Incoming spatial shape modes

Summary: A specified incoming spatial moment and an applied force have different response maps; individual shape helicities obey exact hyperplane restriction bounds which clarify the frequency-dependent cancellation mechanism.

Distinguish an incoming pattern from a force that creates it

Dynamical shape opening follows a fixed force applied to a higher spatial field. The higher response converts that force into a changing moment, which then drives the lower partner. A freely propagating particle must also satisfy its own field equation, so its actual moment cannot automatically be identified with that forcing vector.

The two objects occupy nested submanifolds of the same ordinary macroscopic spatial manifold. Both polarizations, the actual spatial inclusion and the response of each object remain in the following equations. Moment and helicity labels describe spatial patterns; they are not additional spatial dimensions or physical particle assignments.

Retain both responses without processing the incident mode twice

Let $\mathcal R(C)=I^TCI$ restrict a higher spatial moment into the lower sector. Its adjoint uses the full Frobenius moment norm, including lower breathing. Let $\chi_a,\chi_b$ be the corresponding complete response operators at a specified frequency and normal momentum, and let $\epsilon$ denote the common contact coefficient.

For a prescribed unperturbed incoming moment $b_{\rm in}$, the coupled linear response equations are

$$a=\epsilon\chi_a\mathcal R b,\qquad b=b_{\rm in}+\epsilon\chi_b\mathcal R^\dagger a. \tag{1}$$

They give, wherever the indicated inverse exists,

$$\boxed{a= [I-\epsilon^2\chi_a\mathcal R\chi_b\mathcal R^\dagger]^{-1} \epsilon\chi_a\mathcal R b_{\rm in}.} \tag{2}$$

Both objects readjust, and the denominator retains repeated feedback. If the incident moment is instead produced by an applied force $f_b$, then $b_{\rm in}=\chi_bf_b$. Substituting it into (2) recovers the previous leading response $\epsilon\chi_a\mathcal R\chi_b f_b$. An actual incoming moment is already the result of its own spatial dynamics; applying $\chi_b$ to it again would change the physical input.

Consequently $\mathcal R b_{\rm in}=0$ gives zero lower response in this specified channel, while $\mathcal R f_b=0$ need not. The latter force can produce $\mathcal R\chi_bf_b\ne0$.

For a scattering problem the incoming field and outgoing response use their specified propagation boundary conditions. At a discrete pole or resonance, use the full coupled operator or its appropriate limiting resolvent. A finite pair of free bound oscillators has hybridized mode frequencies; an inverse evaluated directly at an uncoupled pole does not replace that eigenproblem. At finite nonlinear contact, the background must also be the readjusted nested pair.

Individual rotating shapes preserve their geometric restriction

Take an even-dimensional higher sector with orthogonal circulation $J^T=-J$, $J^2=-I$. On complex symmetric traceless spatial tensors define

$$\mathcal K(C)=\frac12[J,C],\qquad T=i\mathcal K. \tag{3}$$

The eigenvalues of $T$ are $0,\pm1$. A shape helicity is an eigenvector $Tc=\lambda c$; it labels rotation of the actual spatial pattern relative to $J$. For the full radial Dirac response,

$$\chi(z)c=\chi_\lambda(z)c, \qquad \chi_\lambda(z)= \frac{A(z)+\lambda B(z)}{1-g_0[A(z)+\lambda B(z)]}. \tag{4}$$

Therefore

$$\boxed{\mathcal R\chi(z)c=\chi_\lambda(z)\mathcal Rc.} \tag{5}$$

Changing frequency changes the amplitude of an individual helicity, while its relative spatial restriction stays the same. Degenerate modes can be combined within a helicity; (5) holds for that whole subspace. Combinations of different helicities can instead acquire different relative responses.

A hyperplane retains a fixed fraction of every pure helicity

Let the containing sector have $d=2m$, with $m\ge2$, and let the lower sector be any hyperplane of dimension $d-1$. Choose coordinates in which $J$ rotates the planes $12,34,\ldots$ and the omitted normal is axis $d$. This is a common change of coordinates for the geometry and inclusion, not a preferred division of the higher particle's spatial dimensions.

The singular values of the restriction map on each normalized helicity subspace are:

To derive the nonzero-helicity rows, take the orthonormal complex spatial vectors $v_j=(e_{2j-1}+ie_{2j})/\sqrt2$. Their symmetric squares and products span one helicity, with the conjugates spanning the other. Restriction preserves $v_j$ for $j<m$ and gives $I^Tv_m=e_{d-1}/\sqrt2$ at the vector level. A square of $v_m$ therefore loses a factor $1/2$ in tensor norm; a normalized mixed product loses $1/\sqrt2$; the remaining products retain norm one.

