phi_energy_rung_step
plain-language theorem explainer
The energy at rung n+1 equals the energy at rung n multiplied by φ. Device physicists modeling photobiomodulation would cite this to step along the discrete energy ladder from the base coherence quantum. The proof unfolds the rung definition and reduces via the real exponent addition law followed by algebraic simplification.
Claim. $E(n+1) = E(n) · φ$ where $E(n) := E_{base} · φ^n$ and $E_{base} = φ^{-5} · eV_{to J}$.
background
The module formalizes a photobiomodulation device operating on the Recognition Science φ-ladder. E_base is the base energy φ^{-5} eV converted to joules, the recognition coherence quantum. The rung energy is defined as E(n) = E_base · φ^n for real-valued n, mapping rung index to photon energy in joules.
proof idea
The proof unfolds the definition of phi_energy_rung to expose E_base · φ^{n+1}. It rewrites the power via the real exponent addition rule φ^{n+1} = φ^n · φ, then simplifies the product with ring.
why it matters
This scaling relation enables traversal of the φ-energy ladder in the photobiomodulation model, directly supporting the result that rung 6 yields φ eV near 766 nm. It instantiates the self-similar fixed point φ forced at T6 of the unified chain and the eight-tick octave structure. No downstream uses are recorded yet.
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