A Moving Spatial Contact Is Not Its Static Orbit Average

The Physical Distinction

The source-saturation theorem concerns a specified static source operator. It does not say that an electron following a deterministic orbit has the interaction or mass of a stationary source spread along that orbit. A moving contact generally fails to commute with itself at different times. Averaging its source before solving the field equation can therefore erase coherent transport.

Here a prescribed great circle on the same ordinary spatial sphere gives an exact, finite calculation. The point contact and its strength are retained assumptions. A second version supplies a reciprocal positive collective rotor, explicitly retaining its energy and conserved charge. Both are conditional inverse models, rather than reductions already derived from Psi_infinity. They introduce no mediator particles or electron probability clouds. A classical spatial field and its coefficients remain distinct from the physical position of the source.

The sphere is a d-dimensional ordinary spatial control. The great circle is a controlled deterministic path, not a recovered six-direction cylindrical electron orbit.

Exact Rotation Harmonics of the Actual Spinor Band

Use the positive Dirac shells 0≤l≤N on S^d, with

$$g_l=\binom{l+d-1}{d-1},\qquad S_N=\binom{N+d}{d},\qquad \alpha=\frac{d-1}{2}.$$

The ground-Killing-spinor frame used in the source calculation supplies, on a great circle, the shell kernel

$$K_l(\theta)=a_l(\cos\theta)I -i\sin\theta\,b_l(\cos\theta)\gamma_1,$$

$$a_l(t)=\frac{l+d-1}{d-1}C_l^\alpha(t),\qquad b_l(t)=C_{l-1}^{\alpha+1}(t),\qquad b_0=0.$$

In this frame its value between two points on the circle depends on their angle difference. Independent projection onto the actual monogenic polynomial modes verifies that statement at l=0,1,2 for every occupied depth, including pairs whose first point is away from the north pole.

Fix a unit polarization with γ₁ eigenvalue σ=±1. Let U_θ be normalized evaluation into the selected spatial band, and b(θ)=U_θ u its unit bright field. Then

$$b(\theta)^\dagger b(0) =\frac{\sum_{l=0}^N[a_l(\cos\theta) -i\sigma\sin\theta\,b_l(\cos\theta)]}{S_N}.$$

The Gegenbauer generating function gives

$$C_l^\alpha(\cos\theta) =\sum_{j=0}^l\frac{(\alpha)_j(\alpha)_{l-j}}{j!(l-j)!} e^{-i(l-2j)\theta}.$$

Combining the scalar and Clifford parts gives positive joint shell and rotation weights. For k=l−2j,

$$\boxed{w_{lk}^{(\sigma)} =\frac{l+d-1+\sigma k}{(d-1)S_N} \frac{(\alpha)_j(\alpha)_{l-j}}{j!(l-j)!}.}$$

The rising factorial is (α)_j=α(α+1)…(α+j−1), with (α)_0=1. The numerator is at least d−1, so every present weight is positive. Their sum at fixed l is g_l/S_N and their total sum is one. The Gram kernel is therefore realized on the joint cyclic space by

$$b(\theta)=e^{iM\theta}b_0,\qquad (b_0)_{lk}=\sqrt{w_{lk}},\qquad M_{lk,lk}=k.$$

This uses actual shell and spatial rotation data; the coefficient dimension is not physical spatial depth. Orthogonal mode degeneracies that never couple to this circle contact remain decoupled under the shell-diagonal bare Dirac operator. The two σ channels have mirrored weights. They are not identified with electric charge, particle/antiparticle, or an observed chirality. A frame change can shift the rotation generator and Floquet representative together, so M is not an independently established electron angular-momentum observable.

One Geometric Coefficient Controls Width and Motion

The exact weighted moments are

$$\langle k\rangle=\frac{\sigma N}{d+1},\qquad \langle l\rangle=\frac{dN}{d+1},$$

$$\boxed{\operatorname{Var}(k)=G_N =\frac{(d^2+d-1)N(N+d+1)}{(d+1)^2(d+2)}.}$$

G_N is precisely the directional coefficient of the previously calculated spatial geometric tensor. That name describes field-mode geometry; it does not impose quantum position probabilities. The same coefficient gives the leading finite-separation overlap defect and the moving bright-field derivative:

$$1-|b(\theta)^\dagger b(0)|^2=G_N\theta^2+O(\theta^4), \qquad \|(1-P)\dot b\|^2=\dot\theta^2G_N, \quad P=bb^\dagger.$$

Thus the geometry that penalizes a spread source also fixes the missing transport when a point source moves. G_N vanishes at N=0 and is strictly positive at every excited cutoff for all six occupied depths. It supplies no depth-five exception or depth-six preference.

