A Strict Gap Between Relaxed and Partition-Constrained Spectral Compression in a Six-State Lumpable Markov Chain

Oleg Kiriukhin

arxiv: 2604.10820 · v1 · submitted 2026-04-12 · 🧮 math.PR · econ.EM· math.CO· math.ST· stat.TH

A Strict Gap Between Relaxed and Partition-Constrained Spectral Compression in a Six-State Lumpable Markov Chain

Oleg Kiriukhin This is my paper

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

classification 🧮 math.PR econ.EMmath.COmath.STstat.TH

keywords Markov chainlumpable chainspectral compressionpartition constrainedrelaxed orthonormal framesdeterminant maximizationsix-state modelreversible chain

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The pith

In a symmetric six-state lumpable Markov chain, the best determinant from any three-partition is strictly smaller than the best determinant from any orthonormal three-frame.

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

The paper examines a reversible lumpable Markov chain with six states and the operator T equal to the square of its transition matrix. It compares the largest determinant that can be obtained by compressing T with any three-dimensional orthonormal matrix against the largest determinant obtained when the matrix is restricted to normalized indicator vectors of a three-cell partition of the states. Closed formulas for key partitions and an exhaustive check of all ninety possible partitions establish that the partition version always falls short. This matters to a reader because it shows a concrete case where restricting to simple state groupings produces a measurable loss compared with the more flexible relaxed approach.

Core claim

For the symmetric six-state lumpable chain and the positive operator T = P squared, the supremum of det(Q_A(T)) over all three-partitions A is strictly less than the relaxed supremum sup det(U^* T U) over orthonormal U with three columns.

What carries the argument

The relaxed supremum D^rel_3(T) of the determinant of U^* T U for any U with orthonormal columns, set against the partition-constrained supremum of det(H_A^* T H_A) where H_A is built from the normalized indicator vectors of a three-cell partition of the state space.

Load-bearing premise

The specific symmetric six-state reversible lumpable chain with T equal to P squared permits closed formulas for central partitions together with a complete enumeration of all ninety partitions that together prove the strict inequality.

What would settle it

An explicit three-partition in this six-state model whose determinant equals or exceeds the relaxed supremum would falsify the claimed strict gap.

read the original abstract

This paper studies a finite reversible lumpable Markov chain for which relaxed spectral compression yields a larger determinant than partition-constrained compression. For a symmetric six-state lumpable chain and the positive operator $T=P^2$, I compare the relaxed benchmark \begin{equation*} \mathfrak D^{\mathrm{rel}}_3(T):=\sup_{U^*U=I_3}\det(U^*TU) \end{equation*} and the partition-constrained benchmark \begin{equation*} \sup_{\mathcal A\,\mathrm{3\text{-}partition}}\det Q_{\mathcal A}(T), \qquad Q_{\mathcal A}(T)=H_{\mathcal A}^*TH_{\mathcal A}. \end{equation*} Here the partition-constrained benchmark is the compression induced by normalized indicator vectors of genuine partitions of the state space. I derive closed formulas for the two analytically central partition families, prove strict upper bounds for both in a local-mode-dominated regime, and combine these bounds with an exhaustive enumeration of all $90$ partitions into three nonempty cells in an explicit six-state model. For this model, one obtains a strict global gap: \begin{equation*} \sup_{\mathcal A}\det Q_{\mathcal A}(T)<\mathfrak D^{\mathrm{rel}}_3(T). \end{equation*} Thus, in this model, indicator-based partition frames are strictly weaker than relaxed orthonormal frames even after global partition-constrained optimization.

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.

Desk Editor's Note private letter to a colleague

This paper gives a clean, explicit six-state counterexample where the relaxed spectral sup strictly beats the best partition-constrained determinant for T=P².

read the letter

The punchline is that the authors exhibit a symmetric reversible lumpable chain on six states where the relaxed benchmark D^rel_3(T) is strictly larger than the supremum over all three-cell partitions of det(Q_A(T)). They reach this by closed formulas on two central partition families, local-mode upper bounds, and a complete sweep of the remaining partitions. That combination produces a verifiable strict gap rather than an asymptotic statement. The setup is finite and parameter-free once the transition rates are fixed, so the claim is in principle falsifiable by re-running the enumeration or checking the determinants. The comparison itself is direct: one side optimizes over orthonormal frames, the other over normalized indicator vectors of actual partitions, with no self-referential fitting. This is the sort of concrete data point that can be cited when someone asks whether partition methods are always optimal for lumpable chains. The math looks internally consistent and the citation pattern is ordinary for the subfield. The main soft spot is the dependence on the enumeration being both complete and numerically accurate. Ninety partitions is manageable, but any single miscomputed determinant or missed cell assignment would erase the strict inequality. The local-mode bounds are also tied to the specific rates chosen; if those rates sit outside the regime where the bounds are valid, the argument needs adjustment. I would want to see the explicit transition matrix and either the full list or reproducible code for the sweep before treating the gap as settled. This paper is for people working on spectral compression, lumpability, or model reduction in finite Markov chains. A reader who needs a counterexample to partition optimality will get immediate value; someone looking for broad new theory will find less. It deserves peer review because the claim is sharp, the model is small enough to check, and the gap, if real, is worth having on record.

Referee Report

2 major / 2 minor

Summary. The paper constructs an explicit symmetric six-state reversible lumpable Markov chain and the operator T = P². It defines the relaxed benchmark 𝔇^rel_3(T) as the supremum of det(U* T U) over orthonormal 3-frames U and the partition-constrained benchmark as the supremum of det(Q_A(T)) over all 3-partitions A, with Q_A = H_A* T H_A using normalized indicator vectors. Closed formulas are derived for two central partition families, strict upper bounds are proved in the local-mode-dominated regime, and these are combined with an exhaustive enumeration of all 90 partitions to establish the strict inequality sup_A det Q_A(T) < 𝔇^rel_3(T) for the chosen model.

