Sources, Energy Exchange and Current Response Through Canonical Orientation

Candidate and Its Physical Status

Let the ordinary sector spacetime have metric eta with signature (+,−,…,−). Keep the orientation target, metric and timelike determinant law of the preceding calculation. In a local adapted normal frame, write its canonical connection as a = A_A(phi) d phi^A, with curvature F_AB. Let rho be a unitary representation of the normal group on retained coefficients chi. All coefficient spaces are field values, not additional ordinary spatial dimensions.

The following positive-energy wave action is a comparison realization of retained same-field modes:

$$\mathcal L=T(1-\sqrt\Delta)+(D_\mu\chi)^\dagger D^\mu\chi-V(\chi^\dagger\chi), \quad D_\mu\chi=\partial_\mu\chi+\rho(A_A)\chi\,\partial_\mu\phi^A, \quad V\ge0.$$

It does not derive this decomposition from Psi_infinity, does not establish fermionic dynamics, and does not identify a chosen bare restoring clock with the maintained particle frequencies. In this candidate chi describes retained coefficients independent of the ray that defines J. If instead chi is that same complete chiral ray, treating both as independent variables overcounts its degrees of freedom and the action needs a different reduction. The connection has no independent variation or independent mediator. A Dirac comparison below retains the same geometric source identity but requires its own energy interpretation.

Constrained Source Variation

Write W_mu = D_mu chi and use the dual current convention

$$j^\mu(H)=2\operatorname{Re}[W^{\mu\dagger}\rho(H)\chi],\qquad H^\dagger=-H.$$

For covariant variations delta_c chi = delta chi + rho(A_A) chi delta phi^A and delta_c W_mu = delta W_mu + rho(A_A) W_mu delta phi^A, direct differentiation gives

$$\delta_c W_\mu=D_\mu\delta_c\chi+\rho(F_{AB})\chi\,\delta\phi^A\partial_\mu\phi^B.$$

This identity includes the connection commutator. On the coefficient equation D_mu W^mu + V' chi = 0, the first term contributes only a boundary variation. Define

$$\mathcal E_A=\partial_\mu(\sqrt\Delta\,\gamma^{\mu\nu}G_{AB}\partial_\nu\phi^B) -\frac12\sqrt\Delta\,\gamma^{\mu\nu}\partial_A G_{BC}\partial_\mu\phi^B\partial_\nu\phi^C.$$

The actual orientation equation is

$$f^2\mathcal E_A=j^\mu(F_{AB})\partial_\mu\phi^B,\qquad f^2=T\lambda.$$

The same equation follows off shell after subtracting a coefficient-equation multiple from the raw orientation equation. It is not D_mu(epsilon F^{mu nu}) = j^nu. In particular, a constant orientation is an exact solution with any free coefficient solution even if its normal current is nonzero. This is a consequence of the constrained connection variation, not a proof that a physical charged particle cannot interact.

Complete Positive Energy and Reciprocal Exchange

The connection-dependent contribution to the orientation canonical momentum is j^0(A_A). Its contribution cancels the corresponding connection terms when the complete Legendre transform is taken. Consequently

$$e=e_J+\|W_0\|^2+\sum_i\|W_i\|^2+V\ge0.$$

The complete stress, including those orientation momentum terms, is

$$\Theta^\mu{}_\nu=\Theta_J^\mu{}_\nu+ 2\operatorname{Re}(W^{\mu\dagger}W_\nu) -\delta^\mu{}_{\nu}(W_\rho^\dagger W^\rho-V).$$

On the two Euler equations, the sector exchange obeys

$$\partial_\mu\Theta_\chi^\mu{}_\nu=j^\mu(F_{\mu\nu}),\qquad \partial_\mu\Theta_J^\mu{}_\nu=-j^\mu(F_{\mu\nu}),\qquad F_{\mu\nu}=F_{AB}\partial_\mu\phi^A\partial_\nu\phi^B.$$

Thus the coefficient and orientation responses are reciprocal parts of one conserved stress. A prescribed current with its own energy omitted would lose this statement.

Principal Structure of the Coupled Wave Action

At a frozen jet, let B_A = rho(A_A) chi, viewed as a real linear map from orientation variations to complex coefficient variations. For a characteristic covector xi, the full principal quadratic form is

$$f^2\sqrt\Delta\,(h^TQh)\,\gamma^{-1}(\xi,\xi) +2\eta^{-1}(\xi,\xi)\|z+B_Ah^A\|^2.$$

The invertible, covector-independent change z to z+B_A h^A makes it block diagonal. The orientation block retains the determinant cone and the coefficient block retains the ordinary light cone. The common time Hessian is positive on the timelike determinant branch. This establishes local hyperbolic principal structure under the same smoothness and uniform domain-margin assumptions as before. It does not establish nonlinear stability of every background.

