Foundation Qrft Smlagrangian Skeleton Sector Cost Reciprocal Symm

The symmetry of sector cost

The Standard Model Lagrangian, the master equation of particle physics, separates into four sectors: gauge kinetic, fermion kinetic, Yukawa, and Higgs potential. In the Recognition Science framework, each sector carries a cost, a real number that measures how far that sector's state sits from its recognition vacuum, the balanced point where the cost is zero. The declaration sectorCost_reciprocal_symm is a proved theorem inside the framework's machine-checked library of formal theorems. It states that for any positive deviation ratio r, the cost at r equals the cost at its reciprocal 1/r. In plain terms, stretching a sector away from balance by a factor of two costs exactly the same as compressing it by a factor of two. The theorem is not an assumption; it is derived from the framework's core cost function, which itself is forced by five plain conditions.

The cost function in question is J(x) = (x + 1/x)/2 - 1, the unique function satisfying those five conditions: reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity. The reciprocal symmetry of the sector cost is therefore not a separate postulate but a direct consequence of this uniqueness. The theorem holds for every positive real r, and the framework's certificate bundles it with three other properties: zero cost at the vacuum, nonnegativity for all positive deviations, and positivity everywhere except the vacuum. Together these properties say that the cost is a well-behaved measure of deviation, one that cannot go negative and that penalizes any departure from balance, while treating over- and under-shooting symmetrically.

In Recognition Science, this symmetry carries a structural meaning. The framework models the Standard Model's four sectors as a single skeleton, and the cost at each sector is the same J-cost. The reciprocal symmetry means the framework's notion of deviation does not care about direction, only magnitude. This is not a claim about the physical content of the Standard Model; it is a claim about the mathematical structure the framework imposes on it. The theorem does not say that gauge kinetic and Yukawa sectors have equal costs, nor that the sectors interact symmetrically. It says only that for any single sector, the cost function is even under inversion.

What the theorem does not claim is equally important. It does not derive the Standard Model Lagrangian from scratch; the skeleton is a structural opening, not a complete derivation. It does not assert that the four sectors are physically interchangeable, only that they share the same cost form. And it does not claim that the reciprocal symmetry is observed in nature; it is a theorem about the framework's cost function, not a measurement. The theorem's reach is exactly its statement: for any positive deviation r, the cost at r equals the cost at 1/r, and that is all.

theorem sectorCost_reciprocal_symm {r : ℝ} (hr : 0 < r) :
    sectorCost r = sectorCost r⁻¹ := Cost.Jcost_symm hr

/-- Per-sector J-cost on the sector's canonical-deviation ratio. -/
def sectorCost (r : ℝ) : ℝ := Cost.Jcost r

/-- 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 theorem does not derive the full Standard Model Lagrangian, only a structural skeleton. The theorem does not claim the four sectors have equal costs or interact symmetrically. The theorem does not assert that reciprocal symmetry is an observed physical law.

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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