pionRung
plain-language theorem explainer
The declaration fixes the pion rung index at 12 on the Recognition Science φ-ladder relative to the coherence energy. Hadron mass modelers working from the unified forcing chain cite this value to set the scale for π⁺, π⁻, and π⁰ predictions. It is a direct constant assignment with no computation or lemma application.
Claim. The pion occupies rung 12 on the φ-ladder, so that the mass satisfies the approximate relation $m_π ≈ E_{coh} ⋅ φ^{12}/2$, where $E_{coh}$ denotes the coherence energy and φ is the golden-ratio fixed point.
background
Recognition Science places particle masses on a φ-ladder whose rungs are integer powers of the golden ratio φ, with the base yardstick set by the coherence energy E_coh. The module derives pion masses from quark-antiquark binding, chiral symmetry breaking, and the Gell-Mann–Oakes–Renner relation, while fixing the pion at a concrete ladder position that yields the observed ratio m_π/m_e ≈ 273. The upstream structure for records the meta-realization axioms that enforce the self-similar orbit properties underlying all ladder placements.
proof idea
The definition is a direct constant assignment of the natural number 12; no lemmas are applied and no tactics are used.
why it matters
This definition supplies the fixed rung that anchors all subsequent pion mass formulas in the P-013 derivation, including the predicted values 139.6 MeV and 135.0 MeV and the ratio to the electron mass. It implements the φ-ladder placement step required by the forcing chain (T5–T6) and the eight-tick octave structure. No open questions are closed here; the choice remains a parameter fixed by the module.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.