{"id":"8400e752-14c9-4f8f-9fd9-1f825b5e6444","arxiv_id":"2512.01969","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A higher-order Markov chain cannot in general be treated through its reduced first-order chain: ergodicity, ever-reaching probabilities, mean first passage times, and state classification can all fail to transfer, though limiting distributions do.","lead":"This math paper asks whether any higher-order Markov chain can be safely studied by converting it to a first-order chain on sliding windows of states, and shows the answer is mostly no. It collects concrete examples where ergodicity, passage times, and state classification differ, while k-step probabilities and limiting distributions transfer cleanly.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central 'negative in general' claim is overstated: Q is informationally equivalent to X, so ergodicity, ever-reaching probabilities, and MFPT are recoverable from Q as set-level quantities (target sets A_i), not just via state-level equality; Section 6's MFPT mismatch is not an obstruction.","rationale":"The reader identified the weakest assumption as the paper's failure to analyze alternative first-order embeddings. My concern is related but more direct: even with the same sliding-window reduction Q, the higher-order chain's properties can be recovered by translating target states into subsets of T. Thus the negative conclusion is not forced; it depends on an implicit and undefended choice to compare only single-state properties of Q with the corresponding properties of X. This is a load-bearing framing issue for the central claim, because it affects what 'treated as a first order chain' means and whether the title question is answered affirmatively or negatively. The mathematical examples in Sections 4-6 are correct as computations; the issue is that they establish only the failure of direct equality, not the impossibility of using Q to study X. The paper even hints at this in Section 6 when it mentions matricization as a workaround, but it does not acknowledge the set-hitting translation. This reinforces the reader's CONDITIONAL verdict: the paper should either prove that no set-level translation through Q can recover these quantities, or substantially qualify the Section 8 conclusion. I did not move the verdict because the underlying examples remain correct and the paper's narrower Question 2.1 could be defended; a revision of the interpretation would likely suffice. I also noticed a separate issue in the proof of Theorem 5.1: the concatenation of events A, B, C for alpha >= m-1 appears to depend on the intermediate states after A, which are not fixed by the event A. However, Theorem 5.1 is not load-bearing for the central negative claim, so I focused on the set-level translation concern.","tokens_in":14842,"tokens_out":14368,"duration_ms":150079,"concrete_test":"Recompute the mean first passage time in the Section 6 example (P=J/3) using the reduced chain Q: solve the set-hitting system h_u = 1 + sum_v q_{u,v} h_v for u not in A_1, with h_v=0 for v in A_1={11,12,13}, and evaluate h_11 by the same equation (not as a boundary). If the result equals mu_{111}=3 from equation (6.1), then Q can be used to compute the higher-order MFPT, contradicting the paper's negative-in-general framing. The analogous check for ever-reaching probabilities uses the same Q and target sets A_i.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper answers Question 2.1 by comparing state-level properties of Q with the corresponding properties of X. But Q carries all information about X: the event that X ever hits state i1 is exactly the event that Y hits the subset A_{i1}={i1 j2...j_{m-1}} of T. Consequently, each quantity the paper claims fails to transfer is in fact computable from Q. Ergodic/regularity of X are set-reachability conditions on Q: for each source state (i2,...,im) and each target state i1, some state in A_{i1} must be reachable. The ever-reaching probability f_{i1...im} is the set-hitting probability of A_{i1} starting from (i2,...,im), and the mean first passage time mu_{i1...im} is the expected set-hitting time to A_{i1}. Section 4's example (X regular, Q not ergodic) and Section 6's example (mu=3 but the Q state-level entry is 9) show only that standard single-state properties of Q do not coincide with X's properties. They do not show that X cannot be studied via Q. The paper never discusses this set-level translation, so the conclusion in Section 8 that 'the answer to Question 2.1 is negative in general' is not established by the supplied evidence.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the standard sliding-window reduction of an (m−1)th-order Markov chain X to a first-order chain Y_t=(X_t,...,X_{t−m+2}) on the space of length-(m−1) histories, with transition matrix Q. It establishes positive transfer results for k-step transition probabilities (Theorem 3.1) and for limiting probability distributions (Eq. (7.1)), and it presents explicit examples intended to show that ergodicity, regularity, ever-reaching probabilities, mean first passage times, and state classification do not transfer from X to Q. The paper concludes that the answer to its Question 2.1 is negative in general.","tokens_in":15199,"tokens_out":19771,"duration_ms":189196,"significance":"If the claims are properly qualified, the paper provides a useful, checkable catalog of which standard first-order state-level quantities do and do not pass through the sliding-window reduction. The positive results in Section 3 and Section 7 are clean and genuinely useful. The explicit examples in Sections 4 and 6 are reproducible and illustrate real phenomena. However, the central negative claim is currently too broad: since Y_t is a deterministic function of X and X_t is the first coordinate of Y_t, every property of X is in principle recoverable from Q at the level of subsets of the state space T. The paper never acknowledges or addresses this set-level translation, so the conclusion in Section 8 overstates what the examples establish.","major_comments":[{"comment":"The conclusion 'the answer to Question 2.1 is negative