Static Averaging and Exact Time Ordering

Take the canonical classical mode equation

$$i\dot\psi=[D+\lambda P(\Omega t)]\psi,\qquad D_{lk,lk}=\kappa(l+d/2),\qquad \lambda=\mu_d S_N.$$

Here κ is the bare spatial Dirac frequency unit, equal to c/R if that particular spatial wave-speed identification applies. Its relation to the sector frequency unit μ_d remains open. The linear contact and its normalization are assumed. This first-order action is not inferred from an arbitrary second-order Hessian; the separate kinetic bridge specifies the extra structure such an inference would require. The positive-band restriction is part of this model, not a proof of invariance under a full nonlinear field equation containing both Dirac signs.

The second-order moving-contact calculation separately retains both oscillator frequency sectors. It derives their different quadratic clock shift, resolved source-work instabilities and a complete autonomous augmented-energy stability bound. Those results use the inverse squared restoring law with bare dispersion omitted, rather than inheriting the present first-order equation.

For κ=0, averaging uniformly around the circle gives contact eigenvalues λW_k, where W_k=Σ_l w_lk. All excited cutoffs have max W_k<1; N=0 has W_0=1. This is the static source-saturation theorem in an explicit form.

Instead set ψ=e^(iMΩt)χ. The exact time-dependent equation becomes

$$\boxed{i\dot\chi=[D+\lambda P_0+\Omega M]\chi.}$$

No time averaging is used. After one orbit the transport e^(2πiM) is the identity in the chosen frame. The eigenvalues z of this rotating generator are therefore Floquet representatives, defined modulo Ω. They are not automatically intrinsic particle mass-frequencies. For a unit eigenstate,

$$z=E_{\rm field}+\Omega\langle M\rangle,\qquad E_{\rm field}=\langle D\rangle+\lambda|b_0^\dagger\chi|^2.$$

The source can supply work in the prescribed-path model. Removing its transport term from z by verbal interpretation would not calculate a complete particle energy.

The unique eigenvalue above every diagonal entry a_lk=κ(l+d/2)+Ωk obeys the rank-one secular equation

$$1=\lambda\sum_{lk}\frac{w_{lk}}{z-a_{lk}},\qquad \chi_{lk}\propto\frac{\sqrt{w_{lk}}}{z-a_{lk}}.$$

For κ=0 and sufficiently slow motion,

$$z=\lambda+\Omega\langle k\rangle +\frac{\Omega^2G_N}{\lambda}+O(\Omega^3), \qquad |b_0^\dagger\chi|^2 =1-\frac{\Omega^2G_N}{\lambda^2}+O(\Omega^3).$$

The actual expansion parameter includes the spread of Ωk relative to λ; these formulas are not uniform in cutoff for arbitrary fixed λ. They show why a coherent moving state can retain nearly the whole instantaneous contact while its static orbit average does not. They also show why the unchanged contact alone cannot give exact bright following at nonzero Ω and N≥1.

Bare kinetics requires the joint (l,k) space. Combining all l at fixed k before retaining D would delete real spatial frequency differences. Even with a correct transport connection, the bright field has bare kinetic residual

$$\|(D-\langle D\rangle)b_0\|^2 =\kappa^2\operatorname{Var}(l),\qquad \operatorname{Var}(l)=\frac{dN(N+d+1)}{(d+1)^2(d+2)}.$$

Reciprocal Motion and the Complete Energy

A minimal autonomous comparison assigns the contact's position angle a positive collective inertia I and canonical momentum P_θ:

$$H=\frac{P_\theta^2}{2I}+\psi^\dagger D\psi +\lambda|b(\theta)^\dagger\psi|^2.$$

The rotor would have to be realized by collective motion of the same field, not a mediator. Its inertia and restriction to a great circle are new premises, not derived electron–nucleus dynamics. Direct variation gives

$$\dot\theta=P_\theta/I,\qquad \dot P_\theta=-2\lambda\operatorname{Im} [(b^\dagger\psi)^*b^\dagger M\psi].$$

The canonical norm Q=ψ†ψ, the complete H, and

$$J=P_\theta-\psi^\dagger M\psi$$

are conserved. Q is a classical phase-space action normalization here, not a probability postulate or a calibrated electron amplitude. The time-dependent field energy and rotor energy exchange reciprocally.

The change ψ=e^(iMθ)χ contributes −χ†Mχ dθ to the canonical one-form. At fixed J, the exact reduced Hamiltonian is consequently

$$\boxed{H_J=\frac{(J+\chi^\dagger M\chi)^2}{2I} +\chi^\dagger D\chi+\lambda|b_0^\dagger\chi|^2.}$$

Its field equation contains +ΩM, with Ω=(J+χ†Mχ)/I. A prepared relative equilibrium can use the secular eigenstate, but its complete energy is IΩ²/2+E_field. The rotating eigenvalue z alone is not that energy per norm. This calculation establishes reciprocal finite-dimensional dynamics and its conservation laws. It does not establish localized support, capture, long-term stability, a relativistic boost law, or a complete particle.

