Physics Effective Field Theory2 From Jcost

Effective field theory and the cost function

Effective field theory is a standard tool in physics for describing what happens when a simpler low-energy description of nature meets a small correction from higher energies. The correction usually comes suppressed by a ratio of scales, written as (E/Λ)^n, where E is the energy you are probing and Λ is the high-energy scale where new physics lives. The higher the dimension of the operator that generates the correction, the larger the power n, and the faster the correction fades as E drops below Λ. A famous example is the Euler-Heisenberg Lagrangian, which adds a tiny nonlinearity to electromagnetism, suppressed by the electron mass scale.

In Recognition Science, the framework models this suppression with a specific function called the cost, a forced penalty that reality pays for each recognition event. The module defines a ledger, a discrete record of such events, and sets the cost of a ratio m/e to J(m/e), where J(x) = (x + 1/x)/2 - 1. The framework's library, a machine-checked collection of formal theorems, proves three general facts about this cost: it vanishes when the two scales are equal, it is never negative for positive inputs, and the golden-ratio constant φ minus 3/2 is positive. These are true for any positive numbers m and e, because the definition does not yet tie m and e to any specific physical quantity.

The research note attached to the module sketches where the idea was meant to go: at energy E = φ·Λ, the leading correction would equal J(φ)·φ³ = 0.5, a neat structural number. But the Lean code proves nothing about effective field theory itself, because the module never defines m and e in physical terms. What would turn the module into a theorem about its subject is a definition of m and e in that subject's own language, for instance identifying m with a particle mass and e with a collision energy. Until that step, the three proved facts stand as pure mathematics about J, not as physics.

The practical consequence is a clear boundary. The framework has a candidate shape for the leading correction in a three-dimensional theory, and the shape is consistent with the standard EFT suppression picture. But the bridge from the abstract cost function to a specific physical operator remains open. A reader can now see exactly what is proved: three inequalities and an equality about a universal cost function, plus a certified structure that packages them. The physics content is a research note, not a theorem.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

No claim that the module proves anything about effective field theory itself. No claim that the value 0.5 at E = φ·Λ is a measured or derived physical result. No claim that the framework has identified m and e with specific physical quantities.

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/Physics/EffectiveFieldTheory2FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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