main_resolution
plain-language theorem explainer
The declaration exhibits an explicit complete model in which ledger computation yields constant-zero internal complexity while recognition requires linear steps, producing a strict separation that renders the classical P versus NP formulation ill-posed under the dual-cost projection. Researchers examining ledger-style or recognition-theoretic complexity would cite the construction when testing information-hiding barriers from double-entry structure. The proof proceeds by direct assembly of concrete LedgerComputation, RecognitionComplete andClay
Claim. There exists a complete model $CM$ extending a ledger computation such that flux is conserved by definition, the computation complexity satisfies $T_c(n)=0$ while recognition complexity satisfies $T_r(n)=n$, and the Clay bridge projection maps only onto the computation component, rendering the standard P versus NP question ill-posed whenever $T_c$ differs from $T_r$.
background
LedgerComputation supplies a state space, an evolution rule via double-entry updates, an encoding map, a measurement predicate, and a flux-conservation proof. RecognitionComplete augments this with dual functions $T_c$ (internal evolution steps) and $T_r$ (observation operations) together with sub-polynomial and linear bounds. CompleteModel further includes a Turing special case that discards recognition cost and a ClayBridge that projects onto the classical Turing model, making P versus NP ill-posed when the two complexities diverge. The module is explicitly scaffold and exploratory, outside the verified certificate chain.
proof idea
The proof assembles a concrete LedgerComputation with Unit states and identity evolution, a RecognitionComplete with constant-zero $T_c$ and identity $T_r$, and a ClayBridge whose ill-posed map returns reflexivity. These are packaged into a CompleteModel; the existential witness is then supplied by reflexivity on flux conservation, a decision tactic establishing the inequality at $n=1$, and simplification on the ill-posed projection.
why it matters
The declaration supplies the principal constructive witness for the hypothetical ledger-based separation of computation and recognition scales inside the ComputationBridge module. It relies on the ClayBridge and RecognitionComplete structures to illustrate how balanced-parity encoding creates an information-theoretic barrier. Because the module is scaffold, the result remains exploratory and does not belong to the core Recognition Science certificate chain; it touches the open question whether classical complexity statements remain well-posed once observation cost is made explicit.
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