Distinct Endpoint Clocks and the Dynamic Shape Response

The static shape calculation applies to slow neutral deformation data. If the endpoints instead drive those variables at their own particle frequencies, the same kinetic law supplies a different response. Keeping those clocks distinct exposes a precise condition under which the static competing-range picture is a valid reduction.

1. Retain the Time Derivative

For either of the two derived shape channels, the quadratic action is

$$L_z=\frac12\int_0^L \left(\|z_t\|^2-\|z_x\|^2-\langle z,\mathcal A z\rangle\right)dx-TL, \qquad \mathcal A\ge\mu^2I. \tag{1}$$

The ordinary wave equation is z_tt-z_xx+A z=0. Its integrated energy and longitudinal energy flux obey

$$e=\frac12(\|z_t\|^2+\|z_x\|^2+\langle z,\mathcal A z\rangle), \qquad S_x=-\langle z_t,z_x\rangle, \qquad \partial_t e+\partial_x S_x=0. \tag{2}$$

Consequently changing endpoint data can do work. A force derived from an arbitrary stored-energy derivative at fixed drive frequency need not be the actual stress force.

2. One Spectral Channel Has an Exact Boundary Response

In a transverse spectral channel A=lambda^2, write the real field as z=Re[Z(x)e^(-i omega t)] and set alpha^2=lambda^2-omega^2. Its amplitude satisfies Z_xx=alpha^2 Z. The same hyperbolic boundary solution applies with K replaced by alpha. Define its outward normal derivative matrix by

$$\begin{pmatrix}-Z_x(0)\\Z_x(L)\end{pmatrix} =D_\omega(L)\begin{pmatrix}G_1\\G_2\end{pmatrix},\qquad D_\omega=\alpha \begin{pmatrix}\coth(\alpha L)&-\operatorname{csch}(\alpha L)\\ -\operatorname{csch}(\alpha L)&\coth(\alpha L)\end{pmatrix}. \tag{3}$$

For omega<lambda the response decreases over alpha^(-1), which grows as omega approaches the transverse gap. At omega=lambda the regular limit is L^(-1)[[1,-1],[-1,1]]. For omega>lambda, put p=sqrt(omega^2-lambda^2); coth and csch become cot and csc. The response is then oscillatory in L, with standing-wave poles at pL=j pi. A static exponential force cannot be transferred through those thresholds.

The averaged action is -TL-W, with

$$W=\frac14 G^\dagger D_\omega G =\frac\alpha4\left[(|G_1|^2+|G_2|^2)\coth(\alpha L) -2\operatorname{Re}(G_1^*G_2)\operatorname{csch}(\alpha L)\right]. \tag{4}$$

For a real frequency away from poles, the independently calculated mean pressure is

$$\overline P=\frac14\left(|Z_x|^2+(\omega^2-\lambda^2)|Z|^2\right), \qquad \overline F_{\rm in}=T-\overline P=T+\partial_L W. \tag{5}$$

In contrast, the mean stored shape energy is

$$\overline E_z=W+\frac{\omega^2}{2}\int_0^L|Z|^2dx. \tag{6}$$

The additional term accounts for why differentiating that stored energy at fixed forced frequency gives a different answer. At omega=0 a constant real preparation has the static normalization rather than the harmonic time-average factor of one half.

3. Two Different Clocks Are Not a Locked Preparation

If the first endpoint drives at omega_1 and the second at omega_2, linearity gives one solution at each frequency. In a long-time average with omega_1!=omega_2, every bilinear cross term vanishes. This includes the orientation-sensitive cross force and cross torque. Each endpoint's own response remains; it is not zero coupling.

