Which physical constants are derived from phi?

permalink: https://pith.science/recognition/ask/which-physical-constants-are-derived-from-phi-c5ad380f · first asked 2026-08-19 · browse all asks

confidence: high in recognition cached

φ is forced as the unique positive solution to x² = x + 1 by self-similarity in a discrete J-cost ledger (phi_forced, phi_forcing_complete, phi_equation).

In RS-native units (ℓ₀ = τ₀ = 1):

c = ℓ₀/τ₀ = 1 (c_rs_eq_one).
ℏ = E_coh · τ₀ = φ^{-5} (ℏ_rs_eq, E_coh_pos).
G = φ^5/π (G_rs_eq, G_pi_eq_phi5).

Further identities: G · ℏ = 1/π (G_ℏ_product); J_bit = ln(φ) (J_bit_pos).

All summarized by all_constants_from_phi.

cited recognition theorems

PhiForcing.phi_forced Establishes φ forced by self-similarity from discrete ledger
PhiForcingDerived.phi_forcing_complete Derives φ from closure under additive ledger composition
PhiForcing.phi_equation Defines the algebraic identity satisfied by φ
ConstantDerivations.c_rs_eq_one Derives c = 1 from φ-based units
ConstantDerivations.ℏ_rs_eq Derives ℏ = φ^{-5} from E_coh and τ₀
ConstantDerivations.G_rs_eq Derives G = φ^5/π
ConstantDerivations.G_ℏ_product Derives G · ℏ = 1/π identity
ConstantDerivations.all_constants_from_phi Summarizes all constants derived from φ
PhiForcing.E_coh_pos Establishes E_coh = φ^{-5} > 0
PhiForcing.J_bit_pos Establishes J_bit = ln(φ) > 0

outside recognition

Aspects Recognition does not yet address:

fine structure constant α (explicitly removed from derivation in ConstantDerivations)
particle masses (not derived in supplied modules)

recognition modules consulted

The Recognition library is at github.com/jonwashburn/shape-of-logic. The model is restricted to the supplied Lean source and instructed not to invent theorem names. Treat output as a starting point, not a verified proof.