{"id":"a6de9e20-349b-4500-a0f9-36b264f09ab3","arxiv_id":"2509.04172","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"This paper builds Welschinger-Witt invariants and proves they match quadratic Gromov-Witten invariants for k-rational del Pezzo surfaces of degree at least 6, conjecturing agreement in general.","lead":"Researchers define new 'Welschinger-Witt invariants' that package signed counts of real rational curves into algebraic objects over arbitrary fields. They conjecture these equal quadratic Gromov-Witten invariants, which would let all such counts of rational surfaces be computed from real-geometric data, and they prove the conjecture for del Pezzo surfaces of degree at least 6.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 6.7's removal of the large-characteristic assumption is load-bearing; without a verified floor-diagram identity for all fields, Corollary 6.8 and Theorem 6.1 lack proof.","rationale":"The reader's weakest-assumption analysis identifies Theorem 6.7 as the main load-bearing point, and I agree. The construction of V W in Section 4 is self-contained and convincing, and the abstract Witt-invariant framework is solid. The proof of the main theorem, however, depends on two external inputs: the quadratic Abramovich–Bertram formula from [BW25] and the floor-diagram computation from [JPMPR25]. Of these, Theorem 6.7 is the more delicate because it explicitly claims to remove a large-characteristic hypothesis from the external result. The paper's own proof of this removal is a two-step argument: match in characteristic 0, then use unramifiedness away from {2,3}. This is a standard spreading principle, but it requires the unramifiedness of both sides to be checked in the precise functorial sense, and it requires the characteristic-0 comparison to be valid for all relevant fields, not merely for Q-valued points. The text does not carry out these checks in detail. This is not an accusation of error; the cited preprints may well be correct. But because Corollary 6.8 is the only bridge from multireal values to β-integrality, and β-integrality is the only mechanism that upgrades the known agreement over R to an equality over all fields, the central theorem has a genuine soft spot here. The recommended remedy is an explicit verification in a small positive-characteristic case. I do not think this concern changes the reader's verdict: CONDITIONAL remains appropriate, pending verification of the external inputs or a fully detailed proof of Theorem 6.7.","tokens_in":50390,"tokens_out":17297,"duration_ms":170929,"concrete_test":"Compute both sides of Theorem 6.7 for d=(4,2,0,0), s=1 over F_5: evaluate Q'^{(m-1)}_d(δ) for δ∈Sq_1(F_5) using the definition in §5.1 (or the enumerative formula [KLSW23a, Thm 3]) and compare with the sum over essential 1-marked floor diagrams of class (4,2,0,0) as in Definition 6.6. If the two Witt classes in W(F_5) differ for any δ, the large-characteristic removal is invalid and Theorem 6.1 is unproven. If they agree, re-run the same check for d=(5,2,1,1), s=2 to include nontrivial twin trees.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 6.1 is proved by reducing β-integrality of Q_{n,d} to the toric case, where Theorem 6.7 asserts that Q'^{(m-s)}_d equals the sum of quadratic multiplicities of essential s-marked floor diagrams. This is a generalization of [JPMPR25, Thm 10.13] with the 'sufficiently large characteristic' assumption removed. The removal is argued by saying both sides are unramified away from S={2,3} and that equality is known in characteristic 0. That strategy is valid only if: (1) the left side is unramified in the precise sense of Definition 2.15 — for the toric blow-up X_{k^3} this is supposed to follow from Theorem 5.7, but the required smooth proper model over Z[1/6] and the verification of Hypothesis 5.1 for D are not spelled out; (2) the right-hand sum is an S-integral Witt invariant on Sq_s, which is plausible because it is a Z-polynomial in the trace forms t_j, but it is not demonstrated; and (3) the characteristic-0 comparison from [JPMPR25] covers all fields K containing Q, not just Q, so that specialization to F_p via a complete DVR is legitimate. If any of these three conditions fails, the equality over F_p — and hence the β-integrality of Q' (Corollary 6.8) — is unsupported. Since Corollary 6.8 is the only input that upgrades the multireal agreement over R to an equality over all fields, Theorem 6.1 would not be established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a conjectural relationship between Welschinger invariants, which are signed counts of real rational curves, and quadratic Gromov–Witten invariants, which are counts taking values in Witt–Grothendieck rings over general fields of characteristic not 2 or 3. The authors construct multivariable unramified Witt invariants from Welschinger invariants (Theorem 4.3), prove that quadratic Gromov–Witten invariants are Witt invariants and are unramified under appropriate hypotheses (Theorems 5.4 and 5.7), and formulate Conjecture 5.14 asserting equality of these two types of invariants for rational del Pezzo surfaces. The