The linked carrier's induced geometric response

The positive collective kinetic coefficient used in the inverse source model must have an origin in the same field. It does not follow merely from allowing the occupied carrier wave to respond. That wave's exact frequency-dependent response can be computed with all its signed modes retained, and its dominant low-frequency time-derivative coefficient has the opposite sign.

This is a calculation of a forced wave response. It is not negative particle inertia. The complete positive-frequency particle and its sustaining motion still require their joint energy and momentum.

The retained geometric mode and the full response

Use the actual finite projected linked normal operator, including its spatially computed hopping integrals and finite-hopping frequency corrections. The constant-polynomial deformation has the two Hermitian source matrices $O_1,O_2$ derived in the reciprocal source calculation. In its stated scaled coordinate,

$$H(a)=H_0+\sum_j a_jO_j. \tag{1}$$

Let $H_0\psi_n=\lambda_n\psi_n$, with unit norm and $\lambda_n>0$. For every other positive or negative eigenstate define

$$\nu_l=\lambda_l-\lambda_n,\qquad b_{lj}=\langle\psi_l,O_j\psi_n\rangle. \tag{2}$$

The mean $g_{j,n}=\langle\psi_n,O_j\psi_n\rangle$ contributes the diagonal phase response. Subtracting that mean from the forcing removes the common phase direction from the shape calculation.

For a real drive with Fourier convention $e^{-i\omega t}$, solve both the ordinary and conjugate wave equations. Their combined current response is

$$\boxed{\delta g_j(\omega)=\sum_k\chi_{jk}(\omega)\delta a_k(\omega), \qquad \chi_{jk}(\omega)=\sum_{l\ne n} \left[ \frac{b_{lj}^*b_{lk}}{\omega-\nu_l} -\frac{b_{lj}b_{lk}^*}{\omega+\nu_l} \right].} \tag{3}$$

An upper-half-plane frequency specifies the retarded response. The poles are the actual rotating wave frequencies. No signed partner has been replaced by a positive-frequency approximation.

Static, Berry and time-derivative coefficients

Below the first nonzero response pole, expansion of (3) gives

$$\chi(\omega)=K+i\omega F-\omega^2 I+O(\omega^3), \tag{4}$$

$$\boxed{ \begin{aligned} K_{jk}&=-2\sum_l\frac{\operatorname{Re}(b_{lj}^*b_{lk})}{\nu_l},\\ F_{jk}&=-2\sum_l\frac{\operatorname{Im}(b_{lj}^*b_{lk})}{\nu_l^2},\\ I_{jk}&=2\sum_l\frac{\operatorname{Re}(b_{lj}^*b_{lk})}{\nu_l^3}. \end{aligned}} \tag{5}$$

Here $K$ is the static frequency Hessian of the retained linear deformation. $F$ is its antisymmetric Berry curvature in the convention $A_j=i\langle\psi_n,\partial_j\psi_n\rangle$. $I$ is the coefficient of $\dot a_j\dot a_k/2$ in the wave-induced local effective Lagrangian. The entire expression carries the wave's common canonical norm when that norm is not one.

These coefficients are not independent phenomenological parameters. They follow from the same signed resolvent. Nor can the derivative expansion replace (3) near a pole. A signed coefficient in that expansion does not establish an instability of the full Hermitian forced-wave problem.

The source matrices are derivatives with respect to the scaled $a$. For the physical polynomial coordinate $\epsilon$, the conversion $a=\gamma\kappa_d\epsilon/C_d$ must also be applied to (5). Both quadratic coefficients acquire the corresponding squared coordinate factor. Their units cannot be inferred from an untransformed numerical table.

The transverse partner fixes the leading sign

In the large-hopping two-endpoint limit, $O_1$ is diagonal on the selected positive state and $O_2$ connects it to its negative partner. That partner has $\nu=-2\lambda_n$ and unit transverse matrix element. Therefore

$$\boxed{ \chi_{22}(\omega)=\frac{4\lambda_n}{4\lambda_n^2-\omega^2},\quad K_{22}=\frac1{\lambda_n},\quad I_{22}=-\frac1{4\lambda_n^3}.} \tag{6}$$

The exact finite carrier retains the small fast-band contributions as well. At $\gamma=10000$, 130-digit calculations give the following dimensionless values in the common coordinate convention:

The depth-five neutrino controls and all remaining maintained quark controls obey the same sign and inverse-cube leading law. Both surface signs are included. Across the full scan, $-4\lambda_n^3I_{22}$ remains close to one; the largest departure is about $2.2\times10^{-4}$. The exact finite frequency is used in that ratio, rather than substituting the ideal simplex count.

The complete calculation also retains the much smaller longitudinal coefficient and nonzero Berry curvature. Real and imaginary source quadratures make the symmetric cross coefficient zero in this convention; their antisymmetric response is retained in (3)–(5).

Consequence for source reconstruction

Eliminating this occupied wave does not generate the positive, band-independent geometric inertia assumed in the collective completion. It supplies (5), with the dominant contribution (6). A positive complete geometric kinetic term would therefore need the sustaining modes' own contribution, or a reconstruction retaining the full response instead of assigning this coefficient an independent positive value.

This conclusion applies to the retained linear geometric deformation and the full finite carrier. The full normal complement, the physical kinetic law of the supporting texture, higher derivatives of the deformed spatial operator and any additional moving-basis terms have not been included in that claim. They must be computed before identifying a complete geometric source or a physical particle's inertia.

Verification

carrier_induced_response.py reconstructs every signed carrier eigenvector from the actual normal-mode hopping integrals. An independent inversion of the full ordinary and conjugate matrices agrees with (3), component by component.

carrier_induced_checks.py also extracts (5) from finite-frequency responses and checks every maintained control at depths 3, 4, 5, 6 and 10 on both surface signs. Its 130-check report passes at 130-digit precision. No amplitude was fitted and no physical relative sector scale was inferred from these dimensionless coefficients.