phi_forcing_principle
plain-language theorem explainer
Researchers tracing the forcing chain cite the result that self-similarity in a J-cost ledger forces the golden ratio φ to satisfy φ² = φ + 1 as its unique positive solution, with the bit cost and coherence energy positive. The principle occupies level 4 between ledger construction and dimension forcing. Its term proof assembles the equation, a uniqueness lemma, and two positivity facts without additional hypotheses.
Claim. Let φ = (1 + √5)/2. Then φ² = φ + 1, φ is the unique positive real solution to r² = r + 1, the minimum cost satisfies 0 < ln φ, and the coherence quantum satisfies 0 < φ^{-5}.
background
The PhiForcing module derives the golden ratio from self-similarity in a discrete ledger equipped with J-cost. J_bit denotes the minimum non-trivial cost, defined as the natural log of φ. E_coh is the coherence quantum given by φ to the minus five. From the module, self-similar scales satisfy the equation x² = x + 1, with φ the unique positive solution. The upstream phi_equation theorem states that φ² = φ + 1 by unfolding the definition and applying algebraic identities. Positivity lemmas for J_bit and E_coh are imported from Constants and related modules, relying on φ > 1.
proof idea
This is a term proof that builds the conjunction explicitly. The first conjunct comes from the phi_equation theorem in the PhiRing algebra. The uniqueness conjunct applies the golden_constraint_unique lemma to the hypothesis that r > 0 and r² = r + 1. The remaining conjuncts are the imported positivity lemmas J_bit_pos and E_coh_pos.
why it matters
The declaration fills the φ step in the forcing chain from the Recognition Science primer, after J-uniqueness and before the eight-tick octave and D = 3. It connects the composition law to the self-similar fixed point. No parent theorems are listed in the used-by graph, indicating it serves as a foundational interface for constant matching and stellar assembly derivations.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.
papers checked against this theorem (showing 5 of 5)
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DESI hints at gravity coupling beyond Einstein at 3 sigma
"the signal for non-minimal coupling of gravity arises from requiring a theoretically stable phantom crossing cosmology, as indicated by the data, within the EFT framework"
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Two solar-neutrino channels can corner MeV sterile neutrinos
"The flux of νH from such decay chain is proportional to the 8B solar neutrino flux and the mixing parameter |UαH|², scaled by a mass-dependent phase-space factor"
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Black-hole scalar yields null-state equations for 4D CFTs
"There is a decoupling mechanism in the bulk perturbative series at certain conformal dimensions, resulting in isolated lower-order equations."
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A √N detuning rule shields collective quantum batteries from decay
"∆*(N) = g√(N/(2γ₀)) ∝ N^{1/2}"
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A Noble-Gas-Centered Coordinate for Within-Period Atomic Property Trends
"The choice φ = (1+√5)/2 as the argument-rescaling factor is presently a modeling ansatz, not a derived chemical constant"