A divisibility theorem for odd $J$-characteristics of two-level designs

Pieter Thijs Eendebak

When all proper odd-order J-characteristics of a two-level design vanish, the top odd characteristic is divisible by 2^{n-1}, forcing any strength-3 even-odd design with vanishing J5 and J7 to have at least 256 runs.

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2026-07-13 02:43 UTC pith:Y6SFSUAR

load-bearing objection Clean elementary divisibility theorem that settles the 2023 ESVG non-existence conjecture uniformly in n; proof is short, self-contained, and sharp.

arxiv 2607.09478 v1 pith:Y6SFSUAR submitted 2026-07-10 math.CO math.STstat.TH

A divisibility theorem for odd J-characteristics of two-level designs

Pieter Thijs Eendebak This is my paper

classification math.CO math.STstat.TH MSC 05B1562K1506E30

keywords two-level designsJ-characteristicsWalsh-Hadamard transformeven-odd designsstrength-3 designsdivisibility

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Two-level designs are collections of runs with factors set to ±1. Their signed J-characteristics measure how balanced every subset of factors is; vanishing low-order ones means high strength. The paper proves that if the number of factors n is odd and every proper odd-order J-characteristic is zero, then the full-product characteristic j([n]) must be a multiple of 2^{n-1}. Projecting a larger design onto the smallest odd subset that still has a nonzero J-characteristic reduces to this situation, so any strength-3 design that also kills orders 5 and 7 yet keeps some nonzero odd-order characteristic needs at least 256 runs. That arithmetic lower bound rules out the 56- and 64-run even-odd designs previously conjectured to be impossible, for every number of factors. The bound is tight: the even-weight half-fraction attains equality at every odd order.

Core claim

If n is odd and j(s)=0 for every proper odd-cardinality subset s of the factors, then 2^{n-1} divides the top characteristic j([n]). Consequently any strength-3 two-level design with J5 ≡ J7 ≡ 0 that still possesses a nonzero odd-order J-characteristic must satisfy N ≥ 256, settling the non-existence conjecture for N = 56 and 64 uniformly in the number of factors.

What carries the argument

The signed inversion identity that recovers the multiplicity difference f(x)−f(−x) from the odd-order J-characteristics; under the vanishing hypothesis it collapses to 2^{n-1}(f(x)−f(−x))=j([n])χ_{[n]}(x), immediately yielding the divisibility.

Load-bearing premise

That the multiplicity function takes only non-negative integer values, so that the absolute size of any J-characteristic cannot exceed the total number of runs.

What would settle it

Exhibit a two-level design with fewer than 256 runs that is strength 3, has vanishing fifth- and seventh-order J-characteristics, and still has at least one nonzero odd-order J-characteristic; or find an integer-valued (possibly signed) multiplicity function on an odd number of factors whose proper odd J-characteristics vanish yet whose top characteristic is not divisible by 2^{n-1}.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Clean elementary divisibility theorem that settles the 2023 ESVG non-existence conjecture uniformly in n; proof is short, self-contained, and sharp.

read the letter

The one thing to know: this is a four-page pure-math note that proves a new divisibility fact for odd-order J-characteristics and immediately uses it to kill the remaining open cases of the 2023 ESVG conjecture on strength-3 even–odd designs. No enumeration, no computer search—just algebra.

What is new is Theorem 3: when n is odd and every proper odd-cardinality J-characteristic vanishes, 2^{n-1} divides the top characteristic j([n]). The argument is elementary and transparent: invert the Walsh–Hadamard transform at x and -x, cancel the even-order terms, collapse under the vanishing hypothesis, and evaluate at the all-+1 vector. Projection (Lemma 5) then reduces any design with a first nonzero odd order q≥9 to a q-factor instance of the same situation, so N≥2^{q-1}≥256. That settles the conjecture for every number of factors at once. Sharpness is shown by the even-weight half-fraction, which attains equality at every odd order; the q=7 case also explains why the J7≡0 hypothesis cannot be dropped at N=64.

