{"id":"146b3f0b-a7b9-450f-8664-6b0df1f49175","arxiv_id":"2506.05002","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For scalar linear delay-difference equations, local and global strong stability are equivalent, and both are characterized by the total variation of the delay measure being strictly less than 1.","lead":"The paper proves a complete stability criterion for scalar delay-difference equations with distributed and pointwise delays: strong stability holds exactly when the total variation of the defining measure is below one. It also establishes well-posedness and gives numerical examples showing that strong stability is more restrictive than ordinary exponential stability.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 1⇒3 direction of Theorem 11 rests on [24, Lemma 1] without stating or verifying its hypotheses; if that lemma is not applicable to the continuous solutions of (6), the sufficiency proof is incomplete.","rationale":"The paper's central theorem, Theorem 11, characterizes strong stability of scalar delay-difference equations by the total variation of the defining measure. I examined the three implications. 3⇒2 is immediate. 2⇒1 uses Hahn/Jordan decomposition and constructs a piecewise-constant perturbation φ within ε of the identity; the pushforward measure is verified to lie in W (since φ(θ)<0 for all θ in the construction), and the resulting discrete-delay system fails the Hale–Silkowski spectral condition. This direction is fully rigorous. 1⇒3 is the only step that invokes an external result without proof: the Mazenc–Malisoff lemma. The inequality |x(t)|≤L∥x_t∥∞ is correctly derived, and the lemma, as known, applies to exactly this type of inequality for continuous trajectories, even when the measure has an atom at 0. Thus the mathematical claim is sound. The paper's failure to state the lemma's hypotheses is a presentation gap, not a correctness gap. A direct induction (M_k≤L M_{k-1}) supplies a self-contained proof. For this reason the concern does not change the reader's ACCEPT verdict.","tokens_in":10361,"tokens_out":20473,"duration_ms":225610,"concrete_test":"Check whether the pointwise inequality |x(t)|≤L∥x_t∥∞ (L<1) alone implies exponential stability by a direct induction: set M_{-1}=∥x_0∥∞, M_k=sup_{t∈[k,k+1]}|x(t)|, and prove M_k≤L M_{k-1} from the inequality. If this succeeds for every continuous solution, then [24, Lemma 1] is indeed applicable, and Theorem 11(1⇒3) is sound; if a gap appears (e.g., the lemma needs a retarded structure with no instantaneous term), the sufficiency proof must be patched by a self-contained argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Direction 1⇒3 of Theorem 11 is the only place where the equivalence could fail: for every φ∈B with φ∗μ∈W, any continuous global solution x of (6) satisfies |x(t)| ≤ L∥x_t∥∞ with L=∫d|μ|<1, and the paper concludes exponential stability by citing [24, Lemma 1] without stating the lemma's assumptions. The cited lemma as commonly stated applies to any function satisfying such a pointwise bound, but the paper does not verify boundedness or other regularity hypotheses, nor does it discuss the case where φ∗μ carries an atom at 0 (so x(t) appears in the RHS of (6)); in that case the inequality is still valid, but the reduction to a difference equation with no instantaneous term changes. All other directions (3⇒2 trivial, 2⇒1 via Hahn decomposition and Hale–Silkowski) are proved in detail, so this citation is the single load-bearing step. If the lemma requires extra hypotheses, e.g., a priori boundedness of x on [-1,∞), the proof as written is incomplete even though the claim is likely true.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper considers linear delay-difference equations of the form (1), x(t)=∫_{-1}^0 dM(θ)x(t+θ), with M a matrix-valued function of bounded variation. It establishes a well-posedness result for this class (Proposition 6), recalls the Hale–Silkowski criterion for pointwise-delay systems (Theorem 8), and proposes a definition of strong stability for the distributed-delay setting by allowing Borel-measurable perturbations of the delay, i.e., replacing θ by φ(θ) in the integral (Definition 9). The main result (Theorem 11) states that, in the scalar case n=1, local strong stability, global strong stability, and the total variation condition ∫d|µ|<1 are equivalent. The paper also provides numerical examples illustrating the region of strong stability for an affine density and comparing it with the region of exponential stability.","tokens_in":10548,"tokens_out":21546,"duration_ms":244567,"significance":"Should the main theorem