Hidden Historical Structure in a Minimal Discrete Rewrite System

We study a minimal parallel discrete rewrite system, A → AB or BA and B → A or B. Different histories can reconverge to the same terminal word, creating information that is invisible to the endpoint alone. We define history fibers and additive path observables, derive an exact parity constraint, and show that two explicit reconvergences generate a primitive rank-two historical lattice. Local parsing ambiguities provide the elementary interpretation of these historical differences. After an integral change of basis this lattice is ℤ². Under terminal conditioning, the large-scale counting dynamics approaches the Fibonacci matrix, with dominant eigenvalue φ = (1+√5)/2. The associated local boundary recursion has eigenvalues 1 and -1/2 giving a contractive antisymmetric boundary/orientation mode. Assuming a sufficiently strong limiting historical measure and summable correlations, the hidden historical sector admits a Green-Kubo covariance. A two-boundary contraction hypothesis gives the candidate exponent α = 2 log 2 / log φ ≈ 2.88084. These large-scale statements are explicitly conditional. The paper does not claim to derive physical spacetime, gauge symmetry, electromagnetism, or a fundamental physical theory of information.