Explain the Lean structure `PublicCostLayer` in module `IndisputableMonolith.Foundation.DimensionalConstraints.CostLayer`. 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

PublicCostLayer is a bundled mathematical proposition that lists seven essential properties of the Recognition Science cost functional $J(x)$. It asserts that:

• The cost function is uniquely determined for positive real numbers.
• In logarithmic coordinates, it takes the exact form $\cosh(t) - 1$.
• The cost of exactly balanced recognition ($x = 1$) is $0$.
• It is symmetric for reciprocal ratios ($J(x) = J(1/x)$).
• It is non-negative for any positive input.
• Zero cost occurs if and only if the input is exactly $1$.
• As the ratio approaches zero, the "defect" diverges beyond any finite bound (an absolute floor that prevents $x=0$).

2. Why It Matters in Recognition Science

In RS, the physical universe is a downstream consequence of minimizing $J$-cost. PublicCostLayer serves as an epistemic firewall. It isolates the exact cost-theoretic properties required for the "dimensional constraints rebuttal" into a compact, paper-specific namespace. By exposing only these seven properties, the framework provides a clean "public API" for reviewing the geometric and dimensional proofs without requiring the reader to navigate the full internal cost derivation pipeline.

3. How to Read the Formal Statement

In Lean, a structure : Prop acts as a logical conjunction of multiple statements.

• The ∀ symbol means "for all".
• x : ℝ restricts the domain to real numbers.
• 0 < x → ... means the property assumes the input is strictly positive. For example, the field reciprocal : ∀ {x : ℝ}, 0 < x → Cost.Jcost x = Cost.Jcost x⁻¹ formalizes exact reciprocal symmetry. To a programmer, this is an interface defining required properties. To a mathematician, it reads like a list of axioms. However, the RS framework does not assume them; it proves them upstream.

4. Visible Dependencies and Certificates

The declaration relies on upstream modules IndisputableMonolith.Cost, Cost.AczelClassification, and Foundation.LawOfExistence. Immediately below the structure, the file provides the constructor theorem public_cost_layer. This theorem acts as a certificate, proving that the RS cost function satisfies the interface by plugging in actual proofs from the framework (such as Cost.Jlog_as_cosh for the log-closed form and Foundation.LawOfExistence.nothing_cannot_exist for the null barrier).

5. What This Declaration Does Not Prove

This module acts as a specification bridge, meaning its scope is strictly limited:

• It does not prove the foundational cost equation. The heavy lifting of the Aczel classification and d'Alembert equations is performed upstream and merely imported here.
• It does not prove the emergence of dimension or spacetime. While it packages the cost properties used for dimensional forcing, the actual proofs that $D=3$ and that spacetime is 4-dimensional reside in separate, downstream modules.