An Explicit Additional Transport Law

The prescribed-path counterdiabatic operator

$$H_{\rm cd}=i[\dot P,P] =-\Omega(MP+PM-2\langle M\rangle_bP)$$

cancels the off-bright derivative exactly. It is an added term, not a consequence of point contact. A full convective connection has an especially transparent autonomous canonical realization:

$$L_{\rm conv}=\frac i2\psi^\dagger \overleftrightarrow{\partial_t}\psi +\dot\theta\,\psi^\dagger M\psi +\frac I2\dot\theta^2-\psi^\dagger V(\theta)\psi.$$

For V(θ)=e^(iMθ)V₀e^(−iMθ), the same rotating transformation removes the connection from the whole action. The canonical source momentum is now Π_θ=Iθ̇+ψ†Mψ, so

$$H_{\rm conv}=\frac{(\Pi_\theta-\psi^\dagger M\psi)^2}{2I} +\psi^\dagger V(\theta)\psi\ge0$$

whenever V is positive. This retains the complete source kinetic energy; it does not turn a prescribed negative transport term into missing energy. With V₀=λP₀ and κ=0, the bright field follows with rotating phase frequency λ exactly. The positive complement choice V₀=λ(2I−P₀) also has bright frequency λ, now the minimum field frequency. Both choices stipulate the desired count through λ. The latter adds a restoring law already used in the canonical kinetic bridge; it is not obtained here from the spatial source.

This constructive action demonstrates what an exact transport repair would have to include. It still needs a same-field derivation of the connection and source inertia, plus the bare-kinetic repair if κ≠0. In this controlled circle model the changed action decouples the rotor from the field in co-moving variables. It prescribes compatible transport; it does not demonstrate capture, binding or spontaneous synchronization. Coordinates alone do not add this physical connection to an unchanged action. Its group generator and constant-angle-speed solution do not select an occupied cutoff, a sector unit or a full electron orbit.

Common Comparisons at Every Occupied Depth

The executed calculation retains all fourteen maintained massive assignments. It uses the same Ω/μ_d=0.1,1,10 and κ/μ_d=0,1,10 controls at every assignment, with no particle-specific tuning. Bare κ=0 is the contact-only control. Comparison Q=1 and Iμ_d=1 are declared units, not physical amplitude or inertia measurements. No identification of orbital Ω with the sector fundamental is assumed.

Representative results at Ω/μ_d=1 are:

The columns compare distinct operators or contributions, not measured particle masses. The coherent contact fraction comes from the continued largest rotating eigenbranch. For large Ω, choosing a Floquet branch is itself a physical selection problem; the largest unfolded eigenvalue is not generally the largest eigenvalue of the averaged contact.

Depth five exhibits the same nonzero geometry and smooth response as the other depths. No coefficient was inserted to suppress it. A contact that moves coherently can avoid the loss associated with a static spread source, but this occurs broadly and supplies no new favorable evidence for the assigned sector dimensions. The missing source, spatial kinetic completion and dynamical selection remain essential.

Reproduction and Accuracy

moving-contact-checks.py and its recorded results retain exact rational low-cutoff normalization and moments, every assigned kernel, 126 joint shell/rotation secular cases, translated kernels from independent spatial polynomial modes, and both polarization signs. The final run passes 1,025 evaluations across thirty-six checked conditions.

Direct integrations of the moving field at all six occupied depths agree with the exact rotating solution to 1.1×10⁻¹¹. In the same controls, replacing the operator by its time average changes the one-period state by norm 0.049–0.072. These are finite numerical examples, not a general error bound. The secular roots also agree with independent complete matrices at small cutoffs. Eight autonomous rotor integrations include all six depth controls and the actual electron/muon cutoffs. They exchange nonzero source momentum while conserving complete energy and combined charge to better than 7.1×10⁻¹⁰ relative and 8.9×10⁻¹¹ absolute respectively over their stated short intervals. They are conservation checks, not stability experiments.

The large-count residual uses the operator's actual frequency scale. The initial absolute-residual threshold failed for the tau count 64,512,240 with an absolute residual 8.64×10⁻⁷, corresponding to only 1.34×10⁻¹⁴ of that scale. The corrected report states the relative residual; the equations and numerical eigenstate did not change. Unchanged previous successful code and reports retain their exact source hashes.

The convective action is checked at the canonical one-form, positive Hamiltonian, Noether charge and bright evolution levels, for the rank-one and positive-complement potentials. The bare spatial residual is retained. No claim of complete particle inertia or a derived master interaction follows from these finite-dimensional checks.