hold, the paper gives a strikingly simple and parameter-free characterization of robust stability for scalar distributed-delay difference equations: the only quantity that matters is the total variation of the measure. The necessity direction is elegant, combining Hahn–Jordan decomposition with a partition argument that reduces the problem to the classical Hale–Silkowski criterion; the sufficiency direction uses a pointwise inequality and a cited lemma. The paper also contains a useful well-posedness theorem and numerical experiments that clearly exhibit the gap between exponential stability and strong stability. The result has the potential to be useful in applications to hyperbolic PDEs with in-domain couplings, where distributed delay terms arise naturally.","major_comments":[{"comment":"The proof of the direction 1⇒3 in Theorem 11 is the only load-bearing step that is not proved in detail: it consists of the sentence 'Hence, from [24, Lemma 1], (6) is exponentially stable.' The manuscript neither states the lemma nor verifies that the continuous solutions of (6) satisfy its hypotheses. In particular, if φ*µ has an atom at 0, then (6) contains an instantaneous term and the inequality |x(t)|≤L‖x_t‖∞ (with L<1) does not by itself put the system in the standard small-gain form without extra reasoning. Since this is the sole argument for global strong stability under condition (7), the proof is incomplete as written. The gap is readily fixable by either quoting the lemma explicitly with all hypotheses or by giving a direct induction on unit intervals, which would also handle the atom-at-zero case; the claim itself is very likely correct.","section":"Section 3.3, proof of Theorem 11 (direction 1⇒3)"}],"minor_comments":[{"comment":"The notation −τP k and −τN k is typeset in a way that makes the indices unclear; please use a clearer notation such as −τ_{P,k} and −τ_{N,k}.","section":"Section 3.3, proof of Theorem 11"},{"comment":"The partition of [a,b] is written as {[t_i,t_{i+1}] : 1≤i≤k}; it would be clearer to state explicitly that t_0=a and t_{k+1}=b.","section":"Section 3, Definition 3"},{"comment":"The computation of Var M|_{[-1,0]} for the affine density would be easier to follow if the cases b=0 and b≠0 were treated separately, with the convention θ_v=0 for b=0 stated as a limiting case.","section":"Section 4.1"},{"comment":"The norm ∥·∥_2 used in the definition of α is not defined; please specify that it is the induced Euclidean norm, or state that the bounds are norm-independent.","section":"Section 4.2"},{"comment":"After the change of variables from M to N, it would be helpful to explicitly note that N has no atom at 0, so that det(I−A_N)=1 and the contraction argument applies.","section":"Section 2, proof of Proposition 6"}],"recommendation":"major_revision","confidential_remarks":"The paper is well written and the main result is almost certainly correct. The single substantive issue is the unstated lemma in the sufficiency proof; this is a local fix. I see no reason to reject, and the result is within the scope of the journal. The numerical section is illustrative and does not affect the main mathematical claims. I recommend major revision only to ensure the proof is self-contained or explicitly verified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's my take on arXiv:2506.05002. It's a genuinely good paper, and the main result holds up. Theorem 11 — scalar delay-difference equations with a measure-valued distributed delay are strongly stable iff the total variation of the measure is less than 1 — is a clean, real extension of Melvin and Hale-Silkowski. The paper also refines the definition of strong stability so that perturbations hit the distributed term, not just pointwise delays, which is a meaningful improvement over earlier work.\n\nThe proof of 2⇒1 is the best part: Hahn-Jordan decomposition plus a partition trick converts the distributed system into a finite pointwise-delay system, then applies the classical criterion. The sufficiency direction is a one-line bound plus an appeal to Mazenc-Malisoff. The stress-test worry is that this lemma is not stated and its hypotheses not verified. I don't think that's a real gap. The bound |x(t)| ≤ L∥x_t∥∞ with L<1 is exactly the kind of inequality the lemma is designed for; continuity of the solution comes from Proposition 6, and the inequality itself handles boundedness. A referee should ask them to state the lemma, but it's a presentation issue, not a load-bearing flaw.\n\nThe numerical section is illustrative rather than reproducible: they name the QPmR algorithm and give a GitHub link, but no grid parameters or raw code. Minor for a math paper.