Spatial currents and orientation-dependent coupling

The exterior-form parent gives explicit spatial eigenfields. A local quartic comparison can therefore test their phase and orientation without replacing the fields by an isotropic density. In this comparison a real standing pattern has zero self-current in the specified channel, while two such patterns can have a nonzero joint current. Orthogonal supporting planes produce an inverse-binomial overlap factor. The complete nonlinear source also leaves the original pair of modes, which prevents treating their projected interaction as a closed encounter equation.

The surface, parent and quartic action here are declared inverse choices. They have not been derived as the physical IDWT interaction. No electric or weak charge is assigned to these mathematical currents.

The complete spatial current

Use the unit sphere S^m, with m=d-2, in the same ordinary spatial sector as in Exterior-form realization of the curvature chain. Radius factors can be restored by normalizing with the physical surface measure. Put

$$k=\frac{m-1}{2},\quad \lambda_\ell=\ell(\ell+m-1),\quad \beta_\ell=\ell+k,\quad \chi_\ell=c_{0,\ell}Y_\ell+ \frac{c_{1,\ell}}{\sqrt{\lambda_\ell}}d_SY_\ell, \tag{1}$$

where Y is a unit scalar harmonic and c_0^2=(beta+k)/(2 beta), c_1^2=(beta-k)/(2 beta). This is the complete positive scalar/exact-one-form eigenfield of B. Retain every exterior component in the operator, with Hermitian Clifford matrices gamma_a=i(epsilon_a-iota_a). Direct exterior algebra gives

$$\boxed{\chi_\ell^\dagger\gamma_a\chi_\ell =\frac{1}{\beta_\ell}\operatorname{Im} (Y_\ell^*\nabla_aY_\ell).} \tag{2}$$

For a real harmonic all components in (2) vanish. For a complex circulating harmonic Y=C(x_1+i x_2)^ell, let V generate rotation in its supporting plane. Then VY=i ell Y and

$$J(V)=\frac{\ell}{\beta_\ell}|Y|^2. \tag{3}$$

The two patterns have the same linear frequency. Their currents differ because of their actual complex spatial polarization. The full normalized chain multiplies the endpoint current by its calculated endpoint weight, which is near one in the large-gap regime. A vanishing expectation in this channel is not electric neutrality or annihilation by a charge operator.

A specified reciprocal quartic action

Let P select both endpoint fields of the full canonical chain. The comparison adds

$$J_a=\sum_{e\in\mathrm{ends}}\Psi_e^\dagger\gamma_a\Psi_e, \qquad U=\frac g2\int\sum_aJ_a^2\,dA, \qquad \mathcal N_e=g\sum_aJ_a\gamma_a\Psi_e. \tag{4}$$

The last expression is the exact variation with respect to the complex conjugate field. Intermediate fields retain their parent equations. This is a local choice on the same field's proposed components. It does not introduce a new particle, and its algebraic source does not change the first-order propagation speed. Its physical origin remains unspecified.

Take two distinct real normalized eigenfields A and B of the same parent and write Psi=a A+b B. Their individual currents vanish. Define the real tangent vector

$$\kappa_a=\operatorname{Im} \sum_{e\in\mathrm{ends}}A_e^\dagger\gamma_aB_e, \quad y=\operatorname{Im}(a^*b),\quad I_{AB}=\int|\kappa|^2dA. \tag{5}$$

All interference terms in (4) then give

$$J=-2y\kappa,\qquad U=2g I_{AB}y^2, \qquad \langle A,\mathcal N\rangle=-2igI_{AB}yb, \quad \langle B,\mathcal N\rangle=2igI_{AB}ya. \tag{6}$$

Both responses follow from one energy. Equal phase makes the joint current zero at that instant; it does not keep it zero when the two unperturbed frequencies differ. Equation (6) is a projection of the complete source, not proof that its two-dimensional span is invariant.