paper proves this conjecture for del Pezzo surfaces of degree at least 6 (Theorem 6.1), equivalent to blowing up at most three points, by reducing first to toric surfaces via a quadratic Abramovich–Bertram formula and then using a floor-diagram computation. The central theorem, Theorem 6.1, is thus built on the partially proven Theorem 6.7, which removes the large-characteristic assumption from the floor-diagram identity of [JPMPR25].","tokens_in":50738,"tokens_out":8842,"duration_ms":83851,"significance":"If the main theorem and conjecture hold, the paper provides a striking structural link: all quadratic Gromov–Witten invariants of k-rational surfaces would be determined by Welschinger invariants, packaged as β-integral Witt invariants. The construction of multivariable Witt invariants from Welschinger invariants, the multireal-triangle calculus, and the explicit tables of β- and λ-coefficients are valuable contributions in their own right. The paper is ambitious and introduces new tools that are likely to be influential. However, the central claim is conditional on the unproven details of Theorem 6.7 and on the availability and correctness of several unpublished preprints. The paper is not fully self-contained, and the proof of the key floor-diagram identity is only sketched.","major_comments":[{"comment":"Theorem 6.7 is load-bearing: Corollary 6.8 and therefore Theorem 6.1 rest on the claim that the floor-diagram sum computes the quadratic Gromov–Witten invariant Q'^{(m-s)}_d. The proof given is a single paragraph asserting that both sides are unramified away from S={2,3} and then invoking the characteristic-0 result [JPMPR25, Theorem 10.13]. This is not sufficient as written. To make the spreading-out argument work, the authors must verify: (1) the left-hand side is unramified, which requires checking that the toric surface X_{k^3} has a smooth proper model over Z[1/6] and that Hypothesis 5.1 holds for the relevant divisor class; (2) the right-hand side is an S-integral Witt invariant on Sq_s — this is plausible since μ(D,φ) ∈ Z[t_1,…,t_s], but it is not demonstrated; and (3) the cited [JPMPR25, Theorem 10.13] indeed proves the identity as an equality of Witt invariants over Q, i.e., for","section":"Section 6.1, Theorem 6.7"},{"comment":"The positive-characteristic reduction in the proof of Lemma 6.14 is too compressed. The sentence 'Since the lemma holds in characteristic 0, it follows that the lemma holds for k perfect of positive characteristic as well' is only valid if both sides of the asserted equality are known to be unramified over the chosen DVR R, so that equality over the fraction field descends to the residue field. The proof should explicitly invoke Proposition 5.6, identify the model over R (the blow-up Bl_{E_{\\tilde δ}×R} P^2_R), and check that the hypotheses of Proposition 5.6 are satisfied. As written, the descent step is asserted rather than demonstrated, which is a gap in a key reduction used in the proof of Theorem 6.1.","section":"Section 6.2, Lemma 6.14"}],"minor_comments":[{"comment":"There are several typos and inconsistencies: 'Weslchinger' for 'Welschinger' in Remark 4.12, 'Gomov' for 'Gromov' in Conjecture 5.14, and a garbled name in the acknowledgments ('BenoîtV WBertrand'). The notation Et_n(K) / Etn(K) is used inconsistently; please unify.","section":"Throughout"},{"comment":"The diagram in Definition 2.15 is hard to read: the placement of the isomorphism signs (∼=) is ambiguous. It should be clarified which arrows are isomorphisms and which are merely functoriality maps.","section":"Section 2.3, Definition 2.15"},{"comment":"After proving β-integrality of Q_{n,d,Q}, the proof states that the invariant 'must equal V W_{n,d}' by Theorem 4.3. This also uses Lemma 5.13 to identify the multireal values of Q_{n,d,Q} with those of V W_{n,d}; this step should be stated explicitly.","section":"Section 6.2, proof of Theorem 6.1"},{"comment":"The proof uses the identities t_j^2 = 2t_j and the identification of the basis {t_J} with the β-basis. These facts are used without reference; a pointer to Theorem 2.5 or Lemma 2.14 would help the reader.","section":"Section 6.1, proof of Corollary 6.8"},{"comment":"The paper relies heavily on the unpublished preprints [KLSW23a], [BW25], and [JPMPR25]. While this is not a mathematical error, the authors should state in the introduction or in a remark exactly which results from these preprints are used and whether they are available in final form. This is particularly important for Theorem 6.7, which is a nontrivial generalization of [JPMPR25, Theorem 10.13].","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The core mathematical idea is appealing and the paper is likely to be influential if the external dependencies are resolved. However, the proof of Theorem 6.7 is a sketch that leaves the unramifiedness and characteristic-0 comparison insufficiently verified. The authors should be asked to make Theorem 6.7 fully rigorous, including a precise statement of the hypotheses inherited from [JPMPR25] and the verification that the floor-diagram side is S-integral. The heavy reliance on unpublished preprints by overlapping sets of authors is a matter for the editor's judgment; it would be advisable to request that the authors clearly delineate which parts are established in the preprints and which are new here."