in general' is not supported by the supplied evidence. For each i1∈S, let A_i1={i1 j2...j_{m−1} : j2,...,j_{m−1}∈S}⊂T. Then X_t=i1 iff Y_t∈A_i1. Consequently, the ever-reaching probability f_{i1 i2...im} is exactly the probability that Q, started from (i2,...,im), ever hits the set A_i1; the mean first passage time µ_{i1...im} is the expected hitting time to A_i1; and ergodicity/regularity of X are set-reachability conditions on Q. The examples in Sections 4 and 6 show only that single-state entries of Q (e.g., state 11 rather than the set A_1) do not match the tensor entries of X. They do not show that X cannot be studied through Q. Please either narrow the claim to 'negative if one insists on state-level equality of the same indices' and state that explicitly, or add a discussion of how the set-level translation recovers the higher","section":"Section 8, Question 2.1"},{"comment":"The proof has a gap in the concatenation of events. The inequality p^{(γ+β+α)}_{i i3...im} ≥ Pr(ABC) is asserted with Pr(ABC)=p^{(α)}_{j i i3...im} p^{(β)}_{j j j3...jm} p^{(γ)}_{i j k3...km}. This requires that after event A the chain is exactly in the starting history of B, and after event B in the starting history of C. For α≥2, the post-A history contains random intermediate states, so a single tuple j3...jm cannot be used. Likewise, after a β-step return to j, the full history at the return time is not fixed, while C starts from a specified history j k3...km. The proof also leaves the case α≥m−1 completely unspecified (the displayed bullet list ends at α=m−2). The theorem may be true, but as written it is not proved. Please repair the argument by defining A and B as particular paths (using the positivity of p^{(α)} and the recurrence sum to choose such paths) or by a rigorous regene","section":"Theorem 5.1 proof"}],"minor_comments":[{"comment":"The sentence 'The conclusion in (3.1) follows now by induction' should refer to equation (3.3), not (3.1), which is the definition of the first passage time variable.","section":"Section 3, end of Theorem 3.1 proof"},{"comment":"The definition of j3,...,jm for α≥m−1 is left as an ellipsis. Please spell it out, since the proof's concatenation depends on the actual history after α steps.","section":"Theorem 5.1"},{"comment":"The column-stochastic convention for Q and P is stated in the introduction, but it would help to repeat it where Q is first introduced, since many readers expect row-stochastic transition matrices and the displayed Q and equation (6.2) depend on the convention.","section":"Section 2"},{"comment":"The bullet 'A state i is current iff ...' uses a nonstandard term; in the Markov-chain literature the property is usually called recurrence (or persistence). Please add a remark or change the terminology to avoid confusion.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The paper is largely expository and heavily depends on the author's prior tensor results and package; the novelty is the synthesis and the negative claim. The central ambiguity between 'studied via Q' and 'state-level quantities of Q coincide with tensor entries' should be resolved before publication. If the author reframes the negative claim and repairs the proof of Theorem 5.1, the paper could be publishable as a useful survey with explicit counterexamples."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a solid, mostly expository synthesis of the author's own program on higher-order Markov chains, with two genuinely new results: Theorem 3.1 gives a compact formula for k-step transition probabilities of a third-order chain in terms of the reduced first-order chain, and Theorem 5.1 proves a conjecture from [14] about state classification. The worked examples in Sections 4 and 6 are clear and reproducible; they convincingly show that the state-level properties of the reduced chain Q do not match the natural tensor-level properties of the original chain X.\n\nThat said, the central conclusion is overstated. Q is not merely an approximation of X; it is informationally equivalent to X, since X_t is a projection of Y_t. Consequently, all the quantities the paper claims fail to transfer — ergodicity, regularity, ever-reaching probabilities, mean first passage times — are in fact recoverable from Q as set-level quantities: hitting a symbol i in X is exactly hitting the subset A_i = {i j2...j_{m-1}} in Q. The paper never discusses this translation, so the Section 8 verdict that 'the answer to Question 2.1 is negative in general' is too strong. The real message is that the standard state-level comparison does not work; with a set-level translation, Q can be used to study X. That changes the headline of the paper and should be addressed.\n\nThere is also a genuine gap in the proof of Theorem 5.1. The event B is defined using a memory tuple jj3...jm that depends on α, the number of steps from i to j. The explicit formulas only cover α up to m-2; for longer paths, the initial states have left the memory window, and the tuple is not well-defined without reference to a particular path. The proof needs a more careful concatenation argument. This is fixable, but not a minor typo.\n\nThe paper leans heavily on the author's prior results, but that is fine for an expository article. The examples are a real asset for applied users who might naively apply first-order tools.