{"id":"29dc7bce-c260-49f8-98f4-06e578843364","arxiv_id":"2607.07283","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A multi-time Markov renewal theory on partially ordered lattices is developed, yielding stratified inverse-renewal limits that are Gaussian on single-coordinate cells and non-Gaussian minima on interfaces, plus exact-time local theorems and Markovian augmentation criteria.","lead":"This paper extends Markov renewal theory to multi-dimensional time, where each transition carries a vector of durations and observation is over lower rectangles in a lattice. It yields stratified renewal theorems where inverse fluctuations are Gaussian on single-coordinate cells and minima of correlated Gaussians on interfaces, with applications to reliability and warranty modeling.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified. The stratified inverse limits are proved correctly via regenerative cycle inversion; a notational overload of N(k) in Section 5 is confusing but not erroneous.","rationale":"The reader correctly identifies Assumption 3.3 as the strongest structural hypothesis, but I assess it as a standard, clearly stated condition rather than a load-bearing correctness risk. The joint minorization is conventional in Nummelin splitting theory, and the paper provides a drift criterion (Theorem 3.11) that verifies both the minorization and the moment assumptions. The functional inverse theorem does rely entirely on the regenerative structure (no operator-theoretic alternative exists for the rectangular/inverse limits, unlike the exact-time local theorem which has both proofs), but this is a limitation of scope, not a flaw in the argument. My detailed trace of the proof of Theorem 5.6 found the inverse comparison argument, the uniformity over Λ via compactness and modulus of continuity, and the incomplete-cycle lemma all to be correct. The critical-interface theorem (5.7) correctly retains the √t·h displacement. The large deviation bounds (Theorem 5.3) correctly handle the rectangular asymmetry via Cramér's theorem with ε-enlargement. The lumpability criterion and the memorylessness obstruction are verified. The only issue I found is a notational overload of N(k) in Section 5, where the same symbol denotes both the cycle count and the renewal count. The proof correctly handles the distinction (via Lemma 5.1 and the D_N replacement), but the theorem statement does not flag the switch. This is a presentation issue that could confuse readers but does not affect the validity of the results. The paper's central claims hold under the stated assumptions. Confidence in ACCEPT is moderate because, as the reader notes, there is no independent formal verification of pure mathematics.","tokens_in":57837,"tokens_out":12449,"duration_ms":2632758,"concrete_test":"Verify the notational consistency in Theorem 5.6: confirm that the first convergence (centered at tκ(λ)) concerns the regenerative cycle count N(k) = sup{n: C_n ≤_d k}, while the second convergence (centered at tℓκ(λ)) concerns the renewal count from Section 2, obtained by replacing D_{N(k)} via Lemma 5.1. If both convergences were about the same quantity, the centering would be inconsistent by a factor of ℓ.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I traced the proof of Theorem 5.6 (functional inverse theorem) and Theorem 5.7 (critical-interface theorem) in detail. The core argument is sound: the invariance principle for i.i.d. regenerative cycle vectors (Y_j, L_j, R_j(g)) gives a Brownian limit; the inverse comparison sandwiches N(k_t(λ)) between n_t^+(λ) and n_t^-(λ) using the modulus of continuity of the partial-sum process on O(t^{-1/2}) intervals; inactive coordinates have a uniform deterministic margin of order t by compactness of Λ ⊂ Δ_I, which dominates the √t fluctuations. The Skorokhod representation subsequence argument is standard and correctly applied. The incomplete-cycle lemma (Lemma 5.1) correctly bounds the boundary contribution by max_{j≤ct} L_j = o_P(√t) under E_a[L_1^2] < ∞. The critical-interface theorem follows the same structure with the deterministic √t·h displacement retained, yielding the drifted minimum limit. The operator-theoretic local theorem (Theorem 6.5) uses standard Fourier-analytic saddle-point methods with correct contour shifting and exponential domination. The lumpability criterion (Theorem 7.2) and the memorylessness obstruction (Proposition 7.5) are correct — I verified the mixed finite difference gives negative mass for the product-form survival. The reader's concern about Assumption 3.3 (joint minorization on Q rather than P) is about applicability, not correctness: it is standard in the Nummelin splitting literature, clearly stated, and verified by the multiplicative drift criterion (Theorem 3.11). One genuine presentation issue: in Section 5 and Theorem 5.6, the symbol N(k) is used for both the regenerative cycle count (first convergence, centered at tκ(λ)) and the renewal count from Section 2 (second convergence, centered at tℓκ(λ)). The proof correctly distinguishes these — the second convergence replaces D_{N(k)} by the renewal count via Lemma 5.1 — but the theorem statement does not flag the switch. This is confusing but does not constitute a mathematical error,","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper develops a discrete Markov renewal theory on a standard Borel state space with vector-valued sojourn times and lower-rectangle observation on N^d. The core objects are a kernel-valued convolution resolvent (the Markov renewal potential) and a semi-Markov field observed on the partially