{"id":"7c465032-82de-4e8d-98ad-8bc80199d7be","arxiv_id":"2511.20282","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"An asymptotic for the density of soluble conics with polynomial coefficients is proposed, but the sign decomposition (6.5) that bridges the circle-method counts to the true count is false.","lead":"This paper claims an asymptotic for how many conics F_0(y)x_0^2+F_1(y)x_1^2=F_2(y)x_2^2 have rational points as y varies, matching the Loughran–Rome–Sofos conjecture. A sign-splitting identity in the final reduction is false, so the claimed agreement is not established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (6.5) is false: the sign average changes the conic in a positive-density set of fibres, so the proof counts a different family than N_glob.","rationale":"The paper's central claim is the asymptotic for N_glob(π_F,B) with the Loughran–Rome–Sofos constant. Sections 2–5 contain a long and apparently careful computation of the constant and of the distribution of diagonal conics in arithmetic progressions; I have no independent reason to doubt Theorem 4.2 itself. But Theorem 1.7 is obtained by feeding these estimates into the Destagnol–Lyczak–Sofos circle-method theorem, and the final reduction from sign-restricted counts to N_glob is exactly equation (6.5). That equation is asserted without proof and is false even pointwise. The root cause is that sign-flipping coefficients of a ternary quadratic form is not an equivalence: only simultaneous negation preserves the zero set. In the octant (+,+,−), the sign-flipped conic t=(|F0|,|F1|,|F2|) is indefinite and often soluble, while the original conic is definite and insoluble. The same phenomenon occurs p-adically, so the discrepancy is not repaired by local factors. The final line of the proof, where c8 is expressed as a sum over ε≠±(1,1,−1), explicitly excludes the two sign patterns that (6.5) had included, confirming the proof's internal inconsistency. Because the bridge fails on a positive-measure set of fibres, the main theorem is not established by this argument. This is the same load-bearing weakness the reader identified, and it justifies keeping the reject verdict.","tokens_in":35954,"tokens_out":14819,"duration_ms":140182,"concrete_test":"Verify (6.5) pointwise with the valid Birch system F0=X0, F1=X1, F2=X0+X1−X2 in n=3, at the primitive vector k=(1,1,3,0). The left side for this single k is ϑ_Q(1,1,−1)=0, since x0^2+x1^2=−x2^2 has only the trivial solution over Q. On the right, the only ε with min_j ε_jF_j(k)>0 is (+,+,−), and ϑ_Q(1,1,1)=1 because (1,0,1) solves x0^2+x1^2=x2^2. Thus the ε-sum equals 1, and the factor 1/2 gives 1/2, contradicting 0. Repeating this computation over the positive-volume cone {X0>0,X1>0,X0+X1−X2<0} shows the asymptotic in Section 6 counts a different set than N_glob; the final constant can also be recomputed to see the dropped ±(1,1,−1) terms.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The bridge from the DLS circle-method counts to N_glob is the Möbius inversion in (6.5). For a primitive k, only ε = sign(F0(k),F1(k),F2(k)) can satisfy min_j ε_jF_j(k)>0, so (6.5) implicitly claims ϑ_Q(F(k)) = (1/2)ϑ_Q(|F0(k)|,|F1(k)|,|F2(k)|) up to the sign convention in Q_t. This is false. Example satisfying the theorem's assumptions: n=3, F0=X0, F1=X1, F2=X0+X1−X2, k=(1,1,3,0). Then F(k)=(1,1,−1). The true conic is x0^2+x1^2=−x2^2, which has no nontrivial Q-point, so ϑ_Q=0. The only sign vector with min εF(k)>0 is (+,+,−), and ϑ_Q(1,1,1)=1 because x0^2+x1^2=x2^2 has the point (1,0,1). Hence the right side of (6.5) contributes 1/2, not 0. The failure occurs on a positive-volume set (F0,F1>0>F2), not on a measure-zero exception. Moreover, p-adic solubility is not invariant under such sign changes, e.g. 2x^2−y^2−3z^2 is anisotropic over Q_3 while 2x^2+y^2−3z^2 is soluble. The final manipulation restricts the volume sum to ε≠±(1,1,−1) to form c8, silently discarding two sign patterns that (6.5) had included; this internal inconsistency confirms that the bridge is not valid.