{"id":"99f156d7-7cdc-45b5-97b2-3e555e866720","arxiv_id":"2606.10058","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves absorption properties, nonsingularity of fundamental tensor equations, and tensor-series links for mean first passage times in higher-order Markov chains, with a MATLAB solver provided.","lead":"The paper proves that making one or more states absorbing in a higher-order ergodic Markov chain turns the chain absorbing, shows the fundamental tensor equation is nonsingular, supplies a MATLAB solver, and links mean first passage time tensor slices to these fundamental tensors via a series representation. A smart generalist might read it for tools to analyze sequence models with memory, such as in time-series or language data.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Modification of a single state to absorbing may not preserve a well-defined higher-order transition structure without explicit construction rules for sequences.","rationale":"The reader's weakest_assumption directly names the setup step that must hold for every later theorem; the abstract-only review correctly flags it as load-bearing. With full text now available the same assumption remains the point least secured by the given information.","tokens_in":1641,"tokens_out":329,"duration_ms":18581,"concrete_test":"Extract the precise definition of the absorbing modification (likely §2 or §3) and the statement that the resulting chain remains order-k; substitute a concrete 2-state, order-2 example with explicit transition tensor, apply the modification, and verify whether the new transition probabilities on pairs still satisfy the Chapman-Kolmogorov relation for order exactly 2 (or collapse to order 1).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claims (resulting chain is absorbing; fundamental tensor equation is nonsingular; MFPT tensor slices equal modified fundamental tensors) all rest on the initial construction: starting from a higher-order ergodic chain on the product space and altering one original state to absorbing. In a k-th order chain the state is a k-tuple; making one coordinate value absorbing requires specifying how every tuple containing that value updates its transition probabilities. The abstract asserts this yields a well-defined absorbing higher-order chain whose fundamental tensor satisfies a nonsingular linear equation, but the weakest link is whether that construction is canonical and preserves the order-k Markov property without additional stipulations on the transition tensor.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript extends mean first passage times and fundamental matrices to higher-order Markov chains. It claims to prove that modifying one or more states of a higher-order ergodic chain to absorbing produces an absorbing chain; that the linear equation satisfied by the fundamental tensor of a higher-order absorbing chain is nonsingular (with a MATLAB solver fund provided); and that each horizontal slice of the mean first passage time tensor equals the fundamental tensor of a chain obtained by making one state absorbing, yielding a tensor series representation for selected mean first passage times.","tokens_in":1770,"tokens_out":504,"duration_ms":21646,"significance":"If the state-modification construction is rigorously defined and preserves the order-k Markov property, the results generalize classical absorbing-chain theory and supply both analytic and computational tools for passage-time analysis in sequence models. The explicit tensor-series representation and the solver constitute concrete, usable contributions.","major_comments":[{"comment":"The load-bearing construction (altering one coordinate value to absorbing in the product-space state space of a k-th order chain) is not shown to be canonical. The manuscript must supply the explicit rule for updating every k-tuple containing the designated absorbing value; without it, the claims that the resulting process remains a well-defined higher-order absorbing chain, that absorption is proved, and that the fundamental-tensor equation is nonsingular cannot be verified.","section":"Section defining the modified transition tensor (likely §2 or §3)"},{"comment":"The asserted proof that the fundamental-tensor equation is nonsingular (abstract and the section introducing the fund function) appears to rest on the absorbing property obtained from the above construction. If the construction is not uniquely specified, the nonsingularity argument must be re-examined for dependence on additional stipulations that are not part of the standard higher-order Markov axioms.","section":"Proof of nonsingularity of the fundamental tensor equation"}],"minor_comments":[{"comment":"The abstract states the result for 'one or more states' while the body description emphasizes a single state; the statements should be aligned.","section":"Abstract"},{"comment":"The MATLAB function is written as {\\tt fund}; consistent use of \\texttt{fund} throughout the text and code listing is needed.","section":"Computational section"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful review and the identification of points requiring greater explicitness. We address each major comment below and will incorporate the necessary clarifications in a revised manuscript.","responses":[{"response":"We agree that the original manuscript did not supply a fully explicit coordinate-wise rule for every k-tuple. In the revision we will add a precise definition of the modified transition tensor that states, for each k-tuple containing the designated absorbing state s, the new probability mass is reassigned according to the original marginal on the remaining coordinates while setting the absorbing coordinate to probability 1 at s. This rule is the natural extension of the classical absorbing-state construction to the product-space representation of a k-th order chain and preserves the order-k Markov property by construction. With this definition the absorption proof and the nonsingularity argument become directly verifiable from the standard axioms.","revision_made":"yes","referee_comment":"[Section defining the modified transition tensor (likely §2 or §3)] The load-bearing construction (altering one coordinate value to absorbing in the product-space state space of a k-th order chain) is not shown to be canonical. The manuscript must supply the explicit rule for updating every k-tuple containing the designated absorbing value; without it, the claims that the resulting process remains a well-defined higher-order absorbing chain, that absorption is proved, and that the fundamental-tensor equation is nonsingular cannot be verified."},{"response":"The nonsingularity claim is indeed derived from the absorbing property that follows once the modification rule is fixed. By inserting the explicit update rule described above, the revised manuscript will make the logical dependence transparent: the resulting transition tensor satisfies the higher-order absorbing-chain axioms, from which the nonsingularity of the fundamental-tensor linear system follows by the same block-triangular argument used in the classical case. We will also add a short lemma confirming that the construction introduces no extraneous stipulations beyond the standard axioms plus the absorbing-state designation.","revision_made":"yes","referee_comment":"[Proof of nonsingularity of the fundamental tensor equation] The asserted proof that the fundamental-tensor equation is nonsingular (abstract and the section introducing the fund function) appears to rest on the absorbing property obtained from the above construction. If the construction is not uniquely specified, the nonsingularity argument must be re-examined for dependence on additional stipulations that are not part of the standard higher-order Markov axioms."