A negative signed wave momentum derivative has a precise phase interpretation when the supporting coefficients undergo an imposed oscillatory translation. This calculation concerns a partial response. Positive inertia of a complete particle must be assessed from the momentum of its whole sustaining field.

Let a self-adjoint stationary wave operator have a normalized isolated mode phi with selected frequency omega. In coordinates moving with the prescribed profile, a small imposed velocity v(t) contributes -v(t)P to that operator. The first variation satisfies

i∂tδϕ=(H−ω)δϕ−v(t)Pϕ.(1) i\partial_t\delta\phi=(H-\omega)\delta\phi-v(t)P\phi. \tag{1}

Both frequency components of a real velocity must be kept. Expanding in the complete spectral resolution of H gives the retarded momentum susceptibility

χPP(ν)=∑j2Δj∣⟨j∣P∣ϕ⟩∣2Δj2−(ν+i0)2,Δj=Ej−ω.(2) \boxed{\chi_{PP}(\nu)= \sum_j\frac{2\Delta_j|\langle j|P|\phi\rangle|^2} {\Delta_j^2-(\nu+i0)^2},\qquad \Delta_j=E_j-\omega.} \tag{2}

The sum includes a spectral integral if the operator has continuum modes. Equation (2) requires the stated matrix elements and resolvent limit to exist. The occupied phase direction is removed in the usual linear normalization condition. Here nu is the imposed disturbance frequency; omega remains the particle's selected mass-frequency.

At zero driving frequency, (2) is the stationary partial momentum derivative. Away from resonance, a real negative susceptibility gives delta P(t)=chi_PP(nu) v(t), with momentum opposite in phase to the velocity. It does not require a negative inertia for the complete moving structure. It also does not imply a dissipative time lag.

An exact unitary two-level control verifies the sign and phase. With Delta=-2 and P=0.7 sigma_x, the linear susceptibilities are -0.50127877 at nu=0.3 and +0.392 at nu=3. Direct time evolution gives respectively phase pi and phase zero. Reducing the velocity amplitude from 0.001 to 0.0002 reduces the nonlinear response error by the expected factor of approximately 0.04, while norm is conserved.

The two-level system is a verification control, not an IDWT particle model. The report partial-momentum-phase-20260914T123325_527778Z.json passes ten checks and retains the trajectories and source versions. An actual joint IDWT translation still has to include the sustaining pattern's own response, total momentum and stress.