2026-09-01
Rotation as a projection of non-accelerating splittings
The operator \(j^{ϖt}\) as the limit of non-accelerating splittings
Let's imagine a setup with a large number of identical linear channels. In each channel, the slider moves only along a short straight line, at a constant speed and without acceleration. Simply adding the slider positions would again produce linear motion. But let's connect the channels differently: the output of each channel will not be added, but rather multiplied by the output of the next.
With two channels, a quadratic trajectory emerges; with four, a fourth-degree polynomial; with eight, an eighth-degree polynomial. Simultaneously, we reduce the speed of each slider so that the total change in the entire system remains the same. As the number of channels doubles, the radial error of their overall projection decreases, the phase approaches \(\omega t\), and the output point moves along a circle with increasing accuracy. In the infinite limit, a set of rectilinear, acceleration-free motions combine to form a precise rotation.
This resembles the image of a circle, emerging not from a single rotating motion, but from the coordinated work of many rectilinear channels. No channel is aware of the circle and has no acceleration; curvature appears only in the law of their multiplicative projection.
In the previous section, the sine and cosine coordinates of circular motion were separately represented by Euler products. Each quadratic factor was constructed from two opposite linear branches, and in the idempotent notation, their union was accomplished using conjugation.
Another version will be constructed below. We start with the full internal operator \(\j^{\varpi t}\), decompose it into identical linear split states, and then re-derive the original operator as the limit of their product. The new scheme requires neither separate sine and cosine expansions, nor opposite branches, nor conjugation.
1. The Original Operator of Internal Motion
Let the first level of the phase expansion be defined by two mutually complementary idempotents:
\[\tag{1} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] The hyperbolic unit is defined by their difference:
\[\tag{2} \j=\ep-\em, \qquad \j^2=1. \] On idempotent components, it acts as two eigenvalues:
\[\tag{3} \j\ep=+\ep, \qquad \j\em=-\em. \] Therefore, for any function defined on these values, the spectral decomposition is valid.
\[\tag{4} f(\j)=\ep f(1)+\em f(-1). \] Choosing the principal branch of the complex power, we have
\[\tag{5} 1^a=1, \qquad (-1)^a=e^{i\pi a}. \] From this it follows directly
\[\tag{6} \boxed{ \j^a=\ep+\em e^{i\pi a}. } \] The internal state of the particle is specified by the parameter
\[\tag{7} a=\varpi t, \qquad \omega=\pi\varpi. \] In the absence of external motion, that is, when \(b=0\), the global operator takes the form
\[\tag{8} \boxed{ J(t)=J(\varpi t,0) =\j^{\varpi t} =\ep+\em e^{i\omega t}. } \] Expanding the complex exponential, we obtain
\[\tag{9} \j^{\varpi t} =\ep +\em\cos(\omega t) +i\em\sin(\omega t). \] Thus, the \(\ep\)-component remains constant, and the \(\em\)-component rotates in the complex phase plane \(\operatorname{span}_{\mathbb R}\{\em,i\em\}\).
2. Acceleration of the Rotating Component
Let's isolate the rotating part of the operator:
\[\tag{10} \Psi(t) :=J(t)-\ep =\em e^{i\omega t}. \] Its first derivative is
\[\tag{11} \dot\Psi(t) =i\omega\em e^{i\omega t}, \] and the second derivative is
\[\tag{12} \boxed{ \ddot\Psi(t) =-\omega^2\Psi(t). } \] Therefore, the original operator contains the usual accelerated rotation. For the complete state
\[\tag{13} \boxed{ \ddot J(t) =-\omega^2\bigl[J(t)-\ep\bigr] =-\omega^2\em e^{i\omega t}. } \] Now we pose the inverse problem: obtain this rotation from states, each of which has a zero second derivative.
3. Binary Splitting Depth
Let \(m=0,1,2,\ldots\) be the splitting depth. We choose the number of elementary channels at this depth to be equal to
\[\tag{14} N=2^m. \] We divide the total phase \(\omega t\) equally between all channels. For each \(k=1,2,\ldots,N\), we introduce a linear split state.
\[\tag{15} \boxed{ L_{m,k}(t) =\ep+\em\left(1+i\frac{\omega t}{2^m}\right). } \] Since \(\ep+\em=1\), it can be written even more briefly:
\[\tag{16} L_{m,k}(t) =1+i\em\frac{\omega t}{2^m}. \] At \(t=0\) each channel startsI'm starting from the unit state:
\[\tag{17} L_{m,k}(0)=\ep+\em=1. \] In the \(\ep\)-component, the state is stationary. In the \(\em\)-component, it moves along the forward direction \(i\em\) with constant velocity.
4. No acceleration in each elementary splitting
Differentiating (15), we obtain
\[\tag{18} \dot L_{m,k}(t) =i\em\frac{\omega}{2^m} =\operatorname{const}. \] Repeated differentiation yields the main, higher-level result:
\[\tag{19} \boxed{ \ddot L_{m,k}(t)=0, \qquad k=1,2,\ldots,2^m. } \] No elementary channel reverses the direction of its velocity. Therefore, in each individual splitting, both the centripetal acceleration and the corresponding centrifugal inertial term are absent.
