Polarization selection of a periodic spatial source

Summary: A positive matrix calculated from full spatial source operators identifies exactly which occupied polarizations cancel a selected frequency sideband, while retaining the incoming object's spatial pattern and common-rotation covariance.

Turn a proposed polarization barrier into a complete calculation

The reciprocal periodic field equation retains the source of both deformed objects. Its bare spatial response also permits an exact question: for a given occupied positive-frequency profile, which second polarizations cancel a specified source sideband?

The answer depends on the spatial source operators, the complete bound Dirac resolvents, normal propagation and the incoming pattern. It can be calculated without selecting a special zero for one particle depth. All fields live over ordinary macroscopic spatial directions in the one nested manifold. Polarization coefficients label field components over those directions; they are not additional physical dimensions.

Let $W$ be an isometric frame of the occupied positive-frequency spatial multiplet:

$$H_bW=MW,\qquad W^\dagger W=I_k,\qquad \phi_1=Wz_1,\quad\phi_2=Wz_2,\quad\|z_1\|=\|z_2\|=1. \tag{1}$$

The dagger in (1) includes integration over the full bound spatial field. The frame contains both upper and lower Dirac components, not merely their rest spinors. In the tested multiplets every unit $z$ preserves the same self-consistent binding moments. This property is checked before their polarizations are varied.

The background is $\Psi_0=aWz_1e^{-iMt}+bYWz_2e^{iMt}$, where $Y$ is its existing normal Clifford matrix. Keep the actual source operators $O_A$, including all scalar second moments and dipoles, and define

$$U_A=O_AW,\qquad V_A=O_AYW,\qquad H_q=H_b+qY. \tag{2}$$

Here $q$ labels the perturbation's Fourier dependence $e^{iqy}$ about the rest background, along the existing containing coordinate $y$.

Keep the full polarization matrix of every source channel

For the positive shifted output at $\omega+2M$, the full ordinary and conjugate response gives a $k\times k$ matrix for each input–output pair:

$$\boxed{F_{AB}(\omega,q)= V_A^\dagger(\omega+M-H_q)^{-1}U_B -V_B^\dagger(\omega+M+H_{-q})^{-1}U_A.} \tag{3}$$

The second term follows by transposing the conjugate propagation term. It retains $H_{-q}$; it must not be replaced by the first resolvent or assigned its sign by analogy. The actual scalar sideband is

$$\boxed{\Pi_{+2,AB}=a\bar b\,z_2^\dagger F_{AB}z_1.} \tag{4}$$

Equation (4) keeps both occupied polarizations rather than replacing their relation by a depth-dependent coefficient. Its cancellation concerns this particular frequency sideband. Same-frequency scalar responses and the spatial current outputs are separate readouts of the full perturbed field.

A positive Gram matrix describes the entire cancelling subspace

For fixed $z_1$, define

$$\boxed{G(z_1)=\sum_{A,B} (F_{AB}z_1)(F_{AB}z_1)^\dagger\succeq0.} \tag{5}$$

Then

$$\boxed{\sum_{A,B}|\Pi_{+2,AB}|^2 =|ab|^2z_2^\dagger G(z_1)z_2.} \tag{6}$$

For $ab\ne0$, all selected input–output sideband amplitudes vanish exactly when $z_2\in\ker G$. The smallest eigenvalue gives the minimum summed response over unit second polarizations. A full-rank $G$ rules out a cancelling second polarization at that tested frequency and for that complete channel set.

The equivalence is immediate from the sum of nonnegative squared amplitudes in (6). Strictly positive weights on the channel terms can change the numerical eigenvalues but cannot change the common kernel. Thus the nullspace criterion does not require adding dimensional quantities with an assumed physical metric. Comparing strengths or rates does require the appropriate source units, kinetic weights and flux normalization, as in canonical coupling normalization.

Retain the actual incoming object's shape

The all-channel condition in (5) is stronger than the condition for one incoming source. For a specified complex spatial force pattern $f_B$, define

$$F_A(f)=\sum_BF_{AB}f_B,\qquad \boxed{G_f(z_1)=\sum_A(F_A(f)z_1)(F_A(f)z_1)^\dagger.} \tag{7}$$

This matrix tests that particular pattern and its phase. It can have a larger kernel than (5), even when (5) has full rank. If the lower object's moment is a tensor $C$ on its own $p$ directions, the containing source uses its actual inclusion $ICI^T$ to construct $f$. The lower pattern must be derived or specified; it is not determined solely by $p$.

In a solved interaction, the appropriate force is the effective force after both objects' response, including self-feedback. A zero in (7) for the initially applied pattern does not prove that the readjusted force stays in that kernel. The full pair equation supplies that additional calculation.

