Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: A conserved electromagnetic source requires a specified current projection, not identification with the detector-averaged orbit image.

Scope

This starts the electromagnetic test of the actual multidimensional orbits in the demo, including the circle and two-, four-, and six-petal cases. It does not replace them with stationary orbitals. The coefficient ray [c], the sector point [Z], and the Dirac field on physical space remain distinct until a physical bridge is specified.

Bound within, free without, section 8, already proposes a fibre-integrated electron current coupled to the photon. Higher-dimensional polarization, sections 3 and 6, adds the invariant representation filter. These specify a route to a vertex, not the numerical source produced by a particular c(t). The depth-two photon sector must also be related to the electromagnetic field seen by an ordinary spatial observer; its spatial depth is not a license to replace ordinary Maxwell propagation by a two-space-dimensional wave equation.

Conserved pushforward

Let a density n and tangent current J on a fixed sector manifold Q obey ∂t n + div_Q J = 0. Assume no boundary flux, or sufficient decay, and a smooth physical charge readout X_t: Q → R³. This is a conditional source construction, not an identification of X_t with the demo's apparatus map. For charge e* (negative for an electron), define distributions

ρ(t,x)=e∗∫Qn(t,u)δ(3)(x−Xt(u)) dμ(u), \rho(t,x)=e_*\int_Q n(t,u)\delta^{(3)}(x-X_t(u))\,d\mu(u),

j(t,x)=e∗∫Q[dXt(J)(t,u)+n(t,u)∂tXt(u)]δ(3)(x−Xt(u)) dμ(u). j(t,x)=e_*\int_Q \left[dX_t(J)(t,u)+n(t,u)\partial_tX_t(u)\right] \delta^{(3)}(x-X_t(u))\,d\mu(u).

For every compactly supported smooth test function f, integration by parts gives

ddt∫ρf d3x=e∗∫Q[J⋅d(f∘Xt)+n∇f(Xt)⋅∂tXt]dμ=∫j⋅∇f d3x. \frac{d}{dt}\int\rho f\,d^3x =e_*\int_Q\left[J\cdot d(f\circ X_t) +n\nabla f(X_t)\cdot\partial_tX_t\right]d\mu =\int j\cdot\nabla f\,d^3x.

Thus ∂tρ + ∇·j = 0, with total charge e*∫n. The term involving ∂_tX_t is required for a moving readout/frame. This proof allows many-to-one maps; it needs neither an inverse map nor a choice of one point on each fibre. Gauge invariance of ∫(j·A − ρφ) dt d³x follows, up to boundary terms, from this continuity equation. Gauge invariance alone does not select X_t.

For a point trajectory U(t) with n = δ_U and J = U̇δ_U, these equations reduce to ρ = e_*δ(x−Γ(t)), j = e_*Γ̇δ(x−Γ(t)), where Γ = X_t(U(t)). This is precisely the extra assumption needed before a displayed curve can be treated as a moving electromagnetic point charge.

An invariant polarization filter acting on a spinor current needs its own covariant conservation check. Arbitrarily deleting current components does not inherit this proof. A concrete projected action must provide the conserved source, its normalization, and its observer-space interpretation.

Why the exposure image is insufficient

The demo's registration density is a full-period average of G_σ(x−Γ(t)). Its being stationary does not make the instantaneous source stationary. For a fixed normalized centered spatial kernel, the conditional source ρ = e_*G_σ(x−Γ), j = e_*Γ̇G_σ(x−Γ) conserves charge, but still has electric dipole d(t) = e_*Γ(t). Spatial broadening alone does not smooth that dipole's time dependence. A physical frequency-dependent form factor can affect high-frequency radiation; that is additional source physics, not a consequence of plotting a blurred exposure.

A complete-period average of a periodic conserved source has divergence-free mean current. It discards nonzero time harmonics. Computing radiated power from that mean is not computing the mean power of the original source.

Calculation to close

Derive the charged projected action and its current; evaluate it on the actual bound orbit; push it to the physical electromagnetic observer with its polarization filter; then compute its multipoles and response to a probe. The source must include the nucleus and any compensating field contribution, not only the electron, when testing a complete bound system.

The follow-up spectrum test gives explicit constraints for several orbit shapes and separates a literal point-source assignment from a fixed bilinear current assignment. Neither assignment is selected by the current V2 text.