DCRP51: Wave–Pseudo-Eikonal Rigidity and the Null-Hessian Survivor Cone

Building on the central perfect-response system DCRP50 reduced to — q_zz=Δ_hq, |∇_hq|^2-3/2(q_z-4a)^2=12(a'+a-2a^2) — and using DCRP41/DCRP44's unique periodic gauge-flat eigenmode and the full-wave-cone/vorticity-realizability frontier of X72 Round43, this round asks whether the wave–pseudo-eikonal system forces q to be affine. The answer does not hold unconditionally, but carries strong signed rigidity: using a Lorentzian Bochner identity and an exact Hessian factorization, it proves that on any connected block where the right-hand side C(s)=12M_a(s) satisfied by the shifted scalar u=q-4a(s)z is non-positive, u must be spatially affine. Combining this with the infeasibility results of DCRP41/DCRP50 shows that M_a(s) cannot be negative for a persistent survivor, yielding a sharp periodic amplitude window 0≤a(s)≤1/2, with every genuinely non-affine point necessarily lying on the rank-one null-Hessian characteristic cone D^2u=κℓ⊗ℓ (ℓ a null vector). STOP-D51 explicitly does not claim general wave–pseudo-eikonal affine rigidity — it only establishes that the active periodic branch is confined to this logistic amplitude band and that non-affine points are constrained to the null-Hessian cone — left to DCRP52 to impose Hessian integrability/Codazzi conditions on D^2u=κℓ⊗ℓ, test whether ℓ must be constant, and connect to the X72 vorticity-stress algebraic cone.