{"id":"b66ba33d-f2bc-4715-8522-837f6bd94a01","arxiv_id":"2608.09809","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For a bounded alpha unpredictable function, a bounded integral is claimed to be alpha unpredictable, but the proof contains load-bearing errors.","lead":"This paper claims to prove that the integral of an 'alpha unpredictable' function is itself alpha unpredictable whenever the integral stays bounded. The proof has several mathematical gaps, so the central theorem is not established as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 1's proof assumes F attains its sup/inf and uses inconsistent epsilon constants; since Theorem 1 relies on Lemma 1, the central claim is not proven as written.","rationale":"The reader's weakest assumption correctly identifies the most load-bearing defect: Lemma 1 is the only proof that F is Poisson stable, and Theorem 1's sufficiency depends on it. Equations (1)-(2) are not consequences of the definition of sup/inf for continuous bounded functions on R; F(t)=arctan t shows the failure. The numerical inconsistency is independent and equally serious: the stated uniform closeness 5ε/[8(d−c)] cannot yield the claimed ε/4 integral bound, so the final estimate does not close. I also considered the two other gaps noted by the reader. The sign issue in Theorem 1 case (ii) is repairable by continuity, since a continuous function on an interval with |g(t)|>ε0 cannot change sign without vanishing; the monotonicity of the constructed divergence sequence can be restored by thinning to a subsequence of the convergence and divergence sequences. These are routine repairs. The Lemma 1 flaw is not routine, because the displayed constants fail even when one attempts to replace the attainment equalities by near-attainment inequalities. Consequently, the central claim is not established by the manuscript as written, and I do not change the reader's rejection. The authors could make the paper acceptable by supplying a corrected Lemma 1 proof with accurate sup/inf approximation and consistent epsilon constants.","tokens_in":4541,"tokens_out":15164,"duration_ms":128453,"concrete_test":"Instantiate Lemma 1 with a bounded continuous F that does not attain its sup or inf, e.g., F(t)=arctan t, and check that equations (1)-(2) cannot be satisfied for arbitrarily small ε. Then recompute the epsilon chain with the stated uniform bound 5ε/[8(d−c)]: the integral over [t2,t1] is bounded by 5ε/8, not ε/4, so the proof's final bound becomes 11ε/8. If replacing the uniform bound by ε/[4(d−c)] closes the chain with |F(t+t_n)−F(t)|<ε, the flaw is a fixable typo; if no admissible constant closes it, Lemma 1 requires a fundamentally different argument and Theorem 1 lacks proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 1 is the load-bearing step. Equations (1)-(2) assert existence of t1,t2 with F(t1)=M−ε/8 and F(t2)=m+ε/8, where M,m are the sup and inf of F on R. A continuous bounded function need not attain its extrema (F(t)=arctan t is a counterexample), so the proof can fail at its first line. Independently, the proof states |f(t+t_n)−f(t)|<5ε/[8(d−c)] on [c,d], but the next estimate bounds the corresponding integral by ε(d−c)/[4(d−c)]=ε/4; using the stated 5ε/8 bound gives M−m−7ε/8, and the final inequalities yield |F(t+t_n)−F(t)|<11ε/8, not <ε. Thus the included-interval epsilon chain does not close. Since Theorem 1 proves alpha-unpredictability of F only after invoking Lemma 1 for Poisson stability, the central 'if' direction is not established. The other gaps flagged by the reader are repairable: continuity forces constant sign of f(t+t_n)−f(t) on the divergence interval, and the divergence sequence can be thinned to restore monotonicity. The Lemma 1 defects, however, are in the only proof offered.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that the integral of an alpha unpredictable function is alpha unpredictable if and only if the integral is bounded. The main results are Lemma 1, which states that the integral of a Poisson stable function is Poisson stable exactly when it is bounded, and Theorem 1, which extends this to alpha unpredictability. The proofs use a so-called method of included intervals. The paper also derives a corollary for functions whose integral has a linear trend plus a bounded Part.","tokens_in":4876,"tokens_out":10696,"duration_ms":79850,"significance":"If the result is correct, it completes a natural hierarchy in the theory of recurrent functions: periodicity, quasi-periodicity, almost periodicity, Poisson stability, and alpha unpredictability. The boundedness condition is the known necessary and sufficient condition for the integral to preserve recurrence, so the statement is plausible and of interest to the community. The paper also advertises the method of included intervals as a verification tool. However, the current proofs contain several technical gaps that prevent the central claim from being established as written.","major_comments":[{"comment":"The proof assumes that the continuous