substrate_ok
plain-language theorem explainer
Any Hilbert space satisfying the Recognition Science axioms admits a healthy substrate formed by a normalized state paired with the identity operator as Hamiltonian. Quantum modelers constructing ILG actions from the quantum bridge would cite this existence result to initialize their substrate. The proof obtains the normalized state via an existence lemma, builds the identity Hamiltonian, and checks the bounded-below condition by direct computation of the inner product equaling one.
Claim. Let $H$ be any Hilbert space equipped with the Recognition Science Hilbert-space structure. Then there exists a normalized state in $H$ together with a self-adjoint Hamiltonian bounded from below that together constitute a healthy substrate on $H$.
background
The module supplies the quantum substrate layer for the Induced Lattice Gravity (ILG) construction, bridging the Hilbert-space and observable definitions to the ILG action. A healthy substrate is the pair consisting of a normalized state and a Hamiltonian that is self-adjoint and bounded below. Upstream lemmas include the shifted cost $H(x) = J(x) + 1$ from CostAlgebra, which converts the Recognition Composition Law into the d'Alembert equation $H(xy) + H(x/y) = 2 H(x) H(y)$, and the nuclear-density and block-offset structures that place physical quantities on the phi-ladder.
proof idea
The tactic proof first invokes the existence lemma for normalized states to obtain a state ψ. It then constructs the Hamiltonian as the identity observable, inheriting self-adjointness directly. The bounded-below clause is discharged by rewriting the inner product to the squared norm, using the norm-one property of the normalized state and the identity inner-product identity. The resulting pair is returned as the healthy substrate.
why it matters
The result supplies the non-emptiness of the quantum substrate required before any ILG action or forcing-chain application can proceed. It therefore sits at the base of the quantum-to-relativity bridge in the Recognition Science program. No downstream uses are recorded, indicating it functions as an interface lemma whose parent is the overall ILG well-definedness claim. It aligns with the T5 J-uniqueness and T7 eight-tick octave landmarks by guaranteeing a Hilbert-space carrier on which those structures can act.
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