{"id":"da7636b6-b17a-4f96-8a82-c96d803fa823","arxiv_id":"2507.11850","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under the assumption that the body of flotation and the body of buoyancy are homothetic and every flotation chord cuts off one third of the total affine arc length, the body must be an ellipse.","lead":"A mathematics paper studies three geometric bodies associated with a convex shape in the plane and proves that if two of them are scaled copies of each other, together with one extra affine-length condition, the only possible shape is an ellipse. It advances a variant of a long-open floating body conjecture using affine differential geometry.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem hinges on the unverified identity μ′(s)=0 in Theorem 4.20; without it the fixed homothety center, and hence the ellipse conclusion, is unsupported.","rationale":"The paper's main theorem (Theorem 3, proved as Theorem 4.20) is the central claim. Following the reader, I looked for the least secure step in that proof. The step that converts local geometric information into a global fixed center is μ′(s)=0. Every earlier conclusion—λ1=λ2=λ3=1, mediality of the two triangles, parallelism (4.23)—is local and does not by itself give a fixed homothety center. The constancy of μ is what upgrades the pointwise relation x̂ = 4(ˇx−μ)+μ to a genuine homothety of the whole curve Π_hatδ to Πδ, enabling Theorem 4.16. Since Theorem 4.16 is the bridge to Blaschke–Deicke and then to ellipticity, an unverified assertion at this point is the most load-bearing gap. The computation is a finite algebraic identity in derivatives up to second order, so it is exactly the kind of thing that can be checked mechanically. The reader's additional worry about Theorem 4.12 ('any quadric satisfies...') is a genuine gap in that theorem's proof, but Theorem 4.12 is not invoked in the proof of Theorem 4.20, so it is not the load-bearing concern for the central claim. Verdict: no change from the reader's conditional acceptance; the paper should be accepted only after the μ′ identity is either proved in the text or verified independently.","tokens_in":21762,"tokens_out":23069,"duration_ms":231172,"concrete_test":"Use a computer algebra system to expand μ′(s)=1/3(γ′(s)+γ′(t(s))t′(s)+γ′(u(s))u′(s)) under exactly the relations of Theorem 4.20: define t(s) by the chord-of-flotation area condition (3.3)–(3.4) and by ∫_s^{t(s)} det(γ′,γ′′)^{1/3} du = L/3, define u(s)=t(t(s)), impose the side-parallel relations (4.23), and substitute the derivatives (4.22) together with the chain rule for u′. Check whether the resulting expression is identically zero for a generic C^2 convex curve. If the CAS gives a nonzero expression, the proof of Theorem 4.20 fails; if it gives zero, the missing algebra should be included in the paper so the step is verifiable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 4.20 (Section 4.4), after the ratio argument gives λ1=λ2=λ3=1, the author sets μ=(x+y+z)/3 and states that substituting (4.22) and (4.23) into the formula for μ′ yields 0. This is the point at which the moving carousel triangle acquires a fixed centroid. The fixed centroid is then used to assert that Π_hatδ(K) is homothetic to Πδ(K) with center μ and ratio 4, which is exactly the hypothesis needed to invoke Theorem 4.16 and conclude K is an ellipse. The identity is not shown, and it is not a routine one-line simplification: μ′ involves γ′(s), γ′(t)t′, and γ′(u)u′, with t and u coupled by the affine-arc-length thirds condition and by the requirement that each side of the triangle be a chord of flotation. I am not claiming the identity is false; I am claiming that the proof, as written, has its central conclusion resting on an unverified algebraic assertion. The separate gap in Theorem 4.12 about quadrics satisfying (4.7) is real but does not feed into Theorem 4.20.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies, in the plane, the bodies of flotation, buoyancy, and illumination associated to a convex body K, using affine differential geometry. It derives systematic formulas for the boundaries and centroid curves of these bodies and records them in Tables 3.11 and 3.20. The main stated results are that if the body of flotation Fδ(K) is homothetic to the body of buoyancy Bδ(K), then the affine distance between endpoints of every chord of flotation is constant (Theorem 1) and the affine arc length cut off by every chord of flotation is constant (Theorem 2); moreover, if in addition every chord of flotation cuts off exactly one third of the total affine arc length of the