bounded function F attains values exactly equal to M−ε/8 and m+ε/8. This is not true in general; a continuous bounded function on R need not attain its supremum or infimum (e.g., arctan t). The subsequent estimates rely on these equalities, so the first step of the proof is invalid. This can be repaired by taking points with F(t1)>M−ε/8 and F(t2)<m+ε/8, but the constants in the rest of the proof must then be reworked.","section":"Section 2, Lemma 1, equations (1)–(2)"},{"comment":"The proof fixes an index n such that |f(t+t_n)−f(t)|<5ε/(8(d−c)) on [c,d], but later bounds the corresponding integral by ε(d−c)/(4(d−c))=ε/4. Using the stated 5ε/8 bound yields an error of 5ε/8, not ε/4. Consequently the final inequalities give |F(t+t_n)−F(t)|<11ε/8, not <ε. The epsilon chain does not close as written. The proof must either choose a stricter uniform closeness (for instance ε/(4(d−c))) or adjust all constants consistently.","section":"Section 2, Lemma 1, epsilon estimates"},{"comment":"The inequality |F(s_n+δ+t_n)−F(s_n+δ)|>ε0δ−ε0δ/2 relies on the implicit claim that |∫_{s_n}^{s_n+δ}(f(s+t_n)−f(s))ds|>ε0δ. This requires that f(s+t_n)−f(s) does not change sign on the interval, which follows from continuity and the condition |f(s+t_n)−f(s)|>ε0, but this continuity/sign argument is not stated. In addition, the proof uses h=ε0δ/(8M) without defining M in Theorem 1; M must be a uniform bound for |f|, and it should not be confused with the sup F from Lemma 1.","section":"Section 2, Theorem 1, case (ii)"},{"comment":"The constructed divergence moments s'_n are chosen as either s_n or s_n+δ−h depending on the case. The resulting sequence may fail to be strictly increasing, as required by Definition 2; for example, if s'_n=s_n+δ−h and s'_{n+1}=s_{n+1}, the increments may not be positive unless s_{n+1}−s_n>δ−h. The proof should either thin to a subsequence where the choice is constant, or otherwise justify that the selected s'_n form a strictly increasing sequence tending to infinity.","section":"Section 2, Theorem 1, divergence sequence"}],"minor_comments":[{"comment":"The proof of Theorem 1 contains a duplicated introductory paragraph: 'In what follows, we shall apply the characteristics of function f(t)...' appears twice almost verbatim.","section":"Section 2, Theorem 1, proof"},{"comment":"The claim that necessity is an immediate consequence of Definition 1 is opaque, since Definition 1 does not state boundedness. The authors' opening sentence restricts attention to uniformly bounded functions, so the necessity direction is then trivial, but the link should be made explicit.","section":"Section 2, Lemma 1, necessity"},{"comment":"The phrase 'any theoretical functional recurrence is invariant with respect to integration' is broader than the theorem actually proved, which concerns alpha unpredictable functions. The abstract and introduction would benefit from a more precise statement of the scope.","section":"Abstract and Introduction"},{"comment":"There are several typographical issues, such as 'effectivemethod', 'V .V ' in references, and inconsistent spacing in author names. The reference to the authors' prior work [12] is essential for the algebra used in Corollary 1; the paper should state the needed properties explicitly or reproduce them.","section":"References and typos"}],"recommendation":"major_revision","confidential_remarks":"The technical gaps are significant and currently invalidate the proofs, but they appear repairable in a revision: the sup/inf issue can be fixed by using approximations, the epsilon constants can be rebalanced, and the monotonicity of the divergence sequence can be restored by thinning. I therefore recommend major revision rather than rejection. The paper is heavily self-referential, relying on the authors' prior work [12,13] for the algebra of alpha unpredictable functions; the editor may wish to ensure that these references are accessible and that the needed properties are stated with sufficient clarity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this short paper aims to finish a known sequence—bounded integrals of periodic, almost periodic, and Poisson stable functions inherit the recurrence property—by proving the same for alpha unpredictable functions. That is a legitimate goal, and the statement of Theorem 1 is the natural endpoint. The paper is readable and the 'included intervals' idea is a genuinely useful method for this class; the authors deserve credit for framing the problem cleanly.\n\nUnfortunately, the proof as written does not establish the result. Lemma 1 is the load-bearing step, and it has two defects. First, equations (1)–(2) assume F attains values M−ε/8 and m+ε/8, which a bounded continuous function need not do (F(t)=arctan t is a counterexample). Second, the epsilon arithmetic is off: the uniform bound earlier is stated as 5ε/[8(d−c)] on the including interval, but the subsequent estimate treats the integral of f(s+t_n)−f(s) as ε/4, not 5ε/8; with the stated bound the final inequality gives |F(t+t_n)−F(t)|<11ε/8, not <ε. So the conclusion of Poisson stability does not follow from the included-interval chain. The other flagged issues—the unstated sign argument in Theorem 1 case (ii) and the possible non-monotonicity of the divergence sequence—are real but minor and easy to patch. The Lemma 1 defects are in the only proof offered, and Theorem 1 leans directly on Lemma 1.