H_AreaQuantization
plain-language theorem explainer
H_AreaQuantization encodes the hypothesis that any ledger eigenstate of an area operator has area expectation value equal to an integer multiple of ell0 squared. Discrete-spacetime and quantum-gravity researchers would cite it when deriving Planck-scale area quanta from simplicial flux bits. The definition is introduced as a direct implication from the ledger-eigenstate predicate without invoking additional lemmas.
Claim. Let $H$ be a Hilbert space with RSHilbertSpace structure, $A$ an area operator on $H$, and $ψ ∈ H$. The predicate holds precisely when, if $ψ$ is a ledger eigenstate, there exists $n ∈ ℕ$ such that $⟨ψ, A.op ψ⟩_ℂ = n ⋅ ℓ₀².
background
The area operator is the structure that assigns to each simplicial surface the recognition flux carried by its faces, with each 3-simplex face contributing one bit of flux potential. A state $ψ$ is a ledger eigenstate when the simplicial faces are activated in a definite pattern. The constant ell0 is the fundamental RS length (voxel size), obtained from the relation ell0 = c ⋅ τ₀ with τ₀² = ℏG/(πc⁵).
proof idea
This is a definition that directly packages the implication: ledger eigenstate implies existence of integer n with area expectation equal to n times ell0 squared. No upstream lemmas are applied inside the definition itself; it functions as the hypothesis interface for the two downstream theorems that consume it.
why it matters
The definition supplies the hypothesis for the area-quantization theorem and the minimal-area-eigenvalue theorem, both of which rest on the discrete nature of recognition bits. It implements the area-quantization prediction of Phase 9.1, which traces back to the eight-tick octave (T7) and the planarity constraints of the simplicial ledger. It leaves open the empirical test in the Planck regime.
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