ordered lattice. The paper establishes: (1) unified potential representations for semi-Markov transitions, first-passage laws, occupation measures, and rewards; (2) a Markovian augmentation of the semi-Markov field via the backward recurrence vector, with a strong lumpability criterion for projecting back to the original state space; (3) a stratified inverse-renewal theory decomposing the direction simplex into rate-determining cells, with Gaussian limits on single-coordinate cells and minima of correlated Gaussian fields on interfaces, including functional inverse limits (Theorem 5.6), critical-interface limits (Theorem 5.7), and logarithmic large-deviation estimates (Theorem 5.3); (4) an operator-theoretic local theorem for exact-time potentials via Fourier-Laplace perturbations of Markov-additive kernels (Theorem 6.5), with a regenerative periodic form (Theorem 6.7); and (5) finite-state specializations and reliability/warranty applications. The proofs use Nummelin splitting for regeneration, the Nagaev-Guivarc'h spectral method for local limits, and a subsequence/Skorokhod representation argument for the functional inverse theorems.","tokens_in":58120,"tokens_out":1682,"duration_ms":225588,"significance":"The paper makes a substantial contribution by extending classical Markov renewal theory to a genuinely multidimensional observation geometry. The stratification of the inverse-renewal problem—where the limit is Gaussian on single-coordinate cells but a minimum of correlated Gaussians on interfaces—is the central new structural insight and is rigorously established. The functional inverse theorem (Theorem 5.6) and critical-interface theorem (Theorem 5.7) are the principal results and are proved with explicit uniform bounds over compact directional sets. The operator-theoretic local theorem (Theorem 6.5) and its regenerative counterpart (Theorem 6.7) provide two independent routes to exact-time asymptotics, with the latter making arithmetic lattice classes explicit. The lumpability criterion (Theorem 7.2) and the memorylessness obstruction (Proposition 7.5) correctly identify why the backward recurrence vector is structural in dimensions d >= 2. The connection to reliability and warranty applications in Section 9 gives concrete finite-state formulas. The paper is largely self-contained, with complete proofs in the supplementary material for all deferred results.","major_comments":[{"comment":"Notational overload of N(k): In Section 2.2, N(k) = sup{n >= 0 : S_n <=_d k} is defined as the renewal count for the original Markov renewal chain. In Section 5, the same symbol N(k) = sup{n >= 0 : C_n <=_d k} is used for the count of complete regenerative cycles. These are different random variables (the latter counts cycles, not individual transitions), and the transition between the two uses is not explicitly flagged. While the proofs in Section 5 are internally consistent under the regenerative interpretation, a reader attempting to connect the asymptotic results of Section 5 back to the semi-Markov field quantities of Sections 2 and 4 may be confused. The authors should clarify the relationship, e.g., by noting that the regenerative cycle count governs the asymptotics and that the transition count D_{N(k)} differs from N(k) by the incomplete-cycle term controlled in Lemma 5.1.","section":null},{"comment":"Assumption 3.3 (joint minorization on Q rather than on P alone) is the load-bearing hypothesis for the entire regenerative framework: the split atom must regenerate the next state AND the following vector increment simultaneously. The paper states this is stronger than minorizing the embedded chain and provides the multiplicative drift criterion (Theorem 3.11) as verification. However, Theorem 3.11 is stated and proved only for the case where the minorization holds at one step. The remark that 'if the minorization holds for the m_0-step joint kernel... the construction is applied to the m_0-skeleton' is brief. Since the applicability of the stratified inverse theorems to concrete models depends on this assumption being verifiable, the authors should briefly indicate which standard model classes (e.g., finite-state irreducible, geometrically ergodic with exponential increments) satisfy it","section":null}],"minor_comments":[{"comment":"Section 2.2: The notation X_{N(k)+1} is used for the next renewal increment after the observation point, but X_n was not previously defined as the increment variable (S_{n+1} - S_n is used in Definition 2.5). Clarify that X_{n+1} = S_{n+1} - S_n.","section":null},{"comment":"Section 3.3: The directional rate rho_lambda and the rate-determining set I(lambda) are defined using mu (stationary mean per embedded transition), while in Section 5 the corresponding quantities kappa(lambda) and I(lambda) use m (mean cycle displacement). The equivalence m/ell = mu is noted, but the switch in notation between sections could be made more explicit.","section":null},{"comment":"Section 5, Theorem 5.6: The Brownian motion W = (W^Y, W^L, W^R) is said to have covariance equal to that of (Y_1, L_1, R_1(g)) under P_a. It would help to state that this is a (d+2)-dimensional Brownian motion and that W^Y is d-dimensional, W^L and W^R are scalar.","section":null},{"comment":"Section 6, Assumption 6.1 (S4): The notation |varrho(zeta)| is used for the modulus of the analytic continuation of the dominant eigenvalue, but varrho was defined as the eigenvalue itself (complex-valued). Clarify that |varrho(zeta)| denotes its modulus.","section":null},{"comment":"Section 7.2: The active-set decomposition uses W_B(u)(x) for the probability that the active set equals B, but W also denotes the Brownian motion in Section 5. This is a minor notational collision across sections.","section":null},{"comment":"Section 8.2: The finite-state periodic structure is defined via return increments to a fixed state i, and Proposition 8.3 shows L_i is independent of i. It would be useful to note that this coincides with the regenerative lattice L* of Section 3.3 when the atom is taken as a singleton state.","section":null},{"comment":"References: The companion works Kordalis and Trevezas (2025, 2026a, 2026b, 2026c) are cited extensively. Since 2026a and 2026b appear to be manuscripts/submissions, their availability should be noted for the reader.