizations raise genus by 2 and one Z/2-destabilization lowers it by 2—H5 must have genus g(Σ)+3, i.e. H5 should be Σ#Σ0#Σ1#Σ2. The printed 'H5=(Σ#Σ0)' therefore appears to be a typo. This is not cosmetic: Axiom 2.26(5) (Trade invariance) is stated for every simple trade pentagon, and Section 6.6 resolves the A4({1}) bifurcation into \"exactly a simple trade loop.\" If the pentagon as defined is empty, the axiom is vacuous; if the correction is silently made, the theorem as written refers to a different object than the one the proof addresses. The reader's conditional verdict rightly flags the trade computation in Section 8; this definitional gap lies upstream of that computation and must be resolved before the computation can be assessed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops naturality for real Heegaard Floer homology. It sets up a graph of real Heegaard diagrams with moves including real handleslides, {1}- and Z/2-stabilizations, a new crossover move, and a new \"simple trade\" loop. It defines weak and strong real Heegaard invariants and proves (Theorem 8) that real sutured Floer homology, real Heegaard Floer homology, and real link Floer homology are strong invariants. Theorems 9–10 convert strong invariance into transitive systems and canonical maps, yielding Theorem 1: functors HFR^◦ : RMan_* → F[U]-Mod, independent of the choice of Heegaard diagram, with isotopic diffeomorphisms inducing identical maps, hence an equivariant mapping class group action. The strategy follows JTZ21: singularity classification in Sections 3–4, diagram construction and loop decomposition in Sections 5–6, the metatheorem in Section 7, and verification in Section 8. The paper also defines involutive real Heegaard Floer homology using naturality of the conjugation involution.","tokens_in":65389,"tokens_out":9607,"duration_ms":88371,"significance":"If correct, the paper is a significant step: it upgrades real Heegaard Floer homology from an isomorphism-class invariant to a natural functor with equivariant mapping class group actions, a prerequisite for equivariant cobordism maps. The paper is well organized and careful about isolating what is new: the only axiom with no unreal counterpart is Trade invariance, Axiom (5), and the authors identify the A4({1}) singularity as its source. The singularity classification and the diagram calculus are presented in detail, and the proof strategy is transparent and falsifiable. The main weakness is that the new simple trade loop is currently not well-defined as printed because the genus data are inconsistent; since Theorem 9 and Section 6.6 rely on this loop being a nonempty, concrete object, the definitional issue must be fixed before the central claim can be accepted.","major_comments":[{"comment":"The genera assigned to the pentagon vertices are incompatible with the declared edge h. Definition 2.25(1) gives H1 = Σ#Σ0 (genus g+1), H2 = Σ#Σ0#Σ1 (genus g+2), H3 = H4 = Σ#Σ0#Σ1#Σ2 (genus g+3), and H5 = Σ#Σ0 (genus g+1). Item (2) then declares h: H4 → H5 to be an equivariant diffeomorphism, but no equivariant diffeomorphism exists between surfaces of different genera. Moreover, the stated combination of two {1}-stabilizations and one Z/2-destabilization does not close the genus count as printed. If H5 is meant to be Σ#Σ0#Σ1#Σ2, the definition should say so explicitly; if the edge assignments or the genus effect of Z/2-stabilization are different, that convention must be stated. As written, Axiom 2.26(5) is either vacuous or refers to a pentagon that is not the one analyzed in Section 6.6. This is load-bearing: Theorem 9 and the resolution of D2 in Section 6.6 depend on the simple trade","section":"§2.4, Definition 2.25 and Axiom 2.26(5)"},{"comment":"The proof of Trade invariance — the only axiom without an unreal counterpart — is described as \"requiring analyzing specific holomorphic curves\" analogous to JTZ21's handleswap verification, but the actual moduli/index computation is not carried out in the text provided for review. Since Axiom 2.26(5) is the single new ingredient that upgrades the weak real invariant to a strong one, the referee needs to see either the full holomorphic-curve count or a precise citation to a place where it is performed. In addition, Section 6.6 states that the six remaining strata around the A4({1}) singularity are glued \"at a point, a loop around which is exactly a simple trade loop,\" but the edge-by-edge identification with Definition 2.25 is not shown. Once the genus mismatch is corrected, this identification should be written out explicitly.","section":"§6.6 and §8"}],"minor_comments":[{"comment":"The definition of a Z/2-stabilization is ambiguous. It says there is a disk D ⊂ Σ1 and a punctured torus T ⊂ Σ2 with Σ1 \\ D = Σ2 \\ T, while also requiring D ∩ τ(D) = ∅ and T ∩ τ(T) = ∅. Since τ(T) is then contained in Σ2 \\ T, the equation Σ1 \\ D = Σ2 \\ T seems inconsistent unless T denotes an orbit of two punctured tori. Please clarify the notation and state explicitly how the genus changes.","section":"§2.3, Definition 2.14"},{"comment":"Remark 2.24 refers to \"Lemma 2.23,\" but the object is Definition 