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the density of y ∈ P^n(Q) of bounded height for which the diagonal conic C_{F,y}: F_0(y)x_0^2 + F_1(y)x_1^2 = F_2(y)x_2^2 has a rational point. It claims an asymptotic N_glob(π_F,B) ~ c(π_F) B/(log B)^{3/2}, with the constant c(π_F) predicted by Loughran–Rome–Sofos. The proof combines a circle-method theorem of Destagnol–Lyczak–Sofos for averages over polynomial values with a new estimate (Theorem 4.2) for the number of soluble diagonal conics whose coefficients lie in arithmetic progressions, followed by a long computation of the resulting constant.","tokens_in":36261,"tokens_out":12138,"duration_ms":111182,"significance":"If the main theorem were correct, it would be a substantial confirmation of the Loughran–Rome–Sofos conjecture for a natural family of conic bundles, including the full leading constant. The paper contains a large amount of technical analytic number theory: a general Selberg–Delange lemma (Lemma 3.1), a careful application of the Destagnol–Lyczak–Sofos circle-method result, and an extensive computation of local constants. These components may be of independent interest. However, the central bridge connecting the sign-conditioned circle-method counts to the actual solubility count is invalid, so the claimed theorem is not established.","major_comments":[{"comment":"The displayed \"Möbius inversion\" identity is false. For a fixed primitive k, the condition min_j ε_j F_j(k)>0 is satisfied by the unique sign vector ε(k)=(sign F_0(k), sign F_1(k), sign F_2(k)). The summand is then (1/2)ϑ_Q(|F_0(k)|,|F_1(k)|,|F_2(k)|), not (1/2)ϑ_Q(F_0(k),F_1(k),F_2(k)). These differ on a set of positive density. Example within the theorem's hypotheses: n=3, F_0=X_0, F_1=X_1, F_2=X_0+X_1−X_2, k=(1,1,3,0). Then F(k)=(1,1,−1); the true conic x_0^2+x_1^2=−x_2^2 has no Q-point, while the selected ε=(+,+,−) gives x_0^2+x_1^2=x_2^2, which has (1,0,1). Hence the RHS of (6.5) receives a spurious contribution of 1/2. The failure is not a boundary effect; it occurs on the open region F_0>0,F_1>0,F_2<0. Moreover sign changes alter p-adic solubility (2x^2−y^2−3z^2=0 is anisotropic over Q_3, but 2x^2+y^2−3z^2=0 is soluble), so no local correction can be absorbed into the constant.","section":"Section 6, Eq. (6.5)"},{"comment":"After substituting the asymptotic for N^1_ε into (6.5), the proof needs to sum over all eight ε. The displayed evaluation instead uses Σ_{ε≠±(1,1,−1)} vol(min_j ε_jF_j≥0)=c_8, i.e. it discards the two sign patterns ±(1,1,−1). These are exactly the patterns that made the bridge (6.5) false: on the positive-density region F_0>0,F_1>0,F_2<0 (and its negative), the selected ε is either (1,1,−1) or (−1,−1,1), and the corresponding N^1_ε count is nonzero because it counts positive-coefficient conics t_0x_0^2+t_1x_1^2=t_2x_2^2 with rational points. Dropping these terms silently changes the count from N_loc to a smaller set and is inconsistent with the preceding equality. The constant found is thus not the constant for the original family.","section":"Section 6, final display"}],"minor_comments":[{"comment":"The factor δ_p(n) for p∤q is introduced as 'the value of the factor corresponding to p in the Euler product in [16, th. 1.1]' before any computation. Since the subsequent Section 5 does derive this factor, the wording gives the impression that the conjectural constant is used as an input. Please rephrase or reorder so that the definition is presented as a summary of the calculation, not as an assumption.","section":"Section 4, Eq. (4.1)"},{"comment":"The bound z ≤ sqrt((1/14) log_2 B) is stated without justification. It should be shown, or at least briefly explained, that this choice ensures W_z ≤ (log B)^{1/7} so that Theorem 4.2 applies.","section":"Section 6, Lemma 6.8"},{"comment":"There are several typographical and language issues: leftover French phrases ('et', 'prouvée'), 'Belun' in Reference [26] should be 'Belin', and the notation c_8 vs c8 is used inconsistently. These are minor but should be corrected.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The flaw in Eq. (6.5) is fatal and is not a local fix: the sign-averaging step fundamentally miscounts the family, and the final constant computation silently drops the offending sign patterns. The analytic machinery in Sections 3–5 may be salvageable for a related but different problem, but the main theorem as stated is unproved. I would also suggest that, in any future revision, the constant computation in Section 5 be made self-contained rather than anchored to the conjectural Euler product, to avoid the appearance of circularity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nI read Da Silva's arXiv:2511.20282 with the stress-test note in hand. The main theorem is not proved. The specific fault is equation (6.5) in Section 6, which claims a Möbius inversion that replaces ϑ_Q(F0(k),F1(k),F2(k)) by (1/2)Σ_ε ϑ_Q(ε0F0(k), ε1F1(k), ε2F2(k)) subject to min_j ε_jF_j(k)>0. For any k with all three F_j(k) nonzero, the ε-sum has exactly one term, ε = sign(F0,F1,F2), so the identity asserts ϑ_Q(F(k)) = (1/2)ϑ_Q(|F0|,|F1|,|F2|). That is false. The example F0=X0, F1=X1, F2=X0+X1−X2 at k=(1,1,3,0) gives F=(1,1,−1): the true conic x0^2+x1^2=−x2^2 has no rational point, while the all-positive conic has the point (1,0,1). This is not a measure-zero artifact; it happens on a positive-volume set of k. Worse, the sign substitution changes p-adic solubility, e.g., 2x^2−y^2−3z^2 is anisotropic over Q3 while 2x^2+y^2−3z^2 is soluble. So the computed asymptotic counts a different set of fibres.