Explain the Lean theorem `unique_cost_on_pos` 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.

1. What the Declaration Says in Plain English

The theorem unique_cost_on_pos establishes that any mathematical function $F(x)$ over positive real numbers that obeys a specific bundle of structural rules must be exactly equal to the Recognition Science cost function $J(x) = \frac{x + x^{-1}}{2} - 1$. The required rules include reciprocal symmetry ($F(x) = F(x^{-1})$), zero cost at $x=1$, strict convexity, a specific calibration, continuity, and a composition law (the "cosh-add" identity).

2. Why it Matters in Recognition Science

This THEOREM corresponds to T5 in the Universal Forcing chain. It demonstrates that the formula for $J(x)$ is not an arbitrary MODEL or a fitted equation, but a mathematical inevitability forced by symmetry and cost-balancing rules. Because $J(x)$ is mathematically unique, the subsequent derivation of the golden ratio $\phi$ and all fundamental physical constants (like $c$, $\hbar$, and $G$) proceeds without any adjustable empirical parameters.

3. How to Read the Formal Statement

theorem unique_cost_on_pos (F : ℝ → ℝ) (hF : UniqueCostAxioms F) :
  ∀ {x : ℝ}, 0 < x → F x = Jcost x

• F : ℝ → ℝ: Considers any real-valued function $F$.
• hF : UniqueCostAxioms F: Assumes $F$ satisfies the property bundle defined in UniqueCostAxioms.
• ∀ {x : ℝ}, 0 < x → F x = Jcost x: Concludes that for all strictly positive real numbers $x$, the value of $F(x)$ is identically Jcost x.

4. Visible Dependencies and Certificates

The declaration directly depends on T5_uniqueness_complete, which performs the heavy lifting: it maps the positive reals into logarithmic coordinates to reduce the composition law into d'Alembert's functional equation, where the unique solution is $\cosh(t) - 1$.

The required properties are bundled in UniqueCostAxioms. Additionally, the module supplies Jcost_regularity_cert, which serves as a formal certificate proving that Jcost itself satisfies the continuity, strict convexity, and calibration requirements demanded by the framework.

5. What This Declaration Does Not Prove

• Pure Math Regularity: The theorem does not prove the foundational real-analysis steps for solving differential equations from scratch. Properties like dAlembert_toODE and ode_bootstrap are supplied as explicit HYPOTHESES inside the UniqueCostAxioms bundle. This isolates the physical uniqueness argument from textbook calculus boilerplate.
• Negative Ratios: The theorem only proves uniqueness on strictly positive numbers (0 < x). It makes no claims about $F(x)$ for $x \le 0$, which aligns with the physical reality that a ratio of recognition sizes must be strictly positive.