The inverse-binomial spatial factor

Let the real harmonics have distinct positive degrees a and b, each formed from the real part of a highest-weight polynomial on an oriented two-plane. Let V_m be the sphere area and define

$$B_m(s)=\frac{s!}{((m+1)/2)_s},\qquad I_0=\int Y_a^2Y_b^2dA,\qquad C=\int Y_aY_b\nabla Y_a\cdot\nabla Y_b\,dA. \tag{7}$$

Here (z)_s is the rising factorial. The squared radius in one plane has the beta distribution with parameters 1 and k. For coincident planes, angular integration of the two distinct real harmonics gives

$$I_{0,\parallel}=\frac1{V_m}\frac{B_m(a+b)}{B_m(a)B_m(b)}, \qquad \frac{C_\parallel}{I_{0,\parallel}}=\frac{abk}{a+b}. \tag{8}$$

For mutually orthogonal planes, their squared radii have the joint Dirichlet distribution. Its mixed moment replaces (a+b)! by a!b!:

$$\boxed{\frac{I_{0,\perp}}{I_{0,\parallel}} =\binom{a+b}{a}^{-1},\qquad \frac{C_\perp}{I_{0,\perp}}=-ab.} \tag{9}$$

The second relation includes the tangential projection of both ambient gradients. Orthogonal planes require at least four ambient directions. This is a geometric preparation of both fields, not a rule selecting one depth as neutral.

The current kernel also includes the derivative and form coefficients. Let K_ab be the exact mixed endpoint matrix element from the full chain, and define

$$\alpha=K_{ab}\frac{c_{0,a}c_{1,b}}{\sqrt{\lambda_b}}, \qquad \beta=K_{ab}\frac{c_{1,a}c_{0,b}}{\sqrt{\lambda_a}}. \tag{10}$$

Then kappa=beta Y_b grad Y_a-alpha Y_a grad Y_b. Integration by parts gives integral(Y_a^2 |grad Y_b|^2)=lambda_b I_0-2C, and the analogous identity with a and b interchanged. Consequently the full result is

$$\boxed{I_{AB}=(\alpha^2\lambda_b+\beta^2\lambda_a)I_0 -2(\alpha^2+\beta^2+\alpha\beta)C.} \tag{11}$$

Thus the inverse binomial is an exact density factor; the current interaction has the additional orientation-dependent factor in (11). For example, with one common gap eta=10,000, the orthogonal-to-aligned current-kernel ratio is 3.14693e-7 for the depth-five degree-11/16 pair and 5.01699e-12 for the depth-six degree-14/36 pair. These are outputs of the stated spatial preparation. They do not imply a preferential depth-five suppression or an observed encounter rate.

The source outside the original pair

For a=sqrt(0.6) and b=i sqrt(0.4), g=1, eta=10,000, direct spatial integration gives the following full source norms. The complementary column subtracts both exact projections in (6).

These are initial forcing coefficients in the declared units. A complementary source norm alone does not measure the resulting field amplitude: its frequencies and reciprocal evolution must be retained. The two-form entries are especially direct evidence that a scalar plus exact-one-form pair is insufficient for the nonlinear evolution.

For fields invariant under rotations of the remaining m-1 ambient directions, the basic forms 1, dtheta, dphi, dtheta wedge dphi close under this local current source. The code checks that closure against all 2^m exterior components at each tested point. This spatial symmetry class is useful for a full response calculation; it is not a restriction on every allowed perturbation or a nonlinear stability theorem.

Verification

The corrected spatial_form_currents_checks.py report spatial-form-currents-20260913T235529_035046Z.json passes 134 checks. These include the complete exterior current and source, common tangent rotation, exact canonical harmonic norm, phase-dependent reciprocal sources, independently refined spatial integrals, and direct ambient gradients for orthogonal planes at the assigned high degrees.

The preceding report failed because its analytic expression used an extra cross contribution in (11). The direct spatial calculation exposed that algebraic error. Its exact code and report are retained as history; the corrected expression agrees with both independent integrations. The inverse-binomial density identity did not require correction.

Related research

• Spatial mass operator: results and remaining dynamics — connects this calculation to its premises, later corrections and the complete source question.