arxiv: 2605.11615 · v1 · submitted 2026-05-12 · 🧮 math.AT · math.CT

Recognition: 2 theorem links

· Lean Theorem

Quillen-McCord theorem for persistence finite posets

Kohei Tanaka, Vitalii Guzeev

Pith reviewed 2026-05-13 01:38 UTC · model grok-4.3

classification 🧮 math.AT math.CT

keywords Quillen-McCord theorempersistence finite posetshomotopy fibersinterleaving distancehomotopy commutativepersistent homologyposet maps

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The pith

A map between persistence finite posets with weakly ε-contractible homotopy fibers induces an upper bound on their homotopy commutative interleaving distance.

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

The paper establishes a persistence version of the Quillen-McCord theorem for persistence finite posets. It shows that if a map f from P to Q has weakly ε-contractible homotopy fibers, then there is an upper bound on the homotopy commutative interleaving distance between P and Q. This would matter to a reader because it gives a way to control the similarity of two persistence structures through the homotopy properties of a map connecting them, which is useful in topological data analysis where such distances quantify persistence features. The result adapts classical homotopy theory to the filtered poset setting.

Core claim

The authors prove that for a map f : P → Q between persistence finite posets with weakly ε-contractible homotopy fibers, the homotopy commutative interleaving distance between P and Q admits an upper bound.

What carries the argument

Weakly ε-contractible homotopy fibers of the map f, which control the size of the homotopy commutative interleaving distance in the persistence setting.

If this is right

The interleaving distance between P and Q can be bounded from the fiber properties of f.
Persistence finite posets connected by such a map differ by at most the scale of the fiber contractibility.
The result supplies a criterion for closeness of filtered posets without computing full persistent homology.
Similar maps preserve the homotopy type of persistence structures up to the given ε.

Where Pith is reading between the lines

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

The bound might be used to compare persistence modules by checking fiber contractibility on simple test posets.
This adaptation suggests how other classical poset theorems could transfer to persistence.
One could check the result by taking chain posets with explicitly contractible fibers and computing their interleaving distances directly.

Load-bearing premise

The homotopy fibers of the map f between the persistence finite posets are weakly ε-contractible.

What would settle it

A concrete map f between two persistence finite posets whose homotopy fibers fail to be weakly ε-contractible and for which the interleaving distance exceeds the stated upper bound.

read the original abstract

In this paper, we establish a persistence version of the Quillen-McCord theorem for persistence finite posets. Given a map $f \colon P \rightarrow Q$ between persistence finite posets $P$ and $Q$ with weakly $\varepsilon$-contractible homotopy fibers, we provide an upper bound for the homotopy commutative interleaving distance between $P$ and $Q$.

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

The paper adapts Quillen-McCord to persistence finite posets by bounding homotopy-commutative interleaving distance under weakly ε-contractible fibers, but the value hinges on whether the proof closes the gap between classical homotopy and the persistence metric.

read the letter

The main new piece here is a direct persistence lift of the Quillen-McCord theorem. Instead of ordinary contractible fibers giving a homotopy equivalence, the authors replace that with weakly ε-contractible homotopy fibers and extract an upper bound on the interleaving distance between the two persistence posets. That bound is the concrete output, and it is stated cleanly in the abstract as the central claim. The setup stays inside finite posets, which keeps the technical overhead manageable and matches the setting where persistence is usually applied in data analysis contexts. If the proof works, this gives a stability-type control that could be useful when comparing filtered structures whose maps have controlled fibers. The classical theorem is cited properly and the adaptation looks consistent with how interleaving distances are handled in the persistence literature. The hypothesis is explicit and necessary, so there is no hidden circularity in the statement itself. The main soft spot is that the abstract alone does not let us verify the key steps: how “weakly ε-contractible” is defined, how the homotopy-commutative interleaving distance is formalized, and whether the fiber condition really propagates through the persistence diagrams without extra constants or assumptions. Those definitions and the inductive or spectral-sequence arguments that would replace the classical Quillen fiber theorem need to be checked line by line. If the authors have carried the original proof over without gaps, the result is solid for its narrow scope; if the persistence version introduces new technical debt, the bound could be weaker than advertised. This is a specialized note in algebraic topology with a TDA flavor. A reader already working on poset homotopy or persistence modules will get the most out of it, while someone outside those two areas will find it too narrow to repay the effort. The paper is coherent on its own terms and shows clear engagement with the relevant literature, so it deserves a serious referee rather than a desk rejection. I would send it out for review with the expectation that the referees focus on the proof details and the precise strength of the bound.

Referee Report

0 major / 2 minor

Summary. The paper establishes a persistence version of the Quillen-McCord theorem for persistence finite posets. Given a map f : P → Q between persistence finite posets P and Q whose homotopy fibers are weakly ε-contractible, it proves an upper bound on the homotopy commutative interleaving distance between P and Q.

Significance. If the result holds, it provides a useful extension of a classical algebraic topology theorem to the persistence setting, enabling bounds on interleaving distances via a natural weakening of the contractible-fiber hypothesis. This strengthens stability and comparison tools for poset-based persistent homology and could support applications in topological data analysis where filtrations are modeled by posets.

minor comments (2)

Abstract: the precise form of the claimed upper bound (e.g., whether it is ε, 2ε, or a function of ε) is not stated explicitly; adding this would clarify the statement for readers.
The definitions of 'weakly ε-contractible' and 'homotopy commutative interleaving distance' should be recalled or referenced in the introduction to make the main theorem self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of our manuscript and for recommending minor revision. The referee's summary accurately describes the main result: a persistence analogue of the Quillen-McCord theorem that bounds the homotopy commutative interleaving distance between persistence finite posets when the homotopy fibers of the map are weakly ε-contractible. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper states a direct theorem adapting the classical Quillen-McCord result to persistence finite posets: given a map f: P → Q with weakly ε-contractible homotopy fibers, an upper bound holds for the homotopy commutative interleaving distance. No equations, definitions, or steps in the abstract or described structure reduce the bound to a self-referential construction, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The hypothesis is stated explicitly as necessary, and the claim is presented as a logical extension of an external classical theorem rather than a tautology of the paper's own inputs. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed from abstract alone; no explicit free parameters, axioms, or invented entities are stated. Standard background assumptions from category theory and persistence theory are presumed but unverified.

axioms (1)

domain assumption Persistence finite posets are well-defined objects in the category of posets with persistence structure
Invoked by the statement of the theorem for P and Q

pith-pipeline@v0.9.0 · 5343 in / 1241 out tokens · 30098 ms · 2026-05-13T01:38:29.412219+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
Given a map f : P → Q between persistence finite posets P and Q with weakly ε-contractible homotopy fibers, we provide an upper bound for the homotopy commutative interleaving distance between P and Q.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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