Sylvester–Möbius attack: the line homotopy closes rank two and misses the top diagonal

Mode: ATTACK
Date: 7 September 2026
Status: exact reduction and stopping result; Conjecture 4.1 remains open

1. Outcome

The first attack on the reverse direction of Fouli–Grate–Kara–LaClair–McCullough–Nguyen–Sobieska–Udanshive, Conjecture 4.1 passes the flat-restriction gate and closes the proposed rank-two identity. It then fails for a structural degree reason.

For a flat restriction $N=M|X$, the spanning, top-diagonal part of the Koszul complex is canonically the cellular chain complex of a graph with triangular 2-cells. Its vertices are pointed hyperplane non-incidences $(H,a)$, its edges are corank-two exchanges, and its triangular cells are rank-three extensions. Precisely,

$$H_2\bigl(K_{\bullet,r}(N;X)\bigr) \cong H_1\bigl(\mathcal T(N);\Bbbk\bigr), \qquad r=\operatorname{rk}X.$$

The symmetric Sylvester-line operator cannot act on this complex. Inserting a third point $c\in\operatorname{cl}(a,b)\setminus\{a,b\}$ makes $\{a,b,c\}$ a rank-two set. A coefficient flat in $K_{3,r}$ has rank $r-3$, so

$$\operatorname{rk}(G\vee a\vee b\vee c) \le (r-3)+2=r-1.$$

Thus every line-average insertion has zero projection to the rank-$r$ spanning block. No choice of coefficients depending on flat ranks or line cardinalities can repair the missing rank.

There is also no remaining propagation step using T-chords alone. Every Sylvester matroid is already T-chordal: for a circuit $C$ of size at least four, choose $a,b\in C$ and a third point $c$ on their line. Minimality forces $c\notin C$, so $\{a,b,c\}$ is an external triangle chord. Replacing the quadratic hypothesis by its consequence “T-chordal” therefore discards all information not already supplied by Sylvester incidence.

The source paper’s own negative control makes the loss visible. Its rank-three truncation $T(PG(3,2))$ is Sylvester, hence T-chordal, but its graded Möbius algebra is not quadratic. An exact rational flat-block calculation gives

$$\dim H_2\bigl(K_{\bullet,4}(T(PG(3,2));E)\bigr)=209,$$

all in the unique rank-three spanning block. This is not a counterexample to Conjecture 4.1 because it fails the quadratic hypothesis. It is a decisive negative control for the proposed proof: line incidence plus T-chordality does not contract nonlinear second homology. A valid proof must use the full coherence of quadratic generation, not merely the known T-chordality consequence.

This disproves the proposed homotopy architecture, not the conjecture. The remaining question is a new global one about the triangle homology of pointed hyperplane incidence; pursuing it would leave the line/T-chord route and enter the classification-style incidence analysis excluded by the Hunt packet.

2. Flat restriction is fully functorial

Let $X$ be a flat of a simple matroid $M$, and put $N=M|X$.

2.1 The Koszul block is the spanning block of the restriction

A basis vector of the Koszul complex of $A(M)$ is $y_F\otimes e_J$. If

$$\operatorname{cl}_M(F\cup J)=X,$$

then $F\cup J\subseteq X$, because $X$ is closed. The flats of $N$ are exactly the flats $F\le X$ of $M$, and for every $U\subseteq X$,

$$\operatorname{cl}_{N}(U)=\operatorname{cl}_M(U).$$

The multiplication rules and the nonzero terms of the Koszul differential therefore agree term for term. Hence

$$K_{\bullet,d}(M;X) \cong K_{\bullet,d}(M|X;X)$$

canonically as based complexes.

2.2 The Sylvester condition descends

If $a,b\in X$, choose a third point $c$ with $\{a,b,c\}$ a circuit. Then

$$c\in\operatorname{cl}_M(a,b)\subseteq X,$$

and the same triangle is a circuit of $M|X$.

2.3 Quadraticity descends

Write $S_E=\Bbbk[y_e:e\in E]$ and let $\pi_X:S_E\to S_X$ kill the variables outside $X$. The standard presentation of the graded Möbius algebra is generated by the squares and the circuit binomials

$$y_{C\setminus i}-y_{C\setminus j}.$$

If $C\subseteq X$, its image is the corresponding relation of $M|X$. If $C\nsubseteq X$, every monomial in such a circuit binomial contains an outside variable. Indeed, if $C\setminus i\subseteq X$ with $i\notin X$, then

$$i\in\operatorname{cl}_M(C\setminus i) \subseteq \operatorname{cl}_M(X)=X,$$

a contradiction. Consequently

$$\pi_X(I_M)=I_{M|X}.$$

If $I_M$ is generated by its quadratic part, its image is generated by quadrics as well. Thus all three ingredients—the block, Sylvester incidence, and quadraticity—pass to a flat restriction.

