phi_energy_rung_zero
plain-language theorem explainer
The theorem establishes that the zero rung on the φ-energy ladder recovers the base energy E_base exactly. Researchers modeling photobiomodulation devices would cite this to anchor the discrete energy spectrum at the recognition coherence quantum. The proof is a one-line simplification that unfolds the ladder definition at n=0.
Claim. $E(0) = E_0$ where $E(n) = E_0 · ϕ^n$ and $E_0 = ϕ^{-5}$ eV (recognition coherence quantum).
background
The φ-energy ladder is defined by E(n) = E_base · φ^n, mapping real-valued rung n to photon energy in joules. E_base is the base energy of the φ-ladder, given by φ^{-5} eV (recognition coherence quantum), converted to joules via multiplication by eV_to_J. The module formalizes Recognition Science foundations for a photobiomodulation device operating in coherence with the φ-ladder, with key results including the discrete energy rungs and therapeutic wavelengths such as rung 6 yielding φ eV.
proof idea
The proof is a one-line wrapper that applies the definition of the φ-energy ladder by simplification at n=0, where φ^0 evaluates to 1 and the product reduces directly to E_base.
why it matters
This anchors the base case of the φ-energy ladder used to derive photobiomodulation energies such as E_PBM at rung 6. It fills the zero-rung step in the discrete spectrum for RS-coherent light therapy, consistent with the φ-ladder structure and eight-tick octave neutrality in the Recognition Science framework.
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