{"id":"ee90d33d-2dca-4956-a1e8-2e13fbaa6f9f","arxiv_id":"2607.27911","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The maximum asymptotic density of semi-induced blue-blue-red paths on four vertices is p*, attained by a large clique together with an asymptotically regular component.","lead":"This paper resolves the last open four-vertex case in the classification of semi-inducible red-blue patterns: it determines the exact maximum asymptotic density p* of ordered blue-blue-red paths in any large graph. The extremal construction is a large clique joined to an asymptotically regular graph.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"I read the paper in good faith. The reader's weakest assumption (Lemma 4.3) is not actually a weak point: the tie-breaking by degree-square is sound. I examined the quotient identities, the half-slope inequality, the edge-perturbation arguments, and the final analytic bounds, and found no error. The proof is long but internally consistent. The only residual risk is a typographical error in a polynomial certificate, which can be checked independently. Consequently the reader's ACCEPT verdict stands.","tokens_in":24730,"tokens_out":37103,"duration_ms":288902,"concrete_test":"Independently expand the Bernstein representations (114) and (144) symbolically (e.g., in SageMath) and verify the coefficients match P(x) and P6(M); also recompute the root x* of (3) to high precision and evaluate F(a*,x*) to confirm p* > 3/20 (as in eq. 37).","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central claim appears sound. The most delicate link, Lemma 4.3, is logically airtight: any nontrivial quotient P with z=d−Pd≠0 strictly reduces the degree-square functional D(W^P)=D(W)−||z||^2, so Φ(W^P)=Φ(W) would contradict the choice of W as a D-minimizer among maximizers. The half-slope inequality (Lemma 4.5) and the perturbation arguments in Lemma 4.9 are justified; the measure-theoretic passages to atoms and non-atomic points are handled via Lebesgue differentiation and do not hide a vanishing quotient loss. The analytic upper bounds in Lemmas 4.14 and 4.15 are supported by explicit Bernstein expansions (eqs. 114, 144) that I spot-checked and found correct. No circularity or missing assumption was found.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper determines the exact semi-inducibility constant for the four-vertex red-blue path H3 with edge colors blue–blue–red. It proves that I(H3)=p*, where x* is the unique root in (0,1/4) of 32x^3-40x^2+13x-1=0, a*=(4x*^2-7x*+2)/(3-8x*), and p*=F(a*,x*); equality is attained by a graph sequence of the form K_{(a*+o(1))n} ⊔ R_n with R_n asymptotically regular. The proof uses graphon compactness, a degree-square-minimizing maximizer, weighted vertex quotients, KKT-style edge perturbations, and a structural reduction to a clique joined to a low-degree part. The final upper bound is reduced to a two-variable optimization, which is solved exactly with explicit polynomial and Bernstein identities.","tokens_in":24943,"tokens_out":41843,"duration_ms":311717,"significance":"This resolves the last open four-vertex case from the Bodnár–Pikhurko classification of non-complete red-blue graphs, giving a new exact semi-inducibility value. The method is of independent interest: the degree-square tie-breaking among maximizers, the half-slope inequality, and the atomization argument form a coherent and largely self-contained structural toolkit. The lower-bound construction is explicit, and the upper-bound proof is independent of it; the analytic estimates are supported by concrete Bernstein expansions. I see no circularity or parameter fitting. If the proof is correct, this is a valuable contribution to the semi-inducibility literature.","major_comments":[],"minor_comments":[{"comment":"The symbol A is used both for the σ-algebra σ(d) in (45) and later for the clique vertex set in Theorem 4.12 and Theorem 4.1. This overload makes the structural section harder to read; consider using a script letter or Σ for the σ-algebra.","section":"§4.1 (notation)"},{"comment":"In the non-atomic part, the transition from Lebesgue–Besicovitch differentiation to the choice of radii satisfying the small L1 condition and then to (60) is compressed. A sentence stating that at a µ-Lebesgue point one first makes the normalized L1 error < η_k^2 and then applies Markov would help.","section":"Lemma 4.5"},{"comment":"The argument from (104) to (106) is terse where R_a crosses from positive to nonpositive. It is correct, but the reader must infer that when R≤0 the bound comes directly from U≤U0≤153/1024, and when R_a remains positive the maximum is V(1/2)=F_aux. Expanding this by a few lines would improve readability.","section":"Lemma 4.14"},{"comment":"The sign checks in (138) and (145)–(146) are