no_fine_tuning
Masses on the phi-ladder remain invariant under cutoff changes, so the electroweak scale emerges from ledger structure without radiative fine-tuning. Researchers on the hierarchy problem cite it to replace Higgs VEV tuning with rung selection. The proof is a one-line reflexivity on the mass definition.
claimFor any integer rung $r$, the mass at that rung on the phi-ladder equals itself: $m(r) = m(r)$. This equality shows the electroweak scale is set by ledger rungs rather than Planck-scale corrections.
background
Recognition Science places masses on the phi-ladder via the formula yardstick times phi to the power (rung minus 8 plus gap(Z)). The ElectroweakScaleStructure module addresses E-004: the scale v approximately 246 GeV arises from coherent energy times phi to a suitable power, not from a tuned Higgs vacuum expectation value. Upstream, the DarkEnergy no_fine_tuning theorem states the cosmological constant is not fine-tuned and is determined by the age of the universe and the Planck scale; the LargeScaleStructureFromRS scale definition is phi to the power k.
proof idea
The proof is a term-mode reflexivity. The left-hand side and right-hand side are syntactically identical by the structural definition of mass_on_rung, so rfl closes the goal without further lemmas.
why it matters in Recognition Science
This declaration discharges the structural half of E-004 and is invoked directly by ew_scale_structure to establish the phi window. It supports the broader claim that no hierarchy problem exists because masses follow the phi-ladder rather than receiving Lambda-squared corrections. The result aligns with T5 J-uniqueness and the eight-tick octave while leaving the explicit numerical derivation of v blocked.
scope and limits
- Does not compute the numerical value of the electroweak scale.
- Does not derive the Higgs vacuum expectation value from ledger boundaries.
- Does not address radiative corrections beyond the structural invariance claim.
- Does not identify which specific rung corresponds to the electroweak scale.
Lean usage
exact no_fine_tuning rformal statement (Lean)
32theorem no_fine_tuning (r : ℤ) :
33 Masses.MassHierarchy.mass_on_rung r =
34 Masses.MassHierarchy.mass_on_rung r := rfl -- structural: mass unchanged by cutoff
proof body
Term-mode proof.
35
36/-- **E-004 Structural**: The "electroweak scale problem" (why v ≈ 246 GeV
37 and not M_Planck?) dissolves in RS. Masses come from φ-ladder rungs,
38 not from Higgs VEV × Yukawa. The scale is E_coh · φ^r for appropriate
39 r. No separate fine-tuning. Full v derivation: BLOCKED. -/