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this paper constructs Welschinger–Witt invariants from classical Welschinger invariants, shows quadratic Gromov–Witten invariants are Witt invariants with controlled ramification, and conjectures they agree. It proves the conjecture for k-rational del Pezzo surfaces of degree at least 6. That's the main new content, and it's good.\n\nThe triangle recipe that packages Welschinger invariants into a β-integral Witt invariant is clean and works. The identification of quadratic GW invariants as Witt invariants (Theorem 5.4) and the ramification control (Theorem 5.7) are substantial. The paper is honest: it explicitly lists the three obstructions to the general conjecture, and the tables in the β, λ, and χ bases are useful.\n\nSoft spots: the proof of the main theorem leans heavily on unpublished preprints—[KLSW23a] for the existence of quadratic GW invariants, [BW25] for the quadratic Abramovich–Bertram formula, and [JPMPR25] for the floor diagram computation. Two of these are by the same authors. That makes the central conclusion conditional. The specific load-bearing step is Theorem 6.7, which removes the large-characteristic assumption from [JPMPR25, Thm 10.13] by an unramifiedness argument. I think the argument works—both sides really are unramified away from {2,3}, and the characteristic-0 comparison from [JPMPR25] covers all fields containing Q—but the proof is terse, and the smooth-proper-model hypothesis for Theorem 5.7 is not spelled out for the toric blow-up. That's a write-more-carefully issue, not a fatal gap.\n\nMinor issues: Example 4.10 leaves an 'exercise' to the reader for a nontrivial Welschinger value, and Table 6 gives approximate coefficients for d=8. Fine for a preprint, but worth polishing.\n\nWho is this for? People working on real enumerative geometry, A1-homotopy, and quadratic forms. It deserves a serious referee, though the referee will need access to the cited preprints. I'd send it to a good journal once the dependencies are clarified.","headline":"Welschinger–Witt invariants: a genuinely new bridge between real and quadratic Gromov–Witten counts, proved for degree ≥6 but conditional on three unpublished preprints.","tokens_in":51296,"tokens_out":3639,"would_cite":true,"duration_ms":35046,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N35","14P99","11E04"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for k-rational del Pezzo surfaces of degree at least 6, quadratic Gromov–Witten invariants equal Welschinger–Witt invariants, and conjectures the equality for all rational surfaces.","keywords":["Welschinger invariants","quadratic Gromov–Witten invariants","Witt invariants","del Pezzo surfaces","β-integrality","floor diagrams","real enumerative geometry","A1-homotopy theory"],"falsifier":"Take k = F7, X = P^2_k, class d = 4 (so n0 = 11), and A = E_2 × E_3 × E_5 × E_7 × F7 in Et_11(F_7). Compute Q_{P^2,d,F7}(A) by enumerating the essential marked floor diagrams of Theorem 6.7 and sum their quadratic multiplicities over F7; compute V W_4(A) = 8β_1(A) + 2β_2(A) + β_3(A) with the β-basis of Section 2. The conjecture (and Theorem 6.1 for this case) holds exactly when the two Witt classes in W(F_7) agree.","tokens_in":50253,"feed_emoji":"🧮","tokens_out":7472,"duration_ms":59566,"temperature":0.7,"pith_summary":"Welschinger invariants are signed counts of real rational curves through point constraints; quadratic Gromov–Witten invariants are their analogues over arbitrary fields of characteristic not 2 or 3, counting curves with values in the Witt ring of quadratic forms. This paper packages Welschinger invariants into β-integral Witt invariants (Welschinger–Witt invariants) and conjectures that every quadratic Gromov–Witten invariant of a k-rational del Pezzo surface is just the evaluation of one of these packages. The conjecture gives a uniform computation of all quadratic Gromov–Witten invariants of rational surfaces from the classical real numbers. The paper proves the conjecture for surfaces of degree at least 6, by showing the quadratic invariants are β-integral using the quadratic Abramovich–Bertram formula and floor diagram counts.","feed_headline":"Real curve counts predict curve counts over every field","feed_subtitle":"One family of signed real counts determines all quadratic curve counts on rational surfaces.","key_machinery":"Witt invariants of étale algebras in the sense of Serre: natural transformations Etn → W valued in Witt rings of quadratic forms. The paper uses the β-basis (βi) of the ring of such invariants; a Witt invariant is β-integral when its β-coefficients are integers. The Welschinger–Witt invariant V Wn,d is built from Welschinger invariants by the triangle recursion (a discrete version of Welschinger's formula), and the proof that quadratic Gromov–Witten invariants are also β-integral passes through essential marked floor diagrams whose quadratic multiplicities are