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a substantial and technically demanding contribution that extends Markov renewal theory to multidimensional observation. The central results (stratified inverse limits, critical-interface theorem, operator-theoretic local theorem) are correctly proved. The companion-paper dependency is significant but not circular: the present paper develops the probabilistic theory on general state spaces, while the companions handle the algebraic finite-state formalism and the unmodulated case. The two major comments are about clarity of presentation (notational overload of N(k)) and applicability transparency (joint minorization verification), neither of which affects correctness. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper builds a complete Markov renewal theory for vector-valued sojourn times observed over lower rectangles in N^d, on a general Borel state space. The headline result is a stratified functional inverse theorem — on each directional cell where a single coordinate determines the renewal rate, you get Gaussian fluctuations; on interfaces where several coordinates tie, the limit is a minimum of correlated Gaussians. This extends Hunter's 1974 two-dimensional scalar results to Markov-modulated cycles, general state spaces, and functional convergence over compact sets of directions. That is genuinely new and the proofs hold up. The critical-interface theorem (Theorem 5.7) giving a drifted minimum on the sqrt(t)-neighborhood of cell boundaries is a clean companion result. The operator-theoretic local theorem for exact-time potentials (Theorem 6.5) uses standard Fourier-analytic saddle-point methods correctly, and the lumpability criterion (Theorem 7.2) plus the memorylessness obstruction (Proposition 7.5) are correct — I checked the mixed finite difference argument. The supplementary material contains complete proofs for all deferred results, and the regenerative cycle inversion argument in Theorem 5.6 is sound: the Skorokhod subsequence reduction, the uniform modulus-of-continuity bounds, and the deterministic margin for inactive coordinates are all correctly handled. The reader flagged Assumption 3.3 (joint minorization on Q rather than just P) as the weakest structural assumption. I think this concern is overstated — it is standard in the Nummelin splitting literature, clearly stated, and verified by the multiplicative drift criterion in Theorem 3.11. It is an applicability condition, not a gap. One genuine presentation issue: in Section 5, N(k) is overloaded — it denotes both the regenerative cycle count and the renewal count from Section 2. The proof correctly distinguishes them, but the theorem statement does not flag the switch. Confusing, not erroneous. The self-citation pattern (four companion papers) is fine here because the cited works cover the algebraic and unmodulated directions, while the central results live in this paper. This is for probabilists working in renewal theory, Markov additive processes, or multiparameter stochastic structure. It deserves a serious referee.","headline":"Stratified inverse-renewal limits for Markov-modulated multi-time chains: genuine new theory, well-proved, one notational glitch","tokens_in":58993,"tokens_out":558,"would_cite":true,"duration_ms":161895,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Multi-time renewal theory partitions directions into Gaussian and non-Gaussian cells","keywords":[],"falsifier":"If a concrete finite-state two-clock semi-Markov model were found where the direction simplex has an interface with |I| >= 2 and the empirical inverse renewal count along directions approaching that interface does not converge to a minimum of correlated Gaussians but instead to a single Gaussian or to a different functional form, the stratified limit theory would fail.","tokens_in":58077,"feed_emoji":"🎯","tokens_out":1068,"duration_ms":187196,"temperature":0.7,"pith_summary":"Classical Markov renewal theory attaches a single scalar waiting time to each transition of an embedded Markov chain. This paper generalizes that picture to vector-valued waiting times on a d-dimensional lattice, where the observation geometry changes from an interval on the line to a lower rectangle in a partially ordered lattice. The central discovery is that this change of geometry stratifies the inverse renewal problem. When one asks how many renewals have occurred before a vector boundary is reached, the answer depends on which coordinate of the boundary is binding. The direction simplex decomposes into cells: on each cell where a single coordinate determines the rate, the inverse fluctuation is Gaussian; on interfaces where several coordinates simultaneously bind, the limit is the minimum of correlated Gaussian processes. The paper builds the full apparatus—kernel-valued convolution potentials, killed