Standard Model Higgs Eftlow Energy Limit

The low-energy bridge

The Standard Model's Higgs field is the reason other particles have mass. In the low-energy limit, physicists describe its effects with an effective field theory (EFT): a simplified set of rules that captures the Higgs boson's behavior at energies far below some new-physics scale. The classical entry point is the Higgs potential, whose shape determines the vacuum state and the masses of the W and Z bosons, the carriers of the weak force. The measured W mass is about 80.4 GeV and the Z mass about 91.2 GeV, a ratio that is set by the weak mixing angle, a key parameter of the theory.

What Recognition Science adds is a claim about where that potential's shape comes from. Its starting point is a ledger, a discrete record of recognition events, and a forced cost function J(x) = (x + 1/x)/2 - 1 that any such ledger must obey. The framework's machine-checked library of formal theorems then builds a bridge: from that cost geometry to an effective scalar coordinate, then to a canonical Higgs EFT, then to the Standard-Model gauge masses and Yukawa couplings. The master certificate, higgsEFTLowEnergyLimitCert, bundles these five modules into one auditable surface. The W/Z mass ratio and the Weinberg-angle structure are proved as theorems on positive gauge couplings. The Yukawa couplings, which set fermion masses, are proved to follow a phi-power ladder, where phi is the golden ratio.

The certificate is honest about its limits. The bridge from cost geometry to the effective scalar coordinate is a theorem only modulo a named NormalizationHypothesis. The longitudinal vector-boson scattering unitarity bound is conditional on RSPreservesLongitudinalUnitarity. Tree-level partial widths and branching ratios are proved, but loop-level channels like h to gamma gamma or h to gluon gluon remain open. The numerical value of the normalization scale Lambda(v) and the Standard-Model rung map from cube combinatorics are both open subproblems. The certificate itself proves no new physics; it is a structural composition that exposes one named, auditable surface for the chain.

What this establishes in plain language is a proof-of-concept: the framework's single cost rule can, through a long chain of formal steps, reproduce the mass and interaction structure of the Higgs sector at low energies. It does not claim a complete derivation of the Standard Model, and it does not claim a prediction for the Higgs mass itself. The value is in the architecture: a small set of axioms, a forced cost function, and a machine-checked path to known physics. The open subproblems are targets, not failures; they mark where the chain still needs work before the bridge is complete.

/-- Master certificate for the cost-geometry → SM-EFT bridge.

    The certificate is a structural composition: each component is the
    master certificate of the underlying module.  No new theorems are
    proved here; the goal is to expose one named, auditable surface for
    Anil's chain. -/
structure HiggsEFTLowEnergyLimitCert where
  /-- THEOREM (modulo `NormalizationHypothesis`):
      RS cost geometry → effective scalar coordinate → canonical Higgs EFT. -/
  bridge       : HiggsEFTBridgeCert
  /-- THEOREM:
      W/Z mass relations and Weinberg-angle structure on positive
      gauge couplings. -/
  ew_mass      : ElectroweakMassBridgeCert
  /-- THEOREM:
      SM-normalised Yukawa couplings on the φ-ladder. -/
  yukawa       : HiggsYukawaBridgeCert
  /-- TREE_LEVEL_ONLY:
      partial widths, branching ratios, signal strengths under
      tree-level matching of amplitudes. -/
  observable   : HiggsObservableSkeletonCert
  /-- CONDITIONAL_THEOREM (modulo `RSPreservesLongitudinalUnitarity`):
      longitudinal vector-boson scattering remains bounded as `s → ∞`. -/
  longitudinal : LongitudinalVectorScatteringCert

/-- Master certificate for the cost-geometry → SM-EFT bridge.

    The certificate is a structural composition: each component is the
    master certificate of the underlying module.  No new theorems are
    proved here; the goal is to expose one named, auditable surface for
    Anil's chain. -/
structure HiggsEFTLowEnergyLimitCert where
  /-- THEOREM (modulo `NormalizationHypothesis`):
      RS cost geometry → effective scalar coordinate → canonical Higgs EFT. -/
  bridge       : HiggsEFTBridgeCert
  /-- THEOREM:
      W/Z mass relations and Weinberg-angle structure on positive
      gauge couplings. -/
  ew_mass      : ElectroweakMassBridgeCert
  /-- THEOREM:
      SM-normalised Yukawa couplings on the φ-ladder. -/
  yukawa       : HiggsYukawaBridgeCert
  /-- TREE_LEVEL_ONLY:
      partial widths, branching ratios, signal strengths under
      tree-level matching of amplitudes. -/
  observable   : HiggsObservableSkeletonCert
  /-- CONDITIONAL_THEOREM (modulo `RSPreservesLongitudinalUnitarity`):
      longitudinal vector-boson scattering remains bounded as `s → ∞`. -/
  longitudinal : LongitudinalVectorScatteringCert

/-- The master certificate is theorem-backed at the level of bundling. -/
def higgsEFTLowEnergyLimitCert : HiggsEFTLowEnergyLimitCert where
  bridge       := higgsEFTBridgeCert
  ew_mass      := electroweakMassBridgeCert
  yukawa       := higgsYukawaBridgeCert
  observable   := higgsObservableSkeletonCert
  longitudinal := longitudinalVectorScatteringCert

theorem higgsEFTLowEnergyLimitCert_inhabited :
    Nonempty HiggsEFTLowEnergyLimitCert :=
  ⟨higgsEFTLowEnergyLimitCert⟩

What this page does not claim

No complete derivation of the entire Standard Model is claimed. No prediction for the Higgs boson mass itself is made. No claim that the low-energy EFT is valid at all energies is made.

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/HiggsEFTLowEnergyLimit.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: