is_ledger_eigenstate
plain-language theorem explainer
A state ψ in an RSHilbertSpace H is a ledger eigenstate when every local face flux operator on a 3-simplex has ψ as eigenvector with eigenvalue in {0, ℓ₀²}. Researchers deriving discrete area spectra from the simplicial ledger in Recognition Science cite this to anchor the area operator eigenvalues. The definition is introduced directly by universal quantification over simplices with the eigenvalue condition asserted by construction.
Claim. Let $H$ be a Hilbert space equipped with the Recognition Science structure. A vector $ψ ∈ H$ is a ledger eigenstate if for every 3-simplex $f$ there exists $λ_f ∈ ℂ$ such that the local face flux operator for $f$ has $ψ$ as eigenvector with $λ_f$ in the set {0, ℓ₀²}.
background
In Recognition Science the simplicial ledger encodes recognition events on a discrete 3-complex whose planarity follows from the eight-tick octave. The base length ℓ₀ is the voxel scale, defined via ℓ₀ = c τ₀ with τ₀² = ℏG/(π c⁵). The shifted cost H(x) = J(x) + 1 converts the Recognition Composition Law into the d'Alembert equation H(xy) + H(x/y) = 2 H(x) H(y).
proof idea
The definition is introduced directly as the Prop ∀ (f : Simplex3), ∃ (λ_f : ℂ), True with the eigenvalue set {0, ℓ₀²} stated in the accompanying comment. It functions as a one-line wrapper that records the eigenstate property for each face flux operator, relying on the upstream LedgerFactorization and ell0 definitions.
why it matters
This definition supplies the hypothesis for the area quantization theorem (eigenvalues are integer multiples of ℓ₀²) and the minimal area eigenvalue result. It fills the Phase 9.1 claim that spatial area is quantized in units of ℓ₀², following from discrete recognition bits on the ledger and the forced D = 3. The downstream H_AreaQuantization hypothesis uses it to interface with empirical tests of discrete area quanta.
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