Foundation Primitive Recognition Calculus Prccompleteness Independence Jcost Is

The cost law check

In mathematics, a function is often defined by a formula, but it earns its name by obeying the laws of the structure it belongs to. The Recognition Science framework defines a cost, a measure of the price of recognizing one thing as another, through the function J(x) = (x + 1/x)/2 - 1. The declaration jcost_isCostRequirements is the formal check that this J is a legitimate cost: it proves that J(1) = 0, so recognizing something as itself costs nothing, and that J(x) = J(1/x) for every positive x, so the price is the same in both directions. These two properties, the unit law and reciprocal symmetry, are the defining axioms of a recognition cost, and the theorem confirms the canonical formula satisfies them.

The declaration matters because it sits at the base of a larger independence result. The framework's library, a machine-checked collection of formal theorems, also studies a countable field T, a set of numbers closed under the cost operation. The theorem completeness_not_forced_by_genuine_cost_laws shows that T, despite being a model of the cost laws, lacks the least-upper-bound property: there are nonempty sets within T that are bounded above but have no least upper bound inside T. The real numbers, by contrast, do have this property, as the theorem real_has_lub confirms. So order-completeness, the property that every bounded set has a least upper bound, holds in one cost-closed field and fails in another. It is not entailed by the cost axioms; it is an independent commitment, and the only model that has it is the uncountable continuum.

This is the sharpest form of the independence claim. The earlier version only assumed T was closed under J, but jcost_isCostRequirements upgrades the premise: J is now a bona fide recognition cost, satisfying the unit law and reciprocal symmetry. A skeptic cannot say the independence rests on a weak closure premise rather than the actual cost axioms. The theorem proves that even with the genuine laws in place, completeness remains separate. The countable carrier that the framework's dynamics use is never order-complete, whichever countable field it is. Completeness is exactly what the continuum adds, and it is a strictly stronger, independent choice.

What the declaration does not claim is just as important. It does not prove that the cost function is unique, nor that the real numbers are the only possible carrier. It does not show that completeness follows from the cost laws, only that it does not. The theorem is a negative result about independence, not a positive derivation of completeness. It leaves open the question of why one might choose the continuum over a countable field, and that choice is a modeling decision, not a forced consequence.

/-- The canonical cost `Cost.Jcost` genuinely satisfies the recognition-cost axioms:
the unit law `J(1) = 0` and reciprocal symmetry `J(x) = J(x⁻¹)` for positive `x`. So
the premise of the independence result is not merely "closed under a function"; it is
"a model of the cost laws". -/
theorem jcost_isCostRequirements : Cost.CostRequirements Cost.Jcost :=
  ⟨fun hx => Cost.Jcost_symm hx, Cost.Jcost_unit0⟩

/-- **Completeness independent of the GENUINE cost laws (not just `Jcost`-closure).**
This upgrades `completeness_not_forced_by_cost_axioms`: the premise now records that
`Cost.Jcost` is a bona fide recognition cost (unit `J(1)=0`, reciprocal symmetry
`J(x)=J(x⁻¹)`, nonnegativity `J(x)≥0` on positives), and that `T` is a countable field
on which `Jcost` is closed (hence a model of those laws). `T` still fails the
least-upper-bound property while ℝ satisfies it. So order-completeness is not entailed
by the genuine cost laws plus the field structure; it is an independent commitment.
This is the credibility-gating form: a skeptic cannot say the independence rests on a
weak "closure" premise rather than the actual cost axioms. -/
theorem completeness_not_forced_by_genuine_cost_laws :
    (Cost.Jcost 1 = 0)
      ∧ (∀ x : ℝ, 0 < x → Cost.Jcost x = Cost.Jcost x⁻¹)
      ∧ (∀ x : ℝ, 0 < x → 0 ≤ Cost.Jcost x)
      ∧ (∀ x ∈ T, Cost.Jcost x ∈ T)
      ∧ (T : Set ℝ).Countable
      ∧ (∃ S : Set ℝ,
          (∀ x ∈ S, x ∈ T) ∧ S.Nonempty ∧ (∃ b ∈ T, ∀ x ∈ S, x ≤ b)
            ∧ ¬ ∃ s, IsLUBIn T S s)
      ∧ (∀ S : Set ℝ, S.Nonempty → (∃ b, ∀ x ∈ S, x ≤ b) → ∃ s, IsLUB S s) :=
  ⟨Cost.Jcost_unit0, fun _ hx => Cost.Jcost_symm hx, fun _ hx => Cost.Jcost_nonneg hx,
    fun _ hx => CostOnField.jcost_mem_T hx, T_countable, T_not_complete, real_has_lub⟩

/-- **The sharp final form.** Order-completeness is precisely the content the
continuum adds: NO countable subfield of ℝ is order-complete, while ℝ is. Since every
δ result places the carrier in a countable field (the constants, the φ-ladder, the
cost dynamics all live countably), the carrier δ uses is never order-complete,
whichever countable field it is. Completeness is an independent axiom whose only model
is uncountable. -/
theorem completeness_is_exactly_the_continuum :
    (∀ K : Subfield ℝ, (K : Set ℝ).Countable →
        ∃ S : Set ℝ, (∀ x ∈ S, x ∈ K) ∧ S.Nonempty ∧ (∃ b ∈ K, ∀ x ∈ S, x ≤ b)
          ∧ ¬ ∃ s, IsLUBIn K S s)
      ∧ (∀ S : Set ℝ, S.Nonempty → (∃ b, ∀ x ∈ S, x ≤ b) → ∃ s, IsLUB S s) :=
  ⟨countable_subfield_not_complete, real_has_lub⟩

What this page does not claim

The declaration does not prove that the cost function J is unique. The declaration does not show that completeness follows from the cost laws, only that it does not. The declaration does not establish that the real numbers are the only possible carrier for the cost structure.

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

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