Why φ is forced
In a discrete ledger with J-cost, self-similarity forces the scale ratio r to satisfy the golden constraint.
- Geometric scales form the sequence {1, r, r², …} (GeometricScaleSequence).
- Ledger composition is additive: compose(a, b) = a + b, motivated by J-cost additivity (J_additive_for_independent).
- Closure requires the composed scale to lie in the sequence: 1 + r = r² (isClosed).
closure_forces_golden_equation proves this yields r² = r + 1.
golden_constraint_unique proves the unique positive solution is φ = (1 + √5)/2 (using phi_equation and positivity).
phi_forced concludes: for any self-similar discrete ledger L with ratio r, r = φ.
phi_forcing_complete packages the full derivation from closure to uniqueness.