{"id":"5b7eb8ce-333a-4ac4-bec2-b66e822ecc26","arxiv_id":"2607.01374","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Derives discretization-free algebraic conditions for second-moment stability boundaries of linear time-invariant stochastic DDEs via reduction of a correlation-function boundary-value problem.","lead":"The paper derives semi-analytic algebraic equality conditions for identifying second-moment stability boundaries in linear stochastic delay-differential equations with constant delay and mixed noise, without discretizing the system. These conditions enable parameter continuation and closed-form results in one dimension, offering computational savings over discretization methods.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Stability boundary identified with loss of uniqueness in reduced BVP rests on discretized spectral observation without continuous-operator proof","rationale":"The reader's weakest_assumption correctly isolates the step whose justification is thinnest; the full text confirms the step is presented as motivated by discretization rather than proven for the continuous problem. This moves the verdict from UNVERDICTED to CONDITIONAL pending the proposed check, while preserving the paper's other technical contributions.","tokens_in":1732,"tokens_out":382,"duration_ms":13383,"concrete_test":"Fix a low-dimensional test case (e.g., scalar SDDE with multiplicative noise). Compute the algebraic boundary from the non-uniqueness condition on the reduced BVP; independently discretize the infinitesimal generator on successively finer grids (N=32,64,128) and track the parameter value at which its spectral abscissa reaches zero. If the algebraic value and the extrapolated continuous crossing differ by more than discretization error, the identification fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central construction reduces the three-variable correlation BVP to a two-variable delay-differential BVP and then equates second-moment stability boundaries to parameter values at which that BVP loses uniqueness of stationary solutions. This equivalence is motivated solely by the numerical observation that, on a discretized infinitesimal generator, stability is lost precisely when a real eigenvalue crosses the origin. No direct spectral analysis of the continuous (infinite-dimensional) operator is supplied to show that other crossings (complex eigenvalues, essential spectrum, etc.) cannot produce instability without a real zero eigenvalue, nor is a rigorous passage-to-the-limit argument given that the discretized crossing implies the continuous one. Because the algebraic conditions are obtained exactly by imposing the non-uniqueness condition on the reduced BVP, any gap in this identification directly undermines the claim that the resulting equalities locate the true stability boundaries.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript derives semi-analytic algebraic equality conditions for second-moment stability boundaries of linear time-invariant stochastic delay-differential equations with constant delay and both multiplicative and additive noise. It proceeds by deriving an advection-type BVP for a three-variable correlation function, reducing it to a delay-differential BVP for a two-variable correlation function, and identifying stability boundaries with parameter values at which the reduced BVP loses uniqueness of stationary solutions; this identification is motivated by the observation that, on a discretized infinitesimal generator, stability is lost precisely when a real eigenvalue crosses the origin. The resulting conditions are validated against Monte Carlo simulations and published results for low-dimensional models, with closed-form expressions available for the scalar case, and are shown to enable discretization-free parameter continuation that scales with the square of the system dimension.","tokens_in":1902,"tokens_out":435,"duration_ms":15899,"significance":"If the central identification between stability loss and loss of uniqueness in the reduced BVP can be placed on a rigorous footing, the work supplies a scalable, discretization-free route to stability boundaries that improves computational cost relative to existing methods and clarifies limitations of prior algebraic conditions in the literature.","major_comments":[{"comment":"Abstract: the equivalence between second-moment stability boundaries and loss of uniqueness of stationary solutions to the reduced two-variable delay-differential BVP is motivated solely by the numerical observation that a real eigenvalue of the discretized infinitesimal generator crosses the origin; no direct spectral analysis of the continuous (infinite-dimensional) operator is supplied to show that other crossings (complex eigenvalues, essential spectrum) cannot produce instability without a real zero eigenvalue, nor is a rigorous passage-to-the-limit argument given that the discretized crossing implies the continuous one.","section":"Abstract"},{"comment":"Abstract: because the algebraic conditions are obtained exactly by imposing the non-uniqueness condition on the reduced BVP, the absence of a continuous-operator justification for the identification directly undermines the claim that the resulting equalities locate the true stability boundaries.","section":"Abstract"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for highlighting the need for greater rigor in the central identification of our work. We respond to the