One Bit Resurrects The Dead Form

Abstract

The four-vertex strongly connected quiver Q's canonical bilinear invariant selects nothing, and its formula has no argument position for a sign. Let Q be the quiver with arrows a₁: L → S, a₂: S → T, a₃: T → C, a₄: C → L, a₅: L → T, a₆: C → T, whose unique in-degree-three vertex is T. On the cycle-support basis {γ₁, γ₂, γ₃} the Tits form has diagonal (−4, −2, 0) and off-diagonal (−4, −3, −2), signature two positive and one negative, determinant −14, and no Cartan matrix of any Dynkin type. The Euler form is constant as a function of any assignment of a sign to the vertices of Q, because its formula contains no term evaluated on a vertex beyond that vertex's own incidences, so no placement of the sign changes any entry of the Euler form or of its symmetrization. The Tits form is blind to the datum a second way, since symmetrization sums each arrow's contribution with its reverse and cancels arrow direction out of the total. The canonical invariant therefore distinguishes Q's four vertices, which have pairwise distinct incidence profiles and admit only the trivial automorphism, and still cannot record which of them is signed. One datum supplied from outside Q closes the gap, its size and shape are fixed. Its location is forced by the graph: a₃ is the unique exit of the unique in-degree-three vertex, which either route to the bilinear form reads off Q's degree sequence with no external input. Its sign-value is what the graph cannot author, the address being internal to the form and the value not (*Signature Deficiency and Rank-One Scale Extension*, Stewart, 2026de). Executing the four axioms of occurrence on that one distinction yields a bilinear form B on H₁(Q; ℝ), unique up to diagonal scaling, whose matrix on the same cycle generators that gave the dead Tits form is the Cartan matrix of A₂ × A₁, positive-definite, with Weyl group S₃ × ℤ/2 and Cartan-Killing image su(3) ⊕ su(2). Every entry of B is a cardinality read off traversal sets uniquely determined by their basepoints, since each vertex on a cycle in Q has a unique outgoing cycle-arrow. The available off-diagonal entries are therefore −1 and 0 alone, which are the two that place the mirrors at π/3 and π/2 and close the reflection group at finite order, where an arbitrary symmetric form on a three-dimensional space generates a group of infinite order. The finiteness of the type is a consequence of the entries being counts, and the type is not reachable by reweighting once the sign is placed. The cycle-generator basis has rank three, which A₂ × A₁ saturates. The fourth generator carrying the u(1) of the full Standard Model gauge algebra is not supplied by H₁(Q; ℝ) and is developed in companion work (*The Temporal Generator*, Stewart, 2026q). The paper proposes that the located-but-unauthorable sign is the structural signature of self-application, a closure computing its own invariant terminating at the one bit it enacts rather than reads (Stewart, 2026g, Consequence 7b), and reads the construction below in that frame. **Keywords:** quiver, path algebra, quiver representations, quotient algebra, Tits form, Euler form, Cartan matrix, Coxeter group, Dynkin diagram, root system, Lie algebra, special unitary group, first homology