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arxiv: 2607.06275 · v1 · pith:77C7CWFE · submitted 2026-07-07 · math.CO · math.PR

Equality conditions for correlation inequalities

Swee Hong Chan , Igor Pak This is my paper

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classification math.CO math.PR MSC 06D0560E1505D05
keywords Ahlswede-Daykin inequalityFKG inequalityequality conditionsdistributive latticecross-factoringcorrelation inequalitySchur positivitylinear extensions
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The pith

Equality conditions for two pillars of correlation inequalities

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

The paper resolves a half-century-old open problem: when does equality hold in the Ahlswede-Daykin (AD) and Fortuin-Kasteleyn-Ginibre (FKG) inequalities? These two inequalities, foundational in combinatorics, probability, and statistical physics, state that increasing events are positively correlated. The authors prove that equality occurs if and only if the underlying distributive lattice splits as a direct product of two sublattices and the four nonnegative functions in the inequality's hypothesis factor into products of functions on those sublattices, with constants satisfying a multiplicative constraint. This structural rigidity — cross-factoring — is shown to be the sole mechanism producing equality. The proof proceeds by first establishing the result for Boolean lattices (via a consistency lemma, an identification lemma, and careful case analysis handling the combinatorial explosion of zero patterns), then lifting to general distributive lattices through a support product lemma and Birkhoff's representation theorem. The authors then derive equality conditions for a cascade of applications: the LPP and Okounkov inequalities for Schur functions, the Fishburn and Björner inequalities for linear extensions of posets, and the recently introduced ADS inequality. Each application requires bespoke analysis to bridge from the abstract AD equality conditions to the concrete combinatorial setting.

Core claim

The central discovery is Theorem 1.3: four nonnegative functions on a finite distributive lattice achieve equality in the AD inequality if and only if they cross-factor, meaning the lattice decomposes as a direct product L1 × L2 and the four functions split into products of functions on L1 and L2 with constants satisfying αβ = γδ. This single result generates all subsequent equality characterizations — for FKG, LPP, Okounkov, Fishburn, Björner, and ADS inequalities — each derived as a corollary, though each requiring substantial additional work to instantiate the abstract cross-factoring condition in a concrete combinatorial setting.

What carries the argument

The cross-factoring condition (Definition 1.2) is the central object. The proof machinery includes: (1) restriction and averaging operations on Boolean lattice functions that preserve the AD condition, enabling inductive arguments; (2) the consistency lemma (Lemma 6.1), which verifies cross-factoring from a reduced set of identities; (3) the identification lemma (Lemma 7.2), which determines which coordinates belong to which factor by classifying each coordinate as type B1 or B2; (4) the support product lemma (Lemma 8.2), which lifts the Boolean lattice result to general distributive lattices via Birkhoff's representation theorem.

If this is right

  • Equality in the FKG inequality requires the lattice to split as a product L1 × L2, the measure to factor as a product measure on L1 × L2, and the two increasing functions to depend on different factors — giving a precise structural characterization of when positive correlation is tight.
  • Equality in the LPP inequality for skew Schur functions occurs exactly when one skew shape is contained in the other (in both outer and inner parts), generalizing the trivial straight-shape dichotomy to the skew setting.
  • Equality in the Okounkov inequality occurs exactly when the difference of the two skew shapes is a coordinatewise 0-1 pattern (up to a global shift), characterizing when log-concavity of Schur products is tight.
  • Equality in Fishburn's inequality for linear extensions is characterized by four connectivity conditions in the comparability graph of the poset: the relevant subsets must be pairwise disconnected in a specific pattern.
  • Equality in the ADS inequality — the Schur-positive analogue of AD — reduces to either trivial zero conditions or a simple proportionality condition between pairs of functions.

Where Pith is reading between the lines

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

  • The cross-factoring structure suggests that equality in correlation inequalities is fundamentally a decomposition phenomenon: the lattice and functions must separate into independent components, analogous to how equality in the Cauchy-Schwarz inequality requires linear dependence. This parallel could guide searches for equality conditions in other inequalities with similar product structure.
  • The difficulty of lifting equality conditions from the strictly-positive case (where there are no equality cases) to the nonnegative case (where numerous cases emerge in the limit) suggests that limit arguments are inherently lossy for equality characterization. This connects to the analogous phenomenon in the Alexandrov-Fenchel inequality for convex bodies, where equality cases also emerge only i
  • The fact that each application (LPP, Fishburn, ADS) requires substantial additional work beyond the black-box AD equality conditions suggests that the cross-factoring condition, while complete, is not always easy to instantiate — the gap between abstract structural characterization and concrete combinatorial verification is itself a nontrivial mathematical challenge.