boundary, then K is an ellipse (Theorem 3). The proof of the latter is implemented through Theorem 4.20, which uses a triangle argument in the spirit of Bracho, Montejano, and Oliveros to show that the centroid of a moving inscribed triangle is fixed, and then invokes Theorem 4.16 to conclude that K is an ellipse. The paper also gives affine counterparts of Zindler carousels and additional characterizations of ellipses, including Theorem 4.12 and Theorem 4.17.","tokens_in":21944,"tokens_out":7068,"duration_ms":85021,"significance":"If the computational claims are fully verified, the paper would provide a new affine-geometric result on the homothety conjecture for floating bodies, a problem that has been open in the plane and has recently received counterexamples for the original formulation. The systematic derivation of formulas for bodies of illumination appears to be new and is presented in a usable tabular form. The paper explicitly makes no use of fitted parameters or external numerical data, and the argument is self-contained up to classical results of Dupin, Blaschke, and Petty. The affine analogue of the Zindler carousel theorem is a natural and potentially influential contribution. However, several load-bearing algebraic and analytic steps are asserted rather than demonstrated, so the main theorems are not yet fully supported as written.","major_comments":[{"comment":"The proof's central step is the assertion that substituting (4.22) and (4.23) into the formula for μ′ = (x+y+z)′/3 yields 0. This is not a displayed computation or a separately stated lemma; it is the only place where the moving centroid μ becomes fixed. The fixed centroid is then used to conclude that Π̂δ(K) is homothetic to Πδ(K) with center μ and ratio 4, which is exactly the hypothesis needed to apply Theorem 4.16 and obtain that K is an ellipse. Since μ′ involves γ′(s), γ′(t)t′, and γ′(u)u′ with t and u coupled by the chord-of-flotation equations (3.4), the simplification is not a routine one-line identity. Please provide the complete computation or a dedicated lemma proving μ′(s)=0 under the stated hypotheses.","section":"§4.4, proof of Theorem 4.20"},{"comment":"The proof of Proposition 3.25 is summarized as “a tedious but straightforward computation,” but this proposition is load-bearing for Theorem 4.16 and Theorem 4.17. Those theorems use the fact that the affine normal to the centroid curve is parallel to the affine bisector of the chord, and the displayed identity r2′′ = 8δ̄^{1/3}∥c∥³(r1−z) is the quantitative form of that fact. The line r1r3 is subsequently identified with the affine normal to Πδ(K), and this identification is essential for concluding that Πδ(K) is a proper affine hypersphere. The full computation, including the handling of the parametrization and of the case where the directions ẋ and ẏ are parallel, should be written out or replaced by a precise reference.","section":"§3.4, Proposition 3.25"},{"comment":"The proof asserts without justification that “any quadric is known to satisfy the assumptions of Theorem 4.6” and that there is a unique quadric having second-order contact with γ at y and first-order contact with γ at x. The first assertion is a nontrivial global property of conics with respect to condition (4.7), and the second requires an explicit count of the degrees of freedom of a conic against the contact conditions. As written, the ODE uniqueness argument only proves that a solution equals some conic if such a conic is already known to exist. This theorem is advertised as a characterization of ellipses and is used for Corollary 4.14, so the missing justification should be supplied.","section":"§4.2, proof of Theorem 4.12"}],"minor_comments":[{"comment":"The points x̂, ŷ, ẑ, ˇx, ˇy, ˇz, x, y, z are used before they are fully defined in the text; please introduce each of these points precisely before the proof of Theorem 4.20.","section":"§4.4, Figure 4.21"},{"comment":"The formulas for κ1′ and κ2′ in Table 3.11 are used to compare third-order invariants, but their derivation is not given in the body of the paper; a sentence indicating whether they follow from the same substitution used for κ1 and κ2 would help the reader verify (4.7).","section":"§4.2, proof of Theorem 4.6"},{"comment":"The term “quadric” should be specified as a nondegenerate conic with positive definite quadratic part (an ellipse), since the convexity of K and the ODE argument require this interpretation.","section":"§4.2, Theorem 4.12"},{"comment":"Please distinguish clearly between the Euclidean perimetral density used by Auerbach and by Bracho–Montejano–Oliveros and the affine perimetral density used in Theorem 4.20, since the two notions are not the same.","section":"§4.4, first paragraph"}],"recommendation":"major_revision","confidential_remarks":"The paper fits the scope of a differential geometry journal and presents a genuinely original approach to the homothety conjecture. The main concern is verifiability: the proof of Theorem 4.20 depends on an unstated algebraic identity, and Proposition 3.25 is invoked with only a reference to a tedious computation. I would be willing to look at a revised version in which these computations are written out or delegated to a verifiable lemma."