phi_unique_self_similar
Self-similar closure forces the golden ratio: r² = r + 1.
plain-language theorem explainer
The golden constraint r² = r + 1 forces the scale ratio to equal the golden ratio φ among all positive reals. Researchers deriving self-similar discrete ledgers from J-cost would cite this uniqueness when closing the forcing chain to φ. The proof is a one-line application of the algebraic uniqueness theorem for the quadratic.
Claim. Let $r > 0$ satisfy the golden constraint $r^2 = r + 1$. Then $r = \phi$, where $\phi = (1 + \sqrt{5})/2$.
background
The PhiForcing module shows that self-similarity in a discrete ledger with J-cost forces φ. The predicate satisfies_golden_constraint r is defined as r² = r + 1; this encodes the requirement that scaling by r composes as r · r = r + 1 in ledger terms, arising from the Fibonacci-like structure of the ledger.
The local setting is given by the module argument: a discrete ledger admits self-similar scales only when the ratio satisfies the golden constraint, and the only positive solution is φ. This rests on upstream results including golden_constraint_unique (the algebraic uniqueness of the positive root) and the definitions of J-cost and LedgerForcing imported from prior modules.
proof idea
One-line wrapper that applies golden_constraint_unique to the hypotheses 0 < r and satisfies_golden_constraint r.
why it matters
This theorem is invoked by phi_harmonic_forced and the matching uniqueness result in FrequencyLadder. It supplies the uniqueness step in the Phi Forcing chain (self-similar ledger implies ratio = φ), directly supporting the framework landmark that φ is forced as the self-similar fixed point (T6). It closes the algebraic part of the argument that scale ratios in the eight-tick octave must be φ.
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papers checked against this theorem (showing 3 of 3)
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Gravity as a broken gauge symmetry, with no Big Bang singularity
"the SSB of the vacuum state in the Wilczek theory is realised when the Higgs-like field acquires a nonzero vacuum expectation value along one of the five directions of the internal space"
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A 38 TeV gauge operator could seed the universe's matter excess
"For the value of g² and λ at finite temperatures, we use g²|_{T≠0} ≃ 0.39 and λ/g²|_{T≠0} ≃ 0.22 obtained by extrapolating from the three-dimensional effective field theory"
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A Noble-Gas-Centered Coordinate for Within-Period Atomic Property Trends
"φ = (1+√5)/2 is presently a modeling ansatz, not a derived chemical constant. ... a first-principles derivation of φ in this chemical context is left to future work."