Orbit electromagnetic spectrum

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: Circle, petalled, and detuned actual orbits give distinct conditional radiation diagnostics and finite-generator constraints on electromagnetic probes.

Two explicit source hypotheses

The current projection audit identifies a missing physical map, not a reason to stop calculating. Here we test two conditional assignments, without claiming either follows from V2:

1. The displayed trajectory is the electromagnetic point-charge position.
2. The current is bilinear in a fixed linear field lift of the coefficient vector c, so its dipole is c†D_i c for fixed Hermitian matrices D_i.

These hypotheses generally disagree. Their predictions are diagnostics for the missing coupling, not two interchangeable descriptions of the same source. The actual orbit family is retained throughout; no stationary orbital model is substituted.

Literal point-source assignment: ideal petals

Use aligned planar readout, θ = νt, fixed nucleus at the origin, fixed R, and the same ν across cases. For L ≥ 1,

$$z(\theta)=\Gamma_x+i\Gamma_y=R|\cos L\theta|e^{i\theta}.$$

The circle is the separate ring sector z = Re^{iθ}. This comparison is not a claim that different physical atomic states have equal radii or clocks. For ordinary nonrelativistic electromagnetic point sources, or the long-wavelength electric-dipole approximation to an extended source,

$$\langle P\rangle=\frac{\langle|\ddot d|^2\rangle}{6\pi\epsilon_0c^3}, \qquad P_{\rm circle}=\frac{e_*^2R^2\nu^4}{6\pi\epsilon_0c^3}.$$

The external electromagnetic formula and its assumptions are derived in UT Austin's dipole-radiation notes, equation 15. All orbit substitutions and spectral results below are calculations here. This is a classical source test, not a quantum spontaneous-emission formula or a derivation of V2's electromagnetic field equations.

The exact Fourier expansion is

$$\frac zR=\frac2\pi e^{i\theta} +\sum_{n=1}^{\infty}a_n \left[e^{i(1+2nL)\theta}+e^{i(1-2nL)\theta}\right], \qquad a_n=\frac{2(-1)^{n+1}}{\pi(4n^2-1)}.$$

Both senses of rotation contribute to the planar acceleration norm. With the first N sideband pairs retained, Parseval gives

$$\frac{P_N}{P_{\rm circle}}=\frac4{\pi^2} +\sum_{n=1}^{N}a_n^2 \left[(1+2nL)^4+(1-2nL)^4\right].$$

Each summand tends to 8L⁴/π², not zero. The formal sum diverges. Equivalently, at each center crossing the one-sided velocity jump has magnitude 2RLν; the raw displayed curve has no square-integrable acceleration. Therefore it cannot be inserted unregularized into a finite-power point-charge calculation. This excludes that unregularized source prescription, not the smooth 6D coefficient orbit. The absolute-value observer readout creates the cusp. For an extended source the dipole approximation cannot be extended to arbitrarily high frequencies; the divergence is not a calculated infinite physical power for V2. The source's form factor and temporal regularity must be specified before taking that limit.

Detuned multi-maximum cases

For the exchange perturbation, put a = Ω/ν and m = (g/Ω)². Then

$$z/R=r(\theta)e^{i\theta},\qquad r=\sqrt{1-m\sin^2(a\theta)},$$

$$r'=-\frac{ma\sin2a\theta}{2r},\qquad r''=-\frac{ma^2\cos2a\theta}{r} -\frac{m^2a^2\sin^22a\theta}{4r^3},$$

$$\frac{\langle P\rangle}{P_{\rm circle}} =\left\langle(r''-r)^2+4(r')^2\right\rangle.$$

Nonzero detuning gives m < 1 and smooth bounded acceleration. Choose the demo's +1-maximum alignments g/ν = L, a = L + 1/2, δ/ν = √(L+1/4):

The detuned images close in one clock turn; their full coefficient rays need two turns when ε = 0. The numbers use quadrature independently checked against Fourier acceleration and a coarser grid. They are not measured atomic emission rates. With a nonzero depth readout Γ_z, its squared acceleration adds to the planar result if it is also identified as physical charge position. No nuclear recoil or radiation reaction is included. Detuning removes the cusp but does not supply a nonradiating bound state.

