Descending photon source

Summary: A higher-sector field sources a photon mode by projection over its bound coordinates while retaining normal propagation, with reciprocal feedback fixed by the same action.

Start with the higher-dimensional field pattern

There is one infinite physical manifold and one nested chain of ordinary macroscopic spatial submanifolds. A higher-sector particle may determine an interaction through its complete spatial field pattern. A photon response is then a particular projection of that interaction. This construction does not assume that the lower particle's bound depth sets the strength of the source.

All retained amplitudes below belong to the single field $\Psi_\infty$. There are no mediator particles. A separation into target and photon amplitudes is a mode decomposition to be derived and normalized, not a postulate of independent physical fields.

The photon geometry separates two bound spatial coordinates from the perpendicular phase directions. In a fixed product chart of a containing depth $b\geq3$, write

$$x=(u,y),\qquad u\in E\simeq\mathbb R^2, \qquad y\in N\simeq\mathbb R^{b-2}. \tag{1}$$

This is a relation to the photon plane. It does not divide the higher particle's dimensions into different physical kinds. A product chart, fixed frame, and product measure are assumptions for the formulas below; curved or moving frames require their Jacobians and connection terms.

Project the bound mode and retain the propagation coordinates

Let $\chi_0(u)$ be a normalized bound profile, $\int_E|\chi_0|^2du=1$. Let $S_\gamma:\mathbb C^2\to\mathcal F_x$ be an isometric map from the two photon amplitudes to the relevant field components in this chart. The bundle $\mathcal F$ is the field's amplitude bundle over physical space. Deriving $S_\gamma$ from the master field is part of the sector construction; its two columns are not extra spatial coordinates. For a vector-wave diagnostic they can be the two spatial polarization vectors in $E$.

Define an isometric inclusion and its Hilbert-space adjoint by

$$(I_\gamma a)(u,y)=\chi_0(u)S_\gamma a(y), \qquad (P_\gamma\Psi)(y)=\int_E\overline{\chi_0(u)} S_\gamma^\dagger\Psi(u,y)\,du. \tag{2}$$

For square-integrable packets, $P_\gamma I_\gamma=I$ and $\|I_\gamma a\|_{L^2(E\times N)}=\|a\|_{L^2(N)}$. Set $Q=I-I_\gamma P_\gamma$. Every field in the domain decomposes as

$$\Psi=I_\gamma a+h,\qquad P_\gamma h=0. \tag{3}$$

The complement $h$ is not automatically one particle sector. A proposed higher-particle family must be embedded into it, or a nonorthogonal family must use the full kinetic Gram normalization.

The photon amplitude $a(y,t)$ retains its propagation phase, for example $e^{i(k\cdot y-\omega t)}$. Integrating over $y$ would erase this dependence or select a particular Fourier component. It is not the projection in (2). The source marginal in nested sector reciprocity serves a different purpose and must not be substituted for a propagating-mode projection.

Exact projected field equation

Write the specified field equation as

$$L\Psi+\mathcal N[\Psi]=0,$$

where $L$ contains the selected linear spatial and temporal operators and $\mathcal N$ contains the remaining terms. All operator products below require a common valid domain. With fixed $I_\gamma$, define $D_\gamma=P_\gamma L I_\gamma$. Substitution of (3) gives the exact row

$$\boxed{D_\gamma a =-P_\gamma Lh-P_\gamma\mathcal N[I_\gamma a+h].} \tag{4}$$

If the linear operator preserves the photon mode and its orthogonal complement, $P_\gamma Lh=0$. That is an intertwining property to check; bound-depth assignment alone does not establish it. Even then, the nonlinear projected term need not close on $a$.

The initial photon-sector forcing of a prescribed higher field $h$ is

$$s_\gamma[h]=-P_\gamma\big(Lh+\mathcal N[h]\big). \tag{5}$$

Thus the source is first computed from the higher-dimensional pattern and its field equations. Photon polarization and the bound profile then select which parts enter the output. Its strength need not be a function of the number two.

A configuration with $s_\gamma[h]\ne0$ is not a complete stationary solution with $a=0$. Its lower-sector background response must be solved before a stability or small-scattering calculation is made. A nonzero static source also does not by itself imply radiation: outgoing waves depend on the source frequency and the propagating spectrum of $D_\gamma$.

One action determines the reverse response

Let $\mathcal E[\Psi]$ be a twice differentiable real energy on a real Hilbert space of field variations. Complex fields may be treated as their real and imaginary components. Around a valid background $\Psi_0$, let $H=D^2\mathcal E[\Psi_0]$ be the self-adjoint Hessian. For fixed isometric target and photon inclusions $I_b,I_\gamma$, the quadratic cross term is

$$\mathcal E_{\rm cross}(a,q)=\langle a,Bq\rangle, \qquad B=I_\gamma^*H I_b. \tag{6}$$

Variation gives the photon force $Bq$ and target force $B^*a$:

$$\boxed{B_{\gamma b}=I_\gamma^*H I_b, \qquad B_{b\gamma}=I_b^*H I_\gamma=B_{\gamma b}^*.} \tag{7}$$

This reciprocal pairing permits a source-led, downward calculation. It does not require equal responses to unequal preparations, occupations, or incident fluxes. A genuinely time-dependent background can exchange energy with perturbations and requires its time-dependent variational equations. Reciprocity of the real Hessian is not automatically a time-reversal symmetry of a scattering experiment.

When coordinates depend on the field, vary the inclusions, frame, and kinetic terms too. Holding an action-dependent projector fixed can omit part of the source. When a selected higher solution already contains photon components, expand around that complete solution rather than counting those components twice.

A direct finite projection check

For a normalized bound vector $\chi$ and discrete normal-coordinate vector $f$, $I_\gamma f=\chi\otimes f$ and $P_\gamma(\chi\otimes f)=f$. A nonconstant phase sequence in $f$ therefore survives exactly. A bound profile $\eta\perp\chi$ obeys $P_\gamma(\eta\otimes f)=0$, independently of that phase.

For a separated linear operator $L=L_E\otimes I+I\otimes L_N$ with $L_E\chi=\lambda_0\chi$, the projected operator is $D_\gamma=\lambda_0I+L_N$. An off-diagonal term connecting $\eta$ to $\chi$ supplies a nonzero $P_\gamma Lh$, as retained explicitly in (4). These finite identities check the projection; they do not prescribe the physical photon dispersion.

Verification

polarization_coupling_checks.py checks the displayed identities with independent spatial quadrature, energy variations, or full finite-system evolution as appropriate. These checks verify the declared mathematics and benchmarks, not empirical couplings.

Related

• Nested sector reciprocity — distinguishes full source marginalization from propagation-preserving mode selection.
• Photon plane coupling — gives the conditional two-form selection at a specified output wavevector.
• Canonical coupling normalization — handles nonorthogonal mode families and physical vertex normalization.
• Generalized sector coupling — organizes the source and response calculation across all sectors.

Dirac source linearization computes the source and its reciprocal response explicitly for the written scalar self-interaction.

Spatial source complementarity gives exact bounds on which vector and rotation descriptors survive a lower projection.

Bilateral polarization coupling retains both objects' polarization and response. A zero static source overlap does not by itself make the mixed response derivative vanish.