Periodic motion, preserved mass ratios and reciprocal support

A supporting pattern can keep moving while retaining a common frequency multiplier across a particle family. This can be demonstrated in the endpoint operator already obtained from the spatial carrier. The same calculation also determines the force exerted on that pattern. Preserving the frequency ratios does not automatically make the supporting motion a solution of its own equations.

The results below retain the complete two-state wave evolution, relative phase, a reacting positive geometric oscillator and the temporal frame terms. The reduction from the entire moving normal field is still open. All coordinates describe proposed modes of the same field; no additional interaction particle or particle-exchange mechanism is introduced.

Mass remains the selected particle frequency in equivalent units. A Floquet phase of a prescribed periodically varying subsystem is a candidate frequency, not yet the frequency of a complete particle state. Its branch, other temporal harmonics, supporting energy and inertia must also be identified. No frequency is chosen modulo the drive frequency merely to obtain a desired mass.

The complete periodically varying endpoint

At a fixed depth, rephase the endpoint basis so the reference coupling $t=t_{n,d}>0$ and the shape coefficient $s>0$. The existing radial calculation gives $t_{n,d}$ proportional to $S(n,d)$, while $s$ is independent of the occupied index at leading order. The complex deformation has two real directions. Retain a static displacement $X$ in one and a periodic displacement in the perpendicular direction:

$$i\dot c=H(t_{\rm time})c,\qquad H=(t+sX)\sigma_1+sA\cos(\Omega t_{\rm time})\sigma_2, \qquad c^\dagger c=1. \tag{1}$$

Here the $\sigma_j$ are Hermitian Pauli matrices, $\Omega$ is the geometric oscillation frequency and $A$ is its shape amplitude. The wave amplitude is normalized; it is not used to assign mass. The first calculation prescribes this geometric history in order to compute its wave frequency and reciprocal force.

Use $\tau=\Omega t_{\rm time}$, $\delta=t/\Omega$, $a=sA/\Omega$ and $x_0=sX/\Omega$. At $x_0=0$,

$$i c'=[\delta\sigma_1+a\cos\tau\,\sigma_2]c. \tag{2}$$

Transform with $R(\tau)=\exp[-ia\sin\tau\,\sigma_2]$. Its value is the identity at both ends of a period. Writing $c=Rv$ gives the exact equation

$$i v'=\delta\left[ \cos(2a\sin\tau)\sigma_1+ \sin(2a\sin\tau)\sigma_3\right]v. \tag{3}$$

The norm of this interaction-picture coefficient is $|\delta|$, independently of the drive amplitude. Its average is $\delta J_0(2a)\sigma_1$, where

$$J_0(2a)=\frac1{2\pi}\int_0^{2\pi}\cos(2a\sin\tau)\,d\tau. \tag{4}$$

The analytic signed Floquet branch continuous from the positive $\sigma_1$ state therefore has

$$\boxed{\epsilon=\delta J_0(2a)+O(\delta^3),\qquad \epsilon_{\rm physical}=tJ_0(2a)+O(t^3/\Omega^2).} \tag{5}$$

Its magnitude supplies the positive branch comparison. Away from a zero of $J_0$, a common $a,\Omega$ preserves the leading simplex ratios. The assigned indices at depths 2, 3, 4, 5, 6 and 10 are all checked with one drive per depth. The two bottom entries remain component controls; this does not derive their geometric-mean particle branch.

An exact half-period identity controls the approximation

Let $U_h$ be evolution from zero to $\pi$. The two identities $H(\pi-\tau)=H(\tau)^T$ and $H(\tau+\pi)=\sigma_1H(\tau)\sigma_1$ imply

$$U_h=a_hI-i(b_x\sigma_1+b_z\sigma_3),\qquad a_h^2+b_x^2+b_z^2=1, \tag{6}$$

with real coefficients, and

$$\boxed{ U_{2\pi}=(\sigma_1U_h)^2 =(1-2b_x^2)I-2ib_x(a_h\sigma_1-b_z\sigma_2). } \tag{7}$$

The first identity makes $U_h$ symmetric: transposition reverses the ordered product, and the reflected Hamiltonian restores its original order. The second identity then determines the full period.

For the branch near $\delta=0$,

$$\boxed{\epsilon=\frac{\arcsin b_x}{\pi}.} \tag{8}$$

Its initial polarization is the positive eigenvector of $a_h\sigma_1-b_z\sigma_2$. This remains a regular choice when the small Floquet splitting approaches zero. It avoids extracting a direction by dividing numerical errors by that splitting.

