Persistence and Source Work in a Second-Order Moving Contact

One Common Inverse Action

The moving spatial contact supplies an exact finite rotation representation of a point evaluation on the ordinary spatial sphere. A complex second-order configuration field has twice the real configuration dimension of a real oscillator. Its two oscillator frequency sectors must both be retained when that contact moves.

Use the earlier inverse positive restoring law

$$L=\dot\xi^\dagger\dot\xi-\xi^\dagger V(\theta)\xi, \qquad V(\theta)=\lambda^2[4I-3P(\theta)], \qquad \lambda=\mu_d S_N, \quad S_N=\binom{N+d}{d}.$$

Here P is the spatial evaluation projector. The stipulated bright frequency is λ and the orthogonal frequency is 2λ at a fixed source. This law is an inverse construction from the desired count clock, not a derived master interaction. The bare spatial Dirac dispersion is omitted in this isolated restoring control; the earlier calculation explicitly retains its mismatch. No cancellation of that dispersion is established here.

The field coefficients describe a classical internal field, not the physical position of an electron or a probability distribution. The source angle is a separate deterministic position parameter. No compactification or mediator is used.

In one Clifford polarization channel σ, the exact circle weights are

$$w_{lk}^{(\sigma)}=\frac{l+d-1+\sigma k}{(d-1)S_N} \frac{(\alpha)_j(\alpha)_{l-j}}{j!(l-j)!}, \qquad \alpha=\frac{d-1}{2},\quad k=l-2j.$$

They are positive and sum to one. Because the present V has no separate l-dependence, define W_k=Σ_l w_lk, b_k=√W_k, and M=diag(k). Then

$$b(\theta)=e^{iM\theta}b,\qquad P(\theta)=b(\theta)b(\theta)^\dagger.$$

Each polarization channel has a cyclic space of dimension 2N+1. The global orthogonal complement to every b(θ) decouples at laboratory frequency 2λ. The two σ signs have mirrored spectra; they are not identified as charge signs or particles and antiparticles. Multiplicity in the full spinor band does not add physical spatial dimensions.

The Complete Quadratic Frequency Pencil

Prescribe θ=Ωt and set ξ=e^(iMΩt)χ. Direct differentiation gives

$$\ddot\chi+2i\Omega M\dot\chi+(V_0-\Omega^2M^2)\chi=0.$$

With χ=u exp(−izt), the exact pencil is

$$\boxed{[V_0-(z-\Omega M)^2]u=0.}$$

In λ units write e=Ω/λ, v=4I−3bb†. All twice-as-many frequency roots are the eigenvalues of

$$\mathcal B=\begin{pmatrix}eM&I\\v&eM\end{pmatrix}.$$

For a nonsingular diagonal factor the rank-one secular equation is

$$F(z,e)=1-3\sum_k\frac{W_k}{4-(z-ek)^2}=0,$$

where z in this formula is also in λ units. Complete matrices retain roots even at possible singular factors. A complex root has amplitude growth rate λ Im z. A real rotating frequency is a Floquet representative modulo Ω, not automatically a complete particle mass-frequency.

The configuration norm ξ†ξ is not the conserved phase charge. With π=ξ̇, the actual signed phase and circle-symmetry charges are

$$Q=i(\xi^\dagger\pi-\pi^\dagger\xi),\qquad j=2\operatorname{Im}(\xi^\dagger M\pi).$$

These are classical Noether quantities, not a derived electric charge. The group generator depends on the chosen evaluation frame; it is not an independently established observed particle angular momentum. On a real frequency mode, with f_k=z−Ωk,

$$Q=2\sum_k f_k|u_k|^2,\quad j=-2\sum_k kf_k|u_k|^2,\quad E_{\rm field}=2\sum_k f_k^2|u_k|^2, \quad E_{\rm field}-\Omega j=zQ.$$

Positive Rotating Energy and Resolved Instability

The rotating conserved quadratic form is

$$E_{\rm field}-\Omega j =\|\pi-i\Omega M\chi\|^2 +\chi^\dagger(V_0-\Omega^2M^2)\chi.$$

