Signature Deficiency and Rank-One Scale Extension

A deficient symmetric bilinear form with a single non-positive eigenvalue is repaired by one datum supplied from outside the space, and no operation internal to a form repairs any deficiency whatever. This paper proves that dichotomy for deficient forms under rank-one extension. A symmetric bilinear form on a finite-dimensional real space realizes a finite reflection group, equivalently is the Cartan matrix of a Dynkin type up to normalization, if and only if it is positive-definite. The form is deficient when it is not, that is, when it has a zero or negative eigenvalue. Where the form is the canonical self-description of a cycle at its scale (the Tits form, Stewart, 2026e, *The Endomorphic Collapse Traverses the Foundations of Mathematics*), the deficiency is the algebraic content of what that cycle reports as "impossible." The theorem: a deficient form with exactly one non-positive eigenvalue is repaired to positive-definite by a single rank-one correction supplied from outside the space (the enclosing scale's datum), and this repair is exactly the passage from the Tits form of the four-gon quiver Q, the four-vertex six-arrow strongly connected quiver on {L, S, T, C} with first Betti number β₁ = 3 (Stewart, 2026da, *Four Mathematical Characterizations of the Four-Gon Quiver*), to the form B that *The Endomorphic Collapse Traverses the Foundations of Mathematics* (Stewart, 2026e) derives. A form admits no repairing rank-one extension precisely when there is no enclosing space to supply the datum, which is the terminal case. The bound is sharp, and the sharpness is proved rather than assumed: a form with two non-positive eigenvalues admits no repairing rank-one extension, by a determinant obstruction. The dichotomy is computable and is verified on Q: the Tits form has signature (2, 0, 1) with determinant −14, nondegenerate with a single negative eigenvalue, the rank-one datum is the sign at the convergence vertex, and B is positive-definite with eigenvalues 1, 2, 3 (A₂ × A₁, repaired). The repairable case is shown to contain an entire classical family rather than isolated examples: the Tits form of every connected affine diagram is positive-semidefinite with a one-dimensional radical, hence has signature (n − 1, 1, 0), and is repaired by a single rank-one datum along the diagram's null root. The theorem states the general condition under which a signature deficiency is a scale-limit (repairable one scale up) versus terminal (no enclosing scale), and gives the test: compute whether a repairing rank-one extension exists. This is the algebraic content of "impossibility is a scale-limit made visible," stated so a reader is forced to it by the signature computation rather than persuaded by the reading. **Keywords:** symmetric bilinear form, signature, Sylvester's law of inertia, rank-one perturbation, Cauchy interlacing, matrix determinant lemma, Tits form, Cartan matrix, positive-definiteness, affine diagram, null root, quiver representation type