Conjugation by $\sigma_2$ reverses $\delta$, so $b_x$ has only odd powers of $\delta$. The first Dyson coefficient from (3) is $\pi\delta J_0(2a)$. Each order has norm at most $(\pi|\delta|)^k/k!$. Setting $z=\pi|\delta|$ gives

$$|b_x-\pi\delta J_0(2a)|\le\sinh z-z. \tag{9}$$

For $z<\operatorname{arsinh}1$, it follows that

$$\boxed{ |\epsilon-\delta J_0(2a)| \le\frac{\arcsin(\sinh z)-z}{\pi}, \qquad \left|\partial_\delta\epsilon-J_0(2a)\right| \le\frac{\cosh z}{\sqrt{1-\sinh^2z}}-1. } \tag{10}$$

The frequency bound begins at $\pi^2|\delta|^3/3$, and the derivative bound begins at $\pi^2\delta^2$. Both are uniform in the prescribed drive amplitude. They control this complete two-state problem; they do not bound an omitted spatial complement.

Direct evolution, an interaction-picture Dyson calculation and (7) independently agree. Near a zero of $J_0$, the first numerical eigenvector extraction was insufficiently resolved. The refined expansion and the regular formula after (8) resolve that issue. A zero of the leading Bessel factor is not an exact zero at every finite $\delta$.

The static shape force follows from the same Floquet operator

Write the periodic state as $c=e^{-i\epsilon\tau}u(\tau)$. The self-adjoint period operator is $H-i\partial_\tau$, on periodic two-component functions with the cycle-averaged inner product. Differentiating its normalized eigenvalue gives

$$\boxed{ \partial_{x_0}\epsilon=\overline{u^\dagger\sigma_1u}, \qquad \partial_a\epsilon=\overline{\cos\tau\,u^\dagger\sigma_2u}. } \tag{11}$$

The overline denotes an average over the full period. Since $x_0$ enters exactly like $\delta$, equations (5) and (10) give

$$\boxed{ \partial_X\epsilon_{\rm physical} =s[J_0(2a)+O(\delta^2)].} \tag{12}$$

For canonical wave charge $\mathcal Q$, the generalized force is minus $\mathcal Q$ times (12). The positive-frequency branch changes the appropriate sign but leaves its leading nonzero magnitude.

Thus a nonzero common frequency multiplier also leaves a nonzero common static source. Cancelling that source by setting the multiplier to zero removes the leading mass frequency. This is a limitation of the common analytic driven branch, not a theorem excluding every self-consistent periodic spatial solution. A different supporting contribution can enter the static shape equation, but must be derived and counted.

Cancelling one temporal harmonic does not cancel the full source

The complete leading source perpendicular to the original coupling is

$$\boxed{ \langle\sigma_2\rangle =-4\delta\sum_{k=1,3,5,\ldots} \frac{J_k(2a)}{k}\cos(k\tau)+O(\delta^3). } \tag{13}$$

This follows by integrating the Bloch equation $\langle\sigma_2\rangle'=-2\delta\langle\sigma_3\rangle$ using the zeroth-order rotated polarization. Half-period symmetry sets the constant of integration to zero. The odd Bessel harmonics are the Fourier coefficients of $\sin(2a\sin\tau)$.

At the first nonzero root of $J_1(2a)$, $a\simeq1.91585$, the fundamental amplitude source vanishes at this order while $J_0(2a)\simeq-0.40276$ remains nonzero. However the third and higher force harmonics survive. They cannot all vanish for a nonzero $a$, since that would require $\sin(2a\sin\tau)$ to vanish for every $\tau$. The static source in (12) also remains.

The time-averaged signed wave energy has the exact derivative expression

$$\overline E_{\rm wave}/\Omega =\delta\,\partial_\delta\epsilon+a\,\partial_a\epsilon, \tag{14}$$

and hence, for unit phase norm,

$$\boxed{\overline E_{\rm wave} =t[J_0(2a)-2aJ_1(2a)]+O(t^3/\Omega^2).} \tag{15}$$

A Floquet phase alone therefore does not account for the wave's energy exchange with its support. A signed partial energy is not negative particle inertia. The complete physical state must satisfy the positive inertia requirement with its sustaining field included.

Allow the geometric oscillator to react

For a stated local test, give the real geometric mode a positive kinetic coefficient $\mathcal M$ and a harmonic restoring frequency $\Omega_g$. Normalize $c^\dagger c=1$, retaining its arbitrary complex polarization, and use

$$L=\mathcal M(\dot y^2-\Omega_g^2y^2) +\frac{i\mathcal Q}{2}(c^\dagger\dot c-\dot c^\dagger c) -\mathcal Qc^\dagger(t\sigma_1+sy\sigma_2)c-E_0. \tag{16}$$

This is an inverse action for retained modes. It does not derive the oscillator or its restoring frequency from the full spatial field. In particular, the earlier curvature calculation supplies no leading pinning coefficient at depths two and three.