2.23. The cross-reference should be corrected.","section":"§2.4, Remark 2.24"},{"comment":"The proof contains hard-to-parse expressions such as \"Fδ(H0)→H0\" and \"Fδ(H0) ◦ F(δH0)\". The arrows and compositions should be rewritten with explicit homotopy classes and source/target objects so the reader can follow the reduction to a single diagram H0.","section":"§2.6, Proof of Theorem 11"},{"comment":"The assertion that \"the space of conformal structures on the rectangle which respect the real structure is zero-dimensional, as the symmetry forces the cross-ratio to be one\" deserves a proof or a reference. The real locus of the moduli space of four-pointed spheres is not automatically a single point, and this dimension statement is used in the claim that real rectangle counting maps are chain maps.","section":"§8.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is potentially very strong: it isolates the genuinely real phenomenon (the simple trade) and structures the proof so that everything else reduces to JTZ21. The definitional issue in Definition 2.25 is almost certainly fixable, but it is not cosmetic — it sits directly under the main theorem. If the authors correct the pentagon definition, spell out the A4({1}) resolution, and provide the promised Section 8 computation, the paper should be acceptable. The scope is appropriate for a top journal in low-dimensional topology."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this paper is the real naturality result for real Heegaard Floer homology, not a routine translation of JTZ21. The new ingredients are real: the A4({1}) fixed-set singularity, the crossover move, and the simple trade loop. The authors also come clean about AI-drafted lemmas and about what is imported from their own prior work. That's honest and useful.\n\nThe soft spot, and it's a real one, is Definition 2.25. The simple trade pentagon as printed has H5 = (Σ#Σ0), while H4 = (Σ#Σ0#Σ1#Σ2). Since h is declared to be an equivariant diffeomorphism H4→H5, no such map exists: the genera differ by 2. So the pentagon is empty as stated, and Axiom 2.26(5) is vacuous for it. The intended fix is obvious from the geometry—H5 should also have genus g(Σ)+3, i.e. Σ#Σ0#Σ1#Σ2—and Section 6.6's resolution of the A4 bifurcation into exactly this loop confirms that reading. But this is not a cosmetic issue: the trade axiom is the one axiom with no unreal counterpart, and the paper's main theorem rests on it. The authors need to correct the definition explicitly and make sure the proof in Section 8 is checking the corrected pentagon, not a different object.\n\nThe other caveats are milder. Section 8, where the trade computation lives, is cut off in the material I saw, so I cannot vouch for it; that computation needs independent checking. Theorem 9's proof is explicitly a sketch deferring to [JTZ21]; that's reasonable for a paper following such a close template, but a referee should ask for enough detail to confirm the real-setting modifications. The AI-drafted classification lemmas deserve a look, but the disclosure itself is a point in the authors' favor.\n\nOverall verdict: the architecture is sound, the new geometric content is genuinely new, and the flaws are local. Send it to peer review. This is exactly the kind of paper that needs a serious referee who will check the trade computation and the corrected definition, not a desk reject.","headline":"Naturality in real Heegaard Floer theory is a serious, substantial paper, but Definition 2.25 has a genus typo that makes the trade pentagon empty as printed; the intended fix is clear and the paper deserves a careful referee.","tokens_in":66006,"tokens_out":2482,"would_cite":true,"duration_ms":24523,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K18"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves real Heegaard Floer homology is natural: diagram-move paths induce canonical maps, making HFR^o a functor with an equivariant mapping class group action.","keywords":["real Heegaard Floer homology","naturality","equivariant mapping class group","strong real Heegaard invariant","A4 singularity","involutive Heegaard Floer homology","real sutured Floer homology","transitive systems"],"falsifier":"Compute the mod-2 count of the holomorphic curves contributing to the simple trade on a small explicit diagram, such as the pentagon of Figure 2.4 on genus-one summands. If the resulting map on CFR is not the identity, Axiom (5) fails and Theorem 8 and Theorem 1 collapse. A complementary check: trace the same A4({1}) loop around the two resolutions of Section 6.6 and compare the induced maps.","tokens_in":64831,"feed_emoji":"🔄","tokens_out":5895,"duration_ms":59230,"temperature":0.7,"pith_summary":"The paper upgrades real Heegaard Floer homology from an invariant defined only up to isomorphism to a natural one. For each flavor, HFR^o is shown to be a functor from based real 3-manifolds to F[U]-modules, with isotopic equivariant diffeomorphisms acting identically. The same argument makes real link Floer and real sutured Floer natural, and yields an involutive version of the invariant. A sympathetic reader should care because