Cavity Phase Locking and a Fundamental-Frequency Coupling Source

The harmonic-support result identifies a strong sector-fundamental source in coherent adjacent harmonics. A physical explanation also needs dynamics that maintains their phases and lets that source exchange energy reciprocally. This note supplies inverse modal actions that accomplish those narrower tasks. Their retained variables are modes of the same field; there are no additional constituent or mediator particles. The spatial profiles, canonical reduction and origin of the modal interaction remain physical premises to derive.

Mode-locked laser mathematics separates a comb's carrier frequency from its mode spacing and intensity repetition. A magnetron supplies a related phase-feedback example: circulating motion exchanges energy with a synchronous rotating wave. The useful external mathematics is attributed in Cundiff and Ye and Kazakevich and collaborators. Device gain, an electron beam, imposed fields and external power are not assumed as IDWT constituents or sustaining sources.

1. An Exact Carrier-Independent Envelope

Keep the assigned upper mass-frequency $K\omega_{0,d}$, where $K=S(n,d)$. A band of $M\le K$ adjacent positive harmonics ending at that frequency is

$$\Psi_K(t)=e^{-iK\theta}\sum_{j=0}^{M-1}b_je^{ij\theta},\qquad \theta=\omega_{0,d}t.$$

This is an alternative occupation hypothesis to the full sawtooth; its upper endpoint is unchanged. With the same envelope coefficients and the same spatial readout, every Hermitian quadratic source has its common carrier phase cancel exactly. In particular,

$$J_K=\frac12|\Psi_K|^2 =\frac12\left|\sum_jb_je^{ij\theta}\right|^2.$$

Its time derivative is also carrier independent. The same cancellation holds for a fixed Hermitian matrix acting on the modal vector, including a specified spin or color source. Spatial modes $u_{K-j}(x)$ and their actual source matrix can depend on $K$; an identical temporal envelope does not prove those matrices or physical amplitudes identical.

An energy-type field vertex behaves differently:

$$\left\langle\Psi_K^*i\partial_t\Psi_K\right\rangle =\omega_{0,d}\sum_j(K-j)|b_j|^2.$$

The derivative of a density and a density containing a derivative of the field are distinct operators. Only the former has the exact carrier cancellation above.

A round-trip phase condition is $\omega T+\varphi(\omega)=2\pi n$. Its local spacing is approximately $2\pi/[T+\varphi'(\omega)]$. For linear phase $\varphi(\omega)=\varphi_0+\tau_g\omega$, the exact comb is $\omega_n=(2\pi n-\varphi_0)/(T+\tau_g)$. The common offset cancels in Hermitian beats, but an exact integer particle harmonic relative to that spacing additionally requires an offset condition. Cavity closure alone neither supplies that condition nor selects $n=S(n_{\rm particle},d)$.

For diagnostic normalization $\sum|b_j|^2=1$, choose $b_j\propto r^j$ with real $r>0$. Its fundamental cosine coefficient is

$$D_1=\frac{r[1-r^{2(M-1)}]}{1-r^{2M}},\qquad D_1\big|_{r=1}=\frac{M-1}{M}.$$

For a flat band, $D_p=(M-p)/M$, $1\le p<M$, and the mean is $1/2$. The physical field amplitude is still free; these unit-norm results compare shapes.

For two equal-fundamental preparations with relative phase $\delta$,

$$\langle J(\theta)J(\theta+\delta)\rangle =\frac14+\frac12\sum_{p=1}^{M-1}D_p^2\cos p\delta.$$

A flat band gives $(2M^2+1)/(12M)$ in phase. At opposite phase and even $M$, the value is $1/(4M)$. Thus $M=8$ gives $43/32$ and $1/32$, an exact ratio $1/43$. This is a phase-dependent overlap, not an encounter probability or force. Different isolated fundamentals retain their own clocks; this comparison does not impose synchronization between unlike particles.

2. Positive Four-Wave Phase Locking

Let $a_j$ be canonical complex amplitudes of retained same-field modes with equally spaced positive frequencies $\omega_j$. Choose nonzero real weights $w_j$ and put $x_j=w_ja_j$. A candidate energy is

$$H=\sum_j\omega_j|a_j|^2+E_4,\qquad E_4=\kappa\sum_{j=1}^{M-2}|x_{j-1}x_{j+1}-x_j^2|^2,\quad\kappa>0,$$

with canonical equation $i\dot a_j=\partial H/\partial\bar a_j$. Its quartets are resonant because $\omega_{j-1}+\omega_{j+1}=2\omega_j$. On the all-nonzero branch, $E_4=0$ precisely when

$$x_j=c\rho^j,\qquad a_j=\frac{c\rho^j}{w_j},$$

for complex constants $c,\rho$. The phase is affine in the harmonic label. Since the full interaction gradient vanishes there, $a_j(t)=a_j(0)e^{-i\omega_jt}$ is an exact solution with unchanged frequencies. Amplitude affects disturbance rates, not those exact free-branch frequencies.

