Foundation Lagrangian From Jcost3

The J-cost action

A Lagrangian is a recipe for turning a physical situation into a single number, the action, whose smallest value selects the behavior nature actually follows. In Recognition Science, the proposed recipe starts from the cost function J(x) = (x + 1/x)/2 - 1, which measures how far a ratio x sits from the value 1. The framework models a field configuration as a ratio m/e between two quantities, and defines the cost of that configuration as J(m/e). The action then integrates this cost over a continuous range of scales and subtracts a discrete sum at special rungs, written L = ∫ J(φ^r) dr - Σ_k J(φ^k) δ(r-rung_k). The action principle states that setting its variation to zero, δS = 0, yields the equations that minimize the total J-cost.

What does the machine-checked library of formal theorems actually establish about this construction? Three general facts, and nothing specific to the subject. First, when the two quantities are equal, m = e, the cost vanishes: J(1) = 0, so the configuration costs nothing. Second, for positive inputs, the cost is never negative; it is always zero or positive. Third, the number φ - 3/2, called the canonical threshold, is strictly greater than zero. These are the three facts bundled into a certificate structure, which exists and is inhabited. The certificate is a formal witness that the three properties hold.

The honest limit matters. The module defines domainCost as J(m/e) without ever saying what m and e are in physical terms. It proves no theorem about a specific field, particle, or force. The docstring itself records that the paragraph above is a research note, not a result. What would turn this into a theorem about its subject is a definition of m and e in that subject's own terms. Without that, the three facts are true of the cost function in general, shared verbatim across 2383 sibling modules, and the Lagrangian remains a proposal awaiting its physical content.

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

The module proves no theorem about any specific physical field or particle. The Lagrangian is a research proposal, not a derived result. The three proved facts are general properties of the cost function, not subject-specific consequences.

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/LagrangianFromJCost3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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