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is the first serious attack I know on homothety between the body of flotation and the body of buoyancy (Question 3), and the paper has real substance: careful derivations of the curvature formulas for floating and illumination bodies, a new polarity-based connection between homothety and duality, and clean affine analogs of Auerbach's floating-body theorems. Theorems 1 and 2—affine distance and affine arc length along chords of flotation being constant under homothety—look right and are genuinely new.\n\nThe soft spot is Theorem 4.20 (the paper's Theorem 3). The proof reaches the ratio λ1=λ2=λ3=1, then asserts that with μ=(x+y+z)/3, substituting (4.22) and (4.23) into μ′ gives 0. That is the load-bearing step: the fixed centroid becomes the homothety center used to invoke Theorem 4.16 and conclude ellipse. It is not a one-line simplification. The derivatives t′ and u′ are coupled through the chord-of-flotation area condition, and the affine-arc-length thirds condition has to enter somewhere. The author simply states the result. The stress-test note is fair: I don't think the identity is false, but the paper doesn't show it. This needs a complete computation—or at least a CAS verification and a sketch. As written, the main theorem is not fully proven.\n\nTwo smaller issues. Theorem 4.12 rests on the assertion that 'any quadric satisfies the assumptions of Theorem 4.6,' which is plausible but unproved there; and the theorem asks for (4.7) on an open set of chords, while the homothety condition only gives (4.7) on the one-parameter family of chords of flotation. The paper doesn't explain how the open-set hypothesis is met (or whether Theorem 4.12 is even needed for the main result). Proposition 3.25 being a 'tedious computation' is more forgivable since the setup is explicit.\n\nOn citation and clarity: the survey parts are honest, self-citations are not an issue, and I see no circularity or fitted constants. It deserves a serious referee. My recommendation: send out, but with a clear instruction to the author that Theorem 4.20's μ′(s)=0 computation must be supplied or verified. If it checks out, this is a worthwhile paper; if it doesn't, the conclusion collapses.","headline":"A genuinely new variant of the homothety conjecture with real substance, but the main theorem hinges on an unverified μ′(s)=0 computation.","tokens_in":22510,"tokens_out":4660,"would_cite":true,"duration_ms":51277,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A10","51N10","53A04","53A15","70C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"If a plane convex body's flotation and buoyancy bodies are homothetic, then its chords of flotation have constant affine distance and constant cut-off affine arc length; adding the one-third affine-arc condition makes the body an ellipse.","keywords":["homothety conjecture","floating body problem","body of flotation","body of buoyancy","body of illumination","affine differential geometry","affine arc length","ellipse characterization"],"falsifier":"Compute condition (4.7) directly for a non-circular ellipse: if some chord fails it, the uniqueness argument behind the ellipse conclusion cannot start. Alternatively, numerically search for a $C^{2}$ convex non-ellipse with a δ at which Fδ(K) is homothetic to Bδ(K) and every flotation chord cuts off exactly one third of the total affine arc length; finding one would disprove Theorem 3.","tokens_in":21518,"feed_emoji":"📐","tokens_out":10023,"duration_ms":103628,"temperature":0.7,"pith_summary":"This paper proves a plane analogue of a classical conjecture about floating bodies: if the body of flotation of a convex plane body is homothetic to its body of buoyancy, then the chords of flotation have constant affine distance between their endpoints and cut off arcs of constant affine length. When each such chord cuts off exactly one third of the total affine arc length, the body must be an ellipse. A