expressed in the same β calculus.","core_discovery":"Central claim (Theorem 6.1): for blow-ups of P^2 in at most three points, the quadratic Gromov–Witten invariant Q_{n,d} — a partially defined Witt invariant — is the restriction of the Welschinger–Witt invariant V W_{n,d}. This is a case of Conjecture 5.14: Q_{X,D,k}(A0) = V W_{X,D}(A0) for every k-rational del Pezzo surface. The authors construct V W_{n,d} as a β-integral Witt invariant whose multireal values are Welschinger invariants, and show quadratic Gromov–Witten invariants are unramified Witt invariants away from 2 and 3. The proof reduces to toric surfaces via the quadratic Abramovich–Bertram formula and concludes β-integrality by floor diagram counts.","pith_inferences":["If Conjecture 5.14 holds, every quadratic Gromov–Witten invariant of a k-rational surface is determined by signed counts of real curves, so computations over finite fields or p-adic fields can be replaced by real enumerative geometry.","The β-integrality of Welschinger–Witt invariants imposes divisibility and congruence conditions on Welschinger invariants themselves—essentially a Witt-valued refinement of Welschinger's formula—that could be tested numerically on existing tables.","The same triangle construction may apply to Welschinger invariants of P^3 and other Fano manifolds (as the paper sketches), predicting quadratic Gromov–Witten invariants of higher-dimensional varieties once the quadratic theory exists."],"forward_implications":["If the conjecture is right, quadratic Gromov–Witten invariants of all k-rational del Pezzo surfaces are computed by the β-integral Welschinger–Witt invariants, whose inputs are only the real curve counts.","The equality implies deformation invariance of quadratic Gromov–Witten invariants for rational del Pezzo surfaces, since V W depends only on the étale algebras of the points.","Quadratic invariants in positive characteristic would be forced to agree with characteristic-zero values, a property the authors note is expected from construction.","The floor-diagram formula provides an effective algorithm: essential marked floor diagrams with Witt-valued multiplicities compute both sides explicitly."],"supporting_citations":[{"why":"Defines quadratic Gromov–Witten invariants as A1-degrees of twisted evaluation maps; the construction this paper extends to Witt invariants.","marker":"[KLSW23a]"},{"why":"Supplies the tropical floor-diagram computation of quadratic Gromov–Witten invariants for toric del Pezzo surfaces that Theorem 6.7 generalizes.","marker":"[JPMPR25]"},{"why":"Provides the quadratic Abramovich–Bertram formula used to reduce blow-ups of P^2 to toric surfaces in Lemma 6.14.","marker":"[BW25]"},{"why":"Introduces Welschinger invariants and Welschinger's formula; these values form the multireal values of Welschinger–Witt invariants.","marker":"[Wel05b]"},{"why":"Establishes invariance of Welschinger invariants and the real Abramovich–Bertram formula, used in Theorem 4.1 to package them into Witt invariants.","marker":"[Bru20]"},{"why":"Supplies Serre's theory of Witt invariants of étale algebras and the basis theorem underlying the β-basis.","marker":"[GMS03]"},{"why":"Relates quadratic Gromov–Witten invariants over R to Welschinger signs, used to match multireal signatures in Lemma 5.13.","marker":"[Lev18]"}],"fun_headline_variants":["From real curves to every field: one count","Witt invariants link real and algebraic curve counts","Real signed counts fix quadratic curve counts","One real count predicts all rational curve counts","Del Pezzo proof: real counts rule all fields"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proof relies on Theorem 6.7: that quadratic Gromov–Witten invariants of toric del Pezzo surfaces are computed by summing the quadratic multiplicities of essential marked floor diagrams, with the large-characteristic assumption of the underlying floor-diagram computation removed.","fun_headline_variants_meta":{"raw":{"variants":["From real curves to every field: one count","Witt invariants link real and algebraic curve counts","Real signed counts fix quadratic curve counts","One real count predicts all rational curve counts","Del Pezzo proof: real counts rule all fields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000626,"raw_usage":{"total_tokens":2720,"prompt_tokens":721,"completion_tokens":1999,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":465,"completion_tokens_details":{"reasoning_tokens":1929}},"tokens_in":465,"tokens_out":1999,"duration_ms":14927,"temperature":1.0,"reasoning_tokens":1929,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:20:06.891121+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take k = F7, X = P^2_k, class d = 4 (so n0 = 11), and A = E_2 × E_3 × E_5 × E_7 × F7 in Et_11(F_7). Compute Q_{P^2,d,F7}(A) by enumerating the essential marked floor diagrams of Theorem 6.7 and sum their quadratic multiplicities over F7; compute V W_4(A) = 8β_1(A) + 2β_2(A) + β_3(A) with the β-basis of Section 2. The conjecture (and Theorem 6.1 for this case) holds exactly when the two Witt classes in W(F_7) agree.","supporting_citations":[],"review_version":1}