Thus V₀−Ω²M²>0 is sufficient for bounded linear motion and real frequency roots. The elementary bound |Ω|N<λ is sufficient at every occupied depth. For N≥1, the exact loss of positive rotating energy occurs at g_E, where g=|Ω|N/λ and

$$1=3\sum_k\frac{W_k}{4-(g_E k/N)^2},\qquad 1<g_E<2.$$

The diagonal factor is positive below g=2 and the deficit is strictly decreasing, proving uniqueness of this threshold. For N=1 the exact result is

$$g_E=2\sqrt{\frac{d+1}{4d+1}}.$$

Loss of this sufficient energy condition does not imply instability. The N=2 controls give a real interval beyond g_E before the first resolved collision in the tested bracket:

Independent equations F=∂F/∂z=0 reproduce the matrix collision. The colliding mode's limiting signed phase charge vanishes. This is a frequency-sector collision, not an established physical decay channel. The finite rotation grid contains stable and unstable windows; it is not an exhaustive chart or proof that all higher speeds remain unstable.

At N=2, g=1.6, every occupied depth has resolved growth. The amplitude e-folding time is 0.469–0.650 target-clock cycles. Direct laboratory integration agrees with the pencil to 6.2×10⁻¹¹ relative over two e-folds. The positive field energy grows at twice the amplitude rate, and its increase equals the work done by the prescribed source to 6.0×10⁻¹¹ relative. A driven finite oscillator is not an isolated decaying particle.

The Same Spatial Metric Determines the Slow Clock Shift

Let m=⟨M⟩=σN/(d+1) and

$$G_N=\langle(M-m)^2\rangle =\frac{(d^2+d-1)N(N+d+1)}{(d+1)^2(d+2)}.$$

The perturbation equation gives, in λ units,

$$u=b-\frac23e(M-m)b+O(e^2),\qquad z=1+em-\frac76 e^2G_N+O(e^3).$$

Indeed v−I is three times the dark projector. Its first-order inverse gives the displayed u correction. The bright second-order equation is −(7/3)G_N−2z₂=0, hence z₂=−7G_N/6. Restoring units,

$$z=\lambda+\Omega m-\frac{7\Omega^2G_N}{6\lambda}+\cdots, \qquad \frac{E_{\rm field}}Q =\lambda+\frac{7\Omega^2G_N}{6\lambda}+\cdots.$$

The same G_N already controls finite-source overlap loss and motion out of the bright subspace. The second-order coefficient differs from the first-order coefficient because the full oscillator sectors respond. The opposite signs of the rotating-frequency and field-energy corrections also prevent identifying one with the other. Symmetric small-speed roots and Richardson refinement verify both coefficients across all excited maintained assignments. G_N has no special zero at depth five.

Reciprocal Source and a Nonlinear Stability Bound

Give θ a positive collective inertia I and retain its canonical momentum P:

$$H=\frac{P^2}{2I}+\pi^\dagger\pi+\xi^\dagger V(\theta)\xi.$$

This rotor must ultimately be a collective coordinate of Psi_infinity; its origin, inertia and restriction to a circle are assumptions here. Direct variation gives

$$\dot P=6\lambda^2\operatorname{Im} [(b(\theta)^\dagger\xi)^*b(\theta)^\dagger M\xi].$$

The complete H, signed Q and J=P+j are conserved. In rotating canonical coordinates the reduced Hamiltonian is

$$H_J=\frac{(J-j)^2}{2I}+\rho^\dagger\rho+\chi^\dagger V_0\chi.$$

There is a stronger finite-model result than spectral stability. For any real pencil root z, put A=z−ΩM and C=V₀−A². Completing squares gives

$$\boxed{H-\Omega J-zQ+\frac{I\Omega^2}{2} =\frac{(P-I\Omega)^2}{2I} +\|\rho+iA\chi\|^2+\chi^\dagger C\chi.}$$

If D=4λ²I−A²>0, the secular equation implies

$$C=D^{1/2}[I-3\lambda^2 vv^\dagger]D^{1/2}\geq0, \qquad v=D^{-1/2}b, \quad 3\lambda^2v^\dagger v=1.$$

The kernel is exactly the continued bright vector. Its other finite eigenvalues are positive. The right side is a conserved nonnegative functional and vanishes on the relative equilibrium P=IΩ, ρ=−iAu. At fixed positive Q the amplitude is fixed; the remaining phase and source angle are symmetry directions. The positive transverse form therefore gives orbital Lyapunov stability of that relative-equilibrium manifold in this finite model, with source backreaction retained. This means that sufficiently small perturbations remain close to the symmetry manifold, rather than that their phase must return to its initial value. Decoupled full-band modes add positive forms when |z|<2λ. Polarization multiplicities enlarge the allowed minimizing manifold.