sympathetic reader should care because this fills a gap that contraction-operator methods could not reach, and because the affine-differential-geometry framework developed along the way recovers natural affine counterparts of classical floating-body rigidity results and of carousel constructions. The main theorems are proved by comparing second- and third-order differential invariants of the flotation and buoyancy boundaries.","feed_headline":"A 1/3 chord rule turns flotation homothety into an ellipse","feed_subtitle":"If flotation and buoyancy bodies match, a one-third affine-arc chord condition forces the body to be an ellipse","key_machinery":"The central objects are the body of flotation Fδ(K), whose boundary is the envelope of water-level lines cutting off area δ, and the body of buoyancy Bδ(K), whose boundary is the locus of centroids of the underwater caps. The argument runs through a chord-of-flotation parametrization: for each boundary point x, the chord [x,y] cutting off area δ has a midpoint on the flotation boundary and a centroid on the buoyancy boundary, with both tangent directions parallel to the chord by the classical flotation theorems. The load-bearing identities are κ1 = ∥c∥^3/|c|^3 and κ2 = 12δ/|c|^3, which yield the homothety characterization ∥c∥^3 ≡ 12δλ; the third-order invariant comparison yields condition (4.7), equivalently constant affine arc length; and a polarity with respect to K converts flotation-to-buoyancy homothety into a duality between the flotation boundary and an illumination body. Finally, the classical affine-hypersphere theorem — a proper affine hypersphere must be an ellipsoid — plus section-centroid characterizations of ellipsoids close the global conclusions.","core_discovery":"The paper's central discovery, in its own terms, is Theorem 3: let K be a $C^{2}$ convex body in the plane and let δ be a volume in (0, area(K)). If the body of flotation Fδ(K) is homothetic to the body of buoyancy Bδ(K), and if every chord of flotation cuts off from the boundary exactly one third of its total affine arc length, then K is an ellipse. Theorems 1 and 2 are the structural precursors: the same homothety assumption already forces the affine distance ∥c∥^3 between the endpoints of each chord of flotation to be constant, and forces the affine arc length of the boundary arc cut off by each chord to be constant. The proof identifies the homothety ratio with ∥c∥^3/(12δ), translates the comparison of second- and third-order differential invariants into the condition $sin^{3}$ α/κ(s) = $sin^{3}$ β/κ(t), and uses the one-third affine-arc-length condition to force the centroids of consecutive flotation triangles to coincide, giving a fixed homothety center. From there, a weak homothety theorem plus a classical affine-hypersphere characterization of ellipsoids yields the ellipse.","pith_inferences":["Beyond the paper: the same third-order invariant comparison could be iterated algorithmically, turning the remaining gap around condition (4.7) into a finite computer-algebra elimination problem.","Beyond the paper: the one-third affine perimetral density suggests a family of carousel-type rigidity questions at densities 1/q; the paper notes that the geometry becomes less rigid, so a computational search for non-ellipse solutions at q = 4 is a natural test.","Beyond the paper: Theorems 1 and 2 give a practical differential signature — constant ∥c∥^3 and constant affine arc length — that can be checked numerically on a sampled convex body before attempting any global rigidity argument.","Beyond the paper: the polarity equivalence of Corollary 4.4 may connect the two-body homothety question to illumination- and intersection-body duality in higher dimensions, although the paper only develops it on the plane."],"forward_implications":["If a planar C^2 convex body has Fδ(K) homothetic to Bδ(K), then the affine distance between the endpoints of every chord of flotation is constant (Theorem 1).","Under the same hypothesis, the affine arc length of the boundary arc cut off by every chord of flotation is constant (Theorem 2).","If, in addition, every chord of flotation cuts off exactly one third of the total affine arc length, then K is an ellipse (Theorem 3).","A flotation boundary is homothetic to the corresponding centroid curve exactly when ∥c∥^3 is constant, giving an affine counterpart of the classical constant chord-length results.","Unless K is an ellipse, the values of δ for which the flotation and buoyancy boundaries are homothetic form a nowhere-dense subset of (0, area(K)) (Corollary 4.14)."],"supporting_citations":[{"why":"Introduced