Canonical response of the actual moving normal modes

A spatial eigenbasis changes when its supporting geometry changes. The canonical response therefore contains more than the change of the finite carrier's eigenvectors. The missing contribution can be computed from the actual normal modes and their complementary spatial inverses, without introducing a fitted kinetic coefficient.

This calculation uses the ordinary unbounded normal plane, the linked geometry and both the original and geometric carriers at depths 3, 4, 5, 6 and 10. It supplies a wave-induced geometric curvature. Its sign is not a sign of particle inertia.

The orthogonal spatial derivative

Let $D\Phi_l=0$, with an isolated chiral kernel and orthonormal modes $\Phi_l$. Transport them locally with $\langle\Phi_i,\partial_a\Phi_j\rangle=0$ at the reference point. Differentiating the kernel equation gives

$$\boxed{\partial_a\Phi_l=-D^{-1}(I-P)(\partial_aD)\Phi_l.} \tag{1}$$

The inverse acts only off the kernel. Since $\partial_aD$ changes chirality, its source lies in the positive channel. The diagonal positive square $L_+$ in the common carrier gives

$$\langle\partial_a\Phi_l,\partial_b\Phi_m\rangle =\langle(\partial_aD)\Phi_l,L_+^{-1}(\partial_bD)\Phi_m\rangle. \tag{2}$$

This avoids differentiating a numerical eigenbasis. It retains the full spatial complement through its regular, decaying Green solutions.

Use $P(z)=z^d+\alpha_kz^{d-k}$, $\alpha_k=x_k+i y_k$, and the polynomial gauge of the linked deformation. Write $\partial_{x_k}u=f_k(r)\cos k\theta$, and let $U_l,V_l$ be the carrier's real normalized radial modes. For $D=D_{t,e}$, its real derivative has two positive-chiral sources, with angular labels $j=l+k$ and $j=l-k$:

$$\begin{aligned} b^+_0&=tw(r^{-k}-f_k/2)U_l +(e+1/2)(f_k'+kf_k/r)V_l/2,\\ b^+_1&=twf_kV_l/2+(e-1/2)(f_k'-kf_k/r)U_l/2,\\ b^-_0&=-twf_kU_l/2+(e+1/2)(f_k'-kf_k/r)V_l/2,\\ b^-_1&=-tw(r^{-k}-f_k/2)V_l +(e-1/2)(f_k'+kf_k/r)U_l/2. \end{aligned} \tag{3}$$

Their spinor components are $(b_0e^{i(j-d)\theta},0,0,i b_1e^{i(j+1)\theta})^T$. The imaginary derivative multiplies the plus source by $-i$ and the minus source by $+i$. The same relation holds after applying (1).

Curvature and positive metric are different contractions

Take the canonical convention

$$A_a=i\langle\Psi,\partial_a\Psi\rangle,\qquad F_{ab}=\partial_aA_b-\partial_bA_a =-2\operatorname{Im}\langle\partial_a\Psi,\partial_b\Psi\rangle. \tag{4}$$

For a single order, angular orthogonality makes the normal curvature diagonal in $l$. Define the nonnegative quantities

$$N_l^\pm=\langle b_l^\pm,L_+^{-1}b_l^\pm\rangle.$$

Then

$$\boxed{(F^{\rm normal}_{x_k y_k})_{ll}=2(N_l^+-N_l^-), \qquad (g^{\rm normal}_{x_k x_k})_{ll}=N_l^++N_l^-.} \tag{5}$$

The curvature is a difference; the positive metric is a sum. Replacing one with the other would change the dynamics. Cross-order entries follow from (2), retaining every pair of matching angular labels. The numerical calculation keeps all centered orders $k=2,\ldots,d$, rather than freezing the other shapes while testing one endpoint.

For the original carrier, the antiunitary map $\Theta=\sigma_y\tau_y K$ sends $\Phi_l$ to $-i\Phi_{d-1-l}$. It implies the exact reflection identity $R F^{\rm normal}_{x_k y_h}R=-F^{\rm normal}_{x_k y_h}$. This identity is checked before it is used to evaluate small differences. The geometric carrier's shifted connection does not have that symmetry.

Add the retained coefficient response

Let $c_{\chi l}$ be the normalized occupied eigenvector of the full finite carrier $H_P$, with eigenvalue $\lambda_n$. Its complete wave curvature at the reference point is

$$\boxed{ F^{\rm wave}_{ab}= \sum_{\chi,l,m}c_{\chi l}^*(F^{\rm normal}_{ab})_{lm}c_{\chi m} -2\operatorname{Im}\sum_{r\ne n} \frac{\langle n,H_a r\rangle\langle r,H_b n\rangle} {(\lambda_r-\lambda_n)^2}.} \tag{6}$$

Here $H_a=\partial_aH_P$ includes the complete spatial derivative. The second term is the retained-mode contribution already studied in induced geometric response. The first term is the actual moving spatial basis addition. Orthogonal transport makes their separation unambiguous at this point. All signed retained modes are included in the spectral expression.

The convention (4) and the differentiated eigenvector formula underlying (6) agree with the original mathematical treatment of adiabatic phase. See M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proceedings of the Royal Society A 392 (1984), equations 6–10. The specific spatial contractions here are calculations for the stated IDWT candidate.

Translation, phase and temporal connection

A common translation has a useful independent check. The coefficient $\alpha_1$ translates the profile center by $-\alpha_1/d$. The normal contribution is exactly

$$F^{\rm normal}_{x_1y_1} =-\frac{i}{d^2}[Pp_1P,Pp_2P]. \tag{7}$$

For a complete translated wave, the retained basis contribution cancels (7). The full canonical translation curvature is zero, as also follows from commuting full spatial translation generators. Complex polarized rays test this cancellation beyond a single real eigenvector. A separate phase–scale test agrees with the derivative of the connection-charge expectation under independently recomputed scaled radial profiles.

The time convention matters. Equations (1)–(6) use the polynomial spatial gauge with $A_0=0$. Under

$$W\mapsto e^{i\zeta}W,\quad A_i\mapsto A_i+\partial_i\zeta, \quad\Psi\mapsto e^{-iQ_A\zeta}\Psi,$$

one must also take $A_0\mapsto A_0+\dot\zeta$. The canonical term $i\Psi^\dagger\dot\Psi-\Psi^\dagger Q_AA_0\Psi$ is then invariant. Changing the spatial representative and setting its temporal connection to zero again describes a different connection history. The physical master action has not yet selected that history or supplied the sustaining geometry's complete kinetic law.

Verification

normal_basis_response.py implements (1)–(6). Independent Cartesian derivatives, direct norms of $D L_+^{-1}b$, phase–scale profiles and the complete translation check are in normal_basis_checks.py. The 475-check report passes at all five maintained depths for both carriers. Raw spatial sources and inverses are preserved separately with their exact source hashes. The resulting curvature is tested as a proposed slow law in the complete motion calculation.