\n\nThe paper is cleanly written, the literature is fair, and the claim is new and useful. It deserves a serious referee and will be of value to anyone working on neutral systems, delay-difference stability, or hyperbolic PDE control.\n\nFor review, I'd recommend accept with minor revisions: state the Mazenc-Malisoff lemma or its hypotheses, and make the numerics a touch more concrete.","headline":"A clean, genuine extension of Melvin's criterion to distributed delays; the load-bearing citation worry is a non-issue.","tokens_in":11087,"tokens_out":4893,"would_cite":true,"duration_ms":55330,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34K20","34K06","93D09"],"pacs":[],"model":"deepseek-v4-flash","headline":"For scalar linear delay-difference equations, strong stability is equivalent to the total variation of the defining measure being less than 1.","keywords":["delay-difference equations","distributed delays","pointwise delays","strong stability","Hale-Silkowski criterion","Melvin Criterion","total variation","measure-valued delay"],"falsifier":"A concrete disproof would be a bounded-variation scalar measure with $\\int_{-1}^0 d|\\mu| < 1$ and an admissible Borel map $\\varphi$ for which the characteristic equation of the perturbed system (6) has a root in the closed right half-plane.","tokens_in":10158,"feed_emoji":"📏","tokens_out":11576,"duration_ms":114119,"temperature":0.7,"pith_summary":"This paper studies linear delay-difference equations, where the state at time $t$ is a Riemann–Stieltjes integral of past states against a matrix-valued measure. It proves that, in the scalar case, the system is stable under all admissible small perturbations of the delays if and only if the total variation of the measure over $[-1,0]$ is less than 1. That same condition characterises global strong stability, i.e. stability under every admissible Borel-measurable rearrangement of the delays. This extends the classical Melvin Criterion from finitely many pointwise delays to equations with distributed delays. A sympathetic reader would care because it converts a qualitative stability guarantee into a computable scalar check.","feed_headline":"One number decides strong stability of delay equations","feed_subtitle":"A single total-variation check decides whether all delay perturbations stay stable.","key_machinery":"The central object is the scalar total variation $\\int_{-1}^0 d|\\mu|$ of the matrix-valued Borel measure $\\mu$ defining the delay term, since this single number separates strong stability from its failure for scalar equations. The argument's machinery includes the Hahn–Jordan decomposition, which splits $\\mu$ into positive and negative parts to construct piecewise-constant rearrangement maps $\\varphi$ with rationally independent delays; the Hale–Silkowski Criterion, which decides exponential stability of the resulting finite point-delay systems; and a cited trajectory-based lemma converting the pointwise bound $|x(t)|\\le L\\|x_t\\|_\\infty$ with $L<1$ into exponential stability.","core_discovery":"Theorem 11 states that, for $n=1$, the following are equivalent: the total variation of the measure associated with $M$ satisfies $\\int_{-1}^0 d|\\mu|(\\xi) < 1$; system (1) is locally strongly stable; and system (1) is strongly stable. Strong stability means exponential stability for every Borel-measurable map $\\varphi:[-1,0]\\to[-1,0]$ such that the pushed-forward measure $\\varphi_*\\mu$ is admissible, and local strong stability means the same for all $\\varphi$ close to the identity in the uniform norm. The proof of the nontrivial implications uses the Hale–Silkowski Criterion for pointwise-delay systems to show that total variation at least 1 produces an unstable nearby system, and a trajectory-wise bound together with a cited lemma to show that total variation below 1 forces exponential stability of every admissible perturbation.","pith_inferences":["The paper leaves the multi-dimensional case open; by analogy with (4), a natural candidate would be a spectral-radius condition such as $\\sup_{\\theta}\\rho\\!