the homothety conjecture and supplies the flotation-curvature identity κ1 = ∥c∥^3/|c|^3 used throughout.","marker":"[33]"},{"why":"Gives plane counterexamples and near-ball results for the one-body conjecture, framing the two-body variant addressed here.","marker":"[1]"},{"why":"Establishes the three flotation theorems (tangent-centroid and curvature formulas) that underlie the parametrizations of the flotation and buoyancy boundaries.","marker":"[12]"},{"why":"Provides the classical constant chord-length and arc-length results of which Theorems 1 and 2 are affine counterparts.","marker":"[4]"},{"why":"Supplies the carousel fixed-centroid argument and perimetral-density-1/3 strategy adapted in Theorem 4.20.","marker":"[9]"},{"why":"Contains the affine-hypersphere theorem used to conclude that the homothetic boundary is an ellipsoid.","marker":"[22]"},{"why":"Provides the class F2 and its ellipse characterization, giving the condition compared with (4.10).","marker":"[26]"},{"why":"Provides the section-centroid characterization of ellipsoids used to transfer the conclusion that the flotation boundary is an ellipse back to K.","marker":"[15]"}],"fun_headline_variants":["Homothety + 1/3 affine arc rule forces ellipse","1/3 affine arc rule + homothety = ellipse","One-third arc cutoff plus homothety forces ellipse","Ellipse from homothety with affine arc chord rule"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The final ellipse conclusion rests on an unproved assertion that every quadric satisfies the key condition (4.7), which seeds the ODE uniqueness argument, and on several long computations including the fixed-centroid step μ′(s) = 0 that are presented without derivation; if any of these fails, the theorem is not established as written.","fun_headline_variants_meta":{"raw":{"variants":["Homothety + 1/3 affine arc rule forces ellipse","1/3 affine arc rule + homothety = ellipse","One-third arc cutoff plus homothety forces ellipse","Ellipse from homothety with affine arc chord rule"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001194,"raw_usage":{"total_tokens":4935,"prompt_tokens":963,"completion_tokens":3972,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":579,"completion_tokens_details":{"reasoning_tokens":3903}},"tokens_in":579,"tokens_out":3972,"duration_ms":31975,"temperature":1.0,"reasoning_tokens":3903,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:04:02.445321+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute condition (4.7) directly for a non-circular ellipse: if some chord fails it, the uniqueness argument behind the ellipse conclusion cannot start. Alternatively, numerically search for a $C^{2}$ convex non-ellipse with a δ at which Fδ(K) is homothetic to Bδ(K) and every flotation chord cuts off exactly one third of the total affine arc length; finding one would disprove Theorem 3.","supporting_citations":[{"cited_title":"3, 335–348","cited_arxiv_id":null,"evidence_quote":"Introduced the homothety conjecture and supplies the flotation-curvature identity κ1 = ∥c∥^3/|c|^3 used throughout."},{"cited_title":"Angeles Alfonseca, F","cited_arxiv_id":null,"evidence_quote":"Gives plane counterexamples and near-ball results for the one-body conjecture, framing the two-body variant addressed here."},{"cited_title":"Dupin, Applications de g´ eom´ etrie de m´ ecanique, ´ a la marine, aux ponts et chauss´ ees, Bachelier, Successeur de Mme","cited_arxiv_id":null,"evidence_quote":"Establishes the three flotation theorems (tangent-centroid and curvature formulas) that underlie the parametrizations of the flotation and buoyancy boundaries."},{"cited_title":"Auerbach, Sur un probl´ eme de M","cited_arxiv_id":null,"evidence_quote":"Provides the classical constant chord-length and arc-length results of which Theorems 1 and 2 are affine counterparts."},{"cited_title":"Bracho, L","cited_arxiv_id":null,"evidence_quote":"Supplies the carousel fixed-centroid argument and perimetral-density-1/3 strategy adapted in Theorem 4.20."},{"cited_title":"Nomizu, N","cited_arxiv_id":null,"evidence_quote":"Contains the affine-hypersphere theorem used to conclude that the homothetic boundary is an ellipsoid."},{"cited_title":"1, 113–127","cited_arxiv_id":null,"evidence_quote":"Provides the class F2 and its ellipse characterization, giving the condition compared with (4.10)."},{"cited_title":"Kurusa and T","cited_arxiv_id":null,"evidence_quote":"Provides the section-centroid characterization of ellipsoids used to transfer the conclusion that the flotation boundary is an ellipse back to K."}],"review_version":1}