Unified State Space
Nine electrical measurement perspectives — arc flash, fault current, loop flow, algebraic connectivity, regime boundaries, transmission line propagation, thermal margin, phase margin, and coupling density — are orthogonal measurements of the same geometric object: the admissible state space 𝒜 of an electrical system and its boundary ∂𝒜.
3 of 9 perspectives independently flag proximity to ∂𝒜. Convergence strengthens the geometric claim.
Θ — Thermal (Irreversibility Margin)
Ω — Control (Phase Margin)
k — Coupling (EMI)
Interior and Topology perspectives use the IEEE 39-bus New England test system (fixed topology).
Nine Perspectives
Distance from the wave-regime Morse critical point where lumped-circuit topology changes to distributed-wave topology
The Lyapunov function V(x) = E_arc(I, t, D) whose level sets define nested closed surfaces. The PPE category boundaries are the level sets where the restoring force (safety margin) changes sign.
The result of the Fortescue projection F: ℂ³ → ℂ³ (positive, negative, zero sequence). The projection is lossy for asymmetric faults — the information loss measures the transmission loss across the network boundary.
The dimension of the cycle space H₁(G, ℝ) — the number of independent loops in the network. This is the topological invariant that Kirchhoff's voltage law conserves. The cycle space is the interior of the admissible region that the system cannot leave without topological surgery.
The second smallest eigenvalue of the graph Laplacian L = D − A. By the Cheeger inequality, λ₂/2 ≤ h(G) ≤ √(2Δλ₂), where h(G) is the edge expansion (minimum cut ratio). λ₂ is the spectral proxy for the topological index of the network manifold.
The real part of the propagation constant γ = α + jβ, which is the eigenvalue of the elliptic transmission operator d²V/dx² − γ²V = 0. The analytical index of this operator (dim ker − dim coker) equals the topological index of the network manifold by the Atiyah-Singer theorem.
The fraction of thermal capacity remaining before irreversible damage begins. Θ = 0 is the boundary ∂𝒜 where the I²t integral equals the conductor's thermal withstand K²S² — the point where the Lyapunov function V = K²S² − I²t changes sign.
The angular distance from the Nyquist critical point (−1+0j). The closed-loop topology changes discontinuously when PM crosses 0° — this is the Morse critical point of the control manifold. PM = 0° is the boundary ∂𝒜 for the control-limited regime.
The dimensionless coupling coefficient k = M/√(L₁L₂) ∈ [0,1]. At k = 0 the subsystems are magnetically isolated; at k = 1 they share a single flux linkage. The boundary ∂𝒜 for the coupling-dominated regime is at k ≈ 0.5, where unintended interaction paths rival the intended signal path.
Atiyah-Singer Bridge
The index theorem (1963) states that for an elliptic operator on a compact manifold, the analytical index equals the topological index. The two measurements below are taken from orthogonal perspectives of the same electrical system.
The discrepancy exceeds 40%. This reflects the gap between the compact smooth manifold idealisation and the actual discrete electrical network. The correspondence is approximate, not exact, for finite graphs.