Aligned coherent pair

Summary: The complete positive-density pair action has an aligned coherent stationary set with an explicit ambient Hessian and finite-mode nonlinear orbital stability.

Stationary branch of the common action

Use the complete canonical component action and notation of coherent pair deformation:

$$E=\frac\kappa2\int(|s_a|^2+|s_b|^2)^2d\nu_{\rm FS}, \quad \lambda_R=\frac{D_R^2}{D_{2R}}, \quad C=\frac{D_ND_M}{D_{N+M}}.$$

This is a declared positive-density action in a mode fibre, with orthogonal bundle components and a specified kinetic metric. It is not the full scalar Dirac action or a physical spatial bound-state theorem. Coefficient indices add no physical spatial dimensions.

For fixed positive norms $J_a,J_b$, choose the same unit ray $z$ in both families. The complete equations admit

$$\begin{aligned} a(t)&=\sqrt{J_a}e^{-i\omega_at}z^{\otimes N}, &\omega_a&=\kappa(\lambda_NJ_a+CJ_b),\\ b(t)&=\sqrt{J_b}e^{-i\omega_bt}z^{\otimes M}, &\omega_b&=\kappa(\lambda_MJ_b+CJ_a). \end{aligned} \tag{1}$$

The self and cross gradients each point along the occupied tensor. Their independent carrier phases are allowed by the two norm symmetries.

All transverse mode blocks

Choose $z=e_0$ and rotate out the two frequencies in (1). Label an occupation by its transverse degree $j$. Define

$$r_{N,j}=\frac{(N)_{\underline j}}{(2N)_{\underline j}}, \qquad p_j=\frac{(N)_{\underline j}}{(N+M)_{\underline j}}, \qquad q_j=\frac{(M)_{\underline j}}{(N+M)_{\underline j}}.$$

For $1\le j\le\min(N,M)$, each transverse occupation carries the two-by-two complex-linear equation

$$i\frac d{dt} \begin{pmatrix}\eta_{a,j}\\\eta_{b,j}\end{pmatrix} =\kappa\begin{pmatrix} J_a\lambda_N(2r_{N,j}-1)+J_bC(p_j-1) &C\sqrt{J_aJ_bp_jq_j}\\ C\sqrt{J_aJ_bp_jq_j} &J_b\lambda_M(2r_{M,j}-1)+J_aC(q_j-1) \end{pmatrix} \begin{pmatrix}\eta_{a,j}\\\eta_{b,j}\end{pmatrix}. \tag{2}$$

For $j$ present in only one family, keep its diagonal entry. The multiplicity of each transverse occupation block is $\binom{j+d-1}{d-1}$. Summing these multiplicities over each family recovers its entire band dimension.

To derive (2), differentiate both factors of the density and retain both components' variations. The self blocks are the single-family Hessian. In the cross term, multiplication by the other coherent density gives the ratios $p_j$ and $q_j$. The mixed derivative gives their geometric mean. Projected phase selection permits anomalous, conjugate-linear terms only in the two $j=0$ carrier components. No such terms are discarded in the transverse equation.

At $j=1$, (2) reduces to

$$\frac{\kappa C}{N+M} \begin{pmatrix} -J_bM&\sqrt{J_aJ_bNM}\\ \sqrt{J_aJ_bNM}&-J_aN \end{pmatrix}. \tag{3}$$

Its eigenvalues are

$$0,\qquad -\frac{\kappa C(J_bM+J_aN)}{N+M}. \tag{4}$$

The zero eigenvector $(\sqrt{J_aN},\sqrt{J_bM})$ rotates both coherent rays together. The other mode changes their relative orientation.

For $j\ge2$, $p_j+q_j<1$. Thus the cross part of (2), divided by $\kappa C$, has negative trace and determinant $J_aJ_b(1-p_j-q_j)>0$. The self diagonal terms also have negative sign because $r_{R,j}<1/2$. All nonzero transverse frequencies are therefore real and have sign opposite to $\kappa$. The only transverse zero modes are the $d$ common coherent rotations. The $j=0$ sector contains the two phase zero modes and carrier drifts caused by varying the conserved norms; fixed norms remove those radial changes.

Nonlinear orbital stability in the full finite space

Orthogonal symmetrization obeys

$$I(a,b)=C\|T_{NM}(a\otimes b)\|^2\le CJ_aJ_b. \tag{5}$$

Equality requires the product tensor to be fully symmetric. Swapping one index between the two normalized factors then gives $\operatorname{tr}(\widehat\rho_a\widehat\rho_b)=1$, where both contractions are positive with trace one. Equality forces both to be the same rank-one projector, hence $a$ and $b$ are coherent tensors on the same ray. The converse is immediate.

Together with the sharp self bounds, this shows that the aligned set is exactly the global energy maximum at fixed norms for $\kappa>0$, and the global minimum for $\kappa<0$. The nonnegative deficit

$$\mathcal D= \frac{\lambda_NJ_a^2+\lambda_MJ_b^2}{2}+CJ_aJ_b -\left[\frac{A(a)+B(b)}2+I(a,b)\right] \tag{6}$$

vanishes precisely on that set. The product of the two fixed-norm spheres is compact. On any closed subset separated from the aligned set, (6) has a positive minimum. Conservation of both norms and the energy therefore proves nonlinear orbital stability of the entire aligned set for either nonzero sign of $\kappa$.

This argument retains all ambient coefficients, including perturbations outside a prepared two-dimensional generating subspace. It supplies a finite-mode result stronger than the absence of growing linear modes. It remains separate from stability of physical spatial profiles, omitted bands and complete Dirac polarizations.

Stability does not imply capture into alignment

The conserved rotation moment is $S=N\rho_a+M\rho_b$. A pair of nonparallel coherent rays has a rank-two $S$, whereas an exactly aligned coherent pair has rank one. The former cannot evolve into the latter under these equations while preserving all its moments.

The deformation calculation shows how contact can nevertheless increase: the patterns can depart from the coherent subset and retain their conserved moments in shape modes. An energy extremum alone does not justify an alignment or relaxation law. Physical time scales require the actual kinetic normalization, and a physical capture process requires its own allowed outgoing modes.

Verification

The normal-source contact calculation uses this aligned family at a large normal degree and derives its complete target operator limit. A separate fixed-norm bound there extends the energy-maximum argument to both scalar Dirac signs when the spatial kinetic operator is zero, retaining its explicitly finite mode scope.

coherent_pair_variation_checks.py compares (2) against the complete differentiated action in all ambient occupation sectors, including the anomalous blocks and both stationary equations. Exact rational inequalities and combinatorial multiplicities cover large degrees without allocating their enormous matrices.