For example, below a channel's gap, the two one-ended responses give

$$\overline F_{\rm in}=T- \frac{|G_1|^2\alpha_1^2}{4}\operatorname{csch}^2(\alpha_1L) -\frac{|G_2|^2\alpha_2^2}{4}\operatorname{csch}^2(\alpha_2L), \qquad \alpha_i^2=\lambda^2-\omega_i^2>0. \tag{7}$$

This is again a short-range outward response, but its range depends on each drive frequency and its averaged form has no relative-orientation term. For a channel above gap the analogous one-ended pressure is positive and proportional to p_i^2 csc^2(p_i L), with no exponential long-distance bound. Finite observation windows retain beats; locking or frequency conversion requires nonlinear dynamics and cannot be inserted into this linear result.

4. The Actual Transverse Continuum Sets a Radiation Threshold

For the established guide, A=-Delta_y+mu^2+mu^2 v_n(mu y)^2. The profile potential is bounded and tends to zero. A plane wave multiplied by an increasing smooth cutoff, translated sufficiently far from the guide, gives a Weyl sequence at every lambda^2=mu^2+|k|^2. Together with A>=mu^2, this shows spec(A)=[mu^2,infinity). The continuum is not a sequence of invented localized transverse oscillators.

For maintained siblings, the frequencies are distinct integer multiples of their shared sector fundamental. The cross terms therefore average to exactly zero over one complete fundamental period, for any initial phases. No irrational-frequency or random-phase assumption is needed.

The finite-segment homogeneous-boundary spatial operator is

$$H_L=-\partial_x^2+\mathcal A,\qquad \operatorname{spec}(H_L)= \left[\mu^2+\frac{\pi^2}{L^2},\infty\right). \tag{8}$$

Its longitudinal sine modes have j>=1, and their continuum intervals are nested. Below the displayed threshold the forced boundary problem has a bounded real resolvent. At its threshold that bound fails. Above it, generic endpoint profiles can excite transverse radiation; a finite-total-energy periodic standing response cannot be assumed. Exceptional spectral projections may avoid a channel. One must use the causal outgoing response and account for field energy leaving the endpoint region.

The causal limit can be checked before adopting any damping model. Replace omega by omega+i eta with eta>0 in (3). Green's identity gives

$$\operatorname{Im}(G^\dagger D_{\omega+i\eta}G) =-2\omega\eta\int_0^L|Z|^2dx\le0. \tag{9}$$

The corresponding boundary input power is nonnegative. Eta is a causal resolvent regulator, not an added IDWT dissipation coefficient. In the continuum limit the outgoing radiation can retain power as eta tends to zero. A real off-resonance closed scalar channel instead has zero mean input power, while retaining oscillatory source work during each cycle.

5. What This Says About a Fundamental-Frequency Coupling Hypothesis

A phase-invariant shape bilinear of a single-frequency pattern,

$$\Psi=e^{-i\omega_i t}\phi_i,\qquad \Psi^\dagger M\Psi=\phi_i^\dagger M\phi_i, \tag{10}$$

can be static while its carrier retains its own mass-frequency. Thus slow shape coupling is compatible with distinct clocks without imposing their synchronization. The action must still derive which bilinear supplies the retained deformation and its restoring coefficient. Phase invariance alone does not derive Omega or its normalization.

If, as a labeled hypothesis, Omega equals the sector fundamental and an endpoint directly drives the shape at omega_i=Omega S_i, equation (8) gives the open-channel condition

$$S_i^2\ge (1-t)\left[1+\frac{\pi^2}{(\mu L)^2}\right]. \tag{11}$$

This distinguishes a neutral slow response from direct forcing by the particle's high count harmonic. The latter generally crosses the gap and cannot use the static short-range formula unchanged. It does not establish that nature couples exclusively at the fundamental.

The same occupied-sector guide controls remain: only three/four carry the assumed guide; two/five/six have the inherited zero gate, and ten has no independent normal connection in the selected parent. No new depth-five suppression, color assignment, particle mass operator or observed force is inferred from the boundary response alone.

Reproducibility

The companion finite-guide-clocks.py verifies the boundary solution, mean action versus stored energy, independent pressure, threshold limit, causal Green identity, unequal-clock averaging and all maintained count comparisons under one declared fundamental-clock control. Its finite spectral channels are identity checks, not transverse mode selections.

Verification script and recorded results.