potentials, first-passage laws, reward functionals, and a Dynkin martingale—on a general Borel state space, then proves a functional inverse theorem giving convergence in the uniform norm over compact sets of directions, a critical-interface theorem for the boundary layer between cells, and logarithmic large-deviation bounds for rare inverse events. An operator-theoretic local theorem for Fourier-Laplace perturbations of Markov-additive kernels provides exact-time asymptotics, while a regenerative splitting argument supplies the arithmetic lattice-class corrections. Separately, the paper identifies the backward recurrence vector as the canonical Markovian augmentation of the observed semi-Markov field and gives a lumpability criterion for when that augmentation can be projected away.","feed_headline":"Multi-time renewal theory partitions directions into Gaussian and non-Gaussian cells","feed_subtitle":"Vector-valued waiting times on lattices produce stratified inverse limits: Gaussian on single-coordinate cells, minima of correlated Gaussi~","key_machinery":"The kernel-valued convolution potential psi = sum of q^{(n)} on the partially ordered lattice (Nd, <=d), with the multi-time convolution (A*B)_k = sum_{r<=k} A_r B_{k-r}; the Nummelin splitting of the joint state-increment kernel Q (not just the embedded chain P) to create a regenerative atom that simultaneously regenerates state and vector increment; the backward recurrence vector U_k = k - S_{N(k)} as the canonical Markovian augmentation; and the Fourier-Laplace kernel Q_z f = sum_k e^{z.k} integral f(y) q_k(x,dy) for the operator-theoretic local theorem.","core_discovery":"The rate-determining partition of the direction simplex is the load-bearing structure. For each direction lambda in the simplex, the set of coordinates achieving the minimum of lambda_r/mu_r determines the renewal rate. When this set has one element, the centered inverse renewal count has a Gaussian limit. When it has several elements, the limit is the minimum of correlated Gaussian coordinates—a non-Gaussian distribution that cannot be obtained by picking one of the adjacent Gaussian cells. The critical-interface theorem shows that on the square-root-of-t neighborhood of a boundary between cells, the limit is a drifted minimum of correlated Gaussians, interpolating between the Gaussian str~","pith_inferences":[],"forward_implications":["The stratified limit structure means that any system modeled by multi-dimensional semi-Markov renewal—reliability with age-usage warranties, queueing with multiple resources, biological exposure models—will exhibit non-Gaussian inverse fluctuations when the operating point sits near a direction where multiple resource constraints bind simultaneously.","The lumpability criterion for suppressing the backward recurrence vector provides a testable condition: the physical state process alone is Markov if and only if the transition probabilities from all admissible ages in the same state fiber coincide, which in dimension one reduces to geometric sojourn times but in higher dimensions has no direct multivariate-geometric analogue.","The operator-theoretic local theorem and the regenerative periodic theorem give two independent routes to exact-time asymptotics, with the regenerative route making lattice-class corrections explicit—useful for finite-state computational models where the period structure matters.","The killed-potential and crossing-kernel decomposition separates first-failure reliability from repeated-failure availability in a single algebraic framework, directly applicable to warranty cost analysis with rectangular coverage regions."],"fun_headline_variants":["Direction simplex governs Gaussian vs correlated-minimum renewal limits","Inverse renewal rate set by active coordinates in the direction simplex","Markov renewal counts: Gaussian on cells, minima of Gaussians on interfaces","Critical interfaces interpolate Gaussian and non-Gaussian renewal limits","Renewal direction partition yields Gaussian and correlated-Gaussian regimes"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The joint minorization condition requires a small set that simultaneously regenerates the next state and the following vector-valued time increment—not just the state alone. If no such set exists, the Nummelin splitting that underpins the regenerative cycles, and hence the entire functional inverse theorem and its stratified limits, cannot be constructed.","fun_headline_variants_meta":{"raw":{"variants":["Direction simplex governs Gaussian vs correlated-minimum renewal limits","Inverse renewal rate set by active coordinates in the direction simplex","Markov renewal counts: Gaussian on cells, minima of Gaussians on interfaces","Critical interfaces interpolate Gaussian and non-Gaussian renewal limits","Renewal direction partition yields Gaussian and correlated-Gaussian regimes"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":672,"prompt_tokens":587,"completion_tokens":85,"prompt_tokens_details":null},"tokens_in":587,"tokens_out":85,"duration_ms":24792,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T15:13:45.043265+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If a concrete finite-state two-clock semi-Markov model were found where the direction simplex has an interface with |I| >= 2 and the empirical inverse renewal count along directions approaching that interface does not converge to a minimum of correlated Gaussians but instead to a single Gaussian or to a different functional form, the stratified limit theory would fail.","supporting_citations":[],"review_version":1}