Locality tests for the geometric current source

Replacing a mode average by a pointwise current changes the full source, even when the same coherent state remains exact. This distinction can be tested without fitting any particle amplitudes. A local oscillator square has an explicit growing harmonic. A different first-order Dirac current law avoids that particular growth in its two-endpoint limit, but does not inherit the result when arbitrary fast carrier bands are added.

These are inverse local comparisons on an ordinary two-dimensional oscillator plane. That plane tests proposed normal dynamics; it has not been identified with the linked carrier's complete spatial mode expansion. Appending another physical plane to that carrier would change the model's dimensionality. A valid embedding must use the same physical coordinates and canonical field. Form and spinor indices below are component labels.

A pointwise completed square

Let $b_j=(y_j+\partial_j)/\sqrt2$ on $\mathbb R^2$, and let $\Xi(y)$ have two carrier components. Define

$$\rho=\Xi^\dagger\Xi,\quad g_j(y)=\frac{\Xi^\dagger O_j\Xi}{\rho},\quad u=\frac1{\sqrt{2K\Omega}},\qquad O_1=\sigma_z,\quad O_2=\sigma_x.$$

The local comparison with the earlier moment action is

$$E_{\rm loc}=\int\left[ \Xi^\dagger D\sigma_z\Xi+ \Omega\sum_j\left|(b_j+u g_j(y))\Xi\right|^2\right]d^2y. \tag{1}$$

Its positive coherent core has carrier vector $(1,0)^T$ and a Gaussian displaced by $-u$ in the first oscillator. Both squares vanish, so its particle frequency and full stationary energy are $D$ and $DQ$. For comparison with the mass construction, use the maintained endpoint count $D=\binom{N+d}{d}$. This supplies the input frequency, rather than deriving it from (1).

At this core the first current has zero linear variation. The second varies with the real part of the opposite carrier component. The full quadratic energy therefore contains

$$\Omega\left|b_2\eta+ 2u\,e_+\operatorname{Re}(e_-^\dagger\eta)\right|^2. \tag{2}$$

Unlike the global moment, this expression acts on every spatial harmonic. It couples the upper oscillator state $n+1$ to the lower state $n$. Keep also the first oscillator's spectator number $r$. In canonical real coordinates $\eta=(q+ip)/\sqrt2$, each exact block has Hessians

$$H_p=\begin{pmatrix}a&0\\0&b\end{pmatrix},\qquad H_q=\begin{pmatrix}a&t\\t&b+2/K\end{pmatrix}, \tag{3}$$

$$a=\Omega(n+r+1),\qquad b=-2D+\Omega(n+r),\qquad t^2=\frac{2\Omega(n+1)}K. \tag{4}$$

The squared disturbance frequencies are the eigenvalues of $H_pH_q$. Every omitted-component residual of this block is zero. Thus its growth cannot be attributed to dropping a responding partner.

For $r=0$, $b<0$, and a complex-conjugate squared-frequency pair, the exact growth rate is

$$\boxed{\Gamma^2=\frac14\left[ \frac{2|b|}{K}-(a-|b|)^2\right].} \tag{5}$$

With the common test parameters $K=16,\Omega=1/4$, every integer $D\ge1$ has two explicit growing harmonics:

$$\begin{array}{c|c} n&\Gamma^2\\\hline 4D-1&(4D-1)/128\\ 4D&(2D-1)/64. \end{array} \tag{6}$$

At $D=1$ the first rate is about $0.153093$ in these frequency units. Large $D$ moves the unstable harmonics to higher normal degree; it does not eliminate them in the unrestricted oscillator comparison. The signed quadratic disturbance energy is not the complete particle's inertial mass, which remains the positive frequency of its proposed core.

Equation (1) also has a second-order spatial principal part. Its canonical time equation is therefore not a fundamental finite-speed Dirac equation. Both issues require attention before identifying this local comparison with a physical source.

A distinct first-order current law

A first-order alternative retains the complete Clifford form representation on the same comparison plane. Let $f_j^\dagger,f_j$ be its two form creation and annihilation matrices, and put

$$\Gamma=(-1)^{\text{form degree}},\qquad \gamma_j=f_j+f_j^\dagger,\qquad \mathcal D_{\rm osc}=\sum_j(f_j^\dagger b_j+f_j b_j^\dagger). \tag{7}$$

For a finite Hermitian carrier matrix $H_0$, set $g_j=\Xi^\dagger O_j\Xi/\rho$ and $t_j=\Xi^\dagger\gamma_j\Xi/\rho$. The complete action uses the canonical time norm and the energy

$$\boxed{E_D=\int\left[ \Xi^\dagger(\Gamma H_0+\Lambda\mathcal D_{\rm osc})\Xi +\Lambda u\sum_j \frac{(\Xi^\dagger O_j\Xi)(\Xi^\dagger\gamma_j\Xi)}{\rho} \right]d^2y.} \tag{8}$$

The nonlinear source is

$$\boxed{S_D=\Lambda u\sum_j (g_j\gamma_j+t_jO_j-g_jt_jI)\Xi.} \tag{9}$$

All three terms follow from the single interaction in (8), including the denominator's variation. The source is degree one in the field, smooth away from zero, and has bounded derivatives on the unit sphere. It therefore extends as a globally Lipschitz algebraic source at zero. The potential itself is degree two. Norm and complete energy accounting use that homogeneity, rather than interpreting an amplitude as mass.

