NormalizationHypothesis
The normalization hypothesis equates the recognition scale Λ to the fourth root of the product of Higgs mass squared and electroweak vev squared. Physicists constructing effective field theory bridges from recognition cost geometry to the Standard Model would cite this as the required matching condition between scales. The declaration is introduced as a direct algebraic equality with no derivation or lemmas.
claimThe normalization hypothesis asserts that for recognition-cost scale Λ, electroweak vacuum expectation value v, and Higgs mass m_H, the equality Λ⁴ = m_H² v² holds.
background
The Higgs EFT Bridge module defines the recognition-cost potential as V_RS(Λ, v, h) := Λ⁴ · J(exp(h/v)), where J(x) = ½(x + x⁻¹) − 1 is the canonical reciprocal cost functional from the J-cost structure. Expansion around the vacuum yields the quadratic coefficient Λ⁴/(2 v²) and quartic coefficient Λ⁴/(24 v⁴). The hypothesis supplies the scale relation needed to match the Standard-Model form ½ m_H² h² + (λ_SM/4) h⁴. Upstream structures include J-cost minimization (convexity and 8-tick dynamics) from PhysicsComplexityStructure and the continuum identification from SimplicialLedger.ContinuumBridge.
proof idea
The declaration is a direct definition of the proposition as the algebraic equality Λ⁴ = m_H² v². No lemmas or tactics are applied; it functions as an interface hypothesis for conditional results in the module.
why it matters in Recognition Science
This hypothesis supplies the defining normalisation map between the recognition-cost scale and the electroweak scale, closing the first two arrows of the RS cost geometry to canonical Higgs EFT chain. It is required by the master certificate HiggsEFTBridgeCert and by the theorems mass_term_matches_SM and quartic_coupling_from_normalization. The open collider-normalisation problem of fixing Λ(v) from the φ-ladder yardstick remains unresolved, as noted in the module documentation. It connects to J-uniqueness in the forcing chain and the eight-tick octave structure.
scope and limits
- Does not derive a numerical value for Λ from the φ-ladder yardstick.
- Does not prove consistency with the measured Higgs mass.
- Does not address higher-order terms beyond the quartic in the potential expansion.
- Does not incorporate gauge boson or fermion mass relations.
falsifier
A concrete falsifier would be an experimental determination of m_H and v that cannot be reconciled with any Λ obtained from the recognition substrate via the φ-ladder yardstick.
Lean usage
theorem mass_term_matches_SM (Λ v m_H : ℝ) (hv : 0 < v) (hΛ : NormalizationHypothesis Λ v m_H) : quadratic_coefficient Λ v = m_H ^ 2 / 2 := by unfold quadratic_coefficient; unfold NormalizationHypothesis at hΛ; simp [hv]formal statement (Lean)
201def NormalizationHypothesis (Λ v m_H : ℝ) : Prop :=
proof body
Definition body.
202 Λ ^ 4 = m_H ^ 2 * v ^ 2
203
204/-- Under the normalisation hypothesis, the SM kinetic-normalised Higgs mass
205 appears as the coefficient of `½ h²` in the RS quartic Taylor potential. -/