Fixed bilinear source: an exact frequency constraint

Suppose a time-independent linear lift ψ(u,t) = Σ_a c_a(t)s_a(u) into an admissible Dirac field on physical space bundle has actually been supplied. Fibre integration, a fixed linear polarization filter, and the dipole integral then turn a bilinear current into d_i(t) = c(t)†D_i c(t). This does not identify s_a with homogeneous coordinates or bypass the spatial spinor-bundle obstruction.

If i ċ = Hc and H has frequency eigenvalues λ_a, direct expansion gives

$$d_i(t)=\sum_{a,b}c_a(0)^*c_b(0)(D_i)_{ab} e^{i(\lambda_a-\lambda_b)t}$$

in its eigenbasis. Only occupied eigenvalue differences can occur. For the ideal four-state orbit the eigenvalues, modulo a common shift, are ν{L,−L,−1,0}. For the separate ring they are ν{0,−1,0}.

These are permitted frequencies, not predicted spectral lines: matrix elements can vanish and ν has not been fixed physically. In particular, the six-petal four-state truncation permits 6ν but not 5ν in such a dipole. These are unforced signal bounds within the stated truncation, not general absorption bounds. A subsequent geometric probe calculation shows that an exactly retained unforced orbit can respond through omitted section modes. The infinite harmonic tail of the displayed petalled position cannot equal any fixed bilinear dipole within this finite model. This remains true for the noncircular detuned square-root readout, which also has infinitely many harmonics (or sidebands for incommensurate Ω/ν).

For general detuning and relative phase, replace the eigenvalues by {ε+Ω, ε−Ω, −ν, 0}. Thus ε, though invisible to every displayed coordinate, shifts cross-block probe frequencies whenever the appropriate D_i elements are nonzero. A photon probe could distinguish identical-looking orbits; whether it does depends on the derived vertex. This is a concrete target for the next physical calculation, not evidence of an observed splitting.

The constraint assumes a fixed finite linear lift and fixed bilinear source. A nonlinear field bridge, time-dependent projector, additional occupied modes, or self-consistent field dynamics needs a new calculation. Arbitrary smooth replacement of the detector map is not a derivation of D_i.

Weak-field response along an actual orbit

Take the preparation time as t₀ = 0. For energy Hamiltonian ℏH and perturbation −Σ_j E_j(t)D_j, define D_i(t) = e^{iHt}D_i e^{-iHt}. Linearizing the evolution gives

$$\delta d_i(t)=\sum_j\int_{t_0}^{t}\chi_{ij}(t,s)E_j(s)\,ds, \qquad \chi_{ij}(t,s)=\frac{i}{\hbar} c_0^\dagger[D_i(t),D_j(s)]c_0.$$

The diagnostic checks this sign and normalization in units ℏ = 1 against an exact small impulse, including detuning and an invisible relative phase. An orbit preparation is generally not stationary, so this is a two-time response, not automatically a function of t−s. Averaging over unknown initial orbit phase must be justified by the preparation. Net absorption, linewidths, and radiative stability need the physical D_i, coupling to field modes, and backreaction; an arbitrary random test matrix supplies none of these.

Outcome and reproduction

The circle alone misses both the readout cusp and the finite-spectrum incompatibility. The more complex cases expose both and supply explicit tests for candidate charge/current bridges. This has not yet produced an independent hydrogen-spectrum prediction or established stable V2 atoms. The next calculation is to derive the dipole/current matrices from the projected action, including the nuclear contribution, and evaluate the source and response rather than fit these matrices to known lines.

orbit_electromagnetic_checks.py checks the Fourier coefficients, formal high-frequency asymptote, allowed bilinear frequencies, three detuned power integrals, impulse response, and a phase-sensitive probe example. These are finite conditional checks, not validation of the master theory.

Related

• Orbit current projection — conservation and the missing charge-readout identification.
• Orbit exchange robustness — exact perturbed flow and distinction between image closure and state recurrence.
• Research status — current proof boundaries and correction routes.