phi_exponent_tau0_tP
plain-language theorem explainer
Recognition Science sets the ratio τ₀ / t_P equal to φ raised to the power 34. Quantum gravity researchers using the phi-ladder to express Planck time would cite this exponent when converting between RS and standard units. The assignment follows from evaluating the base-φ logarithm of the approximate numerical ratio 2.4 × 10^16.
Claim. The natural number $n$ satisfying $τ_0 / t_P ≈ φ^n$ is defined to be 34.
background
The PlanckScale module derives the Planck length, mass, and time from Recognition Science principles by relating them to the fundamental time scale τ₀ and the golden ratio φ. The Planck time satisfies t_P = τ₀ × φ^{-n} for the exponent supplied here, consistent with the RS mechanism that places the Planck scale at the meeting point of quantum mechanics and gravity.
proof idea
The definition is a direct assignment of the constant 34, obtained from the logarithmic relation log_φ(2.4 × 10^16) ≈ 34.
why it matters
This supplies the exponent needed to express the Planck time in terms of τ₀ within the phi-ladder, supporting the target derivations QG-009 and QG-010. It ties the numerical Planck scale to the self-similar fixed point φ from T6 of the unified forcing chain. The module notes the factorization 2 × (8 + 8 + 1) as a structural hint linking to the eight-tick octave.
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