{"id":"c6abafa8-6a58-4d4f-af40-61f4bb463b6a","arxiv_id":"2607.05226","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For mean-zero unit-variance random variables with fourth moment at most κ, the sharp one-sided tail V₁(t,κ) is completely mapped into four explicit regimes, with matching certificates and a proof-degree phase transition.","lead":"The paper gives the exact worst-case one-sided tail probability for any random variable with fixed mean, variance and an upper bound on kurtosis (skewness free). The answer is a four-regime phase diagram with closed forms, extremal distributions and sum-of-squares certificates, plus the observation that one-sided and two-sided tails coincide once the kurtosis bound binds.","discovery_kind":"first_principles","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader's identification of the IIIb solvability gap as the weakest assumption is accurate and matches the paper's own limitations paragraph. That gap does not undermine the strongest claim as written: the map is exact on the three closed-form regimes (with dual certificates and extremals that an independent checker re-verifies in fresh processes), the algebraic characterization of IIIb is rigorous whenever a solution exists, and every downstream application inherits only the closed forms or the t=0 endpoint. The independent exact-arithmetic layer, the symbolic identity battery, and the explicit matched pairs supply stronger support than is typical for classical moment-problem results. Consequently the ACCEPT verdict with low correctness risk stands; no adjustment is warranted.","tokens_in":52923,"tokens_out":572,"duration_ms":13067,"concrete_test":"Independently solve system (S) via high-precision Gröbner bases or certified homotopy continuation at 50 previously unevaluated interior points of the IIIb wedge (e.g., rational (t,κ) with κ>κ* and t∈(0,b(κ))), confirm a feasible non-negative solution exists with primal=dual value matching to ≥30 digits, and re-run the independent verifier on the resulting rationalized certificate; any failure would expose incompleteness of the characterization.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 3.1) is a complete, constructive four-regime description of V1(t,κ) for the skewness-free class C(κ). Regimes I/II/IIIa are fully closed-form with explicit SOS certificates (degree 2 or 4) and matching two-/three-point extremals, all re-checked by an independent exact-arithmetic verifier (Section G) that shares no producer code. Regime IIIb is characterized exactly by the algebraic system (S) plus the matched-pair guarantee of Proposition D.6; global solvability is asserted only as machine-verified on the evaluated grid (Machine-Verified Fact C.12) and is flagged as a limitation. That gap is real but non-load-bearing: the theorem statement scopes it honestly, the closed-form regimes and all applications (quantiles, MoM constant, directional tails) never rely on unproven interior points of IIIb, and attainment of the overall supremum is separately guaranteed by the weak-convergence argument of Lemma D.4. No internal inconsistency, hidden assumption, or unverified identity appears in the matched pairs or the phase diagram of proof degree.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper determines the exact one-sided worst-case tail V₁(t,κ)=sup P(X≥t) over the skewness-free class C(κ) of mean-zero, unit-variance random variables with fourth moment at most κ. Theorem 3.1 gives a four-regime map: a Cantelli tongue where the two-moment bound 1/(1+t²) remains tight; a far-tail closed form (κ−1)/((t²−1)²+κ−1); a plateau (for κ≤3/2) on which the value freezes; and a central wedge characterized by an explicit algebraic system (S). Matching two- and three-point extremals and degree-2/4 SOS certificates are supplied for the closed-form regimes, with independent exact-arithmetic re-verification. The paper also proves a one-/two-sided collapse beyond c(κ), an exact proof-degree phase diagram (Theorem 3.4), the t=0 endpoint recovering He–Zhang–Zhang, and applications to quantiles, median-of-means, margins, and certifiable directional tails.","tokens_in":53160,"tokens_out":1466,"duration_ms":20590,"significance":"If correct, this is a complete, constructive solution of the natural univariate fourth-moment tail problem with free skewness—the form in which kurtosis assumptions typically arrive in robust statistics and SOS estimation. The closed forms, the collapse of Cantelli’s one-sided gain under binding kurtosis, and the exact proof-degree transition along κ=κ_c(t) are new and useful. Strengths that raise the contribution above a pure existence result include: explicit dual certificates and extremal laws for every closed-form regime; an independent exact-arithmetic verifier (fresh process, no shared producer code) that re-checks 47/48 grid instances; recovery of Cantelli, Zelen’s symmetric slice, and the He–Zhang–Zhang constant as special cases; and clean downstream corollaries (exact quantiles, a sharpened MoM block constant 2/11, directional tails under certifiable kurtosis). The machine-checked layer and the openly scoped limitation on IIIb solvability are genuine assets.","major_comments":[{"comment":"Theorem 3.1(IIIb) and Machine-Verified Fact C.12 characterize the central regime by system (S) and assert solvability only at evaluated grid points; the limitations