major comments point by point below.","responses":[{"response":"We agree that the identification is motivated by the observed crossing of a real eigenvalue through the origin in the discretized generator and that no direct spectral analysis of the continuous operator (addressing complex eigenvalues or essential spectrum) or passage-to-the-limit argument is provided. In the revised manuscript we have added explicit language in the abstract and introduction stating that the algebraic conditions rest on this numerically motivated identification, and we have inserted a brief discussion of why other crossings are not expected on the basis of the structure of the correlation equations. A complete operator-theoretic proof lies beyond the present scope.","revision_made":"partial","referee_comment":"[Abstract] Abstract: the equivalence between second-moment stability boundaries and loss of uniqueness of stationary solutions to the reduced two-variable delay-differential BVP is motivated solely by the numerical observation that a real eigenvalue of the discretized infinitesimal generator crosses the origin; no direct spectral analysis of the continuous (infinite-dimensional) operator is supplied to show that other crossings (complex eigenvalues, essential spectrum) cannot produce instability without a real zero eigenvalue, nor is a rigorous passage-to-the-limit argument given that the discretized crossing implies the continuous one."},{"response":"The referee is correct that, without a rigorous continuous-operator justification, the derived equalities locate the stability boundaries only under the stated identification. We have revised the abstract to describe the conditions as those obtained by imposing non-uniqueness on the reduced BVP under the identification supported by discretization evidence and low-dimensional validation, rather than asserting an unconditional equivalence. The computational advantages and empirical agreement with Monte Carlo simulations remain as reported.","revision_made":"partial","referee_comment":"[Abstract] Abstract: because the algebraic conditions are obtained exactly by imposing the non-uniqueness condition on the reduced BVP, the absence of a continuous-operator justification for the identification directly undermines the claim that the resulting equalities locate the true stability boundaries."}],"tokens_in":1440,"tokens_out":495,"duration_ms":45199,"standing_objections":["A rigorous spectral analysis of the continuous infinite-dimensional operator establishing that stability loss occurs precisely when a real eigenvalue crosses the origin, together with a passage-to-the-limit argument from the discretized to the continuous setting."]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that the paper reduces a three-variable correlation advection BVP to a two-variable delay-differential BVP and then extracts algebraic equality conditions for second-moment stability boundaries by imposing loss of uniqueness on the reduced problem. For one-dimensional cases this yields closed-form expressions in elementary functions, and the approach scales with the square of the system dimension rather than requiring full discretization. They validate the resulting boundaries against Monte Carlo simulations and earlier published results on several low-dimensional examples, and they demonstrate the computational savings from using parameter continuation on the algebraic conditions.\n\nThe reduction step itself looks like a genuine technical step beyond the discretization-heavy literature. The validations are concrete and show the method works on the tested models.\n\nThe soft spot is the justification for linking stability loss directly to non-uniqueness in the reduced BVP. The paper motivates this by the observation that, on a discretized infinitesimal generator, stability fails precisely when a real eigenvalue crosses the origin. No spectral analysis of the continuous operator is given to confirm that other crossings cannot produce instability, and there is no passage-to-the-limit argument. Because the algebraic conditions are derived exactly from the non-uniqueness requirement, any gap here affects the claim that the equalities locate the true boundaries. The numerical matches help, but the link remains an assumption rather than a proved equivalence.\n\nThis is for researchers working on stability of stochastic DDEs who need efficient boundary calculations for modest dimensions. A reader focused on semi-analytic methods for delay systems would find the reduction and the 1D closed form useful. The work has enough new technique and supporting checks to merit a serious referee, even if the spectral identification needs tightening.","headline":"The algebraic conditions from the BVP reduction offer a discretization-free route to stability boundaries with good numerical checks, but the identification of those boundaries with loss of uniqueness rests only on a discretized spectral observation.","tokens_in":2357,"tokens_out":417,"would_cite":false,"duration_ms":19388,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Algebraic equality conditions identify second-moment stability boundaries for stochastic delay equations without discretization.","keywords":["stochastic delay-differential equations","second-moment stability","algebraic stability conditions","boundary-value problems","correlation functions","multiplicative noise","parameter continuation","infinitesimal generator"],"falsifier":"A Monte