One first-order operator for the linked geometric Hessian

The linked geometry's second variation and its positive mode metric can be expressed by one first-order matrix operator. Its coefficients follow from differentiating the geometric equations; they are not fitted to particle masses. The same bounded radial function that determines the metric identities also gives an explicit spectral gap.

This is a spatial geometric calculation on an ordinary unbounded plane. The metric remains a stated kinetic comparison, without an assigned physical time coefficient. The calculation does not identify geometric fluctuations with a microscopic spinor or introduce another particle.

The complete quadratic form

Let $D_i=\partial_i-iA_i$, $B=\partial_1A_2-\partial_2A_1$, and use the linked equations

$$(D_1+iD_2)W=0,\qquad B=2(1-|W|^2). \tag{1}$$

For winding $d$, the geometric cost satisfies

$$\begin{aligned} G&=\int\left(|DW|^2+\frac{B^2}{4}+(1-|W|^2)^2\right)d^2y,\\ G-2\pi d&=\int\left(|(D_1+iD_2)W|^2+ \left[\frac B2-1+|W|^2\right]^2\right)d^2y. \end{aligned} \tag{2}$$

Boundary conditions preserve the winding and make the variations decay. For arbitrary complex $\delta W$ and real $\delta A$, define

$$\begin{aligned} F&=(D_1+iD_2)\delta W-i(\delta A_1+i\delta A_2)W,\\ b&=\frac{\delta B}{2}+2\operatorname{Re}(W^*\delta W),\\ g&=\frac12\nabla\cdot\delta A-2\operatorname{Im}(W^*\delta W). \end{aligned} \tag{3}$$

The coefficient of $\epsilon^2$ in $G(W+\epsilon\delta W,A+\epsilon\delta A)$ is $G_2=\int(|F|^2+b^2)$. The positive horizontal metric is

$$\|\delta\|_M^2=\int\left(|\delta W|^2+\frac14|\delta A|^2\right)d^2y, \qquad g=0. \tag{4}$$

The condition $g=0$ follows by orthogonality to the basis-phase variations $(i\zeta W,\nabla\zeta)$. It removes a description redundancy. It is not an independently imposed physical force.

Exact matrix factorization

Use $\alpha_1=\sigma_1\otimes I$, $\alpha_2=\sigma_2\otimes I$, $\Gamma_1=\sigma_3\otimes\tau_1$, $\Gamma_2=\sigma_3\otimes\tau_2$, and $C=\sigma_3\otimes\tau_3$. Package the variation as

$$\Xi=\left(0,\delta W,\frac{\delta A_1+i\delta A_2}{2},0\right)^T, \qquad C\Xi=-\Xi.$$

Then the self-adjoint spatial operator

$$\boxed{D_{\rm geo}=-i\alpha\cdot\nabla+ \frac12\alpha\cdot A(\tau_3-I)+2\Gamma\cdot W} \tag{5}$$

obeys, by direct component differentiation,

$$\boxed{D_{\rm geo}\Xi=(b-ig,0,0,-iF)^T,\qquad \|D_{\rm geo}\Xi\|^2=G_2+\int g^2d^2y.} \tag{6}$$

Here $\Gamma\cdot W=\Gamma_1\operatorname{Re}W+\Gamma_2\operatorname{Im}W$. Moreover $\|\Xi\|^2=\|\delta\|_M^2$. Thus (5) supplies the actual quadratic form and its stated positive norm together. The coefficient two and the scalar connection shift in (5) are fixed by (3).

Writing $D_{\rm geo}=\left(\begin{smallmatrix}0&Q^\dagger\\Q&0\end{smallmatrix}\right)$ with the positive chirality listed first, the linked equation cancels the off-diagonal part of its positive-chiral square:

$$\boxed{Q^\dagger Q= \begin{pmatrix} -\Delta+4|W|^2&0\\ 0&(-i\nabla-A)^2+B+4|W|^2 \end{pmatrix}.} \tag{7}$$

This factorization retains both spatial coefficients and arbitrary angular perturbations. It is not restricted to a radial test function.

An explicit gap from the same bounded weight

At the radial configuration, $W=w(r)e^{id\theta}$, $A=a(r)e_\theta/r$, with $w'=(d-a)w/r$, $a'=2r(1-w^2)$. Set

$$Z=d+1-a,\qquad 1\le Z\le d+1.$$

The metric identity gives $(-\Delta+4w^2)Z=4$. For any compactly supported scalar $f$, integration by parts gives the exact positive-function identity

$$\int\left(|\nabla f|^2+4w^2|f|^2\right)d^2y =\int Z^2\left|\nabla(f/Z)\right|^2d^2y +\int\frac4Z|f|^2d^2y \ge\frac4{d+1}\|f\|^2. \tag{8}$$

Density extends this bound to the form domain. The other channel in (7) has $B+4w^2=2+2w^2\ge2$. Therefore, for $d\ge1$,

$$\boxed{Q^\dagger Q\ge\frac4{d+1},\qquad |D_{\rm geo}|\ge\frac2{\sqrt{d+1}} \quad\text{on }(\ker D_{\rm geo})^\perp.} \tag{9}$$

The second statement follows because the two nonzero chiral spectra coincide. It controls the full normal continuum, not just a finite matrix.

Numerical lowest eigenvalues of the first channel refine to:

Each lies below the second channel's lower bound two. The scalar radial ground state gives a horizontal physical variation: applying $D_{\rm geo}$ to a real first-component ground state produces a negative-chiral fluctuation whose image has $g=0$. The corresponding purely imaginary first-component comparison describes the removed gauge direction. Neither comparison is a particle mass assignment.

The complete zero-mode space

The horizontal polynomial variations have the form

$$\Xi_l=(0,e^{il\theta}U_l, i e^{i(l-d+1)\theta}V_l,0)^T, \qquad l=d-k,$$

where

$$U_l'=\frac{l-a}{r}U_l-2wV_l,\qquad V_l'=\frac{d-1-l}{r}V_l-2wU_l. \tag{10}$$

The $d$ polynomial coordinates $k=1,\ldots,d$ give regular, decaying solutions. Conversely, integrating $[r^2(U_l^2-V_l^2)]'$ gives

$$\int(l+1-a)U_l^2d^2y=(d-l)\int V_l^2d^2y. \tag{11}$$

For $k=d-l\le0$, the left coefficient is strictly positive and the right is nonpositive. For $k\ge d+1$, the left is nonpositive and the right is positive. In each case both fields must vanish. There are no positive-chiral zero modes by (9). The full complex kernel therefore has exactly $d$ dimensions, corresponding to $2d$ real geometric variations, including translation.

Verification and physical scope

linked_geometric_jacobi.py and geometric_jacobi_checks.py verify (6) with independent Cartesian differentiation, (7) on arbitrary positive-chiral fields, the complete uncompleted energy's second variation, and every geometric zero mode at all five maintained depths. The 103-check report retains radial-grid and outer-boundary refinements and their raw modes. The carrier comparison additionally verifies the supersolution identity used in (8).

Equation (5) is now a candidate common differential expression for the geometry and the normal mass carrier. Its carrier comparison distinguishes the geometric rotation representation from an ordinary spinor representation. A common expression is useful structure, but does not by itself derive the same-field kinetic action, the occupied curved equilibrium, or the complete particle's positive inertia.