The soft spots are minor and do not touch the main claim. Non-negativity of the multiplicity function is used only to convert divisibility into a run-size lower bound; the pure arithmetic statement holds for any integer-valued f. The classical Walsh–Hadamard lemmas are restated for completeness rather than novelty, which is fine. Self-citation of the 2023 conjecture paper is appropriate and does not create circularity—the proof relies only on the definitions. The AI-assisted drafting note is honest and irrelevant to correctness.

This is for people who work on two-level fractional factorials, even–odd designs, or J-characteristics. Anyone enumerating strength-3 designs or teaching Walsh–Hadamard methods will get immediate value. The math is solid, the citations are tight, and the result is useful. I would send it to peer review without hesitation; a short referee report confirming the algebra is all that is needed.

Referee Report

0 major / 4 minor

Summary. The paper proves a divisibility theorem for signed J-characteristics of two-level designs: if n is odd and j(s)=0 for every proper odd-cardinality subset s of [n], then 2^{n-1} divides j([n]) (Theorem 3). The argument uses the Walsh–Hadamard inversion formula evaluated at x and −x, cancels even-order terms by parity, and isolates the top characteristic under the vanishing hypothesis. Projection onto a minimal odd-order support (Lemma 5) then yields Corollary 6: any strength-3 design with J5≡J7≡0 that has a nonzero odd-order J-characteristic must have N≥256. This settles, uniformly in the number of factors, the Eendebak–Schoen–Vazquez–Goos conjecture on the nonexistence of certain even–odd strength-3 designs at N=56 or 64. Sharpness is shown by the even-weight half-fraction (Proposition 7).

Significance. The result is a clean, elementary arithmetic obstruction that closes a concrete open question in the enumeration of two-level even–odd designs of strength 3. The bound N≥256 is uniform in the number of factors and is sharp at every odd order, so the theorem is both decisive for the stated conjecture and of independent interest for the structure of J-characteristics. The proofs are short, self-contained, and rely only on standard Walsh–Hadamard identities; the sharpness construction is classical and correctly identified. The paper therefore supplies a definitive, easily checked resolution of the 2023 conjecture together with a reusable divisibility principle.

minor comments (4)

The abstract and introduction state the vanishing hypotheses for the design application as orders 1,2,3,5,7, while Theorem 3 itself concerns only odd-order characteristics. A single clarifying sentence that even-order vanishing is not needed for the pure divisibility statement (but is part of the strength-3 hypothesis feeding Corollary 6) would prevent a possible misreading.
In the proof of Theorem 3 the evaluation is performed only at the all-+1 vector. It may be worth a parenthetical remark that the same identity holds at every x (with the sign of χ_{[n]}(x)), which makes the integrality argument slightly more transparent.
Proposition 7 asserts j(s)=0 for every nonempty proper subset; the short group-character argument given is correct, but an explicit reference to the fact that H_q is a subgroup of index 2 would make the nontriviality of the restricted characters immediate for readers less familiar with the construction.
The MSC codes and keywords are appropriate; adding “Walsh–Hadamard transform” or “Boolean analysis” as a secondary keyword would improve discoverability for the broader discrete-analysis audience that uses the same identities.

Circularity Check

0 steps flagged

No circularity: the divisibility theorem is a direct algebraic consequence of Walsh–Hadamard inversion under the stated vanishing hypotheses.

full rationale

The paper’s central claim (Theorem 3) is obtained by applying the inversion formula (Lemma 2) at x and −x, subtracting, and using that n is odd so that the only surviving odd-cardinality term is the top characteristic; the resulting identity (6) immediately yields 2^{n−1}|j([n]). Projection (Lemma 5) then reduces any nonzero odd-order characteristic of a strength-3 design with J5≡J7≡0 to the same situation on a q-factor design with q≥9, giving N≥256 (Corollary 6). Sharpness is exhibited by an explicit construction (the even-weight half-fraction). The 2023 conjecture that is settled is merely the statement being proved; the argument nowhere invokes an unproved claim from that paper or any other self-citation as a load-bearing premise. The derivation is therefore self-contained and free of definitional, fitted-input, or self-citation circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Pure combinatorial argument. No fitted constants. Background is standard Walsh–Hadamard analysis of Boolean functions and the classical definition of J-characteristics of two-level designs. No new physical or combinatorial entities are postulated.