Load-bearing premise

The reduction from general distributive lattices to Boolean lattices relies on the support product lemma (Lemma 8.2), which assumes that if the extended function on the Boolean lattice factors as a product, then the support of the original function on the general lattice is isomorphic to the product of the supports. This lattice-theoretic bridge is the structural linchpin: if the sublattice structure interacts badly with the factoring, the reduction could fail.

What would settle it

Construct four nonnegative functions on a distributive lattice that satisfy the AD condition and achieve equality in the AD inequality, but do not cross-factor — i.e., no decomposition of the lattice as a direct product L1 × L2 exists such that the functions split into products with constants satisfying αβ = γδ.

read the original abstract

We prove equality conditions for the Ahlswede--Daykin (AD) inequality and the Fortuin--Kasteleyn--Ginibre (FKG) inequality. We then present a number of applications and special cases of these equality conditions. These include Bj\"orner's and Fishburn's inequalities for linear extensions of finite posets, the Lam--Postnikov--Pylyavskyy (LPP) and the Okounkov inequalities for Schur positivity of products of Schur functions. We conclude with equality conditions for the Ahlswede--Daykin--Schur (ADS) inequality recently introduced in Chan--Chen--Pak--Soskin (2026), which is an AD type extension of the LPP inequality.

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

3 major / 5 minor

Summary. This paper proves equality conditions for the Ahlswede–Daykin (AD) inequality and the Fortuin–Kasteleyn–Ginibre (FKG) inequality on finite distributive lattices. The central result (Theorem 1.3) states that four nonnegative functions a, b, c, d on a finite distributive lattice L satisfy equality in the AD inequality if and only if they cross-factor on L, meaning L decomposes as a direct product L₁ × L₂ and the four functions split into products of functions on L₁ and L₂ with constants satisfying αβ = γδ. The FKG equality conditions (Theorem 1.6) follow as a corollary. The paper then derives equality conditions for several applications: the Lam–Postnikov–Pylyavskyy (LPP) inequality, the Okounkov inequality, the Björner inequality, the Fishburn inequality, and the Ahlswede–Daykin–Schur (ADS) inequality. The proof proceeds by first establishing the Boolean lattice case (Theorem 5.1) via a consistency lemma (Lemma 6.1) and an identification lemma (Lemma 7.2), then generalizing to arbitrary distributive lattices via a support product lemma (Lemma 8.2) and Birkhoff's representation theorem.

Significance. The AD and FKG inequalities are foundational results in combinatorics and probability, and characterizing their equality cases is a natural and long-standing problem (tracing back to Daykin–Kleitman–West [DKW79] and Ahlswede–Khachatrian [AK95]). The paper resolves this problem in full generality. The cross-factoring characterization is clean and verifiable, and the applications to Schur positivity inequalities (LPP, Okounkov, ADS) and poset inequalities (Björner, Fishburn) are substantial and non-trivial. The proof is self-contained, using only standard lattice theory and elementary combinatorics. The derivation of each application requires careful analysis beyond a black-box application of Theorem 1.3, which the paper carries out in detail across Sections 10–13. The parameter-free nature of the characterization (no free parameters or ad-hoc axioms) is a notable strength.

