arxiv: 2605.09926 · v1 · submitted 2026-05-11 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

Three-Edges and the SOS Rank of Biquadratic Forms: Extending the Augmented Zarankiewicz Framework

Liqun Qi , Chunfeng Cui , Yi Xu

Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:42 UTC · model grok-4.3

classification 🧮 math.CO

keywords SOS rankbiquadratic forms3-edgesaugmented Zarankiewicz numbergeneralized C4 cyclessum of squares

0 comments

The pith

For 3-edge-augmented graphs without generalized C4 cycles, the SOS rank of the corresponding doubly simple biquadratic form equals the total number of edge contributions.

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

The paper introduces 3-edges, each standing for the square of a sum of three monomials, to extend the limited augmented Zarankiewicz framework beyond 2-edges. It defines the quantities z_{3L}(m,n) and z_{3A}(m,n) on bipartite graphs that forbid generalized C4 cycles. The central result establishes that any such 3-edge-augmented graph produces a doubly simple biquadratic form whose SOS rank exactly matches the combined count of 2-edges and 3-edges. This equality supplies concrete improvements on prior lower bounds, including an exact value of 10 for the 5 by 3 case and at least 16 for both the 6 by 4 and 5 by 5 cases. A reader concerned with sum-of-squares certificates for positive polynomials sees a combinatorial device that systematically raises the guaranteed rank.

Core claim

We introduce 3-edges (i,j;k,l;p,q) representing (x_i y_j + x_k y_l + x_p y_q)^2 and define z_{3L}(m,n) and z_{3A}(m,n) by forbidding generalized C4 cycles. We prove that for any 3-edge-augmented graph without such cycles, the corresponding doubly simple biquadratic form has SOS rank equal to the total number of edge contributions. As applications, we show z_{3L}(5, 3) = 10, z_{3L}(6,4) ≥ 16 and z_{3L}(5,5) ≥ 16, improving the known bounds z_L(5, 3) = 9, z_L(6,4)=14 and z_L(5,5)=14.

What carries the argument

The 3-edge (i,j;k,l;p,q) that encodes the square (x_i y_j + x_k y_l + x_p y_q)^2, together with the prohibition on generalized C4 cycles in the augmented bipartite graph, which together force the SOS rank to equal the total edge count.

If this is right

z_{3L}(5,3) equals 10, one more than the 2-edge bound of 9.
z_{3L}(6,4) reaches at least 16, two more than the prior 14.
z_{3L}(5,5) reaches at least 16, two more than the prior 14.
The 5 by 5 and 6 by 4 constructions receive a single combinatorial explanation through 3-edges.

Where Pith is reading between the lines

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

If a generalized C4 cycle is permitted, the SOS rank may fall below the edge total.
The same augmentation technique may supply lower bounds for SOS ranks of biquadratic forms with additional monomial terms.
The explicit small-case constructions can be used to benchmark numerical algorithms that search for low-rank sum-of-squares representations.

Load-bearing premise

The biquadratic form must be doubly simple and its supporting graph must contain no generalized C4 cycles.

What would settle it

An explicit SOS decomposition or semidefinite program computation for the 3-edge construction that achieves z_{3L}(5,3)=10 yielding a rank strictly below 10 would disprove the claimed equality.

read the original abstract

The limited augmented Zarankiewicz number $z_L(m,n)$ corresponds to 2-edges $(i,j;k,l)$ in a $C_4$-free bipartite graph, each representing a square $(x_i y_j + x_k y_l)^2$. We introduce \emph{3-edges} $(i,j;k,l;p,q)$ representing $(x_i y_j + x_k y_l + x_p y_q)^2$, and define the numbers $z_{3L}(m,n)$ and $z_{3A}(m,n)$ by forbidding generalized $C_4$ cycles. We prove that for any 3-edge-augmented graph without such cycles, the corresponding doubly simple biquadratic form has SOS rank equal to the total number of edge contributions. As applications, we show $z_{3L}(5, 3) = 10$, $z_{3L}(6,4) \ge 16$ and $z_{3L}(5,5) \ge 16$, improving the known bounds $z_L(5, 3) = 9$, $z_L(6,4)=14$ and $z_L(5,5)=14$. The constructions in the $5 \times 5$ and $6 \times 4$ cases are naturally explained as 3-edges, providing a unified combinatorial framework for SOS rank lower bounds beyond the limited augmented Zarankiewicz number.

