pith. sign in
theorem

phi_unique_self_similar

proved
show as:

Self-similar closure forces the golden ratio: r² = r + 1.

module
IndisputableMonolith.Foundation.PhiForcing
domain
Foundation
line
186 · github
papers citing
3 papers (below)

plain-language theorem explainer

The theorem shows that any positive real r obeying the golden constraint r² = r + 1 must equal the golden ratio φ. Researchers deriving forced scale factors from self-similar discrete ledgers in Recognition Science cite this uniqueness step. The proof is a direct one-line application of the algebraic uniqueness lemma for the quadratic.

Claim. Let $r > 0$ satisfy $r^2 = r + 1$. Then $r = φ$, where $φ = (1 + √5)/2$.

background

The Phi Forcing module starts from a discrete ledger equipped with J-cost and self-similarity. The predicate satisfies_golden_constraint encodes the requirement that a scale ratio r must obey r² = r + 1 so that the J-cost remains invariant under scaling; this follows because the next scale must equal the current scale plus the base unit in ledger terms. The upstream golden_constraint_unique theorem states that the golden constraint characterizes φ among positive reals, while FrequencyLadder supplies the IsSelfSimilarRatio predicate and IntegrationGap supplies the active-edge count A that close the forcing argument.

proof idea

The proof is a one-line wrapper that applies golden_constraint_unique to the hypotheses 0 < r and satisfies_golden_constraint r.

why it matters

This declaration completes the self-similarity forcing step inside Phi Forcing and is invoked by phi_harmonic_forced in FrequencyLadder. It realizes the T6 landmark that φ is forced as the self-similar fixed point of the discrete ledger. The module doc-comment records that non-trivial self-similarity requires the unique positive solution to x² = x + 1.

Switch to Lean above to see the machine-checked source, dependencies, and usage graph.

papers checked against this theorem (showing 3 of 3)