self_similar_forces_golden_constraint
plain-language theorem explainer
Self-similar structures in a discrete ledger with J-cost force their scale ratio to satisfy the equation r² = r + 1. Researchers deriving φ from closure under scaling cite this step in the forcing chain. The proof destructures the scale-invariance witness and reduces directly to the closure theorem for geometric sequences.
Claim. If a structure encodes self-similarity via a positive scale ratio r distinct from 1 together with a closed geometric scale sequence, then that ratio obeys the equation $r^2 = r + 1$.
background
The module shows that self-similarity in a discrete ledger with J-cost forces the golden ratio. A self-similar structure consists of a scale ratio r with 0 < r and r ≠ 1, witnessed by a closed geometric scale sequence having that ratio. The golden constraint is the algebraic condition r² = r + 1. This setting rests on the law of existence and discreteness forcing from prior modules. The key upstream result states: 'If a geometric scale sequence is closed under additive composition, then the ratio r must satisfy r² = r + 1.'
proof idea
The proof destructures the scale-invariance witness of the self-similar structure to obtain a geometric scale sequence G, the ratio equality, and the closedness hypothesis. It unfolds the golden constraint definition, rewrites using the ratio equality, and applies the closure theorem directly to G and the closedness hypothesis.
why it matters
This result is invoked by the phi forcing theorem to conclude that the scale ratio equals φ. It fills the step in the forcing chain where self-similarity produces the fixed point satisfying x² = x + 1 (T6). The argument relies on the recognition composition law to ensure cost equivalence under scaling.
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