qm_interpretation_implies_cost_gap
plain-language theorem explainer
QM interpretation from the ledger requires that the J-cost of an entangled N-particle state exceeds the J-cost of the corresponding product state for every N greater than 1. Researchers modeling classical emergence via Recognition Science cost minimization cite this when deriving observable consequences of the interpretation. The proof is a direct one-line instantiation of the universal quantifier inside the hypothesis.
Claim. Let $h$ be the assumption that the QM interpretation structure holds, i.e., that for all natural numbers $N>1$ the entangled J-cost satisfies $J(N,1,1)>J(N,1)$ where $J$ denotes the product or entangled functional. Then for any such $N$, $J(N,1,1)>J(N,1)$.
background
Recognition Science assigns a J-cost to each configuration; the product-state cost is linear in particle number while the entangled cost adds quadratic cross terms. The module states that classical descriptions emerge precisely as J-cost minima. The upstream definition qm_interpretation_from_ledger encodes the interpretation as the proposition that entangled states carry strictly higher cost than product states for $N>1$. The supporting definitions supply the explicit formulas: jcostEntangled$(N,j,α)=N·j+α·N·(N-1)/2$ and jcostProduct$(N,j)=N·j$.
proof idea
The proof is a one-line term-mode wrapper that applies the hypothesis h directly to the parameters N and hN, instantiating the universal quantifier inside qm_interpretation_from_ledger.
why it matters
The result shows that the QM interpretation structure forces a strict cost gap between entangled and product states, reinforcing that classicality appears at J-cost minima within the Recognition framework. It supplies the concrete implication needed to connect the ledger-based interpretation to observable cost differences. No downstream theorems are recorded yet, so the declaration currently serves as a terminal implication step.
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