Explanation of unique_cost_on_pos

(1) In plain English, the declaration states that any function F : ℝ → ℝ meeting a specific list of properties (reciprocal symmetry for positive arguments, normalization F(1) = 0, strict convexity on positive reals, a calibration condition on the second derivative after log substitution, continuity on positive reals, and several regularity hypotheses for an associated functional equation) must coincide exactly with the Jcost function on the positive reals.

(2) In Recognition Science this matters because it supplies the uniqueness half of the cost foundation: once symmetry, convexity, and calibration are granted, the cost functional is forced to be Jcost, supporting the parameter-free derivation of constants and structure from the Law of Logic.

(3) The formal statement is read as follows: the theorem takes an arbitrary cost function F together with a hypothesis hF : UniqueCostAxioms F; it then concludes that for every positive real x the equality F x = Jcost x holds. The proof simply unpacks the fields of hF and delegates to the more general T5_uniqueness_complete.

(4) Visible dependencies inside the supplied source are the structure UniqueCostAxioms, the lemma Jcost_continuous_pos, the theorems Jcost_is_reciprocal and Jcost_is_normalized, and the underlying engine T5_uniqueness_complete that performs the reduction to the d'Alembert equation and cosh solution.

(5) The declaration does not prove that Jcost itself satisfies UniqueCostAxioms (the corresponding definition is commented out), does not establish the physical interpretation of Jcost, and does not connect the uniqueness result to the larger forcing chain or to spacetime emergence.