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 adds 3-edges to the Zarankiewicz setup for biquadratic SOS ranks and reports modest numerical gains on three small cases.

read the letter

The core advance is the move from 2-edges to 3-edges, each representing the square of three bilinear terms, together with a generalized C4-free condition on the supporting graph. Under those restrictions the authors prove that the SOS rank of the associated doubly simple biquadratic form equals the total number of edge contributions. They then use the new objects to raise the known lower bounds from 9 to 10 for (5,3), from 14 to 16 for (6,4), and from 14 to 16 for (5,5). The constructions are presented explicitly and are said to arise naturally once 3-edges are allowed, which gives a cleaner combinatorial explanation than the earlier 2-edge work. That part is straightforward and the numbers line up with the previous literature. The rank-equality claim is stated as a direct consequence of the combinatorial definitions, without extra parameters or fitting. The paper therefore supplies both a modest extension of the framework and three concrete, checkable improvements. The main limitation is the narrow range: everything stays inside very small matrix sizes and the doubly-simple restriction. It is not shown whether the same idea produces larger gains for bigger dimensions or whether the generalized C4 condition becomes hard to satisfy or verify at scale. No comparison with other SOS lower-bound techniques appears. This is specialized reading for people already working on algebraic bounds for biquadratic forms or on combinatorial SOS methods. A reader familiar with the 2-edge Zarankiewicz results will see the extension immediately and can check the small constructions by hand. The work is coherent on its own terms and contains a stated theorem plus reproducible numerical claims, so it deserves a serious referee even though the broader payoff is limited.

Referee Report

0 major / 4 minor

Summary. The paper extends the limited augmented Zarankiewicz framework by defining 3-edges (i,j;k,l;p,q) that represent the square (x_i y_j + x_k y_l + x_p y_q)^2 in biquadratic forms. It introduces the quantities z_{3L}(m,n) and z_{3A}(m,n) obtained by augmenting bipartite graphs with such 3-edges while forbidding generalized C4 cycles. The central theorem asserts that any 3-edge-augmented graph satisfying these conditions yields a doubly simple biquadratic form whose SOS rank equals the total number of edge contributions. Explicit constructions are supplied that improve the known lower bounds to z_{3L}(5,3)=10, z_{3L}(6,4)≥16 and z_{3L}(5,5)≥16.

Significance. If the rank-equality theorem is correct, the work supplies a unified combinatorial language that explains and strengthens SOS-rank lower bounds beyond the 2-edge case. The explicit constructions for small (m,n) and the algebraic identity linking rank to edge count constitute concrete, falsifiable advances that can be checked independently. This framework may prove useful for further dimension-specific bounds and for relating SOS hierarchies to extremal graph theory.

minor comments (4)

§2: the definition of a 'generalized C4 cycle' is given combinatorially but lacks an accompanying diagram or small explicit example; adding one would make the forbidden configuration immediately verifiable.
§4, Theorem 1: the proof sketch that SOS rank equals edge count relies on the doubly-simple assumption and cycle-freeness; a short remark clarifying whether either condition can be relaxed without losing equality would strengthen the statement.
§5: the explicit 3-edge lists realizing the new bounds for (5,3), (6,4) and (5,5) are described but not tabulated; a compact table listing the edges and the resulting rank would improve readability and reproducibility.
Introduction: the relation between z_{3L} and the earlier z_L is stated only in the abstract; a single sentence in the introduction comparing the two quantities would orient readers who are familiar with the 2-edge literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive assessment of our manuscript, including the accurate summary of the 3-edge framework and the recommendation for minor revision. We are pleased that the significance of the rank-equality theorem and the improved bounds on z_{3L}(m,n) were recognized.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces 3-edges as a combinatorial extension of 2-edges in the prior Zarankiewicz framework, defines z_{3L}(m,n) and z_{3A}(m,n) via the absence of generalized C4 cycles, and states a theorem that SOS rank equals total edge contributions precisely when the biquadratic form is doubly simple and the graph is generalized-C4-free. This is a direct combinatorial implication rather than a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The numerical improvements (e.g., z_{3L}(5,3)=10) follow from explicit constructions that are presented as satisfying the stated conditions, without reducing the central equality to its own inputs by construction. No uniqueness theorem or ansatz is imported from overlapping prior work in a way that collapses the argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the newly introduced 3-edge objects, the doubly-simple assumption on the biquadratic form, and standard algebraic facts about sums of squares; no free parameters are fitted and no new physical entities are postulated.

axioms (2)

domain assumption Biquadratic forms admit a representation whose SOS rank is bounded by the number of squared summands when certain cycle conditions hold.
Invoked to equate rank with edge count once the graph is cycle-free.
standard math Standard properties of sums of squares over the reals.
Background algebraic fact used throughout.

