For two weighted four-point square constellations of edge length ℓ, we determine the minimum labeled-pair distance against an uncontrolled relative phase under unit squared-weight normalization. The sharp value is ℓ sin(π/8), attained only at the weights {sin(π/8), cos(π/8)}; the minimum distance then equals this value at every phase. We prove that every phase-independent relay map satisfying the exclusive law has exactly the same worst-phase distance profile, regardless of its output alphabet size. This also determines its worst-phase MAP error exponent in complex AWGN. For a tesseract rotating independently in two fixed coordinate planes, we extend the optimization to all orthographic planar maps and characterize every maximizer. The resulting planar-to-spatial lift is globally optimal within that compression family. Under operator perturbations of norm at most η_P, its sharp separation is ℓ max{sin(π/8) − √3 η_P, 0}. Exact error integrals and reproducible simulations distinguish distance optimality from finite-SNR performance. Appendices give a phase-invariant block receiver, conditional local quadrotor tracking laws, and sparse communication guarantees under explicit delivery assumptions. Single-observation phase ambiguity and the obstruction to uniform separation under arbitrary four-dimensional rotations are proved explicitly. The constellation result is positioned as a normalized specialization of established singular-fade geometry, without a claim of historical priority.
This preprint record includes the 29-page revised manuscript (September 12, 2026) and the complete reproducibility package: LaTeX source, Python verification and simulation code, pinned dependencies, numerical results, reproduction instructions, technical audit and revision notes, and SHA-256 checksums. The simulations are model-based; no field-flight validation is claimed.