A|Ω⟩: The Algebra of Awareness and Volition

We formulate an Awareness–Volition algebra on a bipartite Krein space K = H_Ψ [∔] H_Φ, whose positive sector is interpreted as physical realization and whose negative sector as phenomenal presentation. A densely defined closed transition with a J-invariant domain is grading-odd and Krein-self-adjoint if and only if it has the off-diagonal block form with entries B† and −B for a closed operator B : H_Ψ → H_Φ. Its directional components, Awareness A : H_Ψ → H_Φ and Volition V : H_Φ → H_Ψ, satisfy A² = V² = 0, the Krein-adjoint reciprocity A‡ = V, and −{A, V} = diag(B†B, BB†) ≥ 0. Polar decomposition identifies the two nonzero supports and yields the canonical sector geometry iσ_y ⊗ |B|, paired nonzero cycle spectra, and a normalized return sign of −1. For a distinguished physical vacuum |0⟩ and its phenomenal image |Ω⟩ := A|0⟩, one has A|Ω⟩ = 0, while projective recurrence, when present, belongs to the completed cycle VA. A comparison with the Hilbert-self-adjoint odd counterpart isolates the Krein sign as the source of compact rotation, quadrature, and the nontrivial closed-cycle sign. The construction is the indefinite-metric (pseudo-supersymmetric) counterpart of the Witten supercharge: the same N = 2 pairing of B†B and BB† and the same Fredholm index, but with the skew signature replacing the symmetric one.