For helicity zero, represent the commuting real tensors by Hermitian $m\times m$ matrices. Write their final row and column as a vector $v$ and scalar $a$, with upper block $H_0$ and $\operatorname{tr}H_0=-a$. The full and restricted squared spatial norms are respectively

$$2\|H_0\|^2+4\|v\|^2+2|a|^2, \qquad 2\|H_0\|^2+2\|v\|^2+|a|^2. \tag{6}$$

Split $H_0$ into its traceless part and $-aI/(m-1)$. These orthogonal parts give the final three rows. Complexification preserves the same singular values for complex forcing amplitudes.

In particular, for six dimensions restricted to five,

$$\boxed{\|\mathcal Rc\|\ge\frac12\|c\| \quad\text{on each nonzero helicity subspace},} \tag{7}$$

and the zero-helicity lower bound is $1/\sqrt2$. Every individual helicity therefore retains a nonzero full five-dimensional moment. This is a geometric norm bound, not a lower bound on a physical neutrino interaction probability. The mode's radial source magnitude, the target's allowed preparations, its output map and flux normalization remain essential. In particular, the output here is all symmetric five-dimensional moments, not the physical photon channel.

The full Dirac modes realize these helicities

The spatial Dirac shape calculation gives radial source vectors $u_\lambda,u_{-\lambda}$ and the complete ordinary/conjugate fluctuation matrix

$$\mathcal B_\lambda= \begin{pmatrix} L+g_0u_\lambda u_\lambda^T&g_0u_\lambda u_{-\lambda}^T\\ -g_0u_{-\lambda}u_\lambda^T&-L-g_0u_{-\lambda}u_{-\lambda}^T \end{pmatrix}. \tag{8}$$

An eigenvector $(p,q)$ has moment amplitude $h=u_\lambda^Tp+u_{-\lambda}^Tq$. When $h\ne0$, its moment pattern has helicity $\lambda$. Independently, its frequency obeys

$$1-g_0\chi_{0,\lambda}(\omega)=0, \quad \chi_{0,\lambda}(z) =u_\lambda^T(z-L)^{-1}u_\lambda +u_{-\lambda}^T(-z-L)^{-1}u_{-\lambda}. \tag{9}$$

The diagnostic solves (8) and checks (9), retaining both signs of the radial Dirac spectrum. With one common $g_0=6.5$ and bare scalar coefficient $m_0=0.5$, the first active six-dimensional shape frequencies are approximately:

The bound background frequency is $3.091416$ in these model units. Radial refinement and the independent resolvent condition agree. These are shape excitations of the declared scalar model, not observed particle masses or a complete stability calculation. The analogous five-dimensional calculation uses the same $g_0$ and retains its additional half-helicities.

The slow applied frequencies used in the earlier opening example are below these active shape frequencies. They should not be assigned to a free particle merely because the applied response is well defined there.

Frequency opening describes cancellation between visible components

Decompose a specified force as $f=\sum_\lambda f_\lambda$. Then

$$\mathcal R\chi(z)f =\sum_\lambda\chi_\lambda(z)\mathcal Rf_\lambda. \tag{10}$$

At $z=0$, the odd function $B$ vanishes, so all $\chi_\lambda(0)$ are equal. A force with $\sum_\lambda\mathcal Rf_\lambda=0$ is therefore statically hidden even though its individual components remain visible. As frequency changes, their different responses can spoil that cancellation. This is the mechanism behind linear or quadratic opening in Persistent spatial shape barriers.

An actual occupied particle could realize a coherent combination or a frequency-dependent spatial mode, but its field equation must establish that preparation. A low bound frequency alone does not choose the cancelling combination. Source frequencies also explains why the frequency driving that combination must be derived from its actual spatial source.

Verification

incoming_shape_mode_checks.py compares singular values with independent coordinate tensors, including explicit tensors attaining the lower bounds at depths four, six and ten. Common rotations transform the circulation and inclusion together.

Full ordinary/conjugate Dirac eigenvectors and independent radial resolvents verify the active modes at two radial resolutions and a common interaction coefficient. Independent coupled block equations check (2) with both objects responding, and explicitly distinguish an incident moment from an applied force. All checks remain active under Python optimization.

Related

• Dynamical opening of spatial shape channels — supplies the forced response which (2) specializes.
• Persistent spatial shape barriers — supplies the complete fixed-force frequency classification.
• Polarized Dirac shape response — supplies the actual spatial fluctuation operators and their polarization weights.
• Source frequencies in spatial coupling — supplies the temporal interpretation needed for a particle application.