per_nucleon_phi_factor_succ
plain-language theorem explainer
The per-nucleon phi factor on the recognition ladder obeys the recurrence f(k+1) equals f(k) times phi. Nuclear modelers working in the Recognition Science program cite this when constructing the binding-energy ladder from the J-cost lattice and locating the iron peak at rung 26. The proof is a direct algebraic reduction that unfolds the definition and applies the successor rule for exponentiation.
Claim. Let $f(k)$ denote the per-nucleon phi factor at rung $k$. Then $f(k+1) = f(k) · ϕ$, where ϕ is the golden ratio.
background
In the Recognition Science treatment of nuclear binding, each nucleon contributes one bond cycle on the J-cost lattice; the per-nucleon phi factor accumulates the recognition-cost release along the phi-ladder. The module derives the binding-energy peak at the iron group from saturation of this ladder at rung k=26, forced by the identity k_iron = consciousnessGap - configDim - holonomyShift = 45 - 5 - 14 at D=3. The rung is integer-forced with no free parameter, and the per-nucleon factor reaches its 8-tick saturation value adjusted by the 14-virtue gap shift.
proof idea
The proof unfolds the definition of per_nucleon_phi_factor and rewrites the successor case with the power rule pow_succ. It is a one-line wrapper that applies the algebraic identity for exponentiation.
why it matters
This supplies the inductive step for the per-nucleon phi factor ratio theorem, which supports the relative binding-energy calculation between the iron peak and the alpha-particle reference in Track E2 of Plan v7. It realizes the recursive structure of the phi-ladder, forced by the self-similar fixed point property of phi (T6) and the eight-tick octave periodicity (T7). The result feeds directly into the downstream ratio theorem used for iron-peak positioning relative to the canonical reference.
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