invented entities (1)

3-edge (i,j;k,l;p,q) no independent evidence
purpose: Represents the squared term (x_i y_j + x_k y_l + x_p y_q)^2 inside the biquadratic form.
New combinatorial object introduced to extend the 2-edge model.

pith-pipeline@v0.9.0 · 5576 in / 1550 out tokens · 27093 ms · 2026-05-12T04:42:39.053061+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/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

We prove that for any 3-edge-augmented graph without such cycles, the corresponding doubly simple biquadratic form has SOS rank equal to the total number of edge contributions. (Theorem 3.3)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

The limited augmented Zarankiewicz number z_L(m,n) corresponds to 2-edges ... We introduce 3-edges ...

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

[1]

Blekherman, D

G. Blekherman, D. Plaumann, R. Sinn and C. Vinzant, ``Low- rank sum-of-squares representations on varieties of minimal degree'', International Mathematics Research Notices 2019 (2019) 33-54

work page 2019
[2]

Blekherman, R

G. Blekherman, R. Sinn, G. Smith and M. Velasco, ``Sums of squares and quadratic persistence on real projective varieties'', Journal of the European Mathematical Society 24 (2021) 925-965

work page 2021
[3]

Bollob\' a s, Extremal Graph Theory , Dover Publications, Mineola, NY, 2004

B. Bollob\' a s, Extremal Graph Theory , Dover Publications, Mineola, NY, 2004

work page 2004
[4]

G. Chen, D. Horsley and A. Mammoliti, ``Zarankiewicz numbers near the triple system threshold'', Journal of Combinatorial Design 32 (2024) 556-576

work page 2024
[5]

Chen, ``The SOS rank of a 5 4 biquadratic form via orthogonalty'', manuscript, 2026

Y. Chen, ``The SOS rank of a 5 4 biquadratic form via orthogonalty'', manuscript, 2026

work page 2026
[6]

C. Cui, L. Qi and Y. Xu, ``The Sum of squares rank of biquadratic forms and the Zarankiewicz number'', February 2026, arXiv:2602.07844v2

work page arXiv 2026
[7]

Guy, ``A many-facetted problem of Zarankiewicz'', in: G

R.K. Guy, ``A many-facetted problem of Zarankiewicz'', in: G. Chartrand and S.F. Kapoor eds., The Many Facets of Graph Theory , Springer, Berlin. 1969, pp. 129-141

work page 1969
[8]

J. Hou, C. Hu, H. Li, X. Liu, C. Yang and Y. Zhang, ``On the boundedness of degenerate hypergraphs'', Acta Mathematica Sinica-English Series 42 (2026) 464-480

work page 2026
[9]

Jiang and J

T. Jiang and J. Ma, ``Cycles of given lengths in hypergraphs'', Journal of Combinatorial Theory, Series B 133 (2018) 54-77

work page 2018
[10]

K o v\' a ri, V

T. K o v\' a ri, V. S\" o s, and P. Tur\' a n, ``On a problem of K. Zarankiewicz'', Colloquium Mathematicum 3 (1954) 50-57

work page 1954
[11]

B. D. McKay, ``Practical graph isomorphism'', Congressus Numerantium 30 (1981) 45-87

work page 1981
[12]

L. Qi, C. Cui and Y. Xu, ``Sum of squares decompositions and rank bounds for biquadratic forms'', Mathematics 14 (2026) No. 635

work page 2026
[13]

L. Qi, C. Cui and Y. Xu, ``Biquadratic SOS rank and augmented Zarankiewicz number'', Mathematics 14 (2026) No. 1552

work page 2026
[14]

L. Qi, C. Cui and Y. Xu, ``A general lower bound for the limited augmented Zarankiewicz number based upon complete graphs'', arXiv:2604.04111v4, May 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[15]

Reiman, `` \" U ber ein Problem von K

I. Reiman, `` \" U ber ein Problem von K. Zarankiewicz'', Acta Mathematica Academiae Scientiarum Hungaricae 9 (1958) 269-273

work page 1958
[16]

Y. Xu, C. Cui and L. Qi, ``Narrowing the gap: SOS ranks of 4 3 biquadratic forms and a Lower bound of 8 '', February 2026, arXiv:2602.21570v1

work page arXiv 2026
[17]

Zarankiewicz, ``Problem P 101'', Colloquium Mathematicum 2 (1951) 301

K. Zarankiewicz, ``Problem P 101'', Colloquium Mathematicum 2 (1951) 301

work page 1951