The condition \(\ddot L_{m,k}=0\) applies to each elementary factor. The product of several factors is already a nonlinear function of time and, in general, has a nonzero second derivative.
5. Multiplicative projection
Let's combine all \(2^m\) elementary states not by addition, but by multiplication:
\[\tag{20} J_m(t) :=\prod_{k=1}^{2^m}L_{m,k}(t). \] Since at the same depth all factors have the same law,
\[\tag{21} J_m(t) =\left[\ep+\em\left(1+i\frac{\omega t}{2^m}\right)\right]^{2^m}. \] The orthogonality of idempotents implies a general rule
\[\tag{22} (\ep+\em z)^N =\ep+\em z^N. \] Therefore, the finite multiplicative projection is equal to
\[\tag{23} \boxed{ J_m(t) =\ep+\em \left(1+i\frac{\omega t}{2^m}\right)^{2^m}. } \] For finite \(m\), this is not yet the exact operator \(\j^{\varpi t}\), but its approximation. Exact recovery will occur after passing to the limit.
6. Return to \(\j^{\varpi t}\)
We use the definition of a complex exponential through the limit:
\[\tag{24} \lim_{N\to\infty} \left(1+\frac{x}{N}\right)^N =e^x. \] With \(x=i\omega t\) and \(N=2^m\) we obtain
\[\tag{25} \lim_{m\to\infty} \left(1+i\frac{\omega t}{2^m}\right)^{2^m} =e^{i\omega t}. \] Therefore
\[\tag{26} \begin{aligned} \lim_{m\to\infty}J_m(t) &=\ep+\em e^{i\omega t} \, &=\j^{\varpi t}. \end{aligned} \] The full law of splitting and back projection takes the form
\[\tag{27} \boxed{ \j^{\varpi t} =\lim_{m\to\infty} \prod_{k=1}^{2^m} \left[ \ep+\em\left(1+i\frac{\omega t}{2^m}\right) \right]. } \] Thus, the original accelerated operator is first replaced by a set of linear non-accelerated states, and then exactly reconstructed in their infinite multiplicative projection:
\[\tag{28} \boxed{ \begin{gathered} \j^{\varpi t} \quad\longrightarrow\quad \{L_{m,1},L_{m,2},\ldots,L_{m,2^m}\}, \\ \ddot L_{m,k}=0, \, \xrightarrow{\quad\Pi_{\times},\, m\to\infty\quad} \j^{\varpi t}. \end{gathered} } \] 7. Acceleration during assembly
For a finite number of channels, we denote
\[\tag{29} J_N(t) =\ep+\em \left(1+i\frac{\omega t}{N}\right)^N. \] The first derivative of the finite projection is
\[\tag{30} \dot J_N(t) =i\omega\em \left(1+i\frac{\omega t}{N}\right)^{N-1}. \] The second derivative is no longer zero:
\[\tag{31} \boxed{ \ddot J_N(t) =-\omega^2 \left(1-\frac1N\right) \em \left(1+i\frac{\omega t}{N}\right)^{N-2}. } \] For the product of arbitrary linear factors \(L_k(t)\), the acceleration is formed by the cross products of their constant velocities:
\[\tag{32} \boxed{ \frac{d^2}{dt^2}\prod_{k=1}^{N}L_k =2\sum_{1\le k < l\le N} \dot L_k\dot L_l \prod_{r\ne k,l}L_r, \qquad \ddot L_k=0. } \] As \(N\to\infty\), expression (31) tends to accelerate the original operator:
\[\tag{33} \boxed{ \lim_{N\to\infty}\ddot J_N(t) =-\omega^2\em e^{i\omega t} =\ddot{\j^{\varpi t}}. } \] The acceleration of the projection arises not from the acceleration of any individual channel, but from the cross-interaction of their constant velocities in the multiplication operation.
8. Coordinate Projection and Return to the Circle
To relate the resulting operator to the circular motion of the previous section, consider the state
\[\tag{34} Q=\ep+\em z, \qquad z\in\mathbb C, \] and define its coordinate projection by the rule
\[\tag{35} \boxed{ \mathcal R[Q] :=\ep R\,\operatorname{Im}z +\em R\,\operatorname{Re}z. } \] For \(Q=\j^{\varpi t}\), the complex coefficient is
\[\tag{36} z=e^{i\omega t} =\cos(\omega t)+i\sin(omega t). \] Therefore,
\[\tag{37} \boxed{ \mathcal R\left[\j^{\varpi t}\right] =\ep R\sin(\omega t) +\em R\cos(\omega t) =\mathbf r(t). } \] Thus, the new scheme first restores the global internal operator and then, using a single coordinate projection, returns the law of motion, which in the previous article was constructed using separate products for the sine and cosine functions.
