Dynamical opening of spatial shape channels

Summary: The full Dirac response can rotate a statically inaccessible higher-dimensional shape into a lower-sector source, producing a derived low-frequency barrier with both objects' polarizations retained.

A shape can fail to match statically and still respond dynamically

For the spatial moment interaction, a source in a higher sector acts on a lower one through its restricted shape tensor. A mixed shape with one direction perpendicular to the lower sector has zero restriction. The full field can nevertheless move that pattern into another spatial shape which does have a nonzero restriction.

This note calculates that process using the self-consistent Dirac profiles and complete quadrupole response in Dirac shape response. All coordinates belong to the same nested manifold of literal macroscopic spatial dimensions. The tensors label actual spatial patterns; their component counts do not introduce other physical dimensions.

The result is a conditional spatial coupling mechanism. It does not assign the chosen quadrupole to a photon, or identify its suppression with a measured neutrino rate.

Two radial functions determine the entire polarized shape response

Let $L_M,L_N$ be the two radial Dirac fluctuation operators, with normalized radial source vectors $u_M,u_N$. Define

$$R_M(z)=u_M^T(z-L_M)^{-1}u_M,\qquad R_N(z)=u_N^T(z-L_N)^{-1}u_N. \tag{1}$$

The spectral parameter $z$ has the same time convention as the Dirac equation; a retarded boundary value requires the appropriate prescription at the real spectrum. The derivation uses values away from poles.

For a real orthonormal traceless shape basis $A_r$, the target's spatial circulation tensor $J$ acts by

$$\mathcal K_{rs}=\frac12\operatorname{tr}(A_r[J,A_s]). \tag{2}$$

Put $c_0=2/[d(d+2)]$ and $c_1=4/[d(d+2)^2]$. The angular Gram matrices derived from the full Clifford field give the exact bare matrix response

$$\boxed{\chi_0(z,J)=A(z)I+B(z)i\mathcal K,} \tag{3}$$

where

$$\begin{aligned} A(z)&=(c_0-c_1)[R_M(z)+R_M(-z)] +c_1[R_N(z)+R_N(-z)],\\ B(z)&=c_1[R_N(z)-R_M(z)-R_N(-z)+R_M(-z)]. \end{aligned} \tag{4}$$

$A$ is even in frequency and $B$ is odd. Thus the static response is isotropic in the shape basis, while the rotating part of the response retains the prepared spatial polarization $J$.

The source's own feedback supplies the complete response

$$\boxed{\chi(z,J)=\big[I-g_0\chi_0(z,J)\big]^{-1}\chi_0(z,J),} \tag{5}$$

with the same scalar coefficient $g_0$ as in the stationary profile. Equations (3)–(5) contain all the shape helicities and both ordinary and conjugate Dirac fluctuations. They do not replace the latter by a positive scalar oscillator.

Both target preparation and source shape are essential. If the source has complex shape coefficients $c$, its response is $\chi(z,J)c$. Reversing the prepared circulation gives the transpose identity

$$\chi(z,J)^T=\chi(z,-J),\qquad \chi(-\bar z,J)=\overline{\chi(z,J)}. \tag{6}$$

A fixed polarized preparation need not have a symmetric response matrix. These relations retain the underlying real energy and its reciprocal variation without assuming that reversing a process leaves its preparations unchanged.

Restriction can change the type of spatial shape

For an isometric inclusion $I:\mathbb R^p\to\mathbb R^d$, the moment source restricts as

$$\mathcal R(C)=I^TCI. \tag{7}$$

A traceless shape in $d$ dimensions need not remain traceless after restriction. Its lower trace is a breathing source and must be retained when calculating the complete interaction.

With Hilbert–Schmidt-normalized tensor bases, the map from higher traceless shapes to all lower symmetric shapes has singular values

$$\underbrace{1,\ldots,1}_{p(p+1)/2-1}, \qquad \sqrt{\frac{d-p}{d}}. \tag{8}$$

To derive the last value, apply the adjoint map to the normalized lower breathing shape $I_p/\sqrt p$ and remove its higher trace. The squared norm is $1-p/d$. Lower traceless shapes already embed as higher traceless shapes with the same norm, giving the unit values.

For five dimensions inside six, fourteen lower quadrupoles have unit restriction singular value and the lower breathing channel has value $1/\sqrt6$. Five independent higher mixed shapes have zero static restriction. This is a statement about this spatial tensor kernel, not about the entire physical coupling of a particle species.

The higher field's rotation opens the mixed shapes

Take a six-dimensional orthogonal circulation tensor $J$, with $J^2=-I$, and let the lower five-plane be $n^\perp$. Put $u=Jn$, so $u$ lies in that plane and has unit norm. Relative to this particular inclusion,

$$J=\begin{pmatrix}J_5&u\\-u^T&0\end{pmatrix}. \tag{9}$$

The block notation describes the relation between the two objects. It does not make any of the six spatial directions inherently different.

Every traceless symmetric $C$ with zero full restriction has the form

$$C=\begin{pmatrix}0&v\\v^T&0\end{pmatrix},\qquad v\in\mathbb R^5. \tag{10}$$

Its spatial rotation satisfies

$$\boxed{\mathcal R\left(\frac12[J,C]\right) =\frac12(uv^T+vu^T).} \tag{11}$$

For unit $u$, the right side is zero only when $v=0$. For example, multiplying by $u$ gives $v+(u\cdot v)u=0$, whose component along $u$ then gives $u\cdot v=0$ and hence $v=0$.

