scale_at_two
plain-language theorem explainer
The theorem fixes the energy scale at the second rung of the φ-ladder as exactly φ squared in reference units. Researchers modeling renormalization-group flow via discrete φ-scaling would cite it when anchoring low-rung hierarchies before computing running couplings. The proof is a one-line wrapper that unfolds the ladder-scale definition and normalizes the integer exponent.
Claim. The energy scale at φ-ladder rung 2 equals φ² in units of the reference scale.
background
The QFT module maps running couplings to J-cost changes across φ-ladder rungs, where each rung labels a distinct energy scale and the scale function supplies the exponential hierarchy. The definition states that the energy scale at rung n is phi raised to n in reference units. Upstream, the UniversalForcingSelfReference.for structure records the meta-realization axioms that underwrite the ladder's self-similar construction, while the phiLadderScale definition itself supplies the concrete exponential map.
proof idea
The proof is a one-line wrapper that unfolds the phiLadderScale definition and applies norm_cast to equate the integer power.
why it matters
The result supplies the explicit scale factor at rung 2 that later running-coupling calculations rely on when evaluating J-cost at low energies. It instantiates the φ-forcing fixed point (T6) inside the QFT setting and aligns with the eight-tick octave structure. No downstream theorems depend on it yet, leaving its use in explicit beta-function derivations open.
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