The complete normal carrier has accurate stationary frequencies, but its wave contribution to the motion of the prescribed background has negative inertia. The same neighboring bands that were harmless in the static curvature calculation give a large response to translation.

This calculation includes every component of the linear wave parent. It does not include a dynamical source for the background potential. It therefore does not determine the complete inertia of a self-supported particle, and it does not refute inertia as the proposed origin of mass. It identifies an additional requirement on this inverse construction.

Translate the stated background

Use the complete parent and its canonical norm from A complete finite normal band. The actual axial coordinate of the round feature is z=y_2. Translating its normal background along z at velocity v gives, in its moving coordinates,

H(u)=H0−up2,p2=−i∂x2,u=vνa.(1) H(u)=H_0-u p_2,\qquad p_2=-i\partial_{x_2},\qquad u=\frac{v}{\nu a}. \tag{1}

The operator H and its eigenfrequency w are dimensionless; the physical selected frequency is nu w. A radial normal displacement has an analogous local response with p_1. On the round feature, that radial displacement is a change of its profile, not a Cartesian translation of the whole particle. The axial entry in (1) is the actual global translation tested here. Other Cartesian translations also involve the angular field and are not evaluated by this two-entry normal matrix.

The background is still prescribed. Its own kinetic energy and stress have not been included in H(u). Calling the following result the wave contribution keeps that omission explicit.

Derive the canonical response, including both signs

Let psi be the full normalized positive-frequency field. Its mean normal momentum vanishes by parity. Second-order differentiation of (1) gives

w(u)=w0−12M22u2+O(u4),(2) w(u)=w_0-\frac12\mathcal M_{22}u^2+O(u^4), \tag{2}

where the complete normal displacement matrix is

Mij=−2Re⁡∑s=±1⟨piψ,(w0−Hq+s)−1pjψ⟩.(3) \boxed{\mathcal M_{ij}=-2\operatorname{Re} \sum_{s=\pm1}\left\langle p_i\psi, (w_0-H_{q+s})^{-1}p_j\psi\right\rangle.} \tag{3}

All parent branches and both signs enter each inverse. Normal momentum acts on the additional field components as well as the target. It changes the graded band by one, so (3) includes its complete source. No single guessed excitation gap replaces the inverse.

The canonical stationary reduction has L_eff=-hbar nu w(u), up to the phase term. Therefore its dimensional wave inertia is

Mijwave=ℏνa2Mij.(4) M^{\rm wave}_{ij}=\frac{\hbar}{\nu a^2}\mathcal M_{ij}. \tag{4}

This derivation fixes the sign. It also shows why an arbitrary normalization of the spatial eigenfield is not a mass variable. The field has common full canonical norm one, and its mass-frequency target remains hbar nu w/c^2.

Why the sign is negative

At large gamma, the two target components are concentrated at opposite ends of the normal band: phi_0(x_1)phi_q(x_2) and phi_q(x_1)phi_0(x_2), with equal canonical weights. The frequency in the q+1 band is smaller than w_0 by order 1/gamma.

Applying p_2 to the first end component has a nonzero projection on that lower-frequency band. Both signed parent fields must be included. Their combined squared projection approaches (q+1)/4. Consequently their leading contribution to (3) is

M22∼−q+12w0.(5) \boxed{\mathcal M_{22}\sim-\frac{q+1}{2w_0}.} \tag{5}

The same leading value occurs for the radial profile response. The q-1 band has a larger frequency and does not cancel (5). The finite parent and its complete canonical components preserve this leading behavior in the tested hierarchy.

For q=5, gamma=10000 and M=10^18, the two eigenvalues of w_0 times the normal response matrix are approximately (-3.00154,-2.99845) for L=14, and (-3.00375,-2.99625) for L=36. At q=9, L=24 they are (-5.00275,-4.99725). In particular, the actual axial entry is negative in all three.

The very small stationary residual in Curvature of the complete normal band therefore does not establish a satisfactory translation response. Static curvature and motion of the background excite different combinations of the complementary modes.

Independent moving-frame verification

schwinger-translation-20260914T035835_882431Z.json passes 39 checks. Direct differentiation of the complete Hermite fields verifies the momentum source, including every additional component. Independent 120- and 140-digit resolvents agree.

A separate control at q=3, beta=5, gamma=20 and M=10^6 diagonalizes H_0-u p directly with all three generated normal bands. It gives axial coefficient -174.4794452042. Finite-velocity differences converge to it as u is halved. Independent radial and mixed-direction controls also agree with the complete matrix (3).

The first checker used a negative matrix index that mpmath does not interpret as Python's last-element index. The exact source and its 36-check report are preserved. The revised checker uses an explicit last index and separately tests the actual axial entry; the directly computed frequencies and response coefficients are unchanged.

The physical next step is to derive how the same field sustains and moves this normal background, or to obtain a different isolated normal band with an acceptable complete motion. Adding an arbitrary positive inertia to cancel (5) would not supply that derivation.