section correctly flags the missing global existence proof. Proposition D.6 shows that any solution of (S) yields the exact value, and Lemma D.4 guarantees attainment of the supremum, but without existence of a solution to (S) the identification V₁=w₁+w₂ is not established for every interior (t,κ). This is the only incomplete piece of the map. Please either (i) prove solvability of (S) on the whole central wedge (e.g., via a continuous deformation from the plateau/tongue boundaries together with the gluing identities already verified), or (ii) restate Theorem 3.1(IIIb) so that the claim is explicitly conditional on solvability, with the grid evidence and the sandwich certificates of Section 4 recorded as the current scope. Do","section":null},{"comment":"Section 3.4 and Fact C.10 rule out nested-square-root closed forms for IIIb by exhibiting an irreducible degree-6 minimal polynomial at (t,κ)=(1/2,2). The argument correctly uses Fact C.8 (degrees in iterated quadratic extensions). Higher-order radicals are explicitly not ruled out. For a complete negative statement, either compute the Galois group of the degree-6 polynomial (or of the second sample point (0.4,2.5)) or soften the abstract/introduction phrasing from “provably admitting no closed form in nested square roots” to the precise claim already proved in §3.4. This is load-bearing only for the “no closed form” slogan, not for the positive characterization via (S).","section":null}],"minor_comments":[{"comment":"Figure 1 is the paper’s main visual; the dashed curve t=τ(κ) and the two marked constants κ**, κ* are hard to read at print scale. Enlarge labels and consider a second panel zooming on the plateau region κ∈[1,3/2].","section":null},{"comment":"Table 1’s last column (bounds at (t,κ)=(2,3)) is useful but the caption’s warning that values are not comparable across classes could be strengthened by adding a one-line “class size” note (e.g., Zelen pins m₃; this paper leaves m₃ free).","section":null},{"comment":"Notation clash: in Regime II the statement uses u for (1−p t²)/(1−p) while the global shorthand is u=√(κ−1). The appendix already writes u*; promote that distinction into the main theorem statement.","section":null},{"comment":"Section 4 and Appendix G devote substantial space to LemmaForge and the AI-guided search. The validation battery (Table 2) and the independent verifier are scientifically important; the discovery narrative can be shortened without loss of content, keeping the trusted-base discussion (checker lines, mutation tests, ε-retreat).","section":null},{"comment":"Corollary 5.1(ii): the two-sided quantile formula is stated cleanly, but a one-sentence cross-reference to the gluing at p=1/κ (where the Chebyshev and kurtosis branches meet at √κ) would help readers invert Theorem 3.2 by hand.","section":null},{"comment":"In the worked instance E.1, the weights w±= (9∓4√3)/22 are exact; the approximate decimals are fine for intuition but should be labeled as such to avoid any impression that the witness is numerical.","section":null},{"comment":"References: Selberg (1940) and Guttman (1948) are cited in Table 4 with partial accessibility notes; if the primary texts remain unavailable, a secondary source or a clearer “formula not reproduced” flag would help.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is unusually strong on constructive certificates and independent verification; that layer is rare in probability and should count heavily. The only substantive incompleteness is global solvability of (S) in IIIb, which the authors already flag. I would not reject or send for major revision over it. Fit for a general probability journal is excellent; if the venue emphasizes pure analysis over computational certificates, the LemmaForge material can be trimmed in production without harming the theorems. No concerns about novelty disclosure or citation pattern."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper finishes a classical one-variable moment problem that people in robust stats and SOS estimation keep using as a black box. For mean-zero, unit-variance laws with E X^4 ≤ κ and free skewness, they give the exact worst-case one-sided tail V1(t,κ) for every t>0 and κ≥1. The answer is a clean four-regime map: Cantelli tongue where kurtosis buys nothing, a simple closed-form tail that improves Markov by the factor κ/(κ-1), a plateau for small κ, and a central wedge characterized by an explicit algebraic system that they prove cannot be nested square roots.\n\nWhat is new is the complete map for the free-skewness class (Zelen pins m3; He–Zhang–Zhang only do small deviations), the one-sided/two-sided collapse past c(κ), and the exact proof-degree phase diagram (degree 2 on the tongue, 4 off it). Every closed-form regime ships with an explicit dual polynomial and a two- or three-point extremal; both are re-checked by an independent exact-arithmetic verifier that shares no code with the producer. They recover Cantelli, the He et al. 2√3-3 constant, and Zelen’s symmetric slice as special cases, and the applications (exact quantiles, MoM block constant 2/11, directional tails under certifiable kurtosis) are immediate substitutions.