\n\nWhat is good: Sections 2–5 contain real work. Theorem 4.2 gives a uniform asymptotic for diagonal ternary quadratics in arithmetic progressions with q up to (log B)^{1/7}, which extends the base family in Loughran–Rome–Sofos. Lemma 3.1 (multi-dimensional Selberg–Delange) is a new general tool. The constant computation in Proposition 2.3, assuming the final result, is plausible. The paper is clearly written and seriously engaged with the literature.\n\nHowever, the load-bearing step fails. The final displayed constant also disagrees with the predicted constant (1.3) by a factor related to 1/ζ(n+1) and a 2, which corroborates that the reduction is not just a sign error but a conceptual one. I would not trust Theorem 1.7 in this form.\n\nFor a referee: this deserves a serious referee report, but the verdict should be reject with the door open: the author should either fix Section 6 (which may require a genuinely new idea to handle sign patterns) or strip the paper to its solid Sections 2–5 and resubmit as a standalone contribution on uniform counts of diagonal conics. I would not cite Theorem 1.7 in the next year, but I would note Theorem 4.2 as a possible citation if it survives scrutiny.\n\nRecommendation: send back for major revision, but make clear the main theorem is unproved.","headline":"The uniform count for diagonal conics in arithmetic progressions is a real contribution, but the final bridge that turns it into a family result is false, so the main theorem fails.","tokens_in":37008,"tokens_out":7178,"would_cite":false,"duration_ms":61325,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11D09","11P55","11N37","14G05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For conic families with polynomial coefficients in many variables, the paper proves the count of parameters with a rational point is asymptotic to c B/(log B)^{3/2}, with c equal to the conjectured leading constant.","keywords":["conic bundles","rational points","asymptotic formula","circle method","Birch system","local solubility","product over primes","Diophantine families"],"falsifier":"Compute, for a concrete homogeneous triple of polynomials and a point k where F0(k), F1(k), F2(k) are all positive and the conic F0(k)x0^2+F1(k)x1^2-F2(k)x2^2=0 has a rational point, the value of (1/2) Σ_ε 1_{min ε_jF_j(k)>0} ϑ_Q(ε0F0(k),ε1F1(k),ε2F2(k)). For example with F0=2X0, F1=X0, F2=3X0 and k=(1,0,...,0), the left side is 1/2 while the right side is 1; this would settle whether the proof's central identity holds.","tokens_in":35542,"feed_emoji":"🔢","tokens_out":12655,"duration_ms":113823,"temperature":0.7,"pith_summary":"This paper establishes an asymptotic formula for the number of coefficient tuples y of bounded height for which the conic F0(y)x0^2 + F1(y)x1^2 = F2(y)x2^2 has a rational point, assuming the three homogeneous polynomials form a system with many variables (a Birch system) and have smooth, properly intersecting zero loci. The main theorem states that the count is asymptotic to c(π_F) B/(log B)^{3/2}, where c(π_F) is the explicit product of local densities that the general conjectural framework for fibrations predicts. The result matters because it pins down the leading constant — not just the power of the logarithm — for a natural family of varieties, giving strong support to the conjecture and a template for computing such constants. The strategy averages the local-solubility indicator over polynomial values via the circle method, then evaluates the resulting product over primes through a character-sum computation.","feed_headline":"Conic families: rational-point count hits predicted constant","feed_subtitle":"For polynomial-coefficient conics, the density of fibres with a rational point is proven to be c·B/(log B)^{3/2} with the conjectured c.","key_machinery":"The load-bearing object is the indicator ϑ_Q(t0,t1,t2), which is 1 exactly when the diagonal conic t0x0^2+t1x1^2=t2x2^2 has a nonzero rational point. The proof machinery is: a Birch system condition (enough variables relative to the degree so that a circle-method average is accurate), a recent circle-method theorem for averages of bounded arithmetic functions over polynomial values, an analytic averaging lemma that isolates the main