Stability of reciprocal normal polarization

Summary: The complete finite-gap normal density and reciprocal coherent target have an exact quartic orientation dispersion, with bounded and growing nonlinear controls at equal canonical norm and stationary scalar source.

Test whether the response preparation persists

The balanced normal response has a common leading geometric coefficient across retained grades under specified canonical populations. A stationary preparation with that response need not persist when disturbed. This calculation retains the entire normal density, both normal Dirac signs and the reciprocal target ray to test the finite mode dynamics.

The normal kinetic gap and occupied spectator modes remain conditional reductions of the same field. The target has zero spatial kinetic operator in this control. Its coherent tensor ray is an invariant mode manifold of the degree-one interaction, not an established localized particle. No new physical dimensions, compactification or interaction particle is introduced.

Use the definitions of the linked response note: physical depth $d$, coefficient rank $r=d+1$, target degree $N$, target canonical norm $J$, and common coefficients

$$a=\frac{d+1}{N+d+1},\qquad b=\frac{d+1}{d+2},\qquad B=aJN. \tag{1}$$

Let $P=zz^\dagger$ be the target ray projector and let $Y$ be the complete $2r$-row normal field, including both signs and its explicit occupied spectator columns. Its density and scalar current are

$$\mathcal D=YY^\dagger,\qquad C=\mathcal D_{++}-\mathcal D_{--},\qquad L=(aJ+b\operatorname{Tr}C)I+BP+bC. \tag{2}$$

The exact canonical field equations are

$$i\dot z=gaCz,\qquad i\dot Y=H(L)Y,\qquad H(L)=\epsilon\tau_x\otimes I+g\tau_z\otimes L. \tag{3}$$

The induced target ray kinetic coefficient is $JN$. The full normal Gram matrix $Y^\dagger Y$ and all positive canonical norms are retained. The target's additional scalar phase includes its own self-insertion; it does not change the projector dynamics at fixed $J$.

The complete reciprocal linearization

At a stationary coherent target, $L$, $C$ and the population matrix $U$ have parallel and perpendicular eigenspaces. The positive scalar fixed point of the response note gives the insertion eigenvalues $r_0,r_1$, scalar eigenvalues $c_0,c_1$, and normal density

$$\mathcal D_0=I_2\otimes U+qP_+(L),\qquad U=u_\perp I+\Delta u P, \qquad u_\perp\ge0,\quad u_\perp+\Delta u\ge0. \tag{4}$$

Here $r_1,c_1$ each have multiplicity $d$. Both populations are nonnegative even when $\Delta u<0$. The balanced part has zero scalar current and zero normal energy but a positive canonical norm.

Write $\delta C=\delta\mathcal D_{++}-\delta\mathcal D_{--}$. Linearizing (3) gives the complete finite density system

$$\begin{aligned} i\delta\dot P&=ga\left([C,\delta P]+[\delta C,P]\right),\\ i\delta\dot{\mathcal D}&=[H,\delta\mathcal D] +[\delta H,\mathcal D_0],\\ \delta H&=g\tau_z\otimes \left[B\delta P+b(\operatorname{Tr}\delta C\,I+\delta C)\right]. \end{aligned} \tag{5}$$

The common coefficient-rotation moment is conserved:

$$JN P+\operatorname{Tr}_{\rm normal}\mathcal D \quad\text{is constant}. \tag{6}$$

Indeed, the $\epsilon\tau_x$ term commutes with coefficient rotations. The cross commutators cancel between the target and normal parts of (6), and the scalar self-commutators cancel among themselves. Equation (5) obeys the linearized version for arbitrary complete matrix variations.

At depth six the diagnostic retains all $49$ complexified target matrix entries and all $196$ normal entries. Hermitian variations give the physical density perturbations; the coherent target tangent and fixed normal-population orbit are invariant subspaces. A growing mode below is explicitly realized by canonical fields on those physical subspaces. No instability is inferred solely from an inadmissible density direction.

An independent five-variable orientation block

Choose one perpendicular coefficient direction. Let $p=\delta P_{01}$ and let $D$ be the full two-by-two normal-spin block $\delta\mathcal D_{s0,t1}$. Define

$$\begin{aligned} H_i&=\epsilon\tau_x+gr_i\tau_z,& E_i&=\sqrt{\epsilon^2+g^2r_i^2},\\ D_i&=u_iI_2+\frac q2\left(I_2+\frac{H_i}{E_i}\right),& \delta c&=\operatorname{Tr}(\tau_zD),\quad \ell=Bp+b\delta c. \end{aligned} \tag{7}$$

With the convention $e^{-i\Omega t}$, the exact block is

$$\boxed{\begin{aligned} \Omega p&=ga\left[(c_0-c_1)p-\delta c\right],\\ \Omega D&=H_0D-DH_1+g\ell(\tau_zD_1-D_0\tau_z). \end{aligned}} \tag{8}$$

This has five complex variables and retains every normal-spin transition. Independent matrix embedding checks that it is exactly a block of (5). The common rotation moment supplies its zero-frequency mode.

