Foundation Qrft Smlagrangian Skeleton Sector Cost Pos Off Vacuum

The cost of being away from rest

The Standard Model Lagrangian, the compact expression that encodes the known particle interactions, splits into four canonical sectors: gauge kinetic, fermion kinetic, Yukawa, and Higgs potential. In the Recognition Science framework, each of these sectors is assigned a cost, a measure of how far a field's state deviates from its resting value. The resting value, called the recognition vacuum, is set to unity, where the cost is defined to be zero.

The theorem sectorCost_pos_off_vacuum states a simple positivity fact: for any positive deviation from unity, the cost is strictly positive. In other words, the cost function J(r) = (r + 1/r)/2 - 1 is zero only at r = 1, and is positive everywhere else on the positive real line. This is not a new physical assumption; it is a proved consequence of the framework's foundational cost function, which is forced by five plain conditions including reciprocal symmetry and a composition law. The theorem is checked in the machine-checked library of formal theorems, with no axioms beyond the standard three.

The declaration also bundles this fact into a certificate for the whole Lagrangian skeleton. The certificate records that each sector's cost vanishes at the vacuum, that costs are nonnegative for positive deviations, that the total cost is the sum of sector costs with no cross-sector mixing at tree level, and that there are exactly four sectors. The positivity theorem is the load-bearing piece: it guarantees that the vacuum is a strict minimum, so small deviations from rest carry a real price, and the vacuum is not just one flat point among many.

What the declaration does not claim is just as important. It does not claim that the Standard Model Lagrangian itself has been derived from Recognition Science. The module names the four sectors as canonical structures and proves structural properties of the cost assigned to them, but it does not derive the actual gauge couplings, Yukawa matrices, or the Higgs potential parameters from the framework. The full closure of the program that would tie these sectors to the Wightman / OS bridge is described as multi-month work, and remains open. The positivity theorem is a structural opening, not a complete derivation of particle physics.

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

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

theorem sectorCost_zero_at_vacuum : sectorCost 1 = 0 := Cost.Jcost_unit0

/-- SM-Lagrangian-skeleton certificate. -/
def smLagrangianCert : SMLagrangianCert where
  vacuum_zero := sectorCost_zero_at_vacuum
  reciprocal_symm := sectorCost_reciprocal_symm
  cost_nonneg := sectorCost_nonneg
  total_vacuum_zero := totalCost_zero_at_vacuum
  total_nonneg := totalCost_nonneg
  sector_count := sector_count

What this page does not claim

The declaration does not derive the actual gauge couplings, Yukawa matrices, or Higgs potential parameters from the framework. The declaration does not prove that the recognition vacuum is the unique global minimum of the full Standard Model action, only of the skeleton cost. The declaration does not claim that the Standard Model Lagrangian itself has been derived from Recognition Science.

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)

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.

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