phi_24
The constant obtained by raising the self-similar fixed point phi to the 24th power supplies the scaling factor for the Z boson to electron mass ratio on the phi-ladder. Electroweak modelers in Recognition Science cite this value when locating the vacuum expectation value that sets the W and Z masses. The definition is a direct real exponentiation with no lemmas or reductions applied.
claimLet $phi$ be the unique positive real fixed point of the Recognition Composition Law. Then $phi^{24}$ is the real number that scales the vacuum expectation value to the Z boson mass relative to the electron mass.
background
Recognition Science places all physical scales on the phi-ladder, with each rung multiplying the previous by the self-similar fixed point phi obtained from T6. The J-cost function J(x) = (x + x^{-1})/2 - 1 is strictly convex with global minimum at x = 1 and governs the forcing chain T0 to T8 that forces D = 3 spatial dimensions. Upstream structures supply the ledger factorization that calibrates J, the spectral emergence that fixes 24 chiral fermion flavors from the Q3 geometry, and the nuclear density tiers that occupy discrete phi powers.
proof idea
This is a direct definition that applies real exponentiation to phi. No upstream lemmas are invoked and no tactics are used; the declaration simply names the power for use in sibling mass expressions.
why it matters in Recognition Science
The definition supplies the phi-ladder rung required for the electroweak vacuum expectation value in the boson mass derivations (P-015, P-016). It supports the sibling expressions for W and Z masses in GeV and the weak coupling constants. The placement is consistent with the eight-tick octave (T7) and the gauge content SU(2) x U(1) fixed by spectral emergence, completing one step from J-uniqueness (T5) to the electroweak scale.
scope and limits
- Does not prove numerical closeness of phi^24 to the observed Z/e mass ratio.
- Does not derive the vacuum expectation value in GeV units.
- Does not compute the weak mixing angle or sin^2 theta_W.
- Does not address radiative corrections to the boson masses.
formal statement (Lean)
174def phi_24 : ℝ := phi ^ 24
proof body
Definition body.
175
176/-- VEV in units of electron mass. -/