correct, but the connection between the interval [1/2,1/√2] and the Bernstein variable t∈[0,1] is stated only implicitly; make explicit that the relevant t-range is a subinterval.","section":"Lemma 4.15"},{"comment":"The line 'Put d(x,y):=(d(x),d(y))' after (46) is unused; it can be deleted.","section":"§4.1"}],"recommendation":"accept","confidential_remarks":"I found no significant technical objections. The proof is long but coherent; the most delicate step, Lemma 4.3, is sound, and the final analytic bounds are supported by explicit expansions. The minor comments are presentation-level and do not affect the validity of the main result. The manuscript is suitable for publication in a combinatorics journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is the missing four-vertex case, blue–blue–red path H3, and the paper gives the exact value I(H3) = p*, with a clean extremal construction: a clique disjoint from a regular graph. The result closes the Bodnár–Pikhurko classification, so it's a real milestone for the area.\n\nThe method is the real news: choose a maximizer that minimizes the degree-square functional D(W), then show any nontrivial degree quotient must strictly decrease Φ. That degree-square tie-breaking plus the half-slope inequality (Lemma 4.5) is a genuinely reusable idea, not a one-off computation. The lower bound is simple and exact; the upper bound is a long but explicit chain. I went through the quotient lemmas (4.2, 4.3), the degree-atomization, and the final small-atom/majority estimates. Everything checks out. The constants x* and a* come from solving the calculus problem, not from fitting the construction, so there's no circularity. The citation pattern is normal; the exceptional case is properly attributed to Bodnár and Pikhurko.\n\nThe soft spots are the usual ones for a proof this size. Section 4 is delicate: the measure-theoretic passages (Lebesgue–Besicovitch differentiation, quotient sigma-algebras, coupling to the degree distribution) and the two big polynomial estimates in Lemmas 4.14 and 4.15. I spot-checked the Bernstein expansions (eqs. 114 and 144) and they're correct, but these are exactly the places where a sign error could hide. The paper states it used AI for proof checking and exposition. I don't read that as disqualifying—the arguments are specific and verifiable—but it is another reason to want a careful human referee rather than a quick acceptance.\n\nWho is this for? Extremal graph theorists working on inducibility and semi-inducibility. It deserves a serious referee. I'd send it to someone comfortable with graphon variational arguments and ask them to focus on Section 4.4–4.6, especially the polynomial identities and the measure-theoretic reductions. My own verdict: accept, with moderate-to-high confidence.","headline":"This paper really does settle the last open four-vertex semi-inducibility case, and the proof, though long and intricate, appears sound.","tokens_in":25353,"tokens_out":1555,"would_cite":true,"duration_ms":16599,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper determines the exact maximum limiting density of semi-induced blue–blue–red paths on four vertices, resolving the last open four-vertex case.","keywords":["semi-inducibility","inducibility","red-blue graphs","graphon limits","extremal graph theory","four-vertex graph","blue-blue-red path","degree quotient"],"falsifier":"Evaluate the normalized semi-induced H3 count of the proposed construction for large n—a clique of size a*n disjoint from a regular graph of degree x*n—and compare with p*; any asymptotic excess would refute the upper bound. A direct computer search over small step graphons for Φ > p* would also disprove the theorem.","tokens_in":24656,"feed_emoji":"📐","tokens_out":7034,"duration_ms":70947,"temperature":0.7,"pith_summary":"The paper resolves the exceptional four-vertex case left open in the classification of non-complete red-blue graphs: the semi-induced blue–blue–red path H3. It proves that the maximum limiting density of semi-induced copies in an n-vertex graph is an explicit algebraic number p*, obtained by evaluating a two-variable function F at a distinguished point (a*, x*). Here x* is the unique root in (0, 1/4) of a cubic polynomial, and a* is given by a rational expression in x*. The extremal construction is a disjoint union of a clique of relative size a* and an asymptotically regular graph of relative degree x*. A structural theorem shows that, after a degree-square tie-breaking, every extremal graphon has essentially this split form, reducing the infinite-dimensional maximization to a two-variable calculus check.","feed_headline":"Max density of semi-induced four-vertex path is exact","feed_subtitle":"Extremal graphs split into a clique and a regular remainder; the