This action changes the geometric response law. It is not an equivalent rewriting of the collective oscillator action. Its fixed oscillator coefficient remains a confinement input; (9) supplies a reciprocal current response, not the source of that entire coefficient.

For any normalized carrier eigenvector $H_0\psi_n=\lambda_n\psi_n$, take the form vacuum and the Gaussian displaced by $-u\langle\psi_n,O_j\psi_n\rangle$. Then $t_j=0$, and the normal Dirac term plus the first term of (9) annihilates the Gaussian. The other source terms vanish. Consequently the full stationary solution has frequency $\lambda_n$ and energy $\lambda_nQ$.

The normal principal matrices are $\alpha_j=i(f_j^\dagger-f_j)/\sqrt2$, with $\{\alpha_j,\alpha_k\}=\delta_{jk}I$. The local norm flux is therefore bounded by $\Lambda/\sqrt2$ times the density. The common Hermitian oscillator potential cancels from the difference-density estimate, and the nonlinear difference is bounded by the Lipschitz constant. Integrating outside a forward cone gives finite propagation at this normal speed. This is a statement about the displayed normal equation. Its physical spatial scale, full containing-sector propagation, covariant completion and sustaining geometry still require derivation.

A stability criterion for every endpoint normal block

Take $H_0=D\sigma_z$, $O_1=\sigma_z$, $O_2=\sigma_x$, and the positive core. Let $s=0,1$ be its two carrier labels, $f_1,f_2\in\{0,1\}$ the form occupations and $n_1,n_2$ the oscillator numbers. The complete real-linear equation preserves

$$L_1=n_1+f_1,\qquad L_2=n_2+f_2+s. \tag{10}$$

Each block has at most eight complex spatial components. Its ordinary Hessian $P$ is the full rotating Dirac oscillator matrix. The second real Hessian differs only by the reciprocal coupling

$$Q-P=2\Lambda u \left(|s=0,f=01\rangle\langle s=1,f=00|+\text{adjoint}\right), \tag{11}$$

at identical oscillator indices. Unlike (4), its norm $\varepsilon=2\Lambda u$ does not grow with harmonic number.

Ordering the block by the sign of $\Gamma\sigma_z$ puts its Hessians in the form

$$P=\begin{pmatrix}0&A\\A^\dagger&-2DI\end{pmatrix},\qquad Q=\begin{pmatrix}0&A\\A^\dagger&-2DI+V\end{pmatrix}. \tag{12}$$

In every balanced nonzero block $A$ is invertible, so neither Hessian can acquire a zero eigenvalue as $V$ is turned on. The exceptional first block is rectangular: its remaining negative kernel is untouched by the restriction of $V$, and its determinant is also nonzero. The occupied norm and phase zero directions are treated separately.

The uncoupled Dirac magnitudes are $\sqrt{D^2+\Lambda^2L}$ and $\sqrt{D^2+\Lambda^2(L-1)}$, where applicable. Their opposite-energy imaginary frequencies are separated by at least

$$\delta=3D-\sqrt{D^2+\Lambda^2}. \tag{13}$$

If $\delta>2\varepsilon$, the conserved-Hessian spectral-cluster argument of the collective source applies to every invariant block: the normal uncoupled generator's perturbation disks cannot connect opposite energy signs, and (12) keeps the restricted forms nondegenerate. Each block has imaginary semisimple generator frequencies.

The single choice

$$\Lambda=1,\qquad u=\frac1{2\sqrt2}$$

satisfies the criterion for every $D\ge1$, since $3-\sqrt2>\sqrt2$. This proves spectral stability of every exact endpoint block, not merely the blocks below a numerical cutoff. A bound on the full infinite direct-sum evolution also requires control of its norms; no nonlinear spatial orbital-stability theorem is asserted here.

Fast bands are a separate test

The proof above uses the specific two-endpoint mass matrix. It cannot be transferred to arbitrary carrier bands. A structural control keeps four signed masses $+1,-1,+M,-M$ and one active normal current. The occupied state couples to the $-1$ branch with vertex one and to $+M$ with vertex $0.1$, through the same reciprocal local law.

Without the fast vertex, the primary coupling shifts the negative disturbance frequencies. The crossing must be found using those shifted frequencies, not the bare carrier values. With the fast vertex retained, the complete invariant matrix has growing modes. For example, $M=5$, normal degree 17 gives rate approximately $0.0196789$; $M=20$, degree 8268 gives $0.0129767$. The corresponding controls with the fast vertex set to zero are stable.

This establishes a limitation of a proposed generalization. These assigned four-band controls are not the full linked physical geometry. Their distant resonances may also lie outside a specified effective projection's valid range. The complete physical carrier and that range must be evaluated before making a physical stability claim.

Verification

local_moment_checks.py passes 146 checks. Independent Hermite-Gaussian spatial integration verifies the pointwise energy Hessian, all block residuals vanish, and 70-digit eigenvalues check the exact growth rates across maintained depth controls.

dirac_moment_checks.py passes 388 checks. They include arbitrary complex source variations, the complete normal Clifford representation, the full spatial oscillator matrix, invariant block residuals and high-precision signed spectra at large mass counts.

dirac_fast_band_checks.py passes 24 checks of the full shifted resonances, uncoupled controls and eigenpair residuals, with growth rates independently refined from 65 to 90 digits.