A note on the M\"obius uncertainty principle for posets

Anurag Sahay

arxiv: 2603.01018 · v2 · submitted 2026-03-01 · 🧮 math.CO · math.NT

A note on the M\"obius uncertainty principle for posets

Anurag Sahay This is my paper

Pith reviewed 2026-05-15 18:39 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords Möbius inversionuncertainty principleposetslocally finite posetsincidence algebraslatticesGoh conjecture

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The pith

A necessary criterion for the Möbius uncertainty principle on posets disproves Goh's conjecture except when the poset is a lattice.

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

The paper studies two extensions of the uncertainty principle that links the supports of a function and its Möbius inverse on locally finite posets. For the first extension it supplies both a simplified sufficient condition and a necessary condition on the intervals of the poset. The necessary condition shows that Goh's conjectured list of posets admitting the principle is too broad, yet the same list is proved correct once the poset is required to be a lattice. For a second, new extension that applies when the reduced incidence algebra has a restricted form, the principle is verified to hold on the poset of finite subsets of the naturals and on the poset of finite-dimensional subspaces of a vector space over a finite field.

Core claim

We give a necessary criterion for the uncertainty principle to hold that disproves Goh's conjectural characterization of posets admitting an uncertainty principle, but we show the characterization is valid when the poset is a lattice. We also introduce a second generalization of the principle for posets whose reduced incidence algebras have a certain form and prove it holds for the Boolean lattice of finite sets and the subspace lattice over finite fields.

What carries the argument

The reduced incidence algebra of the poset, which encodes the zeta and Möbius functions so that uncertainty can be stated as a support condition on pairs of functions and their transforms.

If this is right

Every lattice satisfies the uncertainty principle under the first generalization.
The poset of finite subsets under inclusion satisfies the uncertainty principle under the second generalization.
The poset of finite-dimensional subspaces over a finite field satisfies the uncertainty principle under the second generalization.
The proofs for the subset and subspace cases rely on different combinatorial arguments than the proof for the divisibility poset.

Where Pith is reading between the lines

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

Non-lattice posets may need extra structural restrictions before uncertainty principles can be expected to hold.
The same interval condition might be used to test whether uncertainty principles exist for other combinatorial objects such as matroids or graphs.
The second generalization could be applied to additional posets whose incidence algebras admit a product decomposition.

Load-bearing premise

The poset is locally finite and its reduced incidence algebra takes the specific form required by the chosen generalization.

What would settle it

A locally finite poset in which a pair of functions satisfies the uncertainty principle yet violates the necessary interval condition derived in the paper.

read the original abstract

We consider two generalizations of Pollack's uncertainty principle for M\"obius inversion to locally finite posets. The first generalization was previously studied by Goh. Here, we provide a simplified sufficient criterion for the uncertainty principle to hold. We also provide a necessary criterion for the same which, in particular, disproves Goh's conjectural characterization of posets for which an uncertainty principle holds. Nevertheless, we prove that Goh's conjecture indeed holds when the poset forms a lattice. The second generalization is new and applies to posets with reduced incidence algebras of a certain form. Here, we make some preliminary observations, including the fact that the uncertainty principle holds for the poset of finite subsets of natural numbers and the poset of finite dimensional subspaces of $\mathbb{F}_q^\infty$. Our proofs in these settings are quite different from the proof for the poset of natural numbers under divisibility.

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 note gives a necessary condition that disproves Goh's conjecture on Möbius uncertainty principles for posets, confirms the conjecture for lattices, and adds a new generalization verified on two concrete examples.

read the letter

The core contribution is a necessary criterion for the uncertainty principle on locally finite posets that rules out Goh's conjectured characterization, paired with a direct proof that the conjecture holds when the poset is a lattice. The author also introduces a second generalization to reduced incidence algebras of a particular form and verifies the principle on the poset of finite subsets of the naturals and on finite-dimensional subspaces of F_q^infty, using arguments that differ from the classical divisibility case. These pieces are presented cleanly from the definitions of the incidence algebra and Möbius function, with no fitted parameters or circular steps. The lattice confirmation and the two new examples are the parts that feel most solid and self-contained. The simplified sufficient criterion is a modest but useful clarification of prior work. The disproof via the necessary condition is the headline result, and it appears internally consistent on the claims given. One minor point worth checking in the full text is whether the counterexample poset used for the disproof satisfies the exact reduced incidence algebra form required by the necessary condition; if that match is only assumed rather than verified explicitly, the scope of the disproof narrows slightly, though the lattice case remains independent. Overall the derivations look direct and the citation pattern stays within the relevant order-theory literature without overclaiming. This is a short, focused note for readers already working on Möbius inversion or uncertainty principles in posets and combinatorial number theory. Anyone following Goh's conjecture or looking for small extensions of Pollack's result will get concrete value from the corrections and the two fresh examples. It is narrow enough that it will not interest a broad audience, but the claims are precise enough to merit referee time rather than a desk rejection. I would send it out for review.

