Form and Function in Ancient Writing Systems: A Spectral Companion to Computational Paleography

Recent advances in computational paleography (Zemene et al., 2026) applied neural networks to over 28,000 glyph images, identifying significant visual and structural alignments between ancient Ethiopic (Ge'ez) and historical scripts of the Caucasus (most notably Classical Armenian). That work established an objective metric for visual form: stroke curves, angles, and 2D geometric topology. This report delivers the complementary analysis of linguistic function: evaluating writing systems as deterministic discrete dynamical systems. Under the Engineering of Alphabets (EOA) framework, letter names are treated as deterministic recursive sequence engines. Using exact arbitrary-precision integer arithmetic (BigInt Berlekamp–Massey), Frobenius normal form decomposition, and spectral graph theory, we extract the minimal recurrences, characteristic polynomials, and dominant Perron eigenvalues (λ) that govern each script. Across the five target writing systems (totaling 178 individual letters: Ge'ez, Caucasian Albanian, Georgian, Classical Armenian, and Old South Arabian), we establish five distinct structural profiles: The Golden Ratio Core (Ge’ez, φ² ≈ 2.618): Features a 12-letter recurrent core whose sequences satisfy an exact second-order Fibonacci-bisection recurrence, coupled to a 14-letter transient periphery. The Low-Collapse Network (Caucasian Albanian, λ ≈ 2.956): 46 of 49 sequence groups are single letters, yet all 52 letters lock into an identical asymptotic growth constant. The High-Collapse Core (Georgian, λ ≈ 2.521): 33 letters collapse into 11 sequence groups (including an 11-letter consonant block and a 5-letter vowel block) sharing a coupled growth rate. The Decomposed Regime (Armenian): Fragments into 25 distinct internal sequence groups displaying bimodal growth (polynomial and base-2 exponential), demonstrating a divergence between visual form and recurrence dynamics. The Unrecursive Baseline (Old South Arabian): All 29 letters collapse to the constant sequence 1 (λ = 1.000). We propose a three-step collaborative roadmap (including a rank-based Mantel permutation test) to cross-validate these discrete recurrence hubs alongside neural computer-vision paleography models.