ClayBridge
plain-language theorem explainer
ClayBridge defines a structural projection from Recognition Science dual-complexity objects onto Clay's single Turing parameter by mapping every RecognitionComplete instance to its Tc component. Complexity researchers examining ledger-induced information hiding would cite it to show why standard P versus NP statements become ill-posed once recognition costs are omitted. The definition directly encodes the to_clay map together with its equality to Tc and the explicit ill-posedness clause under Tc ≠ Tr.
Claim. A structure ClayBridge consists of a map to_clay from RecognitionComplete to (ℕ → ℕ), a projection axiom ∀ RC, to_clay RC = RC.Tc, and an ill-posedness condition ∀ RC, (RC.Tc ≠ RC.Tr) → to_clay RC = RC.Tc. RecognitionComplete carries a sub-polynomial computation complexity Tc : ℕ → ℕ and an at-least-linear recognition complexity Tr : ℕ → ℕ.
background
RecognitionComplete is the structure that packages dual complexity parameters: Tc records internal evolution steps and is required to be sub-polynomial, while Tr records observation operations and is required to be at least linear. The module is explicitly labeled a scaffold that explores hypothetical ledger-based complexity separations and is not part of the verified certificate chain. Upstream, the RecognitionComplete definition supplies the Tc and Tr fields that appear in the projection and ill-posedness clauses of ClayBridge.
proof idea
This is a structure definition with an empty proof body. It directly declares the three fields: the mapping function to_clay, the projection axiom that forces the image to equal Tc, and the ill-posedness implication that continues to return Tc whenever Tc differs from Tr.
why it matters
ClayBridge supplies the interface consumed by clay_bridge_theorem, which asserts existence of such a bridge rendering the ledger resolution invisible to Clay's framework. It is extended inside CompleteModel and referenced by main_resolution. The declaration fills the compatibility layer in the scaffold exploration of P versus NP under ledger assumptions, showing how omission of Tr renders the problem ill-posed. Module documentation stresses that these constructions remain hypothetical and outside the verified certificate chain.
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