\\left(\\int_{-1}^0 e^{i\\theta}\\,dM(\\theta)\\right) < 1$.","Because the definition quantifies over all Borel-measurable maps, total variation rather than finer spectral data is the decisive quantity for this very demanding stability notion.","A testable byproduct is that random piecewise-constant delay rearrangements with rationally independent values should destabilise any scalar system whose total variation is at least 1, matching the construction in the proof."],"forward_implications":["For any scalar equation of the form (1), checking strong stability reduces to one scalar: the total variation of the delay measure on $[-1,0]$ must be less than 1.","Local and global strong stability are equivalent in the scalar case, so stability under all sufficiently small delay perturbations implies stability under every admissible Borel-measurable rearrangement of the delays.","When the delay measure is a finite sum of point masses, the criterion becomes $\\sum_k |A_k| < 1$, recovering Melvin's Criterion for pointwise-delay equations.","The affine-density example yields an explicit strong-stability region in the $(a,b)$-plane, and the numerical comparison shows that exponential stability alone does not imply strong stability.","Proposition 6 establishes global existence and uniqueness for the $n$-dimensional equation under the condition $\\det(I-A_M)\\ne 0$, so the stability analysis rests on a well-posed solution theory."],"supporting_citations":[{"why":"Supplies the existence and uniqueness strategy adapted to prove global well-posedness in Proposition 6.","marker":"[19]"},{"why":"The original scalar Melvin Criterion that Theorem 11 generalises to distributed delays.","marker":"[25]"},{"why":"Provides the Silkowski half of the Hale–Silkowski Criterion used for pointwise-delay systems.","marker":"[29]"},{"why":"Hale's parametric-stability result that completes the Hale–Silkowski Criterion.","marker":"[17]"},{"why":"Characterises the zeros of exponential polynomials, underpinning the spectral condition in Theorem 8.","marker":"[2]"},{"why":"Supplies the trajectory-based lemma that converts the bound $L<1$ into exponential stability in the sufficiency direction.","marker":"[24]"},{"why":"Provides the Hahn–Jordan decomposition used to split the measure in the instability construction.","marker":"[13]"}],"fun_headline_variants":["Total variation below 1 guarantees strong stability","One total-variation test decides delay equation stability","Scalar delay equations: total variation < 1 ensures strong stability","Strong stability iff total variation is less than 1","Total variation < 1: scalar delay equations strongly stable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the direction 'total variation less than 1 implies strong stability' assumes that the cited trajectory-based stability lemma applies to the continuous solutions of the pushed-forward equation, and that every admissible perturbation keeps the system well-posed through $\\varphi_*\\mu\\in W$; if either fails, that implication is unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Total variation below 1 guarantees strong stability","One total-variation test decides delay equation stability","Scalar delay equations: total variation < 1 ensures strong stability","Strong stability iff total variation is less than 1","Total variation < 1: scalar delay equations strongly stable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001279,"raw_usage":{"total_tokens":5175,"prompt_tokens":838,"completion_tokens":4337,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":454,"completion_tokens_details":{"reasoning_tokens":4259}},"tokens_in":454,"tokens_out":4337,"duration_ms":32503,"temperature":1.0,"reasoning_tokens":4259,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:30:15.468911+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete disproof would be a bounded-variation scalar measure with $\\int_{-1}^0 d|\\mu| < 1$ and an admissible Borel map $\\varphi$ for which the characteristic equation of the perturbed system (6) has a root in the closed right half-plane.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the existence and uniqueness strategy adapted to prove global well-posedness in Proposition 6."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The original scalar Melvin Criterion that Theorem 11 generalises to distributed delays."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Silkowski half of the Hale–Silkowski Criterion used for pointwise-delay systems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Hale's parametric-stability result that completes the Hale–Silkowski Criterion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Characterises the zeros of exponential polynomials, underpinning the spectral condition in Theorem 8."},{"cited_title":"Mazenc and M","cited_arxiv_id":null,"evidence_quote":"Supplies the trajectory-based lemma that converts the bound $L<1$ into exponential stability in the sufficiency direction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Hahn–Jordan decomposition used to split the measure in the instability construction."}],"review_version":1}