limiting density solves an explicit cubic.","key_machinery":"The central machinery is the degree quotient combined with a degree-square tie-break. The functional Φ(W) measures weighted H3 density and can be written as β(1-β) - ∫ψ(d) - E_W(d) in terms of the red degree function d. For any quotient W^P that merges degree information, Lemma 4.3 shows that a nontrivial quotient strictly decreases Φ whenever the chosen maximizer also minimizes D(W) = ||d||²₂; equality would produce a second maximizer with smaller D. This strictness forces the degree distribution of an extremal graphon into a single high-degree atom A of weight > 1/2 that is a clique, with a low-degree remainder, collapsing the problem to the two-variable family F(a,x) and its convex-envelo","core_discovery":"Theorem 1.1 states that I(H3) = p*, where p* = F(a*, x*), x* is the unique root of 32x^3 - 40x^2 + 13x - 1 = 0 in (0, 1/4), and a* = (4x*^2 - 7x* + 2)/(3 - 8x*). Equality is attained by graphs of the form K_{(a*+o(1))n} ⊔ R_n, where R_n is asymptotically regular with every vertex of degree (x*+o(1))n, equivalently relative degree λ*+o(1) inside R_n. The proof works with graphon limits and shows that a maximizer chosen to minimize the degree-square functional must have a clique A of weight > 1/2, no edges between A and its complement, and degree at most 1/2 on the complement. This split structure turns the upper bound into the two-variable inequality F0(a,x) ≤ p*, proved by calculus and conve","pith_inferences":["The degree-square tie-breaking device is likely transferable to other semi-inducibility problems: it converts an infinite-dimensional graphon maximization into a one-dimensional degree analysis whenever degree quotients preserve the objective and the tie-break forces strictness.","One can probe the same problem with a prescribed red-edge density by maximizing F(a,x) under the constraint β(a,x) = c; the explicit formulas here suggest a piecewise algebraic density profile with a phase change near the boundary of the split-regular region.","A computer search over small step graphons could independently test p*: a step graphon with Φ > p* would expose a gap in the quotient argument, while agreement would corroborate the exact value.","The structural dichotomy—either a majority clique atom exists or the density is bounded below 3/20—may generalize to other red-blue trees with a blue prefix path, since the same half-slope inequality (57) governs the high-degree rows."],"forward_implications":["The exceptional four-vertex red-blue graph is now settled, completing the classification of semi-inducibility for all non-complete red-blue graphs on four vertices.","The tie-break argument produces an extremal graphon of a forced split form: a clique of weight > 1/2 plus a remainder whose degrees are at most 1/2.","The upper-bound proof reduces to the explicit inequality F0(a,x) ≤ p*, so any graph's H3-density is controlled by a two-parameter degree summary.","The lower-bound construction is simple and explicit at every n—a clique and a cyclic regular graph—showing the bound is asymptotically sharp."],"fun_headline_variants":["Exact max density for semi-induced 4-vertex path","Semi-induced path density: exact limit found","Clique+regular graph solves semi-induced path extremum","Blue-blue-red path: exact semi-induced maximum","Four-vertex path semi-inducibility: exactly resolved"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is Lemma 4.3: for a chosen maximizer that minimizes the degree-square functional, every nontrivial degree quotient strictly decreases the objective; if equality ever occurred, the proof could not force the clique-plus-low-degree split.","fun_headline_variants_meta":{"raw":{"variants":["Exact max density for semi-induced 4-vertex path","Semi-induced path density: exact limit found","Clique+regular graph solves semi-induced path extremum","Blue-blue-red path: exact semi-induced maximum","Four-vertex path semi-inducibility: exactly resolved"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000243,"raw_usage":{"total_tokens":1358,"prompt_tokens":727,"completion_tokens":631,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":471,"completion_tokens_details":{"reasoning_tokens":553}},"tokens_in":471,"tokens_out":631,"duration_ms":6116,"temperature":1.0,"reasoning_tokens":553,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T23:11:20.257410+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the normalized semi-induced H3 count of the proposed construction for large n—a clique of size a*n disjoint from a regular graph of degree x*n—and compare with p*; any asymptotic excess would refute the upper bound. A direct computer search over small step graphons for Φ > p* would also disprove the theorem.","supporting_citations":[],"review_version":1}