naturality is exactly the ingredient needed to move from diagram-by-diagram computations to functorial cobordism maps, and because it gives well-defined equivariant mapping class group actions.","feed_headline":"Real Heegaard Floer homology is natural","feed_subtitle":"Diagram choices no longer matter; equivariant mapping class groups act on the invariant.","key_machinery":"The carrying object is the graph of real isotopy diagrams with edges given by real equivalences, {1}- and Z/2-stabilizations, and equivariant diffeomorphisms. A strong real Heegaard invariant is a weak invariant obeying five axioms; the new axiom is trade invariance for the simple trade loop (Definition 2.25), a pentagon canceling two {1}-stabilizations against one Z/2-destabilization. The paper derives this loop from the codimension-2 A4({1}) singularity—a fixed-point singularity with normal form x1^2 - x2^2 + x3^5 in a 2-parameter family—classified via an equivariant splitting lemma, and verifies the axiom by counting rigid real-invariant holomorphic rectangles in a cylindrical reformulati","core_discovery":"The central result, Theorem 8, states that real sutured Floer homology, real Heegaard Floer homology, and real link Floer homology are strong real Heegaard invariants: they satisfy functoriality, commutativity, continuity, handleswap invariance, and the new trade invariance. Theorem 1 then states that HFR^o is a functor RMan_* to F[U]-Mod whose value is isomorphic to the groups of [GM], with isotopic diffeomorphisms inducing identical maps. In effect, any sequence of real Heegaard moves connecting two diagrams of the same manifold gives the same canonical isomorphism; the paper proves this by showing that the associated graph has no monodromy and assembling the groups into a transitive syste","pith_inferences":["One consequence the authors leave implicit: if the trade count holds, naturality transfers to any construction built functorially from the real Floer complex, so future equivariant cobordism maps will automatically be independent of decompositions.","The mapping class group action may let real link Floer homology distinguish symmetries of strongly invertible links that ordinary link Floer ignores; this is testable on simple examples by comparing U-action and grading shifts.","A concrete independent check: on a small genus-one model diagram with three handles, compute the Floer map around the simple trade pentagon; the identity result is a finite mod-2 count.","The involutive real invariant HFRI^o could provide an equivariant homology-cobordism obstruction, an extension the paper frames but does not develop."],"forward_implications":["HFR^o carries a well-defined action of the based equivariant mapping class group.","The conjugation involution is defined up to chain homotopy, yielding involutive real Heegaard Floer homology HFRI^o as an invariant.","Real link and real sutured Floer homology inherit naturality, giving mapping class group actions on those invariants.","The canonical isomorphisms allow different Heegaard diagram computations to be glued into one transitive system, so the invariant is path-independent.","This is the first step toward functoriality of real Heegaard Floer homology under equivariant cobordisms."],"fun_headline_variants":["Real Heegaard Floer homology is now provably natural","Equivariant mapping class group acts on real Heegaard Floer","Diagram choice irrelevant for real Heegaard Floer homology","Naturality established for real Heegaard, link, and sutured Floer","Involutive real Heegaard Floer homology defined"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the simple trade loop—the pentagon of moves in Definition 2.25, whose proof is checked in Section 8 by counting holomorphic curves—induces the identity map on real Heegaard Floer homology; if that count were wrong, the transitive system of Section 2.5 would fail and Theorem 1 would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Real Heegaard Floer homology is now provably natural","Equivariant mapping class group acts on real Heegaard Floer","Diagram choice irrelevant for real Heegaard Floer homology","Naturality established for real Heegaard, link, and sutured Floer","Involutive real Heegaard Floer homology defined"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000503,"raw_usage":{"total_tokens":2218,"prompt_tokens":592,"completion_tokens":1626,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":336,"completion_tokens_details":{"reasoning_tokens":1536}},"tokens_in":336,"tokens_out":1626,"duration_ms":11651,"temperature":1.0,"reasoning_tokens":1536,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T00:53:38.425661+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the mod-2 count of the holomorphic curves contributing to the simple trade on a small explicit diagram, such as the pentagon of Figure 2.4 on genus-one summands. If the resulting map on CFR is not the identity, Axiom (5) fails and Theorem 8 and Theorem 1 collapse. A complementary check: trace the same A4({1}) loop around the two resolutions of Section 6.6 and compare the induced maps.","supporting_citations":[],"review_version":1}