scale_at_one
plain-language theorem explainer
The theorem establishes that the phi-ladder scale at rung 1 equals the golden ratio phi. Researchers deriving renormalization-group flows from Recognition Science phi-scaling would cite this as the base case for energy-dependent couplings. The proof is a direct term-mode reduction that unfolds the phiLadderScale definition and normalizes the resulting expression.
Claim. The scale factor on the $phi$-ladder at rung 1 equals $phi$.
background
The module derives running couplings in QFT from phi-ladder scaling, where each integer rung labels a distinct energy scale and J-cost optimization varies across rungs. phiLadderScale is the function that maps a rung to its corresponding scale factor in RS-native units. This construction sits inside the Recognition Science forcing chain, with phi as the self-similar fixed point (T6) and the ladder supplying the discrete structure for RG evolution of alpha, alpha_s, and alpha_W.
proof idea
One-line term proof: unfold the definition of phiLadderScale then apply norm_num to discharge the equality at rung 1.
why it matters
The result supplies the base case for the entire phi-ladder construction in QFT-011. It anchors the scale-dependent J-cost mechanism that produces the observed running of couplings and connects directly to the phi-forcing chain (T5-T8) and the eight-tick octave. No downstream uses are recorded yet, but the theorem is required before any concrete beta-function or alpha-value statements can be stated.
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