At Ω=0 this reduces to the especially simple global identity

$$H-\lambda Q=\frac{P^2}{2I} +\|\pi+i\lambda\xi\|^2 +3\lambda^2\|(1-P(\theta))\xi\|^2\geq0.$$

Eight short integrations verify conservation and nonzero reciprocal momentum exchange, including the actual electron and muon cutoffs. Configuration norms change while Q stays constant. Energy and phase-charge drifts are below 1.5×10⁻¹⁰ relative; combined-rotation drift is below 7.2×10⁻¹⁰ absolute. These integrations check the equations, whereas the stability statement follows from the conserved bound and its positive transverse form. Neither establishes stability to omitted spatial modes or proves that the model supplies a localized physical particle.

Electron Persistence and the Shared Comparison

All fourteen maintained massive assignments use the same Ω/μ_d=0.1,1,10 and comparison inertia Iμ_d=1. Neither Ω nor that inertia is measured or identified with the sector fundamental by these tests. At Ω/μ_d=1:

| Branch | d | N | g=|Ω|N/λ | g_E | E_field/(λQ) | |---|---:|---:|---:|---:|---:| | W | 2 | 75 | 0.0256323 | 1.711665 | 1.000111773 | | strange | 3 | 3 | 0.15 | 1.417343 | 1.009033819 | | up | 4 | 2 | 0.133333 | 1.296526 | 1.009719418 | | ν₁ | 5 | 9 | 0.00449550 | 1.751657 | 1.000004526 | | electron | 6 | 12 | 0.000646412 | 1.854330 | 1.000000080738 | | muon | 6 | 34 | 0.00000885790 | 1.990167 | 1.000000000012 | | tau | 10 | 22 | 0.000000341021 | 1.995233 | 1.000000000000015 |

Every branch at this shared speed satisfies the stronger augmented-energy condition. The electron remains comfortably inside the sufficient region at all three speeds. At Ω/μ_d=10, strange and up have resolved fixed-source growth, with e-folding times 1.53 and 2.07 target-clock cycles. No observed lifetime is inferred from those chosen controls.

An electron candidate needs much stronger persistence than a candidate for a short-lived particle. Finite-lived branches remain admissible when their timescale, outgoing channels and conserved balance fit the particle being modeled. Here the same slow-source law protects the muon and tau even more strongly than the electron. It therefore does not derive their lifetime hierarchy. There is no outgoing continuum or physical decay vertex in this model. The result supports the internal viability of an inverse count action, not the preferred sector assignments or an explanation of decay.

A Different Exact Transport Action

The full second-order convective repair is

$$L_{\rm conv}=\|\dot\xi-iM\dot\theta\xi\|^2 -\xi^\dagger V(\theta)\xi+\frac I2\dot\theta^2.$$

It includes both the velocity cross term and the term quadratic in θ̇. Under ξ=e^(iMθ)χ the entire action decouples into a fixed oscillator and free rotor. With π=ξ̇−iMθ̇ξ and j=2 Im ξ†Mπ, the canonical source momentum is P=IΩ−j, and

$$H_{\rm conv}=\frac{(P+j)^2}{2I} +\pi^\dagger\pi+\xi^\dagger V(\theta)\xi\geq0, \qquad J=P+j=I\Omega.$$

This changed action transports the stipulated λ clock exactly. Adding the physical connection changes the dynamics; merely changing coordinates in the previous action does not. Exact decoupling is not spontaneous binding, capture or synchronization. The master-field origin of the connection and the bare spatial kinetic completion remain open.

Reproduction

second-order-contact-checks.py and its results retain 42 assigned-speed cases, 224 rotation-grid spectra, six independently located collision controls, six direct growing solutions and eight reciprocal integrations. The final run passes 1,002 evaluations across thirty-eight conditions. It verifies the complete augmented-energy identity and transverse gaps wherever a slow bright branch is reported. Prior successful code and its dated report retain their actual hashes.

The established general canonical map and inverse restoring square belong to the earlier kinetic bridge. The present calculation applies them to the actual spatial circle weights, with full frequency-sector mixing, source work, reciprocal energy and persistence conditions. It does not supply a local master source, occupied-index selection, sector normalization, complete spatial support or a relativistic particle solution.