Cost Jcost Logic Jcost L Nonneg

The nonnegative cost

The cost function J(x) = (x + 1/x)/2 - 1 measures the price of recognizing a positive number x. For x = 1, the cost is zero: recognizing the unit costs nothing. For any other positive x, the cost is positive. The theorem JcostL_nonneg states this formally: for every positive x, J(x) is greater than or equal to zero. This is not an assumption or a hope; it is a proved consequence of the function's definition, verified in the framework's machine-checked library of formal theorems.

The proof is a transport. The framework works with two copies of the real numbers: a logical copy and the usual real numbers. The declaration shows that the cost on the logical copy, when translated to the usual reals, is exactly the same formula. Since the usual real version is known to be nonnegative for positive inputs, the logical version inherits that property. The theorem also gives a sharp zero condition: the cost is zero if and only if x equals 1. This pins down the unit as the unique free point in the recognition ledger.

In Recognition Science, the ledger is a discrete record of recognition events, and the cost is what the framework forces any such record to pay. The nonnegativity theorem is the floor under that ledger: it says recognition never pays negative prices. This matters because the framework's later results, such as the golden ratio as the unique self-similar scaling, build on the cost function's shape. A negative region would break the ordering that those results rely on.

The theorem does not claim that the cost function is the only one possible. It does not derive the fine-structure constant, and it does not prove the Riemann Hypothesis. It only establishes one property of one function, within the framework's own definitions. The classical cost formula itself is a definitional choice, not a proved law; what is proved is that this choice has the nonnegativity property.

What the reader can now see is a clean floor: the framework's cost of recognition is never negative, and it vanishes exactly at the unit. That single fact, machine-checked, is the foundation on which the framework's larger claims about scaling and dimension rest.

theorem JcostL_nonneg {x : LogicReal} (hx : (0 : LogicReal) < x) :
    (0 : LogicReal) ≤ JcostL x := by
  rw [le_iff_toReal_le, toReal_zero, toReal_JcostL]
  have hx' : 0 < toReal x := by simpa [lt_iff_toReal_lt] using hx
  exact Jcost_nonneg hx'

theorem JcostL_zero_iff {x : LogicReal} (hx : (0 : LogicReal) < x) :
    JcostL x = fromReal 0 ↔ x = fromReal 1 := by
  constructor
  · intro h
    rw [eq_iff_toReal_eq]
    have hx' : 0 < toReal x := by simpa [lt_iff_toReal_lt] using hx
    have hx0 : toReal x ≠ 0 := ne_of_gt hx'
    have hreal : Jcost (toReal x) = 0 := by
      have := congrArg toReal h
      rwa [toReal_JcostL, toReal_fromReal] at this
    rw [Jcost_eq_sq hx0] at hreal
    have hden : (0 : ℝ) < 2 * toReal x := by nlinarith
    have hsq : (toReal x - 1) ^ 2 = 0 := by
      have := congrArg (fun y : ℝ => y * (2 * toReal x)) hreal
      field_simp [ne_of_gt hden] at this
      simpa using this
    have hsub : toReal x - 1 = 0 := sq_eq_zero_iff.mp hsq
    rw [toReal_fromReal]
    linarith
  · intro h
    rw [h, JcostL_unit0]

@[simp] theorem toReal_JcostL (x : LogicReal) :
    toReal (JcostL x) = Jcost (toReal x) := by
  simp [JcostL, Jcost, toReal_fromReal]

What this page does not claim

The cost function is the only possible cost function. The framework derives the fine-structure constant or proves the Riemann Hypothesis. The classical formula J(x) is a proved law rather than a definitional choice.

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/Cost/JcostLogic.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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