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arxiv: 2604.10792 · v1 · submitted 2026-04-12 · 🧮 math.PR · econ.EM· math.CT· math.ST· stat.TH

Variable-Length Markov Chains on Finite Quivers: Boundary-Window Identifiability, Exact Depth, and Local Rank Comparison

Oleg Kiriukhin This is my paper

Pith reviewed 2026-05-10 15:16 UTC · model grok-4.3

classification 🧮 math.PR econ.EMmath.CTmath.STstat.TH
keywords variable-length Markov chainsfinite quiversboundary-window identifiabilityvisible depthexact depthlocal rank comparisoninformative maps
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The pith

Exact context depth in variable-length Markov chains on quivers is identifiable by rank comparison of their one-step informative maps.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a first-order theory of visible-depth identifiability for variable-length Markov chains on finite quivers observed through boundary windows. It shows that under the representation hypothesis in the edge-homogeneous regime with fixed local visible support, all admissible depths share the same first-order rank for the stationary one-step informative map, yet in the exact-depth regime of context length r the depth-r map achieves full rank while coarser maps lose rank strictly because they factor smoothly through it. A sympathetic reader would care because the resulting coordinate-rank and subspace criteria recover the minimal depth m* directly from observable transition laws and their differentials. The work also supplies a global coordinate-rank theorem that pins down m* under full fine-depth rank and strict losses at smaller depths.

Core claim

Under the representation hypothesis in the edge-homogeneous regime with fixed local visible support, the stationary one-step informative map q_Q^{(m)} has the same first-order rank for all admissible m. In the exact-depth regime with context length r, the depth-r boundary process is the canonical finite-state Markov chain, smaller windows are deterministic truncations, and every coarser informative map factors C^1-smoothly through the depth-r map on the relevant affine transition-array neighborhood, so rank cannot increase beyond depth r. After quotienting a tangent block by directions already invisible at depth r, strict coarse-depth loss equals coarse rank deficiency, equivalently a strict

What carries the argument

The stationary one-step informative map q_Q^{(m)} and its restricted differentials on prescribed tangent blocks, whose rank at different window sizes encodes visible-depth identifiability.

If this is right

  • In the exact-depth regime every coarser informative map factors C^1-smoothly through the depth-r map.
  • Rank cannot increase beyond depth r.
  • Strict coarse-depth loss is characterized exactly by coarse rank deficiency or strict rank drop from depth r to m.
  • Under full fine-depth rank and strict coordinate-rank loss at every smaller depth the global coordinate-rank theorem yields m_*(T, θ0) = r.
  • First-order criteria are invariant under C^1 reparameterization once reduced local coordinates remove stochastic redundancies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The rank-based criteria suggest that depth recovery may be possible from empirical transition frequencies in long observed sequences, provided sample sizes suffice for reliable rank estimation.
  • The factorization property may extend to other context-dependent processes on graphs whenever boundary windows yield analogous stationary maps.
  • The result strengthens the link between variable-length chains and ordinary finite-state Markov chains by showing the depth-r boundary process behaves exactly like the latter.

Load-bearing premise

The representation hypothesis holds and the process is in the edge-homogeneous regime with fixed local visible support.

What would settle it

A concrete counterexample of an exact-depth-r quiver chain where the first-order rank of the informative map at some depth m > r strictly exceeds the rank at depth r would falsify the claim that rank cannot increase beyond r.

