Reciprocal response with two spatial clocks

Summary: The full response of a readjusted nested pair retains two background frequencies, reciprocal shape coupling and both scalar centers, while spatial parity identifies which deformation-induced sidebands can survive.

Start from both nonlinear spatial fields

The two-frequency nested fields solve their common scalar source while their spatial currents carry independent clocks. The next variation must keep both full field perturbations. It cannot replace one object's geometry by a prescribed force and then infer the response of the whole pair.

Use the projected channel action and spatial measures of the nested Dirac pair. Its coordinates describe $Q_p\subset Q_D$ in the one nested manifold of ordinary macroscopic spatial dimensions. A common normal coordinate $y$ belongs to an existing containing sector. The two channel labels and the frequency indices below are descriptions of the same field; they introduce no additional physical space or interaction particle.

For $i=a,b$, write

$$\Psi_{i0}=\psi_{i+}e^{-iM_it}+\psi_{i-}e^{iM_it},\qquad \psi_{i+}=u_i\phi_{i1},\quad \psi_{i-}=v_iY_i\phi_{i2},\qquad H_i\phi_{ij}=M_i\phi_{ij}. \tag{1}$$

The stationary profiles and their deformations must solve the source of both occupied channels. Each channel retains its two rest polarizations.

The common energy fixes the complete moment curvature

Let $A_i$ be the self-consistent quadratic source matrix in channel $i$, and $Q_i=\int\Psi_{i0}^\dagger\beta_i\Psi_{i0}\ne0$. Define source operators for all symmetric second moments and scalar dipoles:

$$O_{iA}=\beta_i(x_i^TAx_i),\qquad O_{i\mu}=\beta_i x_i^\mu. \tag{2}$$

Use Frobenius-orthonormal symmetric tensor bases including breathing. If $I:\mathbb R^p\to\mathbb R^D$ is the spatial inclusion, the higher-to-lower restriction matrix is

$$R_{AB}=\operatorname{tr}(A I^TB I). \tag{3}$$

For the common energy $g_0\operatorname{tr}(IM_aI^T+M_b)^2/2$, its second-moment curvature is

$$\boxed{\kappa_{\rm shape}=g_0 \begin{pmatrix}I&R\\R^T&I\end{pmatrix}.} \tag{4}$$

Separately centering each scalar density also gives

$$\boxed{\kappa_{{\rm dip},i}=-\frac{2A_i}{Q_i}.} \tag{5}$$

For an anisotropic field this is a spatial matrix, not a single radial coefficient. At the centered background there is no mixed first-order shape–dipole block or cross-channel dipole block in this particular energy. Both statements follow by differentiating $M_i=M_{i,\rm raw}-d_id_i^T/Q_i$ at $d_i=0$. The scalar-center derivation explains why (5) must remain even when the background center does not move.

The full field equation keeps both clocks

Let $C_i(t)$ have columns $O_{iA}\Psi_{i0}(t)$, including the dipoles, and put

$$S_i(t)=\begin{pmatrix}C_i(t)\\-C_i(t)^*\end{pmatrix},\qquad T_i(t)=(C_i(t)^\dagger,C_i(t)^T),\qquad B_{i0}(q)=\operatorname{diag}(H_i+qY_i,-(H_i-qY_i)^*). \tag{6}$$

Assemble these as channel block diagonal matrices $S,T,B_0$. For $Z=(\eta_a,\eta_a^*,\eta_b,\eta_b^*)^T$, the complete response is

$$\boxed{i\partial_tZ=[B_0+S(t)\kappa T(t)]Z+S(t)f, \qquad h=T(t)Z.} \tag{7}$$

Here $\kappa$ contains (4)–(5), so source motion in either channel feeds back through the same energy. Independent complex variations of both nonlinear field equations reproduce (7), including both centers.

The cross-channel field blocks contain mixed carrier frequencies $\pm M_a\pm M_b$, in addition to the self-block frequencies $0,\pm2M_i$. The moment lattice below organizes these terms using each channel's own carrier; it does not omit the mixed field harmonics.

The canonical identity is $\Sigma B=(\Sigma B)^\dagger$, with one ordinary/conjugate signature pair for each channel. This retains the real action; it does not assert positive energy or stability.