}],"tokens_in":1302,"tokens_out":527,"duration_ms":16441,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper lifts the classical fundamental matrix and mean first passage time formulas to higher-order Markov chains by working with tensors instead of matrices. It proves that an absorbing version obtained by turning one state absorbing in an ergodic higher-order chain remains absorbing, shows the fundamental tensor equation is nonsingular, supplies a MATLAB solver called fund, and links horizontal slices of the mean first passage time tensor to fundamental tensors from suitably modified chains, which yields a tensor series for selected passage times.\n\nThe tensor formulation and the explicit link to the series representation look new relative to the first-order results cited in the abstract. The MATLAB function is a practical addition that lets readers compute the objects directly. The claims rest on standard Markov axioms rather than fitted quantities or self-referential definitions, so there is no obvious circularity.\n\nThe soft spot is the construction step itself. A higher-order chain has states that are k-tuples, so declaring one original state absorbing requires specifying how every tuple containing that value updates its transition probabilities while keeping the order-k property intact. The abstract asserts the result is a well-defined absorbing chain whose fundamental tensor satisfies a nonsingular equation, but the letter should check whether the full proofs give the precise update rules and confirm the Markov property survives. If that construction is only sketched, the central claims rest on an unexamined step.\n\nThe work is aimed at researchers already using higher-order chains in stochastic processes. A reader who wants the classical mean first passage time toolkit generalized to tensors will find the explicit solver and series formula useful. It is narrow but formally stated, so it deserves a serious referee to verify the derivations and the absorbing modification details.","headline":"Extends the fundamental matrix and mean first passage times to higher-order Markov chains with a tensor equation, nonsingularity result, and series representation, but the absorbing-state construction needs explicit verification.","tokens_in":2210,"tokens_out":414,"would_cite":false,"duration_ms":20983,"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":"For higher-order Markov chains, altering states to absorbing produces a nonsingular equation for the fundamental tensor that links to mean first passage time tensors.","keywords":["higher-order Markov chains","mean first passage times","fundamental tensor","absorbing Markov chains","ergodic Markov chains","tensor series representation"],"falsifier":"An explicit higher-order absorbing chain whose fundamental-tensor equation is singular, or a numerical counterexample in which the claimed series fails to equal the mean first passage times.","tokens_in":2536,"feed_emoji":"","tokens_out":610,"duration_ms":13884,"temperature":0.7,"pith_summary":"The paper focuses on higher-order ergodic Markov chains in which one or more states are changed to absorbing. It proves that the resulting chain is absorbing and that the linear equation for its fundamental tensor must be nonsingular. A MATLAB routine called fund is supplied to solve this equation. Each horizontal slice of the mean first passage time tensor is shown to equal the fundamental tensor of a chain in which exactly one state has been made absorbing. This identification produces an infinite series expression for selected mean first passage times.","feed_headline":"Higher-order absorbing chains have nonsingular fundamental tensor equations","feed_subtitle":"Each slice of the mean first passage time tensor equals the fundamental tensor of a single-state absorbing modification, yielding a series f","key_machinery":"The fundamental tensor of a higher-order absorbing Markov chain, which satisfies a nonsingular linear equation and equals each horizontal slice of the mean first passage time tensor.","core_discovery":"When one or more states of a higher-order ergodic Markov chain are modified to be absorbing, the resulting chain is absorbing. The fundamental tensor of any such higher-order absorbing chain satisfies a nonsingular linear equation. Each horizontal slice of the mean first passage time tensor equals the fundamental tensor obtained by making exactly one state absorbing, which supplies a tensor series representation for the corresponding mean first passage times.","pith_inferences":["The same slice-to-fundamental-tensor link may allow recursive or iterative computation of mean first passage times without enumerating all paths.","The construction suggests a systematic way to obtain analogous tensor identities for continuous-time higher-order processes.","The nonsingularity result could be used to certify uniqueness in numerical schemes that solve tensor equations for higher-order chains."],"forward_implications":["The fundamental tensor equation admits a unique solution.","The MATLAB function fund computes this tensor for any qualifying higher-order absorbing chain.","Selected mean first passage times admit an explicit infinite tensor series.","Each horizontal slice of the mean first passage time tensor corresponds to a single-state absorbing modification of the original ergodic chain."],"fun_headline_variants":["Higher-order absorbing chains satisfy nonsingular fundamental tensor equations","Mean first passage tensor slices match single-state absorbing fundamental tensors","Fundamental tensor equations are nonsingular for higher-order absorbing chains","Tensor series represents mean first passage times via single-state absorptions"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The transition structure of the original higher-order ergodic chain stays well-defined after selected states are changed to absorbing.","fun_headline_variants_meta":{"raw":{"variants":["Higher-order absorbing chains satisfy nonsingular fundamental tensor equations","Mean first passage tensor slices match single-state absorbing fundamental tensors","Fundamental tensor equations are nonsingular for higher-order absorbing chains","Tensor series represents mean first passage times via single-state absorptions"]},"model":"grok-4.3","cost_usd":0.009904,"raw_usage":{"total_tokens":4354,"prompt_tokens":570,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":99037000,"prompt_tokens_details":{"text_tokens":570,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3719,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":570,"tokens_out":65,"duration_ms":23134,"temperature":1.0,"reasoning_tokens":3719,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T15:01:10.622443+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit higher-order absorbing chain whose fundamental-tensor equation is singular, or a numerical counterexample in which the claimed series fails to equal the mean first passage times.","supporting_citations":[],"review_version":1}