read the original abstract

Variable-length Markov chains on finite quivers provide a natural framework for context-dependent stochastic growth under incidence constraints. I study quiver-valued variable-length Markov chains observed through finite boundary windows and develop a first-order theory of visible-depth identifiability via stationary visible one-step transition laws and their restricted differentials on prescribed tangent blocks. For visible depth $m$, the main object is the stationary one-step informative map $q_{\mathcal{Q}}^{(m)}$. In the edge-homogeneous regime, once the local visible support is fixed and the representation hypothesis holds, all admissible visible depths encode the same edge-level extension law and hence have the same first-order rank. In the exact-depth regime of context length $r$, the depth-$r$ boundary process is the canonical finite-state Markov chain, smaller visible windows are deterministic truncations, and every coarser informative map factors $C^1$-smoothly through the depth-$r$ informative map on the relevant affine transition-array neighborhood. Hence rank cannot increase beyond depth $r$. After quotienting a tangent block by directions already invisible at depth $r$, I characterize strict coarse-depth loss exactly by coarse rank deficiency, equivalently by strict rank drop from depth $r$ to depth $m$ on the original block. I also give subspace-based and global selected-coordinate criteria, a global one-coordinate branching criterion, and an explicit depth-two example. Under full fine-depth rank and strict coordinate-rank loss at every smaller depth, a global coordinate-rank theorem yields $m_*(T,\theta_0)=r$. Reduced local coordinates remove stochastic redundancies, first-order criteria are invariant under $C^1$ reparameterization, and the statistical and LAN consequences remain conditional on additional estimation and likelihood-level hypotheses.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a first-order theory of visible-depth identifiability for variable-length Markov chains on finite quivers, centered on the stationary one-step informative map q_Q^(m) and its restricted differentials on tangent blocks. In the edge-homogeneous regime with fixed local visible support and under the representation hypothesis, all admissible visible depths share the same first-order rank. In the exact-depth regime of context length r, the depth-r boundary process is the canonical finite-state Markov chain, smaller windows are deterministic truncations, coarser informative maps factor C^1-smoothly through the depth-r map, and rank cannot increase beyond r. After quotienting tangent blocks by invisible directions, strict coarse-depth loss is characterized by rank deficiency; subspace-based, global selected-coordinate, and one-coordinate branching criteria are given, with an explicit depth-two example. Under full fine-depth rank and strict coordinate-rank loss at smaller depths, a global coordinate-rank theorem yields m_*(T, θ_0) = r. Reduced local coordinates remove redundancies, first-order criteria are invariant under C^1 reparameterization, and statistical/LAN consequences are conditional on further hypotheses.

Significance. If the central claims hold, the work supplies a precise algebraic and differential framework for identifiability and rank comparison in context-dependent processes on quivers, including explicit criteria (subspace, coordinate, branching) and the global coordinate-rank theorem that directly identifies exact depth. The depth-two example and the clean separation of visible versus invisible directions via tangent-block quotienting are concrete strengths that could support downstream statistical applications once the representation hypothesis is verified.

major comments (2)
  1. [Abstract and statements of the representation hypothesis] The representation hypothesis is invoked throughout (abstract; statements on rank equality and edge-level extension laws) to guarantee that admissible visible depths share the same first-order rank, yet no verification procedure, sufficient conditions, or counterexample analysis is supplied for general finite quivers. This assumption is load-bearing for the claim that rank is independent of visible depth and for the subsequent global coordinate-rank theorem.
  2. [Exact-depth regime and C^1 factorization claims] The precise scope of the C^1 factorization of coarser informative maps through the depth-r map is stated for the relevant affine transition-array neighborhood, but the argument does not address whether the factorization remains valid when the full parameter vector p is not observed or when the process leaves the edge-homogeneous regime. This directly affects the claim that rank cannot increase beyond depth r.
minor comments (2)
  1. [Introduction and notation] Notation for the stationary one-step informative map q_Q^(m) and the tangent blocks should be introduced with an explicit diagram or table relating them to the underlying quiver incidence structure.
  2. [Depth-two example] The depth-two example would benefit from an explicit numerical matrix for the transition array and the computed rank drop to illustrate the strict coordinate-rank loss.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments. We respond point by point to the major comments below.

read point-by-point responses

  1. Referee: [Abstract and statements of the representation hypothesis] The representation hypothesis is invoked throughout (abstract; statements on rank equality and edge-level extension laws) to guarantee that admissible visible depths share the same first-order rank, yet no verification procedure, sufficient conditions, or counterexample analysis is supplied for general finite quivers. This assumption is load-bearing for the claim that rank is independent of visible depth and for the subsequent global coordinate-rank theorem.

