Cost Uniqueness Unique Cost On Pos

The uniqueness theorem

The declaration unique_cost_on_pos is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It states that any cost functional F satisfying five plain conditions must equal the function J(x) = (x + 1/x)/2 - 1 on all positive real numbers. The five conditions are reciprocal symmetry (F(x) = F(1/x)), zero cost at unity (F(1) = 0), strict convexity, a calibration condition in logarithmic coordinates, and continuity. The theorem also requires the functional to obey a forced composition law, which is the framework's way of saying that recognizing a compound event costs exactly what its parts dictate, no more and no less.

The proof is a chain of implications. It starts from the functional equation that the composition law imposes, then uses the d'Alembert equation, a classical second-order functional equation, to reduce the problem to an ordinary differential equation. The regularity hypotheses, which are explicit in the theorem statement, ensure that the solution is the smooth one and not a pathological alternative. The conclusion is unconditional: there is no other function that meets all the stated conditions. The theorem is axiom-clean, meaning it relies only on the standard logical axioms of the ambient type theory, not on any framework-specific assumptions.

In Recognition Science, this theorem is a central result. It shows that the cost of recognition is not a choice but a forced consequence of the framework's own definitions. The framework models reality as maintaining a ledger, a discrete record of recognition events, and this theorem pins down the exact price of each entry. The number that emerges is not arbitrary; it is the unique function that satisfies the conditions. This is what the framework means when it says the cost is forced, not chosen.

The theorem does not claim that the physical world actually uses this cost function. It proves a mathematical uniqueness result within a specific formal system. Whether that system describes physical reality is a separate question, one that the framework addresses through empirical checks, not through this theorem alone. The theorem also does not derive the fine-structure constant or any other physical constant; it only fixes the form of the cost function. The value of that function at specific points, such as the golden ratio, is a consequence of the formula, not an input to it.

/-- 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

/-- 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

The theorem does not claim that physical reality actually uses this cost function. The theorem does not derive the fine-structure constant or any other physical constant. The theorem does not claim the cost function is unique without the composition law.

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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