DeviationHilbert
plain-language theorem explainer
DeviationHilbert defines the complex vector space ℂ^N for modeling N-bond deviations around equilibrium. Researchers deriving emergent Hamiltonians from the recognition operator cite this space when linearizing J-cost dynamics. The declaration is a direct type abbreviation with no computational content or proof obligations.
Claim. For positive integer $N$, the deviation Hilbert space is the set of all maps from the finite index set to the complex numbers, written $Fin N → ℂ$, equipped with the standard Hermitian inner product.
background
The module shows how the quantum Hamiltonian emerges as the small-deviation limit of the Recognition Operator. Near equilibrium the J-cost reduces to the quadratic form J(1 + ε) = ε²/2 + O(ε³), which supplies the kinetic term. DeviationHilbert supplies the target vector space for embedding real deviations into this complex setting.
proof idea
Direct type abbreviation: DeviationHilbert N is defined as Fin N → ℂ. No lemmas or tactics are invoked; the abbreviation simply renames the function type for use by embed and DiscreteEvolution.
why it matters
The type is consumed by DiscreteEvolution to define the linearized step operator and by embed to map SmallDeviationState into complex vectors. It supplies the Hilbert-space scaffolding required for the conditional emergence statement H_HamiltonianIsGenerator. Within the Recognition framework it bridges the eight-tick octave dynamics to discrete Schrödinger evolution while the phi-ladder and quadratic approximation remain in force.
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