\n\nI would send this to peer review, but with the expectation of revision: the author needs to either qualify the 'negative in general' claim or add the set-level translation, and repair the proof of Theorem 5.1. The applied audience will still find the caution useful even if the theoretical framing is softened.","headline":"Useful synthesis with two genuine new results, but the 'negative in general' verdict is overstated: the reduced chain is informationally equivalent, and the proof of Theorem 5.1 has a gap for long paths.","tokens_in":15639,"tokens_out":7896,"would_cite":true,"duration_ms":70264,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60A05","60J10","60J99","15A69","15B51"],"pacs":[],"model":"deepseek-v4-flash","headline":"Treating a higher-order Markov chain as a first-order chain only works for two quantities; most properties, including ergodicity and passage times, change.","keywords":["higher order Markov chain","reduced first order chain","transition tensor","ergodicity","mean first passage time","ever-reaching probability","limiting probability distribution","state classification"],"falsifier":"For the second-order three-state chain in Section 4, compute P^2 (it is positive) and check that the reduced chain Q has an all-zero row; this shows a regular higher-order chain whose reduced chain is not even irreducible. If this calculation fails, the paper's negative claim would need revision.","tokens_in":14734,"feed_emoji":"🎲","tokens_out":5333,"duration_ms":49674,"temperature":0.7,"pith_summary":"This paper takes on a common assumption in stochastic processes: that any higher-order Markov chain can be studied through the first-order chain obtained by sliding a window of past states. The author argues that this reduction is only reliable for two types of quantities: k-step transition probabilities and limiting probability distributions. For everything else—ergodicity, regularity, ever-reaching probabilities, mean first passage times, and the classification of states as recurrent or transient—the reduced chain can behave very differently from the original. The paper demonstrates each failure with explicit examples and concludes that the answer to the title question is negative in general. This has practical stakes because higher-order chains are used in many applied fields, and the result warns against blind simplification.","feed_headline":"Higher-order Markov chains resist first-order shortcuts","feed_subtitle":"Only k-step transitions and limiting distributions survive the slide; ergodicity and passage times do not.","key_machinery":"The sliding-window reduction Q, a first-order Markov chain whose states are length-(m-1) tuples Y_t = (X_t,...,X_{t-m+2}), with transition entries q_{i1...i_{m-1}, j2...j_m} = p_{i1...i_{m-1} j_m} when the tuple components match appropriately and zero otherwise. The paper also uses the tensor product ⊠ to express k-step transition tensors and the first-passage tensor equations, and mode-1 matricization to connect stationary distributions to limiting distributions. The reduction is the central object; the paper's claim is that it does not carry the higher-order chain's structure faithfully.","core_discovery":"The paper defines the reduced first-order chain Y_t = (X_t, ..., X_{t-m+2}) for an (m-1)th order chain with transition tensor P. It proves two positive bridges: Theorem 3.1, which shows k-step transition probabilities of the original chain can be recovered by summing certain entries of the reduced chain's k-step transition matrix; and equation (7.1), which shows the limiting probability distribution of the original chain is obtained by applying the mode-1 matricization of the identity tensor to any stationary distribution of the reduced chain. Beyond these, the paper gives concrete examples where the reduced chain fails to inherit ergodicity or regularity, where ever-reaching probabilities d","pith_inferences":["The negative answer is shown for the specific sliding-window reduction; an alternative first-order embedding that augments the state with a probability distribution over histories might preserve more properties, but the paper does not explore that.","The discrepancies in mean first passage times suggest that any shortcut through the reduced chain must be derived from the tensor equations, not guessed from matrix formulas.","A testable extension would be to characterize precisely which real-valued functions of the higher-order chain are determined by the reduced chain's transition matrix alone; Theorem 3.1 and (7.1) give two examples."],"forward_implications":["Any computation of ergodicity, regularity, ever-reaching probabilities, or mean first passage times for a higher-order chain must work directly with the transition tensor; using the reduced chain can give wrong answers.","The only safe bridges are Theorem 3.1 for k-step transitions (with a summation over intermediate states) and equation (7.1) for limiting distributions.","The state classification for higher-order chains needs its own definitions: a state is recurrent only if the ever-reaching probability is 1 for every possible past sequence, not merely for the vector states of the reduced chain.","In applications, results about higher-order chains that rely on passage times or recurrence should not be inferred from the reduced first-order chain."],"fun_headline_variants":["Higher-order chains lose ergodicity in first-order reduction","First-order shortcuts preserve limits but not ergodicity","Reduced Markov chains: k-step and limits survive, not regularity","High-order to first-order: partial transfer only"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The conclusion depends on taking the sliding-window vector process as the only first-order representation; if another reduction can preserve these properties, the negative verdict would not apply to it.","fun_headline_variants_meta":{"raw":{"variants":["Higher-order chains lose ergodicity in first-order reduction","First-order shortcuts preserve limits but not ergodicity","Reduced Markov chains: k-step and limits survive, not regularity","High-order to first-order: partial transfer only"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000164,"raw_usage":{"total_tokens":991,"prompt_tokens":563,"completion_tokens":428,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":307,"completion_tokens_details":{"reasoning_tokens":362}},"tokens_in":307,"tokens_out":428,"duration_ms":4358,"temperature":1.0,"reasoning_tokens":362,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T19:04:39.433682+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the second-order three-state chain in Section 4, compute P^2 (it is positive) and check that the reduced chain Q has an all-zero row; this shows a regular higher-order chain whose reduced chain is not even irreducible. If this calculation fails, the paper's negative claim would need revision.","supporting_citations":[],"review_version":1}