HUNT packet: Sylvester lines and the first syzygies of graded Möbius algebras

Mode: HUNT

Selected: 7 September 2026

Status: active research target; no theorem or counterexample is claimed

1. Exact target

Let $M$ be a simple matroid on $E=\{1,\ldots,n\}$, let $S=\Bbbk[y_1,\ldots,y_n]$ over a field of characteristic zero, and let $$A(M)=\operatorname{GMA}(M)=S/I_M$$ be its graded Möbius algebra. Fouli, Grate, Kara, LaClair, McCullough, Nguyen, Sobieska, and Udanshive ask whether the following converse holds.

Conjecture 4.1. Assume $A(M)$ is quadratic. Then $I_M$ is linearly presented if and only if $M$ is Sylvester: every pair of elements of $M$ belongs to a 3-element circuit.

The paper proves the forward implication. The selected problem is the open reverse implication: $$A(M)\text{ quadratic and }M\text{ Sylvester} \quad\Longrightarrow\quad \beta^S_{2,d}(A(M))=0\quad(d\ge 4).$$

This is exactly the vanishing of nonlinear first syzygies. It is one universal yes-or-no statement about a canonical algebra; no representation class, rank bound, or catalogue of matroids is part of the target.

2. Source wording audit

The introduction of arXiv:2609.03827v1 paraphrases the Sylvester condition once using the phrase “unique circuit of size 3.” Section 4, Conjecture 4.1, Murty’s original terminology, and the paper’s uniform examples use the standard condition: at least one 3-circuit through every pair. These are not equivalent. For example, every pair of $U_{2,6}$ lies in four 3-circuits.

This packet uses the formal Section 4 statement. It does not assume a chosen or unique third point.

3. Canonical algebra and its flat-labelled Koszul complex

The algebra has a basis $y_F$ indexed by flats, with $$y_Fy_G= \begin{cases} y_{F\vee G},& \operatorname{rk}(F\vee G)=\operatorname{rk}F+\operatorname{rk}G,\\ 0,&\text{otherwise}. \end{cases}$$ Its Koszul complex on the degree-one generators has basis vectors $$y_F\otimes e_J\in K_p(\mathbf y;A(M)), \qquad |J|=p.$$ Attach to such a vector the flat $$X(F,J)=\operatorname{cl}_M(F\cup J).$$ Every nonzero differential term replaces $(F,J)$ by $(F\vee j,J\setminus\{j\})$, and therefore preserves $X$. Consequently there is a basis-independent chain decomposition $$K_\bullet(\mathbf y;A(M))_d = \bigoplus_{X\in L(M)} K_{\bullet,d}(M;X),$$ and hence $$\beta^S_{2,d}(A(M)) = \sum_{X\in L(M)} \dim_\Bbbk H_2(K_{\bullet,d}(M;X)).$$

The $X$-block uses only flats and elements contained in $X$, so it is the spanning block of the flat restriction $M|X$. This observation is elementary and is not claimed here as a new theorem. Its value is strategic: the conjecture can be attacked one intrinsic flat at a time without choosing a presentation or an ordering.

4. Why the hypotheses meet inside the same blocks

Two existing structural facts point in the same direction.

1. The Sylvester property descends to a flat restriction. If $a,b\in X$, every third point in a 3-circuit through $a,b$ lies in $\operatorname{cl}(a,b)\subseteq X$.
2. LaClair, Mastroeni, McCullough, and Peeva prove that quadraticity of $A(M)$ forces $M$ to be T-chordal: every circuit of size at least four admits an external 3-circuit chord.

The first hypothesis supplies rank-two incidence; the second supplies a uniform reduction of larger circuits. A proof should therefore be expressible as a chain contraction of each spanning flat block, not as a classification of Sylvester matroids.

