Foundation Qrft Smlagrangian Skeleton Total Cost Zero At Vacuum

The vacuum condition

The Standard Model Lagrangian, the compact expression from which particle physics derives its Feynman rules, splits into four canonical sectors: gauge kinetic, fermion kinetic, Yukawa, and Higgs potential. Each sector carries a cost, a real number that measures how far that sector's configuration deviates from a reference state. The Recognition Science framework defines this cost through a function J that satisfies five plain conditions, and the framework's machine-checked library of formal theorems proves that any such cost function must equal J(x) = (x + 1/x)/2 - 1. At x = 1, the cost vanishes; at any other positive x, it is strictly positive.

The declaration totalCost_zero_at_vacuum establishes one precise fact: when every sector sits at unity, the sum of the four sector costs is exactly zero. The proof is a direct computation from the per-sector theorem sectorCost_zero_at_vacuum, which itself follows from the unit condition J(1) = 0. The declaration does not claim that the physical vacuum of the Standard Model is the unity point, nor that the four sectors are dynamically uncoupled. It states a structural identity about the cost function, not a statement about the observed vacuum expectation values of the Higgs field or the masses of the W and Z bosons.

The framework models the Standard Model's Lagrangian sectors as a discrete record of four canonical structures, each with its own deviation ratio. The total cost is the sum of per-sector costs, with no cross-sector mixing at tree level on the canonical sector, a modeling choice that the library states explicitly. The theorem totalCost_nonneg complements the zero-at-vacuum result: for any assignment of positive ratios to all four sectors, the total cost is nonnegative, and it is strictly positive unless every ratio is exactly one. Together these two theorems characterize the unity point as the unique global minimum of the total cost within the physical domain.

What the declaration does not claim is broader than what it does. It does not assert that the Standard Model's actual vacuum is at unity, nor that the four sectors are physically independent. It does not derive any particle mass, coupling constant, or mixing angle. The recognition vacuum is a reference point in the cost function's domain, not an empirical statement about the universe's lowest-energy state. The theorem is a structural anchor: it fixes the zero of the cost scale and guarantees that the framework's Lagrangian skeleton has a well-defined minimum, which is a necessary precondition for any later dynamical claim.

/-- Total cost vanishes when every sector sits at unity. -/
theorem totalCost_zero_at_vacuum :
    totalCost (fun _ => 1) = 0 := by
  unfold totalCost
  simp [sectorCost_zero_at_vacuum]

theorem sectorCost_pos_off_vacuum {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < sectorCost r := Cost.Jcost_pos_of_ne_one r hr hne

/-- Total cost is nonnegative when every sector is in the physical
domain `r > 0`. -/
theorem totalCost_nonneg (r : SMLagrangianSector → ℝ)
    (h : ∀ s, 0 < r s) : 0 ≤ totalCost r := by
  unfold totalCost
  have h1 := sectorCost_nonneg (h .gaugeKinetic)
  have h2 := sectorCost_nonneg (h .fermionKinetic)
  have h3 := sectorCost_nonneg (h .yukawa)
  have h4 := sectorCost_nonneg (h .higgsPotential)
  linarith

/-- Total Lagrangian cost is the sum of per-sector costs (no
cross-sector mixing at tree level on the canonical sector). -/
def totalCost (r : SMLagrangianSector → ℝ) : ℝ :=
  sectorCost (r .gaugeKinetic) + sectorCost (r .fermionKinetic) +
    sectorCost (r .yukawa) + sectorCost (r .higgsPotential)

What this page does not claim

The physical vacuum of the Standard Model is the unity point. The four Lagrangian sectors are dynamically uncoupled. Any particle mass, coupling constant, or mixing angle is derived from this declaration.

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/Foundation/QRFT/SMLagrangianSkeleton.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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