Matrix-Renormalized Canonical Defects for Substitution Cocycles: Symmetric-Power Eigenforms, Norm Forms, and Finite Boundary States

Abstract

We develop a matrix-valued extension of canonical polynomial defect renormalization for finite-boundary substitution cocycles. The basic algebraic object is an affine bulk/boundary branch, and an anchored direct-sum decomposition canonically separates the purely local term from the bulk-dependent counterterm. For a polynomial observable of degree $m$, cancellation of the top bulk degree is equivalent to an eigenform condition for the induced action on the symmetric power $\operatorname{Sym}^m(V^*)$. Thus scalar renormalization is exceptional for a genuinely non-scalar bulk matrix. When scalar normalization fails, the cyclic subspace generated by the top homogeneous form gives the unique minimal invariant observable space supporting a matrix-valued rescue, and its dimension records the smallest number of coupled observables required for degree reduction. For fixed-point prefixes of a substitution, proper-prefix desubstitution supplies a finite boundary set, while the prefix automaton supplies the additional transition structure needed for a genuinely finite-state statement. We prove that a rational cyclic matrix carries a homogeneous determinant norm form; in the irreducible case this is a number-field norm, canonical up to nonzero rational scaling, and it is an eigenform with eigenvalue $\det(M)$. The Fibonacci quadratic and Tribonacci cubic forms are recovered exactly. For the noble-means family $0 \to 0^m 1,\ 1 \to 0$, the local norm-defect alphabet is exactly $\{0, 1, 4, \dots, m^2\}$, hence has cardinality $m+1$, while the minimal boundary dynamics has only two states. Conversely, Fibonacci and Tribonacci have the same two-valued local alphabet but different dynamic state counts. Finally, the natural observable $x^2$ admits no scalar normalization on any noble-means system but has cyclic renormalization dimension three, giving an explicit minimal matrix rescue.