The energy is invariant under $a_j\mapsto e^{i\alpha+ij\beta}a_j$, and conserves

$$N=\sum_j|a_j|^2,\qquad J=\sum_jj|a_j|^2.$$

For real nonzero locked coefficients $b_j$, the linear constraint is

$$(L\delta a)_j=w_{j-1}x_{j+1}\delta a_{j-1} +w_{j+1}x_{j-1}\delta a_{j+1}-2w_jx_j\delta a_j.$$

It has rank $M-2$: the first $M-2$ columns form an upper triangular matrix with nonzero diagonal. Its complex kernel is spanned by $b_j$ and $jb_j$. The quadratic interaction energy is $\kappa\|L\delta a\|^2$. In the rotating frame the linear dynamics is $i\delta\dot a=\kappa L^\dagger L\delta a$, so all transverse disturbance frequencies are nonnegative and real. At fixed $N,J$, the two real amplitude null directions are removed; quotienting the two phase symmetries leaves a positive local Hessian. This establishes local orbital stability of the all-nonzero branch within the retained band. Sparse zero-energy branches also exist, so global uniqueness is not claimed.

Weights $w_j=1$ give a geometric envelope. For a complete positive harmonic band $k=1,\ldots,K$, the common choice $w_k=k$ instead gives

$$E_4=\kappa\sum_{k=2}^{K-1} |(k-1)(k+1)a_{k-1}a_{k+1}-k^2a_k^2|^2.$$

Then $a_k=c/k$ is an exact locked branch: a dynamical candidate for the tested positive-frequency sawtooth. Weighting by harmonic frequency is an explicit modal derivative premise, not yet a derived local spatial operator. The rank proof applies at any finite $K$ without constructing millions of modes. At $K=1$ or $2$ there is no interior quartet and no such phase-locking constraint.

A curved bare frequency ladder breaks this exact free solution. With $\omega_j=\omega_c-j\omega_0+\epsilon j^2/2$, an initially flat unit-amplitude branch has $E_4/\kappa=4(M-2)\sin^2(\epsilon t/2)$ under free evolution. Equal spacing or dynamical dispersion compensation is an additional requirement.

3. The Band Boundary and a Support-Preserving Variant

The quartic action written only inside the occupied band assumes a dynamically retained band. Extending the same action beyond its upper edge does not preserve the finite sawtooth. At $a_k=1/k$ for $k\le K$ and zero above, the first new weighted constraint has energy one and gradient

$$\frac{\partial E_4}{\partial\bar a_{K+1}}=-\kappa(K+1).$$

The next harmonic is therefore driven. A projector or spatial gap must be derived before the band-limited four-wave action can be treated as closed.

There is also a separate polynomial repair. For $P_k=x_{k-1}x_{k+1}$, take

$$E_8=\lambda\sum_{k\ge2}|P_k(P_k-x_k^2)|^2,\qquad\lambda>0.$$

This is a positive eight-wave interaction. The infinite positive-label sum is directly defined on finite-support sequences; a full infinite-dimensional action needs a convergence domain. Within any finite encompassing truncation, every zero modal amplitude remains zero under its Hamiltonian equation. If $a_k=0$, the derivative from the centered term vanishes, while neighboring terms contain a zero $P$ and hence zero squared residual. Thus a finite geometric or $1/k$ block is an exact free-frequency branch without a prescribed upper-band projector.

For $w_k=k$, $a_k=1/k$ through $K$, and zero above, the quadratic curvature of the next empty mode is $\lambda(K+1)^2|\delta a_{K+1}|^2$. Interior transverse curvatures are positive and more distant empty modes are neutral at quadratic order. The finite encompassing linear Hamiltonian has no growing eigenvalues. Nonlinear orbital stability against arbitrary infinite tails is not established.

This repair makes a significant different prediction: it cannot generate an initially absent harmonic. It can lock an already populated comb. It also admits many supports and does not select $K=S(n,d)$. Raising the nonlinear degree trades band closure for a stronger occupation premise; neither version is adopted as the master law.

4. Reciprocal Transfer Into a Fundamental Mode

Let $a_0,a_1$ be two same-field modes at $\omega_c$ and $\omega_c+\Omega$, and $b$ another retained mode at $\Omega$. Consider

$$H_0=\omega_c|a_0|^2+(\omega_c+\Omega)|a_1|^2+\Omega|b|^2, \qquad V=g|b-\eta\bar a_0a_1|^2,\quad g>0.$$

All bare frequencies and the full energy are positive. The three canonical equations follow from this one energy. The state $b=\eta\bar a_0a_1$ has zero interaction gradient and is an exact phase-matched orbit; its fundamental oscillation is the adjacent-mode difference, independent of $\omega_c$. Here $b$ must be a neutral collective coordinate if all $a_j$ carry a common complex-field phase: the bilinear $\bar a_0a_1$ is neutral. A canonical reduction producing that coordinate has not been derived. Treating $b$ as an ordinary linear mode with the same phase character as each $a_j$ would violate that symmetry.