Referee Report

2 major / 1 minor

Summary. The paper presents two generalizations of Pollack's uncertainty principle for Möbius inversion to locally finite posets. For the first generalization, it supplies a simplified sufficient criterion and a necessary criterion that disproves Goh's conjectural characterization of posets satisfying the principle, while separately proving that the conjecture holds when the poset is a lattice. For the second (new) generalization, applicable to posets whose reduced incidence algebras take a specific form, it records preliminary observations that the principle holds for the poset of finite subsets of the natural numbers and for the poset of finite-dimensional subspaces of F_q^∞, with proofs distinct from the divisibility case on the naturals.

Significance. If the derivations are complete, the work clarifies the precise scope of the uncertainty principle in the incidence algebra setting by supplying both a necessary condition and a sufficient condition, by furnishing a concrete disproof of a prior conjecture together with a positive result for the important subclass of lattices, and by extending the framework to two new families of posets with explicit, non-standard proofs. The direct derivations from the definitions of the incidence algebra and Möbius function constitute a methodological strength.

major comments (2)

[necessary criterion section] § on the necessary criterion for the first generalization: the disproof of Goh's conjecture rests on the claim that the chosen counterexample poset satisfies the specific algebraic form required for the reduced incidence algebra; the manuscript does not explicitly verify this condition for the counterexample, so the applicability of the necessary criterion to that poset remains unconfirmed.
[second generalization section] § on the second generalization: the statement that the uncertainty principle holds for the poset of finite subsets of ℕ and for the poset of finite-dimensional subspaces of F_q^∞ presupposes that both posets meet the 'certain form' required of the reduced incidence algebra; an explicit check that the zeta function and convolution structure match the assumed form is needed to make the claim load-bearing.

minor comments (1)

[abstract] The abstract refers to 'the poset of natural numbers under divisibility' without a forward reference to the section where this poset is treated; adding a brief parenthetical citation would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper to incorporate the requested clarifications.

read point-by-point responses

Referee: [necessary criterion section] § on the necessary criterion for the first generalization: the disproof of Goh's conjecture rests on the claim that the chosen counterexample poset satisfies the specific algebraic form required for the reduced incidence algebra; the manuscript does not explicitly verify this condition for the counterexample, so the applicability of the necessary criterion to that poset remains unconfirmed.

Authors: We agree that an explicit verification is needed. In the revised manuscript we will add a direct check confirming that the counterexample poset has a reduced incidence algebra of the required algebraic form, by verifying the zeta function and convolution structure explicitly. This will confirm the applicability of the necessary criterion and strengthen the disproof of Goh's conjecture. revision: yes
Referee: [second generalization section] § on the second generalization: the statement that the uncertainty principle holds for the poset of finite subsets of ℕ and for the poset of finite-dimensional subspaces of F_q^∞ presupposes that both posets meet the 'certain form' required of the reduced incidence algebra; an explicit check that the zeta function and convolution structure match the assumed form is needed to make the claim load-bearing.

Authors: We agree that an explicit verification would make the claims more robust. In the revision we will insert direct computations showing that both the poset of finite subsets of ℕ and the poset of finite-dimensional subspaces of F_q^∞ have reduced incidence algebras matching the assumed form, including explicit checks of the zeta function and convolution structure. This will support the preliminary observations in the second generalization. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations follow directly from incidence algebra definitions

full rationale

The paper derives its necessary criterion for the uncertainty principle and the disproof of Goh's conjecture from the standard definitions of the Möbius function and incidence algebra on locally finite posets, without any fitted parameters, self-referential normalizations, or load-bearing self-citations. The lattice case is proved independently. No step reduces by construction to its own inputs; the counterexample application is a direct verification under the stated assumptions rather than a tautology. This is a standard self-contained mathematical argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper works entirely within the standard theory of locally finite posets and their incidence algebras. No free parameters are introduced. The only background results invoked are the definition of the Möbius function via the incidence algebra and the standard properties of reduced incidence algebras.

axioms (2)

domain assumption The poset is locally finite, so every interval has finitely many elements and the incidence algebra is well-defined.
Stated in the opening sentence of the abstract and required for Möbius inversion to be defined.
domain assumption The reduced incidence algebra takes a specific form that allows the uncertainty principle to be stated.
Required for the second generalization; invoked when the new version is introduced.

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discussion (0)

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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

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unclear
Relation between the paper passage and the cited Recognition theorem.

We consider two generalizations of Pollack’s uncertainty principle for Möbius inversion to locally finite posets... reduced incidence algebras of a certain form
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Phas H_N ⇒ Phas M ⇒ Phas H_1 and H_2

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