phiRung
plain-language theorem explainer
Recognition Science organizes scales on the φ-ladder. The definition returns φ raised to integer power n for any rung. It is cited when building coherence energies or synchronization gaps from the base quantum. The implementation is a direct real exponentiation abbreviation.
Claim. For integer rung $n$, the φ-ladder scaling factor is $φ^n$.
background
The RS-native units module sets tick as the discrete time quantum and voxel as the spatial step with c=1. Derived quanta are the coherence energy E_coh = φ^{-5} and action ħ = E_coh · τ₀. The φ-ladder supplies all scalings as φ^n for integer n, per the module documentation on native measurement without SI anchoring.
proof idea
This is a one-line definition that directly applies real exponentiation to φ with integer exponent n.
why it matters
It supplies the scaling used to define E_coh_rs = phiRung(-5) and gap45 = phiRung(45), which enter coherence and synchronization calculations with period lcm(8,45)=360. Additive and negation lemmas built on it realize the group law of the ladder. This fills the φ-ladder scaling step in the Recognition framework, linking to the self-similar fixed point phi and the eight-tick octave.
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