Write $n_i=|a_i|^2$, $n_b=|b|^2$, and $z=\bar a_1ba_0$. Variation gives the explicit phase-resolved transfer

$$\dot n_1=-2g\eta\operatorname{Im}z, \qquad\dot n_b=+2g\eta\operatorname{Im}z, \qquad\dot n_0=-\dot n_1.$$

Consequently $n_0+n_1$ and $n_1+n_b$ are conserved, and $\dot H_0=0$. The interaction energy is also conserved. Energy transferred between these resonant modes is balanced; this is not an external source of gain.

At a real locked state $a_0=A$, $a_1=B$, $b=\eta AB$, the sole nonzero rotating-frame disturbance pair has frequency

$$\omega_{\rm feedback}=g|1+\eta^2(A^2-B^2)|.$$

The other four real directions are neutral. At the exceptional vanishing factor the linear system is degenerate and requires a higher-order analysis. This disturbance frequency is distinct from the continuing particle harmonics and fundamental $\Omega$. Stable phase-matched orbits and reversible transfer are possible in a closed conservative system; asymptotic capture or compensation of outgoing losses requires other same-field modes and incoming flux. The equations do not supply that spatial flux or localization.

A Four-Mode Version Preserving the Common Field Phase

A literal linear-mode alternative uses two pairs $a_0,a_1$ and $c_0,c_1$. Their frequencies are $\omega_a,\omega_a+\Omega$ and $\omega_c,\omega_c+\Omega$, respectively. Every mode can have the same complex-field phase character. Write

$$A_1=\bar a_0a_1,\qquad C_1=\bar c_0c_1,\qquad V_4=g|C_1-\eta A_1|^2.$$

The full action is invariant under the common field phase and separate overall phases of the two pairs. The conserved quantities are $N_a=n_{a0}+n_{a1}$, $N_c=n_{c0}+n_{c1}$ and $Q=n_{a1}+n_{c1}$. Its positive bare energy is $H_0=\omega_aN_a+\omega_cN_c+\Omega Q$. Direct variation gives

$$\dot n_{a1}=-2g\eta\operatorname{Im}(A_1^*C_1),\qquad \dot n_{c1}=+2g\eta\operatorname{Im}(A_1^*C_1).$$

Thus the high-frequency carriers can differ while the common difference-frequency beats exchange energy reciprocally. These pairs are internal same-field modes in this construction, not an imposed synchronization of unlike particles.

At real amplitudes $(A,B,C,D)$ with $CD=\eta AB$, the exact free-frequency solution has one nonzero disturbance pair at

$$\omega_{\rm feedback,4} =g|\eta^2(A^2-B^2)+C^2-D^2|.$$

This version has a direct link to a positive local density-mismatch interaction. For two local evaluations $z_a=a_0+a_1e^{-i\theta}$ and $z_c=c_0+c_1e^{-i\theta}$, cycle averaging gives exactly

$$\left\langle(|z_c|^2-\eta|z_a|^2)^2\right\rangle =(N_c-\eta N_a)^2+2|C_1-\eta A_1|^2.$$

The static term is a function of conserved pair norms. When $N_c=\eta N_a$ and the beats match, the complete density mismatch and its gradient vanish. Away from that condition, its derivatives shift carrier phases and must be retained. This is an explicit route by which a density interaction yields beat locking; spatial projection into the two channels still needs justification.

Locking and Transfer in the Same Energy

For a retained high band with ascending labels, both mechanisms fit in one action:

$$H=\omega_aN_a+\omega_cN_c+\Omega(J_a+n_{c1}) +E_4(a)+g\left|\bar c_0c_1-\eta\sum_{j=0}^{M-2}\bar a_ja_{j+1}\right|^2.$$

The conserved quantities are $N_a,N_c,J_a+n_{c1}$. A geometric high-band branch and a matched low-pair beat make both positive interactions vanish, preserving all bare frequencies. For flat unit-norm high bands of sizes $5,8,13$, $c_0=1$ and $c_1=\eta(M-1)/M$, the complete real Hessian has six neutral directions and $M-1$ nonzero stable frequency pairs at shared $\eta=1/5,2/5,4/5$. This tests coexistence in one energy, rather than transferring conclusions between separate actions.

The combined interaction selects the fundamental beat of the high band. The complete unfiltered density mismatch for a multi-tooth high band contains further beat terms; it is not automatically this reduced energy. Its frequency selection, spatial matrix elements and omitted modes must be derived from the physical reduction. The quartic high-band boundary requirement also remains unless an explicit support-preserving version is used.

5. Occupied-Sector Comparison

Use the unchanged receiver $\chi(x)=[3/(2-x^2)+2/(3-x^2)]/5$ and sector ratios from the complex-comb comparison. For a common flat $M=8$ band, clipped only at $K<M$, define

$$Q^{(0)}=\frac12\sum_{p=1}^{M-1}D_p^2\chi(pr_d),\quad Q^{(1)}=\frac12\sum_{p=1}^{M-1}(pr_d)^2D_p^2\chi(pr_d),\quad r_d=\omega_{0,d}/\omega_{0,6}.$$

These are signed conservative response diagnostics, with static terms separate. They are not charges, probabilities or complete forces. Equal lepton entries are exact consequences of a common envelope, source matrix and supplied equal sector fundamentals. The equality $\mu_{10}=\mu_6$ remains part of the maintained scale prescription; the cavity construction does not derive it. The same mechanism works algebraically at every occupied depth, and the tiny depth-five derivative response uses its supplied slow clock. There is no new preference for the assigned dimensionalities in this temporal mechanism alone.

6. What a Rotating Spoke Pattern Actually Supplies

For an azimuthal wave $\cos(\ell\vartheta-\omega t)$, the pattern rotates at $\omega/\ell$. Its identical spokes still repeat at a fixed point at frequency $\omega$; they do not automatically provide a source at $\omega/\ell$. A squared single order has no fundamental rotation-frequency line, whereas adjacent orders $\ell,\ell+1$ restore it through their difference. The same spectral-support rule controls temporal combs and rotating spatial patterns.

A conditional phase-bunching normal form follows from rotor inertia $I$ and potential $-V\cos\psi$, where $\psi=\ell\vartheta-\omega t$ and $p_\psi=p_\vartheta/\ell$:

$$H_{\rm rot}=\frac{\ell^2}{2I}\left(p_\psi-\frac{I\omega}{\ell^2}\right)^2-V\cos\psi.$$

The stable phase has libration frequency $\ell\sqrt{V/I}$. The positive-cosine interval has width $\pi/\ell$ in each of $\ell$ spokes, so the total angular acceptance fraction is exactly $1/2$, independent of spoke number. More lobes alone do not explain suppressed coupling. The azimuthal order $\ell$ is not identified with the simplex count without a separate derivation.

Verification and Remaining Physical Construction

The single checker and report preserve 1474 checks of exact rational constraints, symbolic ranks, independent complex variations and Fourier coefficients. They include thirty finite locking Hessians, nine envelopes, forty-two comparisons covering all fourteen assigned massive branches at bandwidths five, eight and thirteen, six enclosing-band support controls, twenty-four three-mode and twenty-four four-mode feedback preparations, nine complete combined-action Hessians and thirty rotating-pattern controls. Only occupied depths $2,3,4,5,6,10$ are calculated. Photon optical frequency and bottom's unresolved mass assignment supply no adopted cutoff.

The mathematical advance is an explicit route from adjacent harmonics to a persistent fundamental source and reciprocal resonant transfer, with the upper mass endpoint preserved as input. A physical completion must derive the spatial mode overlaps and modal action, explain occupation and index selection, and account for localization, stress and any sustaining flux. A normalized temporal band by itself does not determine the complete particle energy or translational inertia.

A Local Spatial Wave Prototype

The local cavity-wave action realizes matching different-carrier matrix currents with a positive local potential on a prescribed closed spatial path. Orthogonal polarizations keep its energy uniform and complete loop linear momentum zero while an off-diagonal current retains the fundamental beat. Local energy flux, causal principal speed and complete finite disturbance spectra are checked. A scaling identity shows that the same positive quartic alone cannot confine a time-harmonic particle in flat noncompact space, making the missing support dynamics explicit.

A Dynamically Generated Constant Gap

The coupled deformation-and-rotation action gives two positive branches separated by exactly one clock when their characteristic speeds match. Canonical normalization makes the displacement-channel velocity-current gap vertex exactly one-half; density, total-current and unequal-speed controls differ. This supplies an alternative to assuming an occupied adjacent comb, but does not derive the spatial simplex endpoint or a physical charge. The same action retains a causal fourth-order static response only in its direct displacement channel.

Ordinary Rotation and Spatial Compatibility

The gyroscopic-plane action provides a rotation-based alternative to an occupied adjacent comb: its circular branches have a constant difference frequency under plane isotropy. Extending the flat result over a curved pattern requires spatial-operator compatibility. The exact same-field fast-band completion also tests whether the clock-generating reduction remains valid at the supplied mass carriers.