Significance. If the enumeration and bounds hold, the result supplies a concrete, finite, parameter-explicit counterexample showing that even global optimization over partition frames yields a strictly smaller determinant than the relaxed orthonormal benchmark. This demonstrates a genuine gap between indicator-based and relaxed frames in spectral compression for lumpable chains and provides a verifiable test case that could guide the search for tighter relaxations or hybrid constructions. The combination of closed formulas, regime-specific bounds, and complete enumeration is a methodological strength that makes the claim falsifiable by direct computation.

major comments (2)

[§5] §5 (local-mode bounds): the strict upper bounds on det Q_A(T) for the two central families are stated to hold only inside a local-mode-dominated regime whose validity depends on the specific transition probabilities; the manuscript must explicitly verify (e.g., by computing the relevant eigenvalues or mode weights) that the chosen numerical parameters lie inside this regime, otherwise the bounding step does not apply and the gap argument is incomplete.
[§6] §6 (enumeration): the central claim rests on the assertion that none of the 90 partitions achieves or exceeds 𝔇^rel_3(T). The manuscript should report, at minimum, the five largest computed values of det Q_A(T) together with the corresponding partitions, or supply reproducible code/data that lists all 90 determinants, so that the maximum can be independently confirmed and no partition is overlooked.

minor comments (2)

[§2] Notation: the definition of H_A (normalized indicators) and the precise normalization constant should be stated once in a dedicated preliminary subsection rather than repeated inline.
[Abstract] The abstract and introduction both state the inequality; a single forward reference to the theorem number containing the final comparison would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised help strengthen the verifiability and completeness of the argument. We address each major comment below, indicating the revisions we will incorporate.

read point-by-point responses

Referee: [§5] §5 (local-mode bounds): the strict upper bounds on det Q_A(T) for the two central families are stated to hold only inside a local-mode-dominated regime whose validity depends on the specific transition probabilities; the manuscript must explicitly verify (e.g., by computing the relevant eigenvalues or mode weights) that the chosen numerical parameters lie inside this regime, otherwise the bounding step does not apply and the gap argument is incomplete.

Authors: We agree that the local-mode-dominated regime must be explicitly verified for the specific transition probabilities used in the model. In the revised manuscript we will add a short verification subsection (or paragraph) in §5 that computes the relevant eigenvalues of P and the associated mode weights for our chosen numerical values, confirming that the parameters lie inside the regime. This will make the application of the strict upper bounds fully rigorous and complete the gap argument. revision: yes
Referee: [§6] §6 (enumeration): the central claim rests on the assertion that none of the 90 partitions achieves or exceeds 𝔇^rel_3(T). The manuscript should report, at minimum, the five largest computed values of det Q_A(T) together with the corresponding partitions, or supply reproducible code/data that lists all 90 determinants, so that the maximum can be independently confirmed and no partition is overlooked.

Authors: We accept that greater transparency in the enumeration results is desirable. In the revised manuscript we will add a table listing the five largest values of det Q_A(T) together with explicit descriptions of the corresponding 3-partitions. We will also include, as supplementary material, either the complete list of all 90 determinants or reproducible Python code that enumerates and computes them, allowing independent confirmation that the reported maximum is correct. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit enumeration and closed-form bounds establish the gap directly

full rationale

The paper defines two independent optimization problems (relaxed sup det(U*TU) over orthonormal U and partition-constrained sup det(Q_A(T)) over 3-partitions) and proves a strict inequality for one explicit finite reversible lumpable chain by deriving closed formulas for the two central partition families, proving regime-specific upper bounds, and exhaustively checking the remaining 90 partitions. No step reduces to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation; the result follows from direct algebraic manipulation and case enumeration on the given transition matrix without external assumptions that embed the target inequality.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard linear-algebra facts about determinants of compressed operators and the domain assumption that the six-state chain is symmetric, reversible, and lumpable. No free parameters are introduced or fitted; no new entities are postulated.

axioms (2)

standard math Determinant is well-defined and continuous on the space of symmetric positive-semidefinite operators; orthonormal frames achieve the relaxed supremum.
Invoked in the definitions of both D^rel_3(T) and Q_A(T).
domain assumption The six-state chain is symmetric, reversible, and lumpable, allowing the operator T=P² to be analyzed via partitions.
Stated as the setting for the explicit model and the 90-partition enumeration.

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discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[1]

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[1] [1]

D. A. Levin, Y. Peres, and E. L. Wilmer,Markov Chains and Mixing Times, American Mathematical Society, Providence, RI, second edition, 2017

work page 2017

[2] [2]

Aldous and J

D. Aldous and J. Fill,Reversible Markov Chains and Random Walks on Graphs, unfin- ished monograph, available online at https://www.stat.berkeley.edu/~aldous/RWG/ book.html

work page

[3] [3]

J. G. Kemeny and J. L. Snell,Finite Markov Chains, Springer, New York, 1976

work page 1976

[4] [4]

H. A. Simon and A. Ando, Aggregation of variables in dynamic systems,Econometrica29 (1961), 111–138

work page 1961

[5] [5]

C. D. Meyer, Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems,SIAM Review31(1989), 240–272

work page 1989

[6] [6]

L. Duan, D. B. Dunson, and L. Carin, Spectral state compression of Markov processes,Adv. Neural Inf. Process. Syst.32(2019), 7586–7595. 8

work page 2019