Fermion Mass Hierarchies from Prime Stability Conditions

Pre-print presenting an empirical construction in which charged fermion mass ratios are described by the grammar 2^a × p (a ∈ {0,1,2,3}, p a single odd prime) at ten energy scales from M_Z to 10^16 GeV. Eight grammar edges compose into a top-rooted chain linking all nine charged-fermion masses through six within-family ratios and two cross-family bridges. The chain resolves into particle mode decompositions built from eight odd primes {5, 17, 53, 59, 97, 127, 211, 271}. The same eight primes compute all nine CKM mixing magnitudes via a six-term impedance network with no fitted parameters — all within 7% of observed values, with no random assignment in two million trials matching the result (< 0.0001%). The Koide formula for charged leptons reduces, within the mode decomposition, to an algebraic consequence of identity-prime cancellation; the same cancellation predicts Q ≈ 3/4 for down quarks and Q ≈ 8/9 for up quarks. First paper of a triptych. Supersedes v2.0 (May 2026). Key changes from v2.0: Koide formula resolved as algebraic consequence of identity-prime cancellation, with two quark Q-value predictions. Key changes from v1.0: Chain composition links all nine charged fermions back to the top quark as root. Mode decomposition assigns each fermion a unique prime subset with Standard Model quantum number correlations. CKM mixing magnitudes computed from mode impedance network — zero fitted parameters, < 0.0001% null fraction. Neutrino sector excluded on structural grounds. Five new tests (16–19, 21) with scripts for independent verification.