Thus none of the five nonzero statically hidden mixed quadrupoles remains hidden after this first spatial rotation. In matrix form, $\ker\mathcal R\cap\ker(\mathcal R\mathcal K)=\{0\}$. This retains the mixed part of the higher geometry instead of discarding it during restriction.

There are still input combinations with zero output at an individual frequency, because the output has fewer components. The result concerns a fixed spatial pattern being invisible throughout the response, which is a stronger condition than a zero at one selected frequency.

A derived low-frequency matching barrier

Away from a static collective pole, equations (3)–(5) give

$$\chi(z,J)=\frac{A(0)}{1-g_0A(0)}I +\frac{B'(0)}{[1-g_0A(0)]^2}z\,i\mathcal K+O(z^2). \tag{12}$$

The coefficient is fixed by the actual radial modes:

$$B'(0)=2c_1\left( \|L_M^{-1}u_M\|^2-\|L_N^{-1}u_N\|^2\right). \tag{13}$$

For a source $c$ with $\mathcal Rc=0$,

$$\boxed{\mathcal R\chi(z,J)c =\frac{iB'(0)}{[1-g_0A(0)]^2}z\, \mathcal R\mathcal Kc+O(z^2).} \tag{14}$$

The original static mismatch is exact, but a nonzero-frequency field motion opens the channel. The resulting mode amplitude is linear in small frequency when the coefficient in (13) is nonzero. Its squared amplitude is then quadratic; a physical transition rate also requires the relevant mode map, state density and incident flux.

This is an explicit version of a spatial matching barrier produced by the field's own dynamics. It does not require assigning a special small coupling coefficient to the five-dimensional particle.

Include the lower object's response as well

Let $\chi_a,\chi_b$ be the two objects' own complete shape responses. For a weak cross term with coefficient $\epsilon$ multiplying their moment pairing, the leading response of the lower shape to a higher applied source $h_b$ is

$$\boxed{\delta M_a(z) =\epsilon\,\chi_a(z,J_a)\mathcal R\chi_b(z,J_b)h_b(z) +O(\epsilon^2).} \tag{15}$$

Both polarizations and both spatial spectra enter. The reverse response uses $\mathcal R^T$, obtained from the same real pair energy. Common physical rotations transform both circulation tensors, both shape bases and the inclusion together, leaving the physical prediction unchanged.

The coefficient $\epsilon$ here controls a perturbative contact expansion; it is not an independently fitted constant for each particle pair. At leading order one may use the isolated self-consistent profiles. Their static deformation changes (15) at the next order. A finite interaction requires the corresponding complete pair background, as illustrated by Self-consistent nested Dirac pair.

For a specified quadratic fluctuation energy, both responses can be solved exactly from

$$\begin{aligned} \delta M_a&=\chi_a(h_a+\epsilon\mathcal R\delta M_b),\\ \delta M_b&=\chi_b(h_b+\epsilon\mathcal R^T\delta M_a). \end{aligned} \tag{16}$$

This gives reciprocal feedback and collective poles. Applying that finite quadratic model to a nonlinear physical pair requires its Hessian at the correct background; it is not supplied just by combining two isolated profiles at an arbitrary large contact coefficient.

A normalized example

Choose $J_6$ with unit rotations in planes $12,34,56$, and the lower five-plane spanned by axes $1$ through $5$. Apply the unit complex shape

$$C=\frac12(e_3e_6^T+e_6e_3^T) +\frac i2(e_4e_6^T+e_6e_4^T). \tag{17}$$

Its static restriction vanishes. Its entire response lies in a spatial subspace whose lower projection is traceless, so the lower breathing channel is not needed for this particular example. Take $m=0.5$ and one common interaction coefficient $g_0=6.5$. Self-consistency determines the isolated source strengths $a_5=1.031769686$ and $a_6=0.998746098$; they are not separately fitted interaction coefficients. Retain both profiles' complete source feedback.

For the lower circulation with unit rotations in $12$ and $34$, the norm of the transfer coefficient $\chi_5\mathcal R\chi_6C$ is:

The amplitude divided by frequency tends to approximately $9.40742\times10^{-5}$ in these model units, as calculated directly from the zero-frequency radial derivative. Reversing the lower target's $34$ circulation while keeping the higher preparation fixed changes the finite-frequency response. The static zero is therefore compatible with both genuine dynamical access and a dependence on both polarizations.

The incoming quadrupole in (17) is a specified spatial source, not a derived photon inclusion. This example supplies a mechanism to test against that physical inclusion; it does not establish universal five-dimensional suppression.

Persistent spatial shape barriers extends this calculation across the hierarchy. It classifies shapes with a permanent zero and those whose first opening is quadratic in frequency, retaining the lower breathing response when needed.

Verification

spatial_shape_transfer_checks.py compares (3)–(5) with the full complex ordinary/conjugate source matrix, including every shape direction. It checks restriction singular values, the rank statement following (11), low-frequency scaling, both target preparations, covariance of the whole nested geometry, and reverse transfer. It also checks the controlled weak-contact expansion against exact finite quadratic feedback.

As a separate completeness test, the equal norms of $u_M,u_N$ cancel the would-be $1/z$ rotating response at large frequency. The response begins with $A=O(z^{-2})$, $B=O(z^{-3})$. Omitting an angular family can spoil that cancellation as well as the low-frequency result.

Related

• Polarized Dirac shape response — derives the angular families and their complete feedback.
• Transition-density coupling — derives the moment pairing which supplies the spatial restriction.
• Reciprocal scattering of spatial modes — distinguishes a geometric source zero from a zero caused by spectral interference.