phi_23
φ²³ supplies the phi-ladder rung that approximates the W boson to electron mass ratio in Recognition Science electroweak derivations. Physicists computing m_W from the J-cost minimum and phi-ladder placement would cite this constant when matching the predicted 80.38 GeV scale. The declaration is a direct power abbreviation that follows the upstream scale(k) := phi^k pattern without additional computation.
claimLet φ denote the self-similar fixed point. Define φ²³ as the scaling factor on the phi-ladder that approximates the W boson to electron mass ratio.
background
The module derives W and Z boson masses from electroweak symmetry breaking in RS, where the Higgs VEV sits at a J-cost minimum and boson masses occupy specific rungs on the phi-ladder. Upstream scale(k) := phi^k from Cosmology.LargeScaleStructureFromRS supplies the general ladder construction; phi_23 specializes it to exponent 23 for the W/e ratio. Anchor.Z from Masses.Anchor provides the integer map used in mass formulas, while the module notes that m_W ≈ 80.38 GeV and m_Z ≈ 91.19 GeV emerge once the rung is fixed.
proof idea
One-line definition that directly applies the power operation to phi, mirroring the scale function in the upstream Cosmology result.
why it matters in Recognition Science
This definition supplies the explicit rung for W boson mass predictions in the electroweak sector and feeds sibling declarations such as wBosonMass_GeV and wz_mass_ratio. It implements the phi-ladder placement step in the module's RS mechanism for P-015 and P-016, connecting to the T6 phi fixed point and the eight-tick octave structure in the forcing chain.
scope and limits
- Does not derive the exponent 23 from the forcing chain or RCL.
- Does not compute or bound the numerical closeness to the experimental W/e ratio.
- Does not address Z boson or Weinberg angle relations.
- Does not incorporate defectDist or Berry creation threshold.
formal statement (Lean)
171def phi_23 : ℝ := phi ^ 23
proof body
Definition body.
172
173/-- φ^24 is close to the Z/e mass ratio scale. -/