E47 × KKP-RADAR
Spectral Operator Bridge
A calibrated 5×5×5 complex I/Q radar patch is vectorized directly into the existing 125-dimensional spin-2 triple tensor carrier. The unchanged E47 Casimir decomposition then produces seven measurable shell populations, an E47 occupancy, kernel residuals, and a calibrated anomaly statistic.
Operator chain
X(a,r,t)=I+iQ, a,r,t∈{-2,-1,0,1,2}
↓ vec
x ∈ C^125 ≅ V₂⊗V₂⊗V₂
↓
C = Jx² + Jy² + Jz²
↓
K = (C−6I)(C−30I)
↓
Pj , j=0…6
↓
qj = ||Pj x||² / ||x||²
↓
ηE = q₂ + q₅
AE = (q−μ₀)ᵀ Σ₀⁺ (q−μ₀)
Why seven shells, not one scalar?
The exact isotropic baseline is (1,9,25,28,27,22,13)/125. The full Casimir-shell vector records how radar energy redistributes across all irreducible sectors. In the synthetic benchmark it carries substantially more information than scalar deviation of E47 occupancy alone.
| SNR | Casimir 7-shell AUC | random same-size splits | FFT peak / total energy | |ηE−0.376| AUC |
|---|---|---|---|---|
| −10 dB | 0.539 | 0.491–0.513 | 0.623 | 0.501 |
| −5 dB | 0.695 | 0.486–0.506 | 0.904 | 0.534 |
| 0 dB | 0.866 | 0.476–0.516 | 0.996 | 0.634 |
| +5 dB | 0.932 | 0.466–0.517 | 1.000 | 0.696 |
(1,9,25,28,27,22,13) remain near chance at every SNR, while the Casimir seven-shell profile is above all five controls. The standard 3-D FFT peak-to-total-energy statistic is stronger than the Casimir profile at every tested SNR. At 0 dB: 0.996 vs 0.866. This is a comparative synthetic benchmark, not evidence of operational radar superiority.Contraction
Tε = I − εK² ε* = 1/99144 ρ* = 15/17 lim n→∞ Tε*ⁿ = PE 220-step relative error = 6.27e−13 ||PE²−PE||F = 6.59e−15 ||KPE||F = 1.82e−12 null Monte Carlo mean = 0.375602 exact null mean = 0.376
Composite KKP-RADAR vector
existing branch:
X → ρ → Ω → κ → φ
parallel E47 branch:
X → vec(X) → {q0,…,q6}, ηE, RK, AE
fusion:
zRADAR-E47 =
[ρ, Ω, κ, φ, SNR, Doppler,
q0,…,q6, ηE, RK, AE]
Certificate PASS. Exact finite algebra and synthetic numerical bridge validated.
Evidence boundary: no measured operational radar performance is claimed. Random same-size subspaces and the FFT peak/total-energy control are now recorded in the machine certificate; the FFT control wins on this synthetic benchmark. The next empirical gate is a measured/public complex I/Q benchmark against prespecified full CFAR, Doppler-threshold, and supervised baselines.