\n\nThe only soft spot is global solvability of the central system (S): they only claim it on the grid they evaluated, and they flag it themselves. That gap is real but non-load-bearing; the closed forms, the collapse, the degree map, and all applications never need unproven interior points of IIIb, and attainment of the overall supremum is handled separately by a standard weak-convergence argument. The AI-pipeline story is packaging, not a dependency.\n\nThis is for anyone who uses fourth-moment tail bounds or degree-4 SOS estimation. The math is solid, the certificates are reproducible, and the citation pattern is honest. I would send it to referees without hesitation.","headline":"A complete, constructive four-regime map for the skewness-free kurtosis class, with matched SOS certificates and extremals that actually check out.","tokens_in":53811,"tokens_out":572,"would_cite":true,"duration_ms":6747,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60E15","90C22","62G32"],"pacs":[],"model":"grok-4.5","headline":"A kurtosis bound buys an exact four-regime map for the worst one-sided tail, with a phase transition in proof degree.","keywords":["kurtosis","tail bounds","moment problems","Cantelli inequality","sum of squares","proof degree","extremal distributions","median of means"],"falsifier":"Exhibit any single pair (t, κ) inside the claimed central wedge for which the algebraic system has no feasible solution, or any law with mean 0, variance 1 and fourth moment ≤ κ whose tail probability strictly exceeds the value asserted by the map.","tokens_in":53823,"feed_emoji":"📐","tokens_out":823,"duration_ms":6146,"temperature":0.7,"pith_summary":"If you only know that a random variable has mean zero, variance one, and fourth moment at most κ, how large can the probability of exceeding a threshold t become? This paper answers that question completely. The answer is a four-regime phase diagram: inside an explicit tongue the classical two-moment Cantelli bound remains tight and the kurtosis constraint is worthless; far out in the tail a simple closed form improves Cantelli by converting 1/t^{2} decay into 1/t^{4} decay with the optimal constant κ−1; for small κ there is a plateau where the worst-case probability freezes independent of t; and in a central wedge the value is given exactly by an algebraic system that cannot be expressed in nested square roots. Beyond the tongue the one-sided and two-sided worst cases coincide, so Cantelli’s one-sided improvement over Chebyshev is annihilated by fourth-moment information. The same map tells you that the tight bound has a degree-2 sum-of-squares proof exactly on the tongue and requires degree 4 everywhere else. Every closed-form regime ships with an explicit extremal distribution and an explicit dual certificate that an independent exact-arithmetic checker re-verifies.","feed_headline":"Kurtosis buys an exact four-regime map for one-sided tails","feed_subtitle":"Cantelli stays tight inside a tongue; outside, one- and two-sided worst cases collapse and proof degree jumps to 4","key_machinery":"The dual sum-of-squares certificate: a degree-2 or degree-4 polynomial that majorizes the indicator of the tail event, whose expectation under the moment constraints yields a sharp upper bound, matched by an explicit atomic extremal distribution.","core_discovery":"For every threshold t > 0 and every kurtosis budget κ ≥ 1 the worst-case one-sided tail probability over the class of mean-zero, unit-variance, fourth-moment-at-most-κ random variables equals an explicit four-regime map: the Cantelli value 1/(1+t^{2}) on the tongue b(κ) ≤ t ≤ c(κ), the closed form (κ−1)/((t^{2}−1)^{2}+κ−1) for t ≥ c(κ), a t-independent plateau for small κ, and the value of an explicit algebraic system in the remaining central wedge, each attained by a two- or three-point law.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Exact four-regime map for one-sided tails under kurtosis","Kurtosis bound yields four-regime worst-case tail map","Cantelli tongue then closed-form tails under kurtosis","Four regimes: when kurtosis locks one-sided tails","Worst-case tails under kurtosis: tongue, plateau, algebra"],"cache_read_input_tokens":49280,"weakest_assumption_plain":"In the central regime the paper characterises the value by an algebraic system whose solvability is verified on a grid but not proved for every interior parameter point.","fun_headline_variants_meta":{"raw":{"variants":["Exact four-regime map for one-sided tails under kurtosis","Kurtosis bound yields four-regime worst-case tail map","Cantelli tongue then closed-form tails under kurtosis","Four regimes: when kurtosis locks one-sided tails","Worst-case tails under kurtosis: tongue, plateau, algebra"]},"model":"grok-4.5","effort":"low","cost_usd":0.003948,"raw_usage":{"total_tokens":1402,"prompt_tokens":1068,"num_sources_used":0,"completion_tokens":73,"cost_in_usd_ticks":39480000,"prompt_tokens_details":{"text_tokens":1068,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":261,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":1068,"tokens_out":73,"duration_ms":2785,"temperature":1.0,"reasoning_tokens":261,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T07:42:04.710453+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit any single pair (t, κ) inside the claimed central wedge for which the algebraic system has no feasible solution, or any law with mean 0, variance 1 and fourth moment ≤ κ whose tail probability strictly exceeds the value asserted by the map.","supporting_citations":[],"review_version":2}