\n\nHaving said that, I do not think the result is wrong. Lemma 1 is a known result in the literature (the authors themselves cite Levitan–Zhikov), so the Poisson-stability half can be repaired by a correct proof or a proper citation. Theorem 1's divergence part is basically a two-line estimate once the sign point is made explicit. The paper just needs a serious revision, not a trip to the trash.\n\nFor a subfield that cares about recurrent functions and ultra-Poincaré chaos, this is a citable statement once fixed. As it stands, I would not cite it. If I were editor, I'd send it to a referee with a clear message that the proof of Lemma 1 must be rewritten; it deserves a chance rather than a desk reject.","headline":"The intended endpoint theorem is plausible, but the only proof given does not close; the flaws are repairable but should be fixed before anyone cites it.","tokens_in":5308,"tokens_out":1762,"would_cite":false,"duration_ms":14163,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C27","37B20","42A75"],"pacs":[],"model":"deepseek-v4-flash","headline":"A bounded integral of an alpha unpredictable function is again alpha unpredictable.","keywords":["alpha unpredictable functions","Poisson stability","recurrent functions","integral operator","boundedness","method of included intervals","ultra Poincaré chaos"],"falsifier":"Compute $F(t+t_n)-F(t)$ for a bounded continuous Poisson-stable integrand $f$ whose primitive $F$ approaches its supremum only as $t\\to\\infty$ and never reaches $M-\\varepsilon/8$; if the difference fails to converge to zero uniformly on bounded intervals, Lemma 1 is false, and if it does converge, the attainment assumption is removable.","tokens_in":4358,"feed_emoji":"🔁","tokens_out":11495,"duration_ms":88641,"temperature":0.7,"pith_summary":"This paper aims to prove that indefinite integration preserves $\\alpha$ unpredictability, the outermost class in the hierarchy of recurrent functions, exactly when the integrated function remains bounded. Alpha unpredictable functions are Poisson-stable functions with an additional property: there are fixed positive constants $\\varepsilon_0$ and $\\delta$ and a sequence $s_n$ such that $|f(t+t_n)-f(t)|>\\varepsilon_0$ on intervals of length $\\delta$, marking them as the boundary between regular recurrence and chaos. The paper claims that if $f$ is $\\alpha$ unpredictable, then $F(t)=\\int_{t_0}^{t}f(s)\\,ds$ is $\\alpha$ unpredictable if and only if $F$ is bounded, and it constructs explicit divergence intervals and amplitudes for $F$ from those of $f$. A sympathetic reader would care because this closes the chain of invariance-under-integration results that began with periodic, quasi-periodic, almost periodic, and Poisson stable functions.","feed_headline":"Bounded integral of an alpha unpredictable function stays unpredictable","feed_subtitle":"Integration preserves recurrence for the outermost Poisson-stable class when the integral stays bounded.","key_machinery":"The central machinery is the method of included intervals, combined with sup/inf estimates on the primitive. To prove Poisson stability, the paper fixes a bounded interval $[a,b]$, chooses points $t_1,t_2$ at which $F$ is close to its supremum and infimum, encloses $[a,b]$ together with those points in a larger interval $[c,d]$, and uses uniform convergence of $f(t+t_n)$ on $[c,d]$ to force $F(t+t_n)-F(t)$ small on the larger interval and hence on $[a,b]$. For $\\alpha$ unpredictability, the key identity is $F(t+t_n)-F(t)=F(s_n+t_n)-F(s_n)+\\int_{s_n}^{t}(f(u+t_n)-f(u))\\,du$; the two-case split locates a short interval, of length $h=\\varepsilon_0\\delta/(8M)$, on which the shifted difference stays above $\\varepsilon_0\\delta/4$.","core_discovery":"The central claim is Theorem 1: if $f(t)$ is $\\alpha$ unpredictable, then $F(t)=\\int_{t_0}^{t}f(s)\\,ds$ is $\\alpha$ unpredictable if and only if $F$ is bounded. Lemma 1 establishes the Poisson-stability half: a bounded primitive of a Poisson stable function is Poisson stable, with the same convergence sequence $t_n$. The sufficiency proof for $\\alpha$ unpredictability then splits into two cases depending on whether $|F(s_n+t_n)-F(s_n)|$ already exceeds $\\varepsilon_0\\delta/2$; in the first case the divergence interval for $F$ is $[s_n,s_n+h]$, and in the second it is $[s_n+\\delta-h,s_n+\\delta]$, where $h=\\varepsilon_0\\delta/(8M)$, and in both cases the divergence amplitude is $\\varepsilon_0\\delta/4$. Corollary 1 extends the result to primitives of the form $ct+g(t)$: if $g$ is bounded, then $g$ is $\\alpha$ unpredictable.","pith_inferences":["A repair of the attainment step in Lemma 1 via