Rank-One Collapse and the Emergence of Hierarchy (IRMT XII): Koide as a Geometric Crossing in Projected Relational Operators

The charged lepton masses exhibit a remarkable empirical structure captured by the Koide relation, yet the dynamical origin of this relation remains unknown. Within Information Relational Manifestation Theory (IRMT), earlier work showed that the Cl(3) boundary algebra and its Frobenius-norm geometry naturally contain the Koide structure, but subsequent experiments revealed that the low-energy relational Dirac spectrum repeatedly relaxes toward a coherent symmetric triplet with K ≈ 1/3 rather than the physical value K = 2/3. This paper resolves the apparent conflict by showing that the hierarchy does not arise directly in the low-energy spectrum. Instead, it emerges after non-local substrate geometry is projected through the coherent triplet basis via the operator Mₐᵦ = ⟨ψₐ | Φ | ψᵦ⟩. Across the experimental sequence, projected operators were found to generate strong hierarchy naturally, while alternative explanations based on off-diagonal mixing, democratic decomposition, and symmetry-breaking balance were found to possess substantially weaker explanatory power. The dominant control variable was found to be the rank concentration ratio R = σ₁/(σ₁ + σ₂ + σ₃), where σᵢ are the singular values of the projected operator. Experimentally, hierarchy correlated with R at approximately 0.956, identifying rank-one collapse as the primary driver of projected hierarchy. Under residual pair symmetry, p₂ = p₃, the hierarchy parameter satisfies K(R) = R² + (1 − R)²/2. Setting K = 2/3 gives Rₖ = (1 + √2)/3 ≈ 0.8047, matching the observed near-Koide crossing region. The result reframes Koide not as an imposed mass formula or endpoint of symmetry breaking, but as a geometric crossing during rank-one collapse of a projected relational operator. The remaining problem is no longer hierarchy generation, but stabilization: why collapse halts within Koide band rather than continuing toward R → 1.