3. Only three internal diagonals occur in one flat block

Let $X$ have rank $s$. A degree-$d$ basis term in homological degree two has the form

$$y_F\otimes e_a\wedge e_b, \qquad \operatorname{rk}F=d-2, \qquad \operatorname{cl}(F\cup\{a,b\})=X.$$

Since $F\le X$ and two points raise rank by at most two,

$$d-2\le s\le d.$$

Therefore an $X$-block can contribute to $H_2$ only for

$$d\in\{s,s+1,s+2\}.$$

This sharpens the flat decomposition: the proposed proof does not face an unbounded range inside any fixed flat. Its unresolved part nevertheless includes the top diagonal $d=s$, not merely rank-two line blocks.

4. The symmetric line primitive is exact

Let $L$ be a rank-two flat, let $a,b\in L$, and choose $c\in L\setminus\{a,b\}$. Write wedges in the displayed order. Since all three mixed products equal $y_L$ and all squares vanish,

$$\begin{aligned} d(y_a\otimes e_a\wedge e_b\wedge e_c) &=-y_L\otimes e_a\wedge e_c+y_L\otimes e_a\wedge e_b,\\ d(y_b\otimes e_a\wedge e_b\wedge e_c) &= y_L\otimes e_b\wedge e_c+y_L\otimes e_a\wedge e_b,\\ d(y_c\otimes e_a\wedge e_b\wedge e_c) &= y_L\otimes e_b\wedge e_c-y_L\otimes e_a\wedge e_c. \end{aligned}$$

Hence

$$d\bigl((y_a+y_b-y_c)\otimes e_a\wedge e_b\wedge e_c\bigr) =2y_L\otimes e_a\wedge e_b.$$

Averaging over all third points gives the canonical primitive

$$h_L(y_L\otimes e_a\wedge e_b) =\frac{1}{2(|L|-2)} \sum_{c\in L\setminus\{a,b\}} (y_a+y_b-y_c)\otimes e_a\wedge e_b\wedge e_c,$$

with $d h_L=\mathrm{id}$ on $K_{2,4}(M;L)$. This proves the exact rank-two vanishing without choosing a preferred third point. Characteristic zero is used only to divide by $2(|L|-2)$.

5. The top diagonal is a pointed-hyperplane triangle complex

Let $N$ be a simple rank-$r$ matroid on $E$. Define a graph $\Gamma(N)$ as follows.

• A vertex is a pair $(H,a)$, where $H$ is a hyperplane and $a\notin H$.
• An oriented edge is a triple $[F;a,b]$, where $F$ has rank $r-2$ and $F\vee a\vee b=E$. Its endpoints are $(F\vee a,b)$ and $(F\vee b,a)$.

The spanning block in internal degree $r$ has bases

$$K_{1,r}: (H,a), \qquad K_{2,r}: [F;a,b],$$

and its differential is exactly the graph boundary

$$d_2[F;a,b]=(F\vee a,b)-(F\vee b,a).$$

Now attach a triangular 2-cell for every rank-$(r-3)$ flat $G$ and every triple $a,b,c$ that raises its rank by three. Its cellular boundary is

$$[G\vee a;b,c]-[G\vee b;a,c]+[G\vee c;a,b],$$

which is exactly the Koszul differential of $y_G\otimes e_a\wedge e_b\wedge e_c$. Calling the resulting 2-complex $\mathcal T(N)$, there is an identity of chain complexes in degrees three through one, and therefore

$$\boxed{ H_2(K_{\bullet,r}(N;E)) \cong H_1(\mathcal T(N);\Bbbk). }$$

This is a reformulation, not a vanishing theorem. It exposes the precise global invariant that the line homotopy would have to control.

6. No line-supported homotopy reaches this complex

Every term produced by the proposed operator $\eta_{ab}$ adjoins a point $c$ for which $\{a,b,c\}$ is a circuit. Such a triple has rank two. In $K_{3,r}(N;E)$, however, the coefficient flat has rank $r-3$; submodularity gives

$$\operatorname{rk}(G\vee a\vee b\vee c) \le \operatorname{rk}G+\operatorname{rk}\{a,b,c\} =r-1.$$

The term belongs to a proper-flat block and its projection to the spanning block is zero.