A frequency lattice replaces a single assumed period

The single-field periodic response gives bare moment kernels $\Pi_{i,\Delta}(\omega,q)$ at shifts $\Delta=0,\pm2$. Put $\mathbf M=(M_a,M_b)$ and label common source frequencies by

$$\omega_{\mathbf n}=\omega+2\mathbf n\cdot\mathbf M, \qquad \mathbf n\in\mathbb Z^2. \tag{8}$$

Define a bare channel and frequency operator by

$$\boxed{\mathbb P_{i\mathbf n,j\mathbf m}= \begin{cases} \Pi_{i,2(n_i-m_i)}(\omega_{\mathbf m},q), &i=j,\ n_k=m_k\ (k\ne i),\ |n_i-m_i|\le1,\\ 0,&\text{otherwise}, \end{cases}} \tag{9}$$

and $\mathbb K_{i\mathbf n,j\mathbf m} =\delta_{\mathbf n\mathbf m}\kappa_{ij}$. The common source equation is

$$\boxed{h=\mathbb P(f+\mathbb K h),\qquad h=(I-\mathbb P\mathbb K)^{-1}\mathbb P f.} \tag{10}$$

Each bare propagation changes only its own clock index. Cross-channel feedback then permits a sequence involving both, producing frequencies such as $\omega+2M_a+2M_b$. If $M_a/M_b$ is irrational, no single background period exists. If it is rational, coincident physical frequencies must be combined consistently; two labels need not be independent physical harmonics in that case.

A matched finite check retains $-N\le n_a,n_b\le N$ for moments. For fields in channel $i$, retain $-N\le n_i<N$, with the other index unchanged. Their frequencies are $\omega_{\mathbf n}+M_i$. Matching the positive and negative background columns gives

$$\boxed{\mathbb P_N=T_FD_F^{-1}S_F,\qquad Z_F=(D_F-S_F\mathbb K_NT_F)^{-1}S_Ff.} \tag{11}$$

Equation (10) with this same cutoff gives exactly $h=T_FZ_F$. This is a finite identity; convergence of a physical pole or response as the spatial and frequency cutoffs increase remains an additional test.

Spatial parity specifies which rest sidebands can survive

For a centered even scalar potential, the combined spatial parity $\tau_i=\beta_i\mathcal P_x$ commutes with $H_i$ and anticommutes with $Y_i$. Choose positive profiles in the $\tau_i=+1$ sector. Then $\psi_{i\sigma}$ has parity $\sigma$. A scalar second moment has operator parity $\epsilon=+1$ and a scalar dipole has $\epsilon=-1$.

At $q=0$, each frozen resolvent preserves parity. A source column $O_A\psi_\sigma$ has parity $\epsilon_A\sigma$, so its overlap with a readout column can survive only when $\epsilon_A\sigma=\epsilon_B\rho$. Therefore

$$\boxed{\begin{array}{c|cc} &\text{same operator parity}&\text{opposite operator parity}\\\hline \Delta=0&\text{allowed}&0\\ \Delta=\pm2&0&\text{allowed} \end{array}} \tag{12}$$

“Allowed” means parity supplies no zero, not that the amplitude must be nonzero. The central radial examples have a stronger cancellation of the opposite-parity rest sidebands. Deformation can remove that stronger cancellation while preserving (12).

Full self-consistent Cartesian examples at common $g_0=6.5$ and balances $(s_a,s_b)=(0.3,0.55)$ give:

These are responses of the declared finite spatial preparations. The six-dimensional result retains its selected chirality, while the five-dimensional result demonstrates an open channel rather than an inserted neutrino suppression. A quadrupole-only calculation would omit the center response which carries the allowed odd-dimensional sideband.

At nonzero normal momentum, $qY_i$ exchanges combined parity. The equal-parity sidebands can then open too. The complete calculation in (7)–(11) retains this normal propagation and both clocks. Polarization selection of a periodic spatial source calculates the entire cancelling polarization subspace and the normal-momentum opening for a specified lower spatial pattern.

Read each current only after the pair responds

The spatial current readout applies to the solved pair field. For each channel, its integrated density and normal current satisfy at an interior retained harmonic

$$\boxed{\omega_{\mathbf n}\,\delta N_{i,\mathbf n} =q\,\delta J_{i,y,\mathbf n}.} \tag{13}$$

Each projected channel's number remains conserved because its scalar source is Hermitian. The cross response changes its geometry and current without violating that continuity equation. No identification with a physical photon source or an interaction rate is assumed by this readout.

Verification

paired_periodic_response_checks.py solves both Cartesian source profiles at one common coefficient and checks their nonlinear equations at independent carrier phases. Full complex finite variations verify the reciprocal curvature and both scalar-center blocks.

The parity selection is checked in complete spatial resolvents. Matched two-index field and moment equations independently reproduce the full induced response at two harmonic cutoffs, including an output involving both clocks. Current measurements check (13) after feedback. Reports use timestamped filenames and all conditions remain active under Python optimization. The regulated frequencies are not particle mass predictions.