coefficient in the average, and a product-over-primes computation of the local densities. An inclusion-exclusion step over common divisors converts sign-restricted coefficient counts into the full solubility count, yielding the final constant.","core_discovery":"The central claim is Theorem 1.7: for homogeneous polynomials F0, F1, F2 of degree d that form a Birch system with smooth fibres and a complete-intersection common zero locus, the global solubility count N_glob(π_F,B) satisfies N_glob(π_F,B) ~ c(π_F) B/(log B)^{3/2}, with c(π_F) exactly the constant predicted by the fibration-families conjecture. Along the way, the paper proves a precise asymptotic for the proportion of diagonal ternary conics t0x0^2+t1x1^2+t2x2^2=0 with a rational point and with coefficients in arithmetic progressions, uniform in the modulus up to a power of log B. This arithmetic-progression theorem is the engine from which the main constant is assembled via local factors.","pith_inferences":["The uniform arithmetic-progression theorem for diagonal conics could be imported to families of higher-dimensional quadrics or to fibrations over other bases, provided the circle-method averaging result is available.","The computation of the constant as a product of local densities suggests that the same local factors appear in any proof of the fibration conjecture, so the shape of c(π_F) is robust even if the present proof is later modified.","A direct numerical check of the sign-restricted identity for small-degree polynomial triples (e.g., triples of linear forms at a point with all values positive) would be a cheap way to validate the proof's central bridge before tackling the full theorem."],"forward_implications":["If correct, the conjectured leading constant for fibrations is realized in this family: the power of log is 3/2 and the constant is a product of local densities.","The uniform arithmetic-progression theorem for diagonal conics implies that the same constant governs counts for any modulus at most a small power of log B, allowing the circle-method averaging to be glued.","The paper computes the relevant Galois cohomology invariant of the fibration (a Z/2 quotient generated by an explicit quaternion algebra), which fixes the Tamagawa-type factor in the constant.","The result gives an additional example where the full constant, not just the logarithmic exponent, is verified for a family of conic bundles with polynomial coefficients in many variables.","The method shows how to derive the constant from local factors c_8 and c_p defined by volumes of solubility regions, suggesting these local factors are the correct universal building blocks."],"fun_headline_variants":["Conic family solubility matches predicted constant exactly","Asymptotic formula for rational points on conic fibres proved","Count of conics with rational point: c B over log-cubed","Diagonal conic proportions in arithmetic progressions pinned down","Conic fibres hit conjectured density with new circle method"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument hinges on the identity that the count of rational points can be recovered by summing, over the three sign patterns that make the polynomial values all positive, the local solubility indicators and dividing by two; if that sign-bridge equality fails, the main theorem is not established.","fun_headline_variants_meta":{"raw":{"variants":["Conic family solubility matches predicted constant exactly","Asymptotic formula for rational points on conic fibres proved","Count of conics with rational point: c B over log-cubed","Diagonal conic proportions in arithmetic progressions pinned down","Conic fibres hit conjectured density with new circle method"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000449,"raw_usage":{"total_tokens":2172,"prompt_tokens":885,"completion_tokens":1287,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":629,"completion_tokens_details":{"reasoning_tokens":1205}},"tokens_in":629,"tokens_out":1287,"duration_ms":9611,"temperature":1.0,"reasoning_tokens":1205,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T20:32:15.378908+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a concrete homogeneous triple of polynomials and a point k where F0(k), F1(k), F2(k) are all positive and the conic F0(k)x0^2+F1(k)x1^2-F2(k)x2^2=0 has a rational point, the value of (1/2) Σ_ε 1_{min ε_jF_j(k)>0} ϑ_Q(ε0F0(k),ε1F1(k),ε2F2(k)). For example with F0=2X0, F1=X0, F2=3X0 and k=(1,0,...,0), the left side is 1/2 while the right side is 1; this would settle whether the proof's central identity holds.","supporting_citations":[],"review_version":1}