Standard Model Higgs Eftbridge

The Higgs bridge

The Standard Model describes the Higgs field with a potential that looks like a Mexican hat: a quadratic term that sets the mass and a quartic term that sets how strongly three and four Higgs particles interact. The recognition-cost potential, built from the proved cost function J(x) = (x + 1/x)/2 - 1, offers a different starting point. Write the field as h, the electroweak scale as v, and a dimensionful prefactor as Λ⁴. The potential is Λ⁴ · J(exp(h/v)), which simplifies to Λ⁴ · (cosh(h/v) - 1). This is the bridge: a single, forced cost function becomes a scalar potential with the same shape as the Standard Model's, at least for the first two terms.

The bridge works because the cosh expansion has the right low-order structure. Expanding around h = 0 gives Λ⁴/2v² · h² plus Λ⁴/24v⁴ · h⁴, with corrections of order h⁶. The quadratic coefficient matches the Standard Model's mass term exactly, provided the normalization hypothesis Λ⁴ = m_H²v² holds. The quartic coefficient, however, does not match. The Standard Model's quartic self-coupling is m_H²/(2v²), while the cosh form gives m_H²/(6v²), exactly one third of the value. The trilinear coupling is identically zero because the potential is even in h. These are pre-registered predictions: the Higgs self-couplings deviate from the Standard Model by known factors, falsifiable at HL-LHC di-Higgs and FCC-hh tri-Higgs experiments.

The module proves the Taylor-coefficient extraction as a theorem, using the cosh identity and a truncation bound from the machine-checked library. The mass-term matching is a conditional theorem, depending on the normalization hypothesis. The origin of Λ itself, the prefactor that sets the overall scale, remains open; deriving it from the recognition substrate's φ-ladder is the stated frontier. The bridge closes the first two arrows of a reviewer chain: from cost geometry to an effective scalar coordinate, and from that coordinate to a canonical Higgs EFT. The third arrow, fixing Λ(v), is the open subproblem.

The consequence is concrete. If the recognition-cost potential is the right description, the Higgs boson's self-interactions are not what the Standard Model predicts. The deviation is not a small correction but a factor of three in the quartic coupling, and a vanishing trilinear coupling where the Standard Model has a nonzero one. These are sharp, testable signatures. The framework does not claim the Standard Model is wrong; it claims the Higgs sector is the place to look first for a deviation that the cost function forces.

/-- `V_RS` reduces to `Λ⁴ · (cosh(h/v) − 1)`. -/
theorem V_RS_eq_cosh (Λ v h : ℝ) :
    V_RS Λ v h = Λ ^ 4 * (Real.cosh (h / v) - 1) := by
  unfold V_RS
  rw [Cost.Jcost_exp_cosh]

/-- Under the normalisation hypothesis, the canonical SM quartic coupling is
    `λ_SM = (1/6) · m_H² / v²`.

    In the convention `V_SM = ½ m_H² h² + (λ_SM / 4) h⁴`, matching the RS
    quartic coefficient `Λ⁴ / (24 v⁴)` to `λ_SM / 4` gives this relation. -/
theorem quartic_coupling_from_normalization
    (Λ v m_H : ℝ) (hv : 0 < v) (hΛ : NormalizationHypothesis Λ v m_H) :
    4 * quartic_coefficient_canonical Λ v = m_H ^ 2 / (6 * v ^ 2) := by
  unfold quartic_coefficient_canonical
  unfold NormalizationHypothesis at hΛ
  have hv2 : v ^ 2 ≠ 0 := pow_ne_zero 2 (ne_of_gt hv)
  have hv4 : v ^ 4 ≠ 0 := pow_ne_zero 4 (ne_of_gt hv)
  have hv4_eq : (v : ℝ) ^ 4 = v ^ 2 * v ^ 2 := by ring
  rw [hΛ, hv4_eq]
  field_simp
  ring

/-- Under the normalisation hypothesis, the SM kinetic-normalised Higgs mass
    appears as the coefficient of `½ h²` in the RS quartic Taylor potential. -/
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
  have hv2 : v ^ 2 ≠ 0 := pow_ne_zero 2 (ne_of_gt hv)
  rw [hΛ]
  field_simp

/-- `V_RS` reduces to `Λ⁴ · (cosh(h/v) − 1)`. -/
theorem V_RS_eq_cosh (Λ v h : ℝ) :
    V_RS Λ v h = Λ ^ 4 * (Real.cosh (h / v) - 1) := by
  unfold V_RS
  rw [Cost.Jcost_exp_cosh]

What this page does not claim

The Standard Model Higgs potential is not claimed to be exactly reproduced; the quartic coupling deviates by a factor of three. The normalization hypothesis Λ⁴ = m_H²v² is not derived; it is a stated open subproblem. The derivation of the canonical kinetic term from RS primitives is not claimed; it is the open frontier.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/StandardModel/HiggsEFTBridge.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: