A Local Cavity Wave With a Fundamental Current and Uniform Energy

The modal cavity construction identifies positive energies for phase locking and reciprocal beat transfer. A local wave prototype gives a more physical realization of its carrier-independent source. It also separates circulating field motion from a circulating energy lump. This is a construction on a prescribed closed path, not a self-confined particle solution or a derivation of the simplex index.

1. One Field, Two Polarization Blocks, One Local Action

Let $s=R\theta$ run around an ordinary circular spatial path of radius $R$. The closed coordinate describes the path; it does not compactify any surrounding spatial direction. Its placement, thickness and supporting field must ultimately be derived. Write two two-component blocks $a(t,s),c(t,s)$ of one candidate field. The four complex components are field values, not additional spatial dimensions or constituent particles. Their identification within an IDWT spatial carrier is an additional reduction to construct.

Define Hermitian matrix currents

$$Q_a=aa^\dagger,\qquad Q_c=cc^\dagger,\qquad D=Q_c-\eta Q_a,$$

and the local action

$$S=\int dt\,ds\left[ |\partial_ta|^2+|\partial_tc|^2 -v^2(|\partial_sa|^2+|\partial_sc|^2) -\frac g2\operatorname{tr}(D^2)\right],\quad g>0,\ \eta>0.$$

The potential is positive. Common unitary polarization rotations conjugate $D$ and leave its norm invariant; a common complex-field phase also leaves the action invariant. This is a second-order multicomponent wave prototype. Its kinetic normalization is distinct from the first-order Dirac or canonical modal actions, and those actions' Noether charges cannot be transferred to it without a derivation.

Full variation gives

$$\partial_t^2a-v^2\partial_s^2a-g\eta Da=0, \qquad \partial_t^2c-v^2\partial_s^2c+gDc=0.$$

The nonlinearity is local and contains no new principal derivatives. The characteristic speeds are $\pm v$. The energy and flux are

$$\mathcal E=|a_t|^2+|c_t|^2+v^2(|a_s|^2+|c_s|^2) +\frac g2\operatorname{tr}(D^2),$$

$$\mathcal F=-2v^2\operatorname{Re}(a_t^\dagger a_s+c_t^\dagger c_s), \qquad\partial_t\mathcal E+\partial_s\mathcal F=0.$$

Thus this prototype supplies explicit local reciprocal dynamics and conserved energy transport. It does not obtain power from an unspecified independent field.

2. Exact Different-Carrier Solutions

Put $u=\theta-\Omega t$ and $\Omega=v/R$. For positive integer labels $k,l$ and arbitrary constant complex amplitudes $A,B$, take

$$a(u)=\begin{pmatrix}Ae^{iku}\\Be^{i(k+1)u}\end{pmatrix},\qquad c(u)=\sqrt\eta\begin{pmatrix}Ae^{ilu}\\Be^{i(l+1)u}\end{pmatrix}.$$

They obey $c=\sqrt\eta e^{i(l-k)u}a$. Therefore $D=0$ pointwise, the complete interaction gradient vanishes, and every component solves the free traveling-wave equation. This is an exact solution of the nonlinear local action at any amplitude, not only a linearized solution. The two carrier labels may differ while the source beats agree.

The matrix source is

$$Q_a=\begin{pmatrix} |A|^2&A\bar B e^{-iu}\\ \bar A B e^{iu}&|B|^2 \end{pmatrix},\qquad Q_c=\eta Q_a.$$

Its diagonal density is constant. A Hermitian off-diagonal readout such as $B_x=\sigma_x$ gives

$$a^\dagger B_xa=2\operatorname{Re}(\bar AB e^{iu}),$$

at the fundamental $\Omega$, independently of $k$. This uses a specified polarization-sensitive current. Identity density alone would miss the beat. A common rotation of the fields and the readout preserves the result; keeping the readout fixed would describe a changed relative polarization.

For an assigned mass endpoint $K>1$, set $k=K-1$, retain the upper component frequency $K\Omega$, and use the common low pair $l=1$. This preserves the supplied target without asserting that the action selects it. At the down-quark endpoint $K=1$, the comparison uses one populated component and has no off-diagonal beat. Bottom's unresolved mass assignment and the photon's optical frequency supply no adopted endpoint in this test.

The local model therefore provides a concrete physical possibility behind fundamental-frequency coupling: a polarization current can beat slowly even when its underlying field components oscillate much faster and its energy density is uniform. The relevant current operator still has to follow from the sector geometry.

3. Complete Energy and Circulating Momentum

For the exact branch the energy per unit path length is uniform:

$$\mathcal E=2\Omega^2\left[ k^2|A|^2+(k+1)^2|B|^2 +\eta(l^2|A|^2+(l+1)^2|B|^2)\right].$$

Its flux is $\mathcal F=v\mathcal E$. All components continue moving around the path; zero total linear momentum does not mean the field has stopped. With tangent $\hat t(\theta)=(-\sin\theta,\cos\theta)$, the embedded linear momentum is

$$\mathbf P=\int_0^{2\pi}\frac{\mathcal E}{v}\hat t(\theta)R\,d\theta=0.$$

The angular momentum is $L_z=RE/v$, where $E=2\pi R\mathcal E$. This classical orbital quantity is not a derivation of observed spin or its normalization.

A scalar adjacent comb on the same prescribed path behaves differently. For $f=Ae^{iku}+Be^{i(k+1)u}$ with real $A,B$, its energy contains the fundamental interference term. At $v=R=1$, its momentum per unit loop length is

$$\frac{\mathbf P}{2\pi}=2k(k+1)AB \begin{pmatrix}-\sin t\\\cos t\end{pmatrix}.$$

That changing momentum must be balanced by supporting stresses. The orthogonal-polarization construction removes this particular net linear imbalance while retaining a current beat. It does not remove the inward local stress needed to bend the wave flow around a circular path. The total containing field and its energy must supply that confinement before the prototype can represent a free particle.

The prototype's energy depends on amplitude and carrier labels. This calculation does not identify that classical energy or angular momentum with the observed particle mass or spin. Mass remains the supplied selected-frequency target in this coupling test; a complete inertia calculation requires the supporting spatial field too.

4. Disturbances From the Full Local Equations

Let $L_z=-2R\int\operatorname{Re}(a_t^\dagger a_\theta+c_t^\dagger c_\theta)d\theta$ be the path-translation generator. The action's conserved comoving energy is exactly

$$E-\Omega L_z=R\int\left[ |a_t+\Omega a_\theta|^2+|c_t+\Omega c_\theta|^2 +\frac g2\operatorname{tr}(D^2)\right]d\theta\ge0.$$

The exact traveling branch has zero value. In its comoving coordinate $u$, a real eight-component variation $z$ obeys

$$z_{tt}-2\Omega\partial_u z_t+K(u)z=0,$$

where $K=(g/2)L^TL\ge0$ and $L$ maps the field variation into real Hermitian coordinates of $\delta D$. The factor one-half follows from the real kinetic normalization in the written action. The quadratic conserved energy is $\int(|z_t|^2+z^TKz)du$.

The transport operator on velocities is skew-adjoint and $K$ is positive. An exponentially growing finite-energy normal mode would have nonzero velocity and growing positive energy, contradicting conservation. Neutral and polynomially growing directions are not excluded: the energy does not control every change of profile along its zero family. This is not a proof of nonlinear orbital stability or self-localization.

Finite Fourier projections with cutoffs $2,4,6$ preserve this exact energy structure. The full first-order matrices have sizes $80,144,208$. All eighteen occupied-depth preparations have positive-semidefinite potential Hessians and no resolved exponential growth. Exact quadrature of the projected local matrix retains its Fourier products; these are bounded spectral tests, not particle evolution simulations.

5. What the Geometry Test Establishes

The same local algebra is checked at occupied depths $2,3,4,5,6,10$, using all fourteen adopted massive branches and shared values $\eta=1/4,1,2$. The construction does not distinguish those dimensionalities: an ordinary closed path and these field components can be hypothesized at each. Its value is to show compatibility of high mass-frequency endpoints, a carrier-independent fundamental current, positive local dynamics and circulating energy with zero net linear momentum.

If a common wave speed is assumed, the prescribed path radii satisfy $R_d=v/\omega_{0,d}$, so their ratios are inverse sector-clock ratios. This is a consequence of the loop prototype, not a measured particle-size prediction. Deriving the active-depth spatial carrier, the readout operator, finite normal confinement, supporting stress and simplex endpoint selection remains necessary for a sector-specific particle mechanism.

The written positive quartic alone cannot supply ordinary finite-energy time-harmonic confinement in flat noncompact space. For a sufficiently decaying stationary profile with a common temporal phase within each block, let $T$ be its positive gradient energy, $V$ its quartic energy and $W$ its frequency-squared weighted norm. The integrated field equation gives $T+2V=W$. Spatial rescaling gives $(2-d)T-dV+dW=0$. Eliminating $W$ yields $2T+dV=0$, which forces the trivial state at every occupied depth. This exact control explains why the prescribed loop needs additional geometry or supporting dynamics; a useful local current interaction by itself is not the confinement mechanism.

The checker and report preserve 464 checks, forty-two endpoint comparisons, twelve scalar momentum controls, eighteen complete local spectral preparations and six exact ambient-localization controls. They verify the nonlinear potential variation and polarization covariance independently of the exact branch, as well as its source, flux and momentum identities. Overall physical amplitude remains uncalibrated.

Generating the Adjacent Frequencies Dynamically

The bending-clock construction derives a constant separation between two coupled positive geometric wave branches under an equal-speed premise. Its canonically normalized partial velocity current has an exact half vertex, while the conserved total current has no beat. This complements the prescribed-path example without identifying their different field actions or assuming that either supplies the required spatial confinement.