    Authors: The representation hypothesis is introduced as a standing assumption required for the edge-homogeneous regime to ensure that all admissible visible depths encode identical edge-level extension laws and therefore share the same first-order rank. The manuscript develops the identifiability theory conditionally on this hypothesis rather than providing a general verification procedure, sufficient conditions, or counterexamples for arbitrary finite quivers. In the specific setting where the local visible support fixes the transitions without hidden dependencies, the hypothesis holds by the definition of the regime. We will add a clarifying remark in the introduction and the statements of the rank-equality results to make the conditional nature explicit and to note that verification is quiver-specific. The global coordinate-rank theorem is likewise stated under the hypothesis, so its validity is unaffected. revision: partial

  2. Referee: [Exact-depth regime and C^1 factorization claims] The precise scope of the C^1 factorization of coarser informative maps through the depth-r map is stated for the relevant affine transition-array neighborhood, but the argument does not address whether the factorization remains valid when the full parameter vector p is not observed or when the process leaves the edge-homogeneous regime. This directly affects the claim that rank cannot increase beyond depth r.

    Authors: The C^1 factorization is proved inside the edge-homogeneous regime on the affine neighborhood of the transition arrays where the depth-r boundary process is the canonical finite-state Markov chain and coarser windows are deterministic truncations. The informative maps are the visible stationary one-step laws, so the factorization relates these visible maps; it does not require the full unobserved parameter vector p to be available. We agree that the argument is confined to the edge-homogeneous setting and does not extend to regimes outside it. The claim that rank cannot increase beyond depth r is therefore restricted to the exact-depth regime under the stated hypotheses. We will revise the statement of the factorization result and the subsequent rank bound to delineate the assumptions more explicitly. revision: yes

standing simulated objections not resolved
  • A general verification procedure, sufficient conditions, or counterexample analysis for the representation hypothesis on arbitrary finite quivers

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivations of visible-depth identifiability, the C^1-smooth factoring of coarser maps through the depth-r informative map, the characterization of strict coarse-depth loss via rank deficiency after quotienting tangent blocks, and the global coordinate-rank theorem under full fine-depth rank are obtained directly from the algebraic properties of the stationary one-step informative maps q_Q^(m), their restricted differentials on tangent blocks, and the representation hypothesis together with the edge-homogeneous regime. These steps rely on the definitions of the transition laws, deterministic truncations for smaller windows, and rank conditions without any reduction to quantities fitted from data, self-citations that bear the central load, or ansatzes smuggled from prior work. The exact-depth regime statements follow by construction from the fixed local visible support and the canonical finite-state Markov chain property at depth r, rendering the overall argument self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 2 invented entities

The central claims rest on domain assumptions of stationarity, edge-homogeneity, and the representation hypothesis, plus newly introduced objects such as the informative map and tangent blocks; no free parameters are explicitly fitted to data, and no invented entities carry independent evidence outside the model.

axioms (3)
  • domain assumption Representation hypothesis holds
    Invoked to ensure all admissible visible depths encode the same edge-level extension law in the edge-homogeneous regime.
  • domain assumption Process is stationary
    Required for the stationary visible one-step transition laws and informative maps.
  • domain assumption Edge-homogeneous regime with fixed local visible support
    Assumed to obtain equal first-order ranks across depths.
invented entities (2)
  • Stationary one-step informative map q_Q^(m) no independent evidence
    purpose: Encodes visible transition laws at depth m for identifiability analysis
    New central object introduced to study first-order rank and factorization properties.
  • Tangent blocks no independent evidence
    purpose: Prescribed subspaces for restricted differentials in the first-order theory
    Introduced to characterize rank deficiency after quotienting invisible directions.

pith-pipeline@v0.9.0 · 5630 in / 1776 out tokens · 121825 ms · 2026-05-10T15:16:57.383890+00:00 · methodology

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Reference graph

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