The remaining four frequencies are roots of an explicit real quartic. Put

$$\begin{aligned} \Delta r&=r_0-r_1,& R&=r_0+r_1,& \Delta c&=c_0-c_1,& S&=E_0^{-1}+E_1^{-1},\\ Z(\Omega)&=\frac12\left(\Delta c-\frac\Omega{ga}\right),& \Lambda(\Omega)&=\Delta r-\frac{b\Omega}{ga},& F(\Omega)&=\Omega^2-g^2R^2. \end{aligned} \tag{9}$$

Then the dispersion equation is

$$\boxed{\begin{aligned} 0={}&\Omega\left[F(\Omega)-4\epsilon^2\right]Z(\Omega)\\ &+g\left[\Delta u F(\Omega)+q\epsilon^2S\Omega\right] \Lambda(\Omega) +\frac{g\Delta r JN}{2}F(\Omega). \end{aligned}} \tag{10}$$

For a derivation, expand $D=A I+X\tau_x+Y\tau_y+Z\tau_z$. The conserved moment at nonzero frequency gives $A=-JNp/2$; the target equation gives $Z=Z(\Omega)p$. The fixed-point gap relation $\Delta r=B+b\Delta c$ gives $\ell=\Lambda(\Omega)p$. Eliminating $X$ and $Y$ from the remaining normal equations yields (10). The diagnostic compares its coefficients with the characteristic polynomial of the independent block (8). At exceptional degeneracies the matrix equation, rather than a cancelled fraction, defines the complete spectrum.

The zero-gap restriction and the finite-gap difference

At zero kinetic gap, $\Delta c=0$ and $\Delta r=B$. Equation (10) factors into the two normal frequencies $\pm gR$ and the two reciprocal orientation frequencies

$$\Omega_\pm=-gb\Delta u \pm g\sqrt{b^2(\Delta u)^2+a^2JN(JN+2\Delta u)}. \tag{11}$$

The radical is nonnegative for every real $\Delta u$ when $N\ge1$:

$$\begin{aligned} &b^2(\Delta u)^2+a^2JN(JN+2\Delta u)\\ &\quad=b^2\left(\Delta u+\frac{a^2JN}{b^2}\right)^2 +a^2(JN)^2\left(1-\frac{a^2}{b^2}\right)\ge0, \end{aligned} \tag{12}$$

because $a\le b$. The full scalar self-term is essential to this completed-square statement. Real frequencies still allow polynomial growth at a defective degenerate root; this is an exclusion of exponential orientation growth in the stated zero-gap block.

Finite $\epsilon$ couples the orientation motion to the other normal transitions. The full quartic can then have nonreal roots. At $d=6$, $N=12$, $J=1$, $g=0.7$, $q=0.4$, $\epsilon=20$ and $\Delta u=-20$, the growing frequency is

$$\boxed{\Omega=0.1129776272+0.4392350957\,i.} \tag{13}$$

It occurs in the complete density system (5), not just the reduced polynomial. Taking $u_\perp=20.2$ gives $u_\parallel=0.2$, so all occupied canonical eigenvalues are strictly positive.

The specified positive-imbalance controls $\Delta u=0.6$ at $N=12,34$ and gaps $0.1,1,5,20$ have no exponentially growing eigenfrequency in the complete finite linearization. This is a tested set, not a proof for all positive imbalances or all spatial fields.

Nonlinear comparison at equal norm and scalar source

Two controls use the same $d=6$, $N=12$, $J=1$, $g=0.7$, $q=0.4$ and $\epsilon=20$. Fix $\operatorname{Tr}U=121.4$ and compare $\Delta u=0.6$ with $\Delta u=-20$, setting $u_\perp=(121.4-\Delta u)/7$ in each case. Both have normal canonical norm $Q_f=245.6$, the same stationary scalar current and the same stationary normal energy. Only the balanced population distribution differs. These are declared mode controls; none of these quantities is fitted to a particle measurement.

An eigenmode of (8) is embedded into the complete density matrices. To make a valid nonlinear initial field, the perturbation is implemented by unitary rotations of the target ray and the occupied normal columns. Their exact norms and the entire normal Gram matrix are therefore unchanged at initialization. All complex fields then evolve by (3), including their fast phases and the reciprocal change of the target.

For initial density amplitude $10^{-5}$ and $0\le t\le15$, the reversed population amplifies by approximately $726.709$, matching $\exp(15\times0.4392350957)$. The full nonlinear density remains close to the growing linear mode over that interval. A corresponding oscillatory mode of the positive-imbalance control stays bounded.

Independent time refinement checks the complete complex fields, energy, occupied Gram matrix and common rotation moment. The exact source programs and raw trajectories accompany the reports. Conservation of those quantities does not by itself imply stability; the growing solution conserves them while its polarization changes.

The two complete field refinements differ by less than $8.8\times10^{-9}$. The refined growing rate is $0.439235095730$, compared with the linear value $0.439235095738$. Energy drift is below $7.7\times10^{-11}$, occupied Gram drift below $8.8\times10^{-11}$ and rotation-moment drift below $1.3\times10^{-10}$ in these controls.

Consequence for physical selection

The stationary scalar source and a canonical norm do not uniquely select the polarization preparation, even within this one action. Reciprocal finite-gap dynamics distinguishes some preparations that those stationary quantities cannot distinguish. It supplies a concrete stability constraint on the proposed common geometric response.

It does not yet select a unique population, identify the observed coupling currents or produce a spatial particle. The mass-source calculation retains the separate occupation, band matching and total-energy questions. Physical spatial geometry and full Dirac residuals must be supplied before these finite controls can establish particle stability or dimensionality.

Verification

gapped_reciprocal_normal_checks.py compares the complete reciprocal density generator, independent angular block, eliminated quartic, canonical action variations, conserved moments and zero-gap completed square. With the two trajectory reports supplied, it also checks complete-field refinement and nonlinear amplification. The exact finite equations are implemented in gapped_reciprocal_normal.py. gapped_reciprocal_normal_flow.py stores the complete canonical fields for the equal-norm comparison.