Explain the Lean theorem `unique_cost_on_pos_from_rcl` in module `IndisputableMonolith.CostUniqueness`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

Here is an explanation of the Lean theorem unique_cost_on_pos_from_rcl, structured for a mathematically literate reader.

1. Plain English Statement

The THEOREM establishes that there is exactly one mathematically valid cost function for positive numbers that satisfies the core axioms of Recognition Science. Any continuous function $F(x)$ that treats inverse ratios symmetrically, costs zero at a ratio of 1, obeys the Recognition Composition Law, and meets a specific log-scale calibration, must be identical to the RS-native cost function $J(x)$.

2. Why it matters in Recognition Science

In the RS Universal Forcing program, physics derives entirely from the cost of logical recognition. This THEOREM represents Step T5 of the fundamental forcing chain. It guarantees that the mathematical foundation is rigid and has no arbitrary, fitted parameters. By proving that only $J(x) = \frac{x + x^{-1}}{2} - 1$ works, it canonically forces all downstream structures—ensuring physical constants are inevitable rather than empirical choices.

3. How to read the formal statement

The theorem signature takes a generic function F : ℝ → ℝ and requires several hypotheses:

• hRecip and hNorm: The function is reciprocal-symmetric ($F(x) = F(x^{-1})$) and has a root at $1$ ($F(1) = 0$).
• hComp: The function obeys the Recognition Composition Law, the core structural symmetry of RS.
• hCalib and hCont: The function is scaled correctly (its log-parametrization second derivative at $0$ is $1$) and is continuous on $(0, \infty)$.
• h_smooth, h_ode, h_cont, h_diff, h_boot: These five explicit hypotheses act as an analytic bridge, asserting that the function's composition law forms a well-behaved d'Alembert differential equation.
• Conclusion (∀ {x : ℝ}, 0 < x → F x = Jcost x): For all strictly positive real numbers, $F(x)$ precisely matches the RS Jcost.

4. Visible dependencies and certificates

The supplied source demonstrates that the predefined RS Jcost genuinely satisfies these conditions. The module contains specific THEOREM statements proving Jcost_is_reciprocal, Jcost_is_normalized, Jcost_satisfies_composition_law, and Jcost_is_calibrated. These verifications are formally packaged into the Jcost_regularity_cert, confirming that Jcost is an admissible solution to the uniqueness constraints.

5. What this declaration does not prove

• Negative domain: It restricts uniqueness to $x > 0$ (the physical domain of recognition ratios) and makes no claims about $x \le 0$.
• Analytic foundations: It accepts the d'Alembert differential regularity properties as explicit hypotheses rather than proving them from scratch, bridging to upstream functional analysis theorems.
• Downstream constants: It isolates the $J(x)$ uniqueness (T5). It does not natively prove Step T6 (the emergence of the golden ratio $\phi$) or the physical constants, which occur further down the forcing chain.