clay_bridge_theorem
plain-language theorem explainer
The declaration establishes existence of a ClayBridge structure that projects any RecognitionComplete object onto its computation complexity function alone. Researchers working on ledger-based models of complexity would cite it to separate internal evolution cost from observation cost while remaining compatible with Turing-centric formulations. The proof proceeds by direct construction of the bridge record followed by an explicit witness instance whose bounds are verified by norm_num and nonnegativity lemmas.
Claim. There exists a ClayBridge structure $CB$ such that for every RecognitionComplete object $RC$, the map $CB.to_clay(RC)$ equals the computation complexity function $RC.T_c$, and there exists a RecognitionComplete object $RC$ for which the computation complexity is strictly smaller than the recognition complexity.
background
In the scaffold module ComputationBridge, RecognitionComplete is the structure carrying dual functions $T_c : ℕ → ℕ$ (internal evolution steps, required to be subpolynomial: bounded by $c n^k log n$ for some $0 < k < 1$) and $T_r : ℕ → ℕ$ (observation operations, required to be at least linear). ClayBridge is the compatibility record whose to_clay field sends any such $RC$ to $RC.T_c$, with the projection field enforcing that Clay's framework sees only the computation side and the ill_posed field recording that $T_c ≠ T_r$ makes the standard P-versus-NP question ill-posed inside that framework.
proof idea
The proof first builds a ClayBridge record by setting to_clay to the projection onto $T_c$ and discharging the projection and ill_posed fields by reflexivity. It then supplies a concrete RecognitionComplete witness with $T_c$ constantly zero (subpolynomial bound proved via norm_num together with nonnegativity of log and real powers) and $T_r$ the identity (linear bound by direct construction), finally using decide to witness the strict inequality between the two functions.
why it matters
The theorem lives inside an explicitly marked scaffold module whose documentation states it is not part of the verified certificate chain and explores only hypothetical ledger implications. It illustrates the module's central claim that the ledger's balanced-parity encoding creates an information-theoretic barrier between computation and recognition scales, allowing P = NP at computation cost while P ≠ NP at recognition cost. The result directly supports the hypothetical SAT separation and Turing incompleteness statements listed in the module header but remains outside the main Recognition Science forcing chain.
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