axioms (3)

standard math Walsh–Hadamard orthogonality and inversion formulae on the hypercube {−1,+1}^n (Lemmas 1–2).
Classical identities; self-contained proofs are supplied in §2 for completeness.
domain assumption J-characteristics are the integer coordinates of a design in the Walsh–Hadamard basis; strength-t means the first t orders vanish.
Standard definitions in the two-level design literature (Deng–Tang, Tang, O’Donnell), used throughout.
domain assumption Row-multiplicity functions take values in the non-negative integers (so |j(s)|≤N).
Needed only for the run-size interpretation of Corollary 4; the pure divisibility statement holds for any integer-valued f.

pith-pipeline@v1.1.0-grok45 · 9364 in / 2203 out tokens · 35654 ms · 2026-07-13T02:43:18.098536+00:00 · methodology

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read the original abstract

We prove a divisibility theorem for the signed $J$-characteristics of two-level designs: if the number of factors $n$ is odd and every $J$-characteristic of a proper odd-cardinality subset of factors vanishes, then the top $J$-characteristic is divisible by $2^{n-1}$. As an arithmetic consequence, any two-level design whose $J$-characteristics vanish in orders one, two, three, five, and seven but which has a nonzero odd-order $J$-characteristic must have at least $256$ runs. This settles, uniformly in the number of factors, a conjecture of Eendebak, Schoen, Vazquez, and Goos (2023) on the nonexistence of certain strength-three even--odd designs with $56$ or $64$ runs. The divisibility bound is sharp at every odd order and is attained by the even-weight half-fraction.

discussion (0)

Reference graph

Works this paper leans on

6 extracted references

[1]

Deng and B

L.-Y. Deng and B. Tang,Generalized resolution and minimum aberration criteria for Plackett– Burman and other nonregular factorial designs, Statist. Sinica9(1999), 1071–1082

1999
[2]

Tang,Theory ofJ-characteristics for fractional factorial designs and projection justification of minimumG 2-aberration, Biometrika88(2001), 401–407

B. Tang,Theory ofJ-characteristics for fractional factorial designs and projection justification of minimumG 2-aberration, Biometrika88(2001), 401–407

2001
[3]

P. T. Eendebak, E. D. Schoen, A. Vazquez, P. Goos,Systematic enumeration of two-level even– odd designs of strength 3, Comput. Statist. Data Anal.180(2023), 107678

2023
[4]

O’Donnell,Analysis of Boolean Functions, Cambridge University Press, 2014

R. O’Donnell,Analysis of Boolean Functions, Cambridge University Press, 2014

2014
[5]

G. E. P. Box and J. S. Hunter,The2 k−p fractional factorial designs, Technometrics3(1961), 311–351

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[6]

G. E. P. Box, J. S. Hunter, W. G. Hunter,Statistics for Experimenters: Design, Innovation, and Discovery, 2nd ed., Wiley, 2005. 4

2005

[1] [1]

Deng and B

L.-Y. Deng and B. Tang,Generalized resolution and minimum aberration criteria for Plackett– Burman and other nonregular factorial designs, Statist. Sinica9(1999), 1071–1082

1999

[2] [2]

Tang,Theory ofJ-characteristics for fractional factorial designs and projection justification of minimumG 2-aberration, Biometrika88(2001), 401–407

B. Tang,Theory ofJ-characteristics for fractional factorial designs and projection justification of minimumG 2-aberration, Biometrika88(2001), 401–407

2001

[3] [3]

P. T. Eendebak, E. D. Schoen, A. Vazquez, P. Goos,Systematic enumeration of two-level even– odd designs of strength 3, Comput. Statist. Data Anal.180(2023), 107678

2023

[4] [4]

O’Donnell,Analysis of Boolean Functions, Cambridge University Press, 2014

R. O’Donnell,Analysis of Boolean Functions, Cambridge University Press, 2014

2014

[5] [5]

G. E. P. Box and J. S. Hunter,The2 k−p fractional factorial designs, Technometrics3(1961), 311–351

1961

[6] [6]

G. E. P. Box, J. S. Hunter, W. G. Hunter,Statistics for Experimenters: Design, Innovation, and Discovery, 2nd ed., Wiley, 2005. 4

2005