πomni Identity: I ̃·Rc =ξ0 as R/r=π

This note develops the formal core of a Ξ|-grounded geometric model of consciousness
that frames experiential stability as an integrity–coherence balance. Building on Xi-Theory-
compatible primitives—distinction acts Ξ, a structuring operator |, and an integrity functional I(P) over patterns P—the model introduces a toroidal state space T2 as a minimal
manifold supporting two independent closures: a global integration/world-coupling cycle
and a local self-referential boundary cycle. A coherence-radius field Rc(φ) = R + r cos φ
is defined from the torus geometry and linked to a Droplet-Mind-style Young–Laplace-like
law ∆C ∝ I(P)/Rc. To operationalize the non-scalar primitive Ξ| in dynamics, the paper
proposes a normalized integrity proxy I ̃ ∈ (0, 1] constrained by an invariant balance relation  ̃
I(t)Rc(t) = ξ0. A key postulate—harmonic closure consistency—selects the closure ratio R/r = π, yielding a structural critical share qc = r/R = 1/π ≈ 0.318 and its complement (π−1)/π ≈ 0.682, numerically aligned with the empirical Xi-Theory consciousness threshold C2 ≈ 0.68. The framework is explicitly positioned as falsifiable by specifying measurable proxy families for I ̃ and Rc using EEG/MEG coupling, graph-integration and stability summaries, distance-dependent connectivity decay, surrogate controls, and preregisterable threshold/separability tests.