major comments (3)
  1. §8.3, proof of Theorem 1.3: The definition of L₂ on p.28 reads 'L₂ := supp(F₁ + G₁)', which appears to be a typo for 'supp(F₂ + G₂)'. If taken literally, the subsequent claim 'L = supp h ≅ L₁ × L₂' would not follow from Lemma 8.2, since Lemma 8.2 requires the factoring H(x₁,x₂) = [F₁+G₁](x₁)·[F₂+G₂](x₂) and then concludes supp(h) ≅ supp(F₁+G₁) × supp(F₂+G₂). This should be corrected to ensure the application of Lemma 8.2 is valid.
  2. §6.2, Case 1, Eq. (6.4): The definition of d′ is given as d′ := a(0,∗_{n−1}), which appears to be a typo for d′ := d(0,∗_{n−1}). As written, the four functions a′, b′, c′, d′ would not correspond to the restriction operation described in Lemma 5.3, and the subsequent inductive argument would not apply correctly. This typo recurs in Case 2, Eq. (6.6), where d′ := a(0,∗_{n−1}) should presumably be d′ := d(0,∗_{n−1}).
  3. §8.2, Lemma 8.2 (Support Product Lemma): The proof argues that L₁ ∨ L₂ = ϕ(supp f) by showing mutual inclusion. For the direction L₁ ∨ L₂ ⊆ ϕ(supp f), the proof takes x ∈ supp F₁ and argues that (x, z₂) ∈ ϕ(supp f) by first showing (x, w) ∈ ϕ(supp f) for any w ∈ supp F₂ (Eq. 8.4), then using a sequence of join/meet operations on elements of supp F₂ to obtain z₂. The argument is correct but could be streamlined: since (z₁, z₂) is the minimum of ϕ(supp f), we have z₂ ∈ supp F₂, so taking w = z₂ in (8.4) directly gives (x, z₂) ∈ ϕ(supp f). The more elaborate argument with sequences is unnecessary. This is a presentation issue, not a correctness issue.
minor comments (5)
  1. Theorem 9.1: The second lattice in the display is written as 'L₂ = (L₁,∨,∧)' but should be 'L₂ = (L₂,∨,∧)'.
  2. §5.4, Lemma 5.4 proof: In the display equation for d(♢, x∧y), the summation reads 'Σ b(z, x∧y)' but should be 'Σ d(z, x∧y)'.
  3. §13.1: The same L₂ typo as in §8.3 appears — 'L₂ := supp(F₁ + G₁)' should be 'supp(F₂ + G₂)'.
  4. §1.4: The reference [Beck90] is cited as 'Beck' but the bibliography lists the author as 'István Beck'. Consider using a consistent citation key.
  5. The paper would benefit from a notation index or glossary, given the density of notation introduced across Sections 2–13.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful reading and for identifying two genuine typographical errors, both of which we will correct. We also address the suggested streamlining of the proof of Lemma 8.2.

read point-by-point responses

  1. Referee: §8.3, proof of Theorem 1.3: The definition of L₂ on p.28 reads 'L₂ := supp(F₁ + G₁)', which appears to be a typo for 'supp(F₂ + G₂)'. If taken literally, the subsequent claim 'L = supp h ≅ L₁ × L₂' would not follow from Lemma 8.2.

    Authors: The referee is entirely correct. This is a typographical error. The definition of L₂ on p.28 should read 'L₂ := supp(F₂ + G₂)', not 'supp(F₁ + G₁)'. As the referee notes, the application of Lemma 8.2 requires the factoring H(x₁,x₂) = [F₁+G₁](x₁)·[F₂+G₂](x₂), which then yields supp(h) ≅ supp(F₁+G₁) × supp(F₂+G₂). With the corrected definition, the argument goes through as intended. We will fix this in the next revision. revision: yes

  2. Referee: §6.2, Case 1, Eq. (6.4): The definition of d′ is given as d′ := a(0,∗_{n−1}), which appears to be a typo for d′ := d(0,∗_{n−1}). This typo recurs in Case 2, Eq. (6.6).

    Authors: The referee is correct on both counts. In Eq. (6.4) of Case 1, the definition should read d′ := d(0,∗_{n−1}), and the same correction applies to Eq. (6.6) in Case 2. As written with d′ := a(0,∗_{n−1}), the four functions a′, b′, c′, d′ would not correspond to the restriction operation described in Lemma 5.3, and the inductive argument would not apply. With the corrected definition, the functions a′ := a(0,∗_{n−1}), b′ := b(0,∗_{n−1}), c′ := c(0,∗_{n−1}), d′ := d(0,∗_{n−1}) satisfy (AD-cond) by Lemma 5.3, and the induction proceeds as intended. We will correct both occurrences. revision: yes

  3. Referee: §8.2, Lemma 8.2 (Support Product Lemma): The proof of L₁ ∨ L₂ ⊆ ϕ(supp f) could be streamlined by taking w = z₂ directly in (8.4), since (z₁, z₂) is the minimum of ϕ(supp f), so z₂ ∈ supp F₂. The more elaborate argument with sequences is unnecessary.