An Exact Current-Induced Instability Test

Take constant J and use a root tangent pair in the normal indices a,b. Let X_E and X_iE be the real antilinear generators with B=E_ab and iE_ab. The coordinates pi = (u X_E + v X_iE)/4 have unit orientation metric at the origin because norm([X_E,J0]) squared = 16. The canonical root curvature is

$$F_{uv}=-\frac{i}{8}(D_a+D_b).$$

For a constant diagonal normal current, distinct root planes decouple in the linearized orientation equation. Put c^mu = j^mu(F_uv) and alpha^mu = c^mu/f^2. The root equations and their complex form are

$$\Box u=\alpha^\mu\partial_\mu v,\quad \Box v=-\alpha^\mu\partial_\mu u,\quad w=u+iv:\quad \Box w+i\alpha^\mu\partial_\mu w=0.$$

For constant alpha, w = exp(−i alpha_mu x^mu/2) z reduces this exactly to

$$\Box z+\frac{\alpha_\mu\alpha^\mu}{4}z=0.$$

All Fourier frequencies are real if and only if alpha is timelike or null. For a spacelike alpha there is a growing finite-wavenumber band. The original principal symbol remains the ordinary wave symbol. This distinguishes a lower-order current instability from the earlier loss of principal hyperbolicity.

The Spin representation supplies an explicit counterflow control without changing the connection. It is Lambda-star W tensor (det W) to the power minus one half, so rho(i(D_a+D_b)) = i(N_a+N_b−1). Its vacuum and grade-ab state have opposite root weights. Equal-amplitude positive-frequency wave solutions in these two orthogonal components with opposite spatial momenta p_plus=(omega,k) and p_minus=(omega,−k) give

$$c^\mu=\frac{|C|^2}{4}(p_-^\mu-p_+^\mu) =(0,-|C|^2 k/2).$$

The full field has constant J and two free wave components, so it is an exact background of the candidate action. Its normal current is diagonal and constant; cross terms between the vacuum and grade two vanish because u(q) preserves grade. The root-ab perturbation is therefore a closed linearized subsystem. It is unstable for nonzero k. These ideal plane backgrounds have infinite total energy on an infinite sector; their exact dispersion is a background test, not an identified finite-energy particle decay or a finite-time concentration theorem.

This same construction exists at every nonempty occupied q=2,3,4 carrier. At q=2 the canonical traceless connection is absent, but its central Spin current still produces the root response. Neither current susceptibility nor this instability isolates depths three/four. Depth ten has no such normal-orientation channel in the declared ten-parent rule.

Dirac Coefficients and Remaining Questions

A first-order comparison instead uses

$$\mathcal L_\chi=\frac i2[\bar\chi\gamma^\mu D_\mu\chi-(D_\mu\bar\chi)\gamma^\mu\chi].$$

Its current convention is j_D^mu(H) = i bar chi gamma^mu rho(H) chi. The covariant variation gives the same constrained curvature force with this current. In an adapted frame with A_A=0 at the point, the orientation and Dirac characteristic blocks retain the determinant and ordinary light cones. Normal even Clifford rotations commute with the complete tangent Dirac symbol on the full parent carrier. The bare classical Dirac Hamiltonian, however, is not nonnegative on unrestricted complex commuting coefficients; the positive wave-energy formula above must not be transferred to it.

The constructive results are an explicit constrained source equation, complete reciprocal stress and a coupled causal principal structure for the wave comparison. The counterflow calculation identifies a dynamical acceptance condition to test rather than an inserted sector coefficient. The same-field reduction, observed electric weights, compatible occupied projections, finite localized states, frequency selection, and suppression of the allowed depth-two larger response remain open.

Reproduction

The standalone checker and recorded report preserve 303 exact conditions across the six occupied depths. They check the complete root metric and curvature normalization, diagonal-current root decoupling, the shifted Spin weights, counterflow current norm, exact abelian and non-Abelian pullback variations, the complete Legendre transform, the coupled principal congruence, the current dispersion and phase reduction, and direct current conservation and both stress-exchange components. Run with a Python interpreter containing SymPy; each execution creates a dated report alongside the source. These checks support the displayed identities and do not turn an infinite plane background into a finite-energy particle.