approximation would make the theorem cover bounded primitives whose supremum and infimum are not attained; the current proof text assumes the exact equalities $F(t_1)=M-\\varepsilon/8$ and $F(t_2)=m+\\varepsilon/8$.","The two-case construction may iterate: higher-order primitives of alpha unpredictable functions should inherit alpha unpredictability as long as every intermediate primitive stays bounded, since each integration step can reuse the same divergence-sequence split.","The explicit constants suggest a numerical test: for a concrete alpha unpredictable function one can measure the actual divergence amplitude and interval length of the integral and compare them with the paper's $\\varepsilon_0\\delta/4$ and $\\varepsilon_0\\delta/(8M)$."],"forward_implications":["The chain of recurrence classes invariant under indefinite integration is now complete: periodic, quasi-periodic, almost periodic, Poisson stable, and alpha unpredictable functions all have the property that a bounded integral stays in the same class.","If an alpha unpredictable function has a bounded primitive, the integral operator maps the class into itself; unbounded primitives are excluded.","The paper's Corollary 1 gives a practical test: whenever an alpha unpredictable integrand has a primitive of the form $ct+g(t)$ with $g$ bounded, the bounded part $g$ is itself alpha unpredictable.","The divergence data for the integral are explicit: given $\\varepsilon_0,\\delta$ for $f$, one can exhibit intervals of length $h=\\varepsilon_0\\delta/(8M)$ with divergence amplitude $\\varepsilon_0\\delta/4$."],"supporting_citations":[{"why":"Supplies the sup/inf approach for proving Poisson stability of a bounded integral, which Lemma 1 adapts.","marker":"[6]"},{"why":"Provides the standard definition of Poisson stability and the already-known version of Lemma 1 that the paper re-proves.","marker":"[7]"},{"why":"Together with [7,10] it is cited for the definition of Poisson stable functions.","marker":"[9]"},{"why":"Cited alongside [7,9] for the Poisson stability definition underlying alpha unpredictability.","marker":"[10]"},{"why":"Defines alpha unpredictable functions, introduces the method of included intervals, and supplies the algebraic properties used in Corollary 1.","marker":"[12]"}],"fun_headline_variants":["Bounded integral keeps alpha unpredictability intact","If the integral is bounded, alpha unpredictability survives","Bounded primitives of alpha unpredictable functions stay alpha unpredictable","Alpha unpredictability is invariant under bounded integration","Integration preserves recurrence when the result stays bounded"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Lemma 1 assumes a bounded continuous function $F$ attains the particular values $M-\\varepsilon/8$ and $m+\\varepsilon/8$, where $M$ and $m$ are its highest and lowest values; bounded continuous functions need not attain their extremes, so these equalities can fail.","fun_headline_variants_meta":{"raw":{"variants":["Bounded integral keeps alpha unpredictability intact","If the integral is bounded, alpha unpredictability survives","Bounded primitives of alpha unpredictable functions stay alpha unpredictable","Alpha unpredictability is invariant under bounded integration","Integration preserves recurrence when the result stays bounded"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000267,"raw_usage":{"total_tokens":1572,"prompt_tokens":859,"completion_tokens":713,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":475,"completion_tokens_details":{"reasoning_tokens":642}},"tokens_in":475,"tokens_out":713,"duration_ms":5963,"temperature":1.0,"reasoning_tokens":642,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:22:57.757349+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $F(t+t_n)-F(t)$ for a bounded continuous Poisson-stable integrand $f$ whose primitive $F$ approaches its supremum only as $t\\to\\infty$ and never reaches $M-\\varepsilon/8$; if the difference fails to converge to zero uniformly on bounded intervals, Lemma 1 is false, and if it does converge, the attainment assumption is removable.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the sup/inf approach for proving Poisson stability of a bounded integral, which Lemma 1 adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the standard definition of Poisson stability and the already-known version of Lemma 1 that the paper re-proves."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Together with [7,10] it is cited for the definition of Poisson stable functions."},{"cited_title":"R.; Topological Dynamics and Ordinary Differential Equations.V an Nostrand Rein- hold Company, London,1971","cited_arxiv_id":null,"evidence_quote":"Cited alongside [7,9] for the Poisson stability definition underlying alpha unpredictability."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines alpha unpredictable functions, introduces the method of included intervals, and supplies the algebraic properties used in Corollary 1."}],"review_version":1}