mass_rung_step
The theorem shows that advancing one rung on the phi-ladder multiplies mass by phi. Researchers deriving the Recognition Science mass hierarchy from the forced self-similar point would cite it to justify the geometric progression of particle masses. The proof unfolds the exponential definition of mass_rung, applies the real-power addition rule, and closes with ring simplification.
claimLet $m(n) = y · ϕ^n$ denote the mass at rung $n$ for positive yardstick $y$. Then $m(n+1) = ϕ · m(n)$.
background
The mass_rung function supplies the mass-energy at integer rung $n$ on the phi-ladder via the explicit form yardstick times phi to the power $n$. This definition lives inside the Spectral Emergence module, whose setting derives the full Standard Model gauge content, three generations, and mass hierarchy from the single forced datum D=3 together with the binary cube Q3. The phi-ladder itself traces to the self-similar fixed point forced in the UnifiedForcingChain (T6) and the eight-tick octave (T7).
proof idea
The term proof unfolds the mass_rung definition, pushes the integer-to-real cast on the exponent, rewrites via the real-power addition formula under the positivity of phi, reduces the unit power, and finishes by ring simplification of the resulting monomials.
why it matters in Recognition Science
The step is invoked directly by rung_ratio to obtain the exact phi ratio between adjacent rungs and feeds the master spectral_emergence certificate. It supplies the elementary scaling law for the phi-ladder mass hierarchy that the Recognition framework places after the forced phi (T6) and D=3 (T8). No open scaffolding remains at this node.
scope and limits
- Does not fix the absolute value of the yardstick.
- Does not map specific rungs to observed particle masses.
- Does not incorporate the gap(Z) correction term of the full mass formula.
- Does not prove positivity of the yardstick or resulting masses.
Lean usage
rw [mass_rung_step]formal statement (Lean)
288theorem mass_rung_step (ys : ℝ) (n : ℤ) :
289 mass_rung ys (n + 1) = phi * mass_rung ys n := by
proof body
Term-mode proof.
290 unfold mass_rung
291 push_cast
292 rw [Real.rpow_add (by exact phi_pos), Real.rpow_one]
293 ring
294
295/-- **THEOREM**: Mass is positive for positive yardstick. -/