scale_at_zero
plain-language theorem explainer
The phi-ladder scale factor equals unity at rung zero. Physicists modeling renormalization-group flow via Recognition Science phi-scaling cite this base case to fix the reference energy before deriving how couplings vary across rungs. The proof is a one-line term that unfolds the scale definition and reduces numerically.
Claim. Let $S(r)$ denote the scale factor on the phi-ladder at rung $r$. Then $S(0)=1$.
background
In the Recognition Science treatment of quantum field theory, the phi-ladder assigns discrete energy scales to successive rungs indexed by integers. Different rungs correspond to different renormalization scales, and J-cost optimization varies with rung. The scale function maps each rung to a multiplicative factor relative to a reference, with rung zero fixed as the unit scale. This construction appears in the module on running couplings, where the mechanism links phi-ladder transitions to the observed running of alpha, alpha_s, and alpha_W.
proof idea
The proof is a one-line wrapper that unfolds the definition of the phi-ladder scale function and applies numerical normalization to verify the equality at rung zero.
why it matters
This base case anchors the phi-ladder scaling used to derive running couplings from phi-rung transitions. It supports the module goal of expressing how coupling constants change with energy via J-cost variation across the ladder, consistent with the self-similar fixed point phi. No immediate parent theorems are listed, but the result underpins sibling statements on alpha values at low and Z scales.
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