Results from the same operator rule at several depths

The following comparison uses complete readjusted Cartesian profiles, one common $g_0=6.5$, balances $(0.3,0.55)$, and the complex test frequency $\omega/M=0.17+0.19i$. The matrix ranks agree at total Cartesian degrees one and two. These are regulated source-resolvent tests, not measured particle preparations or interaction probabilities.

For the deformed five-dimensional example, no second polarization cancels all these source sidebands at the tested rest frequency. This is not a derived neutrino suppression. For the selected six-dimensional chirality, the rest sideband cancels throughout that occupied frame, while normal propagation opens it.

The three-dimensional cancelling line agrees with the correlated polarization supplied by the bound-space antilinear symmetry. That symmetry does not extend to its retained normal derivative. The five-dimensional matrix conditions differ, so orthogonality of the two polarizations alone supplies no corresponding cancellation theorem.

For an incoming lower traceless tensor $C=\operatorname{diag}(1,-1,0,\ldots)/\sqrt2$, the source-conditioned $G_f$ in (7) instead vanishes at rest in all three sampled geometries. Their complete all-channel matrices are nevertheless different, as the table shows. At the sampled nonzero normal momentum, each $G_f$ is full rank. These results make both the spatial pattern and its propagation essential to the claimed barrier.

Verify a rest cancellation at every pole and derive its opening

A few sampled frequencies cannot prove a frequency-independent barrier. Let $P_E$ be each complete spectral projector of the finite bound operator, and put $U_f=\sum_Bf_BU_B$ and $V_{\bar f}=\sum_B\bar f_BV_B$. Expansion of (3) in its full spectrum shows that the selected source vanishes identically at rest precisely when

$$\boxed{V_A^\dagger P_EU_f =V_{\bar f}^\dagger P_{-E}U_A \quad\text{for every }A,E.} \tag{7a}$$

The diagnostic verifies these matrix residues for the embedded lower traceless tensor above, at both spatial resolutions and all three depths. This establishes a complete finite spectral identity, not merely a zero at the displayed test frequency. A continuum statement would require the corresponding spectral-measure identity and operator domains.

Normal anticommutation gives the resolvents through $\mathcal D=(\zeta^2-H_b^2-q^2)^{-1}$, with $\zeta=\omega+M$. Grouping the positive and negative bound eigenvalues in (7a) cancels the entire part even in $q$. The remaining selected output is exactly

$$\boxed{F_A(f;\omega,q) =q\left[V_A^\dagger Y\mathcal D U_f -V_{\bar f}^\dagger Y\mathcal D U_A\right].} \tag{7b}$$

It is odd in normal momentum, away from its poles. Its leading coefficient is obtained by setting $q=0$ only inside $\mathcal D$, so

$$\boxed{G_f(q)=q^2G_{f,\rm lead}+O(q^4).} \tag{7c}$$

The checked leading matrices are positive definite on each selected occupied frame. Thus every second polarization in those frames acquires this particular bare response when a small nonzero normal momentum is introduced, with an amplitude linear in $q$. Independent full resolvents verify both the sign reversal under $q\mapsto-q$ and (7c).

This is a derived momentum barrier for a stated spatial source. It does not equate the normal wavenumber with the particle's bound clock, and it does not remove the partner's feedback or the normalization needed for an interaction rate.

The result is independent of a coordinate or polarization-frame choice

Under an occupied-frame change $W\mapsto WV$, with $V$ unitary,

$$F_{AB}\mapsto V^\dagger F_{AB}V,\qquad z_j\mapsto V^\dagger z_j,\qquad G\mapsto V^\dagger GV. \tag{8}$$

A common physical rotation of the spatial field, incoming shape and inclusion acts on the source channels through an orthogonal matrix $D$: the symmetric tensors rotate as tensors and the dipoles as vectors. The complete spatial field rotation includes its orbital and spin parts. Its response satisfies

$$\boxed{F'_{AB}=\sum_{C,E}D_{AC}D_{BE}F_{CE}, \qquad f'=Df.} \tag{9}$$

Consequently the complete $G$ is unchanged in the correspondingly rotated occupied frame, and $G_{f'}'=G_f$. A common rotation cannot create or remove the polarization barrier. A relative rotation of the objects can change $f$ and their compatibility, which is the geometric question being tested.

Verification

periodic_polarization_checks.py constructs an isometric frame from the full solved spatial eigenspace and checks its binding moments for independent occupied polarizations. Independent ordinary/conjugate resolvents verify every entry of (4), while extremal Gram eigenvectors reproduce the full measured sideband norm.

Both Cartesian degrees, both normal momenta, strictly positive channel weights and the explicitly embedded lower tensor are checked. Every complete spectral residue verifies the rest cancellation; independent small-momentum resolvents verify its linear amplitude and quadratic Gram. Actual orbital-plus-spin field rotations verify (9), including rotation of the incoming pattern. Reports use timestamped filenames and checks remain active under Python optimization. The physical photon map, the occupied neutrino geometry and normalized rates still require their own derivation.