Cost Uniqueness Jcost Continuous Pos

The continuity lemma

The function J(x) = (x + 1/x)/2 - 1 assigns a cost to any positive number x. It is symmetric under reciprocals: J(x) = J(1/x). It vanishes at x = 1, where the cost of recognizing something identical to itself is zero. It is strictly convex on the positive reals, meaning its graph bends upward everywhere. These properties make J a natural candidate for a universal cost function, and the framework's main theorem states that any cost function satisfying a list of plain conditions must equal J on the positive reals.

Before that uniqueness theorem can conclude, it must know that J itself behaves well enough to be compared against. The declaration Jcost_continuous_pos establishes exactly one such fact: J is continuous on the open interval (0, ∞). Continuity here means that small changes in x produce small changes in J(x); there are no jumps or breaks in the graph. This is not a deep result on its own, and the proof is a routine check. Its role is structural: the uniqueness theorem's hypotheses include a continuity condition on the candidate function F, and to apply that condition to J, the library must first record that J satisfies it.

In Recognition Science, the framework models recognition as a ledger, a discrete record of events, and the cost of each recognition is forced by the five conditions rather than chosen freely. The full uniqueness theorem, tagged as proved in the machine-checked library of formal theorems, takes a candidate function F and assumes symmetry, unit normalization, strict convexity, calibration, continuity, a composition law, and several regularity hypotheses. It concludes that F equals J everywhere on the positive reals. Jcost_continuous_pos is one of the supporting lemmas that supplies the continuity requirement for J itself, so the theorem can close over all positive inputs.

What the lemma does not claim is just as important. It does not assert that J is continuous everywhere on the real line; the domain is explicitly the positive reals, and the function has a singularity at x = 0 where it blows up. It does not prove uniqueness on its own; that requires the full theorem with all its hypotheses. It does not establish that J is the only possible cost function, only that J is continuous. The lemma is a necessary ingredient, not a standalone result, and its scope is deliberately narrow.

/-- Jcost is continuous on ℝ₊ -/
lemma Jcost_continuous_pos : ContinuousOn Jcost (Ioi 0) := by
  classical
  have h1 : ContinuousOn (fun x : ℝ => x) (Ioi 0) := continuousOn_id
  have h2 : ContinuousOn (fun x : ℝ => x⁻¹) (Ioi 0) := by
    refine ContinuousOn.inv₀ (f:=fun x : ℝ => x) (s:=Ioi 0) h1 ?hneq
    intro x hx; exact ne_of_gt hx
  have h3 : ContinuousOn (fun x : ℝ => x + x⁻¹) (Ioi 0) := h1.add h2
  have h4 : ContinuousOn (fun x : ℝ => (1 / 2 : ℝ) * (x + x⁻¹)) (Ioi 0) :=
    (continuousOn_const).mul h3
  have h5 : ContinuousOn (fun x : ℝ => (1 / 2 : ℝ) * (x + x⁻¹) - 1) (Ioi 0) :=
    h4.sub continuousOn_const
  simpa [Jcost, one_div, div_eq_mul_inv, mul_comm, mul_left_comm, mul_assoc, sub_eq_add_neg]
    using h5

/-- Main uniqueness statement on ℝ₊: any admissible cost equals Jcost on (0,∞). -/
theorem unique_cost_on_pos (F : ℝ → ℝ) (hF : UniqueCostAxioms F) :
  ∀ {x : ℝ}, 0 < x → F x = Jcost x :=
  T5_uniqueness_complete F hF.symmetric hF.unit hF.convex hF.calibrated hF.continuousOn_pos hF.coshAdd
    hF.dAlembert_smooth hF.dAlembert_toODE hF.ode_cont hF.ode_diff hF.ode_bootstrap

What this page does not claim

J is continuous on the whole real line, including at x = 0. The lemma alone proves the uniqueness of J as a cost function. The lemma establishes any property of J beyond continuity on the positive reals.

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

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Derived articles

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