    Authors: We agree with the referee's observation. Since (z₁, z₂) is the minimum element of ϕ(supp f), we have z₂ ∈ supp F₂, so taking w = z₂ in (8.4) directly gives (x, z₂) ∈ ϕ(supp f) for any x ∈ supp F₁. The more elaborate argument involving sequences of join/meet operations to obtain z₂ from elements of supp F₂ is indeed unnecessary for this direction of the inclusion. We will streamline the proof accordingly in the revision. We note that this is purely a presentation issue; the mathematical content and correctness of the lemma are unaffected. revision: yes

Circularity Check

0 steps flagged

No significant circularity found. The derivation is self-contained.

full rationale

The paper's central result (Theorem 1.3) is a characterization of equality conditions for the AD inequality. The proof chain is: (1) prove the Boolean lattice case (Theorem 5.1) via elementary combinatorial arguments (Lemmas 5.2, 6.1, 7.2, 7.4, 7.5), (2) lift to general distributive lattices via Birkhoff's representation theorem (Theorem 2.2, a standard external result) and the Support Product Lemma (Lemma 8.2, proved in-house in Section 8.2). The applications (Theorems 3.3, 3.6, 3.8, 4.2, 4.5) are derived from Theorem 1.3 and use external benchmarks: Rajan's theorem [Raj04] for Schur function factorization, the LPP inequality [LPP07], and the RLS inequality (Theorem 12.1, proved in [CP23a]). The self-citations to [CP23a] and [CCPS26] are for prior inequalities and constructions, not for uniqueness theorems that would force the present results. The ADS inequality (Theorem 3.7) is cited from [CCPS26], but its equality conditions (Theorem 3.8) are derived here from Theorem 1.3, not assumed. No step reduces to its inputs by construction: the cross-factoring conclusion (Definition 1.2) is genuinely derived from the AD-equality assumption, not defined into it. The Support Product Lemma (Lemma 8.2) is proved from the factoring of the extension F and the lattice embedding, not assumed. The proof of Theorem 5.1 involves a nontrivial case analysis (Lemmas 7.4, 7.5) that is not a tautology. The paper is self-contained against external benchmarks and the derivation chain has no circular links.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper introduces no new mathematical objects or postulated entities. The 'cross-factoring' condition (Definition 1.2) is a new definition, but it is a characterization of existing objects (functions on lattices), not a new entity. The proof introduces auxiliary operations (restriction, averaging) and technical lemmas, but these are proof tools, not new axioms or entities. No free parameters are fitted or tuned.

axioms (5)
  • standard math Birkhoff's representation theorem: every finite distributive lattice embeds into a Boolean lattice via join-irreducibles
    Invoked in §8.3 to reduce the general lattice case to the Boolean lattice case. This is a standard theorem in lattice theory [Bir33].
  • standard math Rajan's theorem: products of Schur functions are equal if and only if the multisets of indexing partitions are equal
    Invoked in Lemma 13.1 to deduce that either λ^(1) = μ^(1) or λ^(2) = μ^(2) from an equality of products of Schur functions. External result [Raj04].
  • standard math The AD inequality (Theorem 1.1) and FKG inequality (Theorem 1.5) hold as stated
    These are the inequalities whose equality conditions are being characterized. They are external results [AD78, FKG71] used as benchmarks.
  • domain assumption The RLS inequality (Theorem 12.1) holds for products of chains
    Used in the proof of the Fishburn equality conditions (§12). This was proved by the authors in [CP23a, Claim 6.3], but it is a separate inequality used as input, not the target result.
  • domain assumption The ADS inequality (Theorem 3.7) holds
    Used as the external inequality whose equality conditions are characterized in Theorem 3.8. Introduced in [CCPS26], which shares authors, but the inequality is an independent result used as input.

pith-pipeline@v1.1.0-glm · 53717 in / 2954 out tokens · 527125 ms · 2026-07-08T11:10:09.079833+00:00 · methodology

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