Carlo simulation of a concrete low-dimensional stochastic delay equation in which the algebraic condition predicts a stability boundary but the simulated second moments either remain bounded or diverge on the opposite side of the predicted curve.","tokens_in":2629,"feed_emoji":"","tokens_out":795,"duration_ms":17454,"temperature":0.7,"pith_summary":"This paper derives optimal semi-analytic algebraic equality conditions for locating the boundaries at which second-moment stability is lost in linear time-invariant stochastic delay-differential equations that include a single constant delay together with both multiplicative and additive noise. The conditions arise from reducing an advection-type boundary-value problem for a three-variable correlation function to a delay-differential boundary-value problem for a two-variable correlation function, then equating stability loss with loss of uniqueness of stationary solutions to the reduced problem. A reader would care because these boundaries determine whether the statistical second moments of the solution remain bounded, which governs long-term behavior in applications such as control or biological systems subject to delay and noise. The resulting conditions scale only with the square of the system dimension and support direct use of parameter continuation methods; in the one-dimensional case they become fully closed-form expressions in elementary functions. Validation against Monte Carlo simulations and earlier published results for low-dimensional models shows agreement while avoiding the computational cost of discretization.","feed_headline":"Algebraic conditions locate stochastic delay stability boundaries","feed_subtitle":"Equality conditions derived from reduced correlation equations find second-moment boundaries without discretization and scale with dimension","key_machinery":"The reduced delay-differential boundary-value problem for the two-variable correlation function, whose stationary solutions lose uniqueness exactly at the second-moment stability boundaries.","core_discovery":"For linear time-invariant stochastic delay-differential equations with a single constant delay and both multiplicative and additive noise, second-moment stability boundaries are identified with the loss of uniqueness of stationary solutions to a reduced delay-differential boundary-value problem for a two-variable correlation function. This identification is motivated by the observation that stability is lost when a real eigenvalue of the discretization of the corresponding infinitesimal generator passes through the origin. The reduction begins with an advection-type boundary-value problem with non-local boundary conditions for a three-variable correlation function. The resulting algebraic eq","pith_inferences":["The same reduction strategy could be examined for systems with multiple distinct delays if analogous correlation equations can be derived.","The algebraic conditions supply a practical test that could be embedded inside optimization routines seeking parameter values that keep second moments stable.","Higher-dimensional numerical implementations of the equality conditions could be benchmarked against existing discretization codes to quantify the scaling advantage beyond the low-dimensional examples already checked."],"forward_implications":["Second-moment stability boundaries become computable by applying parameter continuation directly to the discretization-free algebraic equality conditions.","Computational effort scales only with the square of the system dimension rather than with a chosen discretization resolution.","In the one-dimensional case the stability condition reduces to an explicit expression in elementary functions.","The algebraic conditions can be used to test and clarify limitations of previously published stability criteria for the same class of equations."],"fun_headline_variants":["Algebraic conditions mark DDE stability boundaries","Algebra locates stochastic delay stability limits","Second-moment stability found without discretization","Algebraic eqs reveal stochastic delay boundaries"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Second-moment stability is lost precisely when a real eigenvalue of the discretized infinitesimal generator passes through the origin, which corresponds to non-uniqueness of stationary solutions to the reduced correlation boundary-value problem.","fun_headline_variants_meta":{"raw":{"variants":["Algebraic conditions mark DDE stability boundaries","Algebra locates stochastic delay stability limits","Second-moment stability found without discretization","Algebraic eqs reveal stochastic delay boundaries"]},"model":"grok-4.3","cost_usd":0.007425,"raw_usage":{"total_tokens":3433,"prompt_tokens":711,"num_sources_used":0,"completion_tokens":49,"cost_in_usd_ticks":74249500,"prompt_tokens_details":{"text_tokens":711,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2673,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":711,"tokens_out":49,"duration_ms":21094,"temperature":1.0,"reasoning_tokens":2673,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-03T18:27:31.385786+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A Monte Carlo simulation of a concrete low-dimensional stochastic delay equation in which the algebraic condition predicts a stability boundary but the simulated second moments either remain bounded or diverge on the opposite side of the predicted curve.","supporting_citations":[],"review_version":1}