The same obstruction can be read edgewise. If $[F;a,b]\in K_{2,r}(N;E)$, then $F\vee a\vee b=E$. No third point on $\operatorname{cl}(a,b)$ lies in $F$, since such a point $c\in F$ would imply $b\in\operatorname{cl}(F\cup a)$ and lower the spanning rank. But a $K_3$ term can produce the edge $[F;a,b]$ only by multiplying its third exterior element $u\in F$ into a rank-$(r-3)$ coefficient flat. The third points supplied by $\eta_{ab}$ are therefore exactly on the wrong side of $F$.

This is independent of every scalar coefficient. In particular, allowing coefficients to depend arbitrarily on $r$ and $|\operatorname{cl}(a,b)|$ cannot change the zero projection.

The failure occurs at chain level, not merely on a hypothetical nonzero homology class. Choose a basis extension of a rank-$(r-3)$ flat by $a,b,c$. The displayed three-edge boundary is a nonzero cycle in $K_{2,r}$. A line-supported degree-two homotopy has zero spanning projection on it, so it cannot satisfy $dh+hd=\mathrm{id}$.

7. Why T-chordal propagation cannot restore the proof

For completeness, the implication used above is elementary:

$$M\text{ Sylvester}\quad\Longrightarrow\quad M\text{ T-chordal}.$$

If $C$ is a circuit with $|C|\ge4$, take distinct $a,b\in C$. Sylvester incidence supplies $c$ with $\{a,b,c\}$ a circuit. If $c\in C$, this triangle would be a proper dependent subset of $C$, contradicting minimality. Hence $c\notin C$.

The Hunt packet proposed to use

$$A(M)\text{ quadratic}\Longrightarrow M\text{ T-chordal}$$

after closing rank two. Under the simultaneous Sylvester hypothesis this adds no new combinatorial constraint. The exact $T(PG(3,2))$ control shows that the retained constraints allow a large nonlinear spanning block. The quadratic hypothesis must instead enter as an algebraic path-coherence statement among circuit reductions, beyond the existence of individual T-chords.

8. What remains open

The attack establishes five exact facts:

1. flat blocks and both conjectural hypotheses pass functorially to flat restrictions;
2. each flat block has only three possible internal diagonals;
3. the canonical characteristic-zero line average contracts the rank-two block;
4. the spanning top diagonal is governed by $H_1(\mathcal T(M|X))$, which the line average cannot touch; and
5. T-chordality is automatic under Sylvester, while the exact $T(PG(3,2))$ control has $\dim H_{2,4}=209$, so T-chordal propagation cannot recover the discarded quadratic coherence.

It does not prove that this triangle homology is nonzero for a quadratic Sylvester matroid. It therefore gives neither a counterexample nor a proof of Conjecture 4.1.

The theorem that quadraticity implies T-chordality in LaClair–Mastroeni–McCullough–Peeva controls circuit decompositions. The complex $\mathcal T(N)$ is instead built from hyperplanes, corank-two flats, and their non-incidences. No cited result turns T-chords into a contraction of this cocircuit-side complex.

Per the precommitted stopping rule, this route ends here rather than fitting operators to individual geometries or cataloguing pointed-hyperplane complexes. A future return would require one new global theorem:

If $N$ is quadratic and Sylvester, is the pointed-hyperplane triangle complex $\mathcal T(N)$ first-homology acyclic over characteristic zero?

9. Reproducibility and credit

The companion standard-library script checks the exact signs in the rank-two primitive, the triangle boundary, $d_2d_3=0$, and the universal rank deficit. A second script applies the already-audited flat-Koszul engine to the single negative control designated by the source paper and verifies $H_{2,4}=209$ over $\mathbb Q$. It performs no matroid search.

The conjecture, its forward implication, and its source examples are due to Fouli, Grate, Kara, LaClair, McCullough, Nguyen, Sobieska, and Udanshive. The graded Möbius presentation and quadraticity-to-T-chordality implication are due to LaClair, Mastroeni, McCullough, and Peeva. The reductions and stopping argument above are an internal FPRD attack and have not received external review.

NEXT MODE: HUNT. Select a target whose canonical global invariant is already acted on by the available hypotheses; do not replace this failed line homotopy with a pointed-hyperplane census.