emergence_scalar_proved
plain-language theorem explainer
The scalar foundation of the Hamiltonian emergence hypothesis holds for any finite dimension N. Researchers deriving quantum mechanics from recognition cost functions would cite this as the completed base case for the quadratic energy functional. The proof is a direct term construction that supplies the required witness by pairing the diagonal Hamiltonian with the quadratic approximation theorem.
Claim. For every natural number $N$, there exists a discrete evolution operator on $N$-dimensional complex space such that for every small-deviation state the absolute difference between total J-cost and quadratic energy is at most twice the sum over all sites of the cube of the absolute deviation.
background
The module develops the emergence of a quantum Hamiltonian from the Recognition Operator in the small-deviation regime. SmallDeviationState parameterizes states near equilibrium by a vector of deviations. The J-cost function reduces to the quadratic form J(1 + ε) = ε²/2 plus higher-order remainder, which supplies the kinetic energy term of the emergent Hamiltonian.
proof idea
The proof is a one-line term wrapper that applies diagonalHamiltonian N to furnish the evolution witness and totalJcost_approx_quadratic to verify the cubic error bound for every small-deviation state.
why it matters
This declaration closes the scalar part of the emergence hypothesis, confirming that the quadratic approximation to J-cost yields a valid energy functional. It directly supports the hypothesis H_HamiltonianIsGenerator, which remains open at the full operator level pending discrete Stone theorem infrastructure. In the Recognition Science framework this corresponds to the small-deviation limit converting recognition dynamics into discrete Schrödinger evolution.
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