Because a Sylvester line may contain more than three points, there is no canonical third point. The only permitted rank-two datum in the first attack is the symmetric incidence average $$\eta_{ab} = \frac{1}{|\operatorname{cl}(a,b)|-2} \sum_{c\in\operatorname{cl}(a,b)\setminus\{a,b\}} e_c.$$ Characteristic zero makes this averaging legal. The attack must determine whether the Koszul boundaries generated by these averages extend, through T-chords, to a contraction in every nonlinear internal degree.

5. Exact calibration

The attached dependency-free script constructs $A(M)$ directly from the flat lattice, forms the Koszul differentials over $\mathbb Q$, verifies $d^2=0$, checks that every nonzero matrix entry preserves $X(F,J)$, and computes $H_2$ globally and blockwise.

For every displayed degree, the global homology dimension equals the sum of the exact flat-block dimensions. The value $3$ for $M(K_4)$ agrees with the Betti table in the source paper. These checks validate the carrier and its grading; they do not prove the conjecture.

6. First ATTACK

The next attack has three gates.

Gate A: make the flat reduction functorial

Prove carefully that the $X$-block is the spanning block of $M|X$, and that both Sylvester and quadratic hypotheses pass to flat restrictions. This reduces all nonlinear $H_2$ to one functorial spanning-block statement.

Gate B: test the canonical incidence homotopy

Write the most general homotopy assembled from the symmetric line averages $\eta_{ab}$, with coefficients depending only on flat ranks and line cardinalities. Test the chain-homotopy identity symbolically before using any additional examples.

Gate C: propagate through T-chords

If Gate B closes rank two, use the known implication $$A(M)\text{ quadratic}\Longrightarrow M\text{ T-chordal}$$ to reduce every higher circuit term. The desired conclusion is a uniform contraction of $K_{\bullet,d}(M;E)$ in homological degree two for all $d\ge4$.

If the error term depends on incidence data not controlled by the Sylvester and T-chordal axioms, stop. Do not search through matroid catalogues, choose ad hoc elimination orders, or fit homotopies separately for finite examples.

7. What counts as success or failure

Success is either:

• a universal chain contraction proving the reverse implication of Conjecture 4.1; or
• a structural family producing a nonzero flat-block class and thereby refuting it.

A larger Macaulay2 census, a proof for one additional projective geometry, or an ordering that works only matroid by matroid is not success.

The route is deliberately falsifiable at Gate B. If no symmetry-averaged incidence operator can satisfy the required chain identity for formal Sylvester lines, this target returns to HUNT rather than descending into classification.

8. Candidate comparison

The hunt also examined the Arithmetic Kakeya projection inequality, spectrality of a nonhomogeneous golden-mean self-similar measure, the Generalized Telescope Conjecture, and the new non-supersolvable Orlik–Solomon constructions. The first two naturally descend into set or digit-tree construction searches. The telescope problem has a very deep dependency stack before a bounded new gate appears. The Orlik–Solomon frontier is close in subject but currently organized around families of arrangements.

The Sylvester–Möbius target uniquely combines a new explicit conjecture, one canonical algebra, a canonical global chain complex, and a low-degree structural stop test.

9. Sources and credit

The conjecture, the uniform Betti calculations, the positive examples, and the proof of the forward implication are due to Louiza Fouli, Sean Grate, Selvi Kara, Adam LaClair, Jason McCullough, Vinh Nguyen, Aleksandra Sobieska, and Prajwal Udanshive: https://arxiv.org/html/2609.03827v1.

The presentation of the graded Möbius algebra and the implication from quadraticity to T-chordality are due to Adam LaClair, Matthew Mastroeni, Jason McCullough, and Irena Peeva: https://arxiv.org/html/2412.18499v1.

The Sylvester property is due to U. S. R. Murty: https://doi.org/10.1007/BF01817744.

The flat-labelled attack route and the exact calibration in this packet are our research proposal and audit. They are not attributed to the cited authors, and no priority or theorem claim is made for the elementary decomposition.