Spatial current response of a two-frequency field

Summary: The spatial number current retains polarization-dependent frequency sidebands that can cancel in the scalar binding source, with an exact continuity identity and the full source feedback preserved.

The source and the measured current are different bilinears

The periodic source equation follows the complete spatial Dirac perturbation through its scalar binding moments. The same perturbation also changes the spatial number current. A cancellation in one readout does not automatically cancel the other.

All spatial coordinates belong to the one nested manifold of ordinary macroscopic dimensions. Let $x$ span the $d$ bound directions and $y$ be the existing containing direction used for the normal partner. For

$$i\partial_t\Psi= [-i\alpha_i\partial_i-i\alpha_y\partial_y+\beta\mathfrak m]\Psi, \qquad \mathfrak m\in\mathbb R, \tag{1}$$

direct differentiation gives

$$\boxed{\partial_t\rho+\partial_i j_i+\partial_y j_y=0, \qquad \rho=\Psi^\dagger\Psi,\quad j_\mu=\Psi^\dagger\alpha_\mu\Psi.} \tag{2}$$

The scalar potential cancels pointwise because it is Hermitian and multiplies the field at that point. Its coefficient may depend on the field's spatial moments without changing this identity. These are the number density and Noether current of the specified field action. Identifying their physical photon projection requires the additional current map discussed in source frequencies.

Derive each current harmonic from the full field

Use the two-frequency background

$$\Psi_0=\psi_+e^{-iMt}+\psi_-e^{iMt},\qquad \psi_+=a\phi_1,\quad\psi_-=b\alpha_y\phi_2, \quad H_b\phi_j=M\phi_j. \tag{3}$$

The profiles have the same radial scalar source and may have different rest polarizations. For scalar input operators $O_A$ and Hermitian output operators $J_\ell$, form columns

$$C_\sigma=(O_A\psi_\sigma)_A,\qquad D_\sigma=(J_\ell\psi_\sigma)_\ell. \tag{4}$$

An integrated number measurement uses $J=I$, and an integrated normal current uses $J=\alpha_y$. A resolved bound-current measurement uses the position-evaluation quadratic form with $\alpha_i$; its finite Galerkin representation is a Hermitian matrix.

Solving the ordinary and conjugate field equations gives the bare output at frequency $\omega+\Delta M$, where $\Delta=0,\pm2$:

$$\boxed{\begin{aligned} \Gamma_\Delta(\omega,q) ={}&\sum_{\sigma-\rho=\Delta} D_\rho^\dagger(\omega+\sigma M-H_q)^{-1}C_\sigma\\ &-\sum_{\rho-\sigma=\Delta} D_\rho^T(\omega-\sigma M+H_{-q}^*)^{-1}C_\sigma^*, \qquad H_q=H_b+q\alpha_y. \end{aligned}} \tag{5}$$

Thus the current readout changes the left columns of the scalar response, while the spatial propagation and input source remain the same. The star means entrywise conjugation, and the conjugate block has opposite normal momentum. A real perturbation also includes its conjugate Fourier component.

After integration over the decaying bound profile, (2) requires

$$\boxed{(\omega+\Delta M)\Gamma_{N,\Delta} =q\Gamma_{y,\Delta}.} \tag{6}$$

This identity holds for every source column and output harmonic. At rest the total number has no response at a nonzero output frequency. The normal current is not constrained to vanish by that statement.

Rest current sidebands measure both occupied polarizations

Define the transition scalar moment

$$\mu_{21,A}=\langle\phi_2,O_A\phi_1\rangle. \tag{7}$$

For the radial background and the centered scalar second moments and dipoles, the integrated normal current has

$$\boxed{\Gamma_{y,+2}=\frac{2a\bar b}{\omega}\mu_{21},\qquad \Gamma_{y,-2}=-\frac{2\bar a b}{\omega}\mu_{21}^*,\qquad \Gamma_{y,0}=0\quad(q=0).} \tag{8}$$

For example, the ordinary positive-sideband term in (5) is $a\bar b\langle\phi_2,(\omega+M-H_b)^{-1}O_A\phi_1\rangle =a\bar b\mu_{21,A}/\omega$. Normal anticommutation and $\alpha_yO_A\alpha_y=-O_A$ give the same contribution from the conjugate term. Their sum yields the first expression in (8). The negative sideband retains the conjugate transition moment and opposite sign.

The $1/\omega$ factor describes the phase change of an existing $2M$ clock under a sustained scalar forcing. It is a linear phase response, not a divergent finite-amplitude current. The readjusted radial clock distinguishes the frozen matrix element from a stationary frequency derivative after the profile changes.

For these radial scalar moments, $\mu_{21}$ is proportional to $\langle w_2,w_1\rangle$. Both rest polarizations therefore enter (8). Orthogonal polarizations cancel the integrated oscillating current. That cancellation does not make all local current matrix elements zero.

Complete Cartesian field calculations at depths three, five and six give the following comparison for orthogonal occupied polarizations:

The local statements concern the declared spatial preparations, source columns and finite regulators. The six-dimensional sideband cancellation uses its selected chirality; it is not a general theorem about every six-dimensional field. The five-dimensional field is treated with the same operators and coefficient as the other depths. No neutrino zero is inserted.

Preserve source feedback before reading the current

Let $h_j$ and $f_j$ be scalar moment and force amplitudes at $\omega+2jM$, and let $\mathbb P,\mathbb K$ be the complete moment response and curvature from the periodic source equation. Define $\mathbb\Gamma_{jk}=\Gamma_{2(j-k)}(\omega+2kM,q)$ for $|j-k|\le1$, with the other bare blocks zero. Then

$$\boxed{h=\mathbb P(f+\mathbb K h),\qquad j=\mathbb\Gamma(f+\mathbb K h) =\mathbb\Gamma(I-\mathbb K\mathbb P)^{-1}f.} \tag{9}$$

The output retains every induced source harmonic. A matched full-field frequency cutoff gives an independent implementation: solve the complete ordinary/conjugate field matrix, then apply the current readout directly. Interior harmonics satisfy (6) after feedback. Cutoff boundaries must use the same retained field harmonics in both calculations.

For two interacting objects, their common action supplies the cross curvature and both responses. Equation (9) describes how to read an output from that solved field; it does not replace the partner's response with a fixed detector or introduce an additional interaction particle.

Verification

periodic_current_checks.py uses the actual centered Cartesian Dirac fields, independent occupied polarizations and one common interaction coefficient. It checks the full normal continuity relation, the analytic rest-current residues and the distinction between integrated and resolved spatial outputs.

A complete ordinary/conjugate harmonic solve independently verifies the current after source feedback. Direct time-dependent spatial bilinears and their Fourier transform reproduce the output sidebands. Reports have timestamped filenames and verification remains active under Python optimization. These results identify geometric selection within the stated model; the physical photon map and normalized interaction rates remain open.