With $\tau=\Omega_gt_{\rm time}$, $z=sy/\Omega_g$, $\delta=t/\Omega_g$ and $\lambda=\mathcal Qs^2/(2\mathcal M\Omega_g^3)$, the exact equations are

$$\boxed{i c'=(\delta\sigma_1+z\sigma_2)c, \qquad z''+z=-\lambda c^\dagger\sigma_2c.} \tag{17}$$

The complete conserved energy, apart from $E_0$, is

$$\frac{E-E_0}{\mathcal Q\Omega_g} =\frac{z'^2+z^2}{2\lambda} +c^\dagger(\delta\sigma_1+z\sigma_2)c \ge-|\delta|-\frac\lambda2. \tag{18}$$

The inequality uses the fixed wave norm and completes the positive geometric square. The wave and geometry are integrated together at $\lambda=0.1,10,100$, without prescribing a cosine trajectory. Their common norm and full energy are conserved. Increasing reciprocal response changes the trajectory substantially. These integrations do not establish a periodic particle or balance the separate static deformation $X$.

For a harmonic geometric motion of dimensionless amplitude $a$, its positive energy obeys the simple identity

$$\boxed{\frac{E_{\rm oscillator}}{\mathcal Q t} =\frac{a^2}{2\lambda\delta}.} \tag{19}$$

At an appreciable drive amplitude, treating $\lambda\delta$ as a small reciprocal-response parameter has a supporting-energy cost. This is not an argument for discarding the geometric kinetic energy.

The original geometric normalization tightens that comparison

When the metric comes from the previously declared positive geometric action, the normalization calculation gives $\mathcal M=E_0M_d/(2\pi d\Lambda^2)$. Here $M_d$ is its horizontal shape metric, $\Lambda=R/L$, $s=\gamma|\kappa_d|$ and $\gamma=g\Lambda$. Put $e_0=E_0/(\mathcal Q t)$. For drive frequency $\Omega$,

$$\lambda\delta= \frac{\pi d\Lambda^2\gamma^2|\kappa_d|^2} {e_0M_d\Omega^4}. \tag{20}$$

The leading normal carrier has interior frequencies no larger than $\gamma\sqrt d$. If the drive is required to be slower by a factor $\eta<1$, then the necessary window $\Omega\le\eta\gamma\sqrt d$ implies

$$\boxed{\lambda\delta\ge \frac{\pi|\kappa_d|^2}{e_0M_d\eta^4 d g^2}.} \tag{21}$$

The separation length cancels from this bound. Both $g=0.02,0.05$ and all six depths are checked using the complete dressed endpoint coefficients. A chosen weak-response budget also imposes a lower bound on $e_0$. This is the cost of those stated approximation conditions, not an exclusion of an exact strongly reacting state. Drive-induced mixing with the spatial complement still needs its own estimate; frequency separation alone is insufficient.

Exact temporal transport is a separate possible structure

Let $M(q)=U(q)M_0U(q)^\dagger$ for a specified unitary mode frame. Instantaneous eigenvalues alone do not determine the actual evolving frequencies. With

$$\mathcal A=\dot U U^\dagger,\qquad D_t=\partial_t-\mathcal A, \tag{22}$$

the positive inverse action

$$L=\|D_t\psi\|^2-\|M(q)\psi\|^2 +\frac I2(\dot q^2-\Omega_g^2q^2)-E_0 \tag{23}$$

reduces exactly under $\psi=Uv$ to fixed wave oscillators plus the positive geometric oscillator. Every eigenfrequency of $M_0$ is preserved under the complete transport. Removing $\mathcal A$ instead changes the frequencies, despite leaving the instantaneous spectrum unchanged. Independent full-period integrations verify both statements for the maintained frequency controls.

This is a useful explicit frequency-preserving structure, with its full time derivatives retained. It is not yet a derived IDWT support. A chosen unitary mixing of modes need not be a physical spatial deformation. If $q$ is only a component-frame choice, it is not physical motion. If it describes a physical geometric mode, its connection, inertia and energy must arise from that mode's action. In (23) the geometric energy is independent of the wave frequency, so their required physical relation has not been selected.

Verification

periodic_endpoint_checks.py passes 193 checks of full-period evolution, reciprocal derivatives, force harmonics, energy and autonomous local motion. Its periodic report preserves the raw trajectories and exact executed sources.

periodic_symmetry_checks.py passes 240 independent half-period, regular-polarization and uniform-bound checks. Its symmetry report tests the analytic bounds at small gaps and at substantially larger frequency ratios.

spectral_transport_checks.py passes 147 checks of complete positive temporal transport and the geometric normalization bound. Its transport report keeps the assumed mode frame distinct from a physical geometry.