This paper introduces the Pi-Residue Cascade as a closure-theoretic mechanism by which rotational coherence-curvature equilibrium stabilizes into ordinary three-dimensional space while conserving its excess curvature as subspatial structure.

 In this framework, π is interpreted as rotational coherence-curvature equilibrium, while 3.0 represents stable SO(3) spatial closure. The difference, ρπ = π − 3, is not a numerical remainder but a conserved curvature residue displaced beneath stable space. This residue first appears as SU(2)-type relational, chiral, torsional, and bivector geometry near d ≈ 2.859. It is then captured as SU(3)-type confinement near d ≈ 2.70.

The refinement advanced here is that confinement is the deepest closure: once curvature is confined, no free curvature excess remains to generate further gauge-bearing infratiers.

The post-confinement layer near d ≈ 2.50 is therefore not another gauge tier, but a coherence-reversal threshold.

The paper also identifies the Euler Gap, ΔE = e − (9 − 2π), as a compound bridge constant linking the geometric Euler surrogate to the true Euler phase baseline.

Keywords

Pi-residue; deepest closure; terminal gauge closure; infratier constants; subspatial closure; SO(3); SU(2); SU(3); chirality; torsion; confinement; coherence reversal; Euler Gap; closure-depth constants.