9. Finite Splitting Error
For finite \(N\), we write the rotating component in polar form:
\[\tag{38} \left(1+i\frac{\omega t}{N}\right)^N =\rho_N(t)e^{i\phi_N(t)}, \] where
\[\tag{39} \rho_N(t) =\left(1+\frac{\omega^2t^2}{N^2}\right)^{N/2}, \qquad \phi_N(t) =N\arctan\left(\frac{\omega t}{N}\right). \] For finite \(N\), the radius is slightly greater than unity, and the phase for \(t>0\) is slightly less than \(\omega t\):
\[\tag{40} \rho_N(t)>1, \qquad \phi_N(t)<\omega t. \] However, over any limited time interval
\[\tag{41} \rho_N(t)\longrightarrow1, \qquad \phi_N(t)\longrightarrow\omega t, \qquad N\to\infty. \] The complex norm of the finite projection also only approaches unity:
\[\tag{42} J_N(t)\overline{J_N(t)} =\ep+\em \left(1+\frac{\omega^2t^2}{N^2}\right)^N \longrightarrow\ep+\em=1. \] This provides a distinctive feature of this hierarchy: if the number of physical levels is finite, consistent radial correction and phase lag, given by formulas (39), should be observed.
10. Why is an infinite limit necessary?
The operator can be exactly factored into a finite number of identical factors:
\[\tag{43} \j^{\varpi t} =\prod_{k=1}^{N} \left[ \ep+\em e^{i\omega t/N} \right]. \] But each such factor already has its own acceleration:
\[\tag{44} \frac{d^2}{dt^2} \left[ \ep+\em e^{i\omega t/N} \right] =-\em\frac{\omega^2}{N^2}e^{i\omega t/N} \ne0. \] Therefore, in the scheme under consideration, it is impossible to simultaneously require a finite number of factors, exact equality to the original operator, and zero acceleration of each factor. The non-acceleration nature of the elementary states is ensured by their linearity, and the exact restoration of rotation is ensured by the infinite limit.
\[\tag{45} \boxed{ \begin{gathered} \text{linearity of each channel} +\text{ infinite multiplicative assembly} \, =\text{ exact accelerated rotation in projection}. \end{gathered} } \] 11. Comparison of two methods
Both constructions lead to the same circular motion, but organize the higher-order representation differently:
| Previous article | This continuation |
|---|---|
| The sine and cosine are constructed separately. | The full operator \(\j^{\varpi t}\) is constructed immediately. |
| Uses Euler products | Uses the limit of the complex exponential |
| Requires opposite branches | All elementary channels have the same direction |
| Applies conjugation | No conjugation required |
| The levels have different coefficients. | At the same depth, all factors are the same. |
| The coordinate law is directly obtained. | First, the internal operator is obtained, then its coordinate projection. |
The first construction is convenient for directly analyzing the coordinates and zeros of the sine and cosine functions. The second is more consistent with the global operator of the model: it preserves its integrity and shows how a single complex rotating component arises from identical linear states.
12. Practical Algorithm
The mathematical construction can already be implemented in a digital or analog model. For a given \(\omega\), operating range \(|t|\le T\), and permissible error, the number of channels \(N=2^m\) is selected. Each channel generates a linear complex signal.
\[\tag{46} L_{N,k}(t) =1+i\em\frac{\omega t}{N}, \qquad \ddot L_{N,k}=0. \] A cascade of multipliers produces an output
\[\tag{47} J_N(t) =\prod_{k=1}^{N}L_{N,k}(t), \] after which the coordinate block \(\mathcal R\) converts it into approximate circle coordinates. The required number of channels can be selected based on the maximum errors. \[\tag{48} \begin{aligned} \varepsilon_{J,N} &=\max_{|t|\le T} \left|J_N(t)-\j^{\varpi t}\right|, \\ \varepsilon_{A,N} &=\max_{|t|\le T} \left|\ddot J_N(t)+\omega^2\em e^{i\omega t}\right|. \end{aligned} \]
This layout is not intended for faster calculation of the exponent: directly calculating \(e^{i\omega t}\) is easier. Its goal is to experimentally and visually demonstrate how nonlinear rotational dynamics emerge at the output of a network consisting only of linearly varying input states and multiplication operations.
13. Possible Future Applications
The first approach is related to distributed oscillation synthesis. A network of identical linear channels and controlled multipliers can be used as a research model in control theory, analog signal processing, and neuromorphic computing. By varying the number of channels and their constant speeds, one can study the occurrence of frequency, phase, curvature, and acceleration at the overall output.
The second approach is a physical hypothesis. If real internal degrees of freedom corresponding to linear split states are ever discovered, it will become possible to check whether the observed acceleration is determined not by the local acceleration of each higher-order channel, but by the law of their joint projection.
In this case, controlling the constant velocities of higher-order states could in the future change the parameters of the observed trajectory:
\[\tag{49} \left\{ \dot L_{N,1},\dot L_{N,2},\ldots,\dot L_{N,N} \right\} \quad\xrightarrow{\, \Pi_{\times}\ } \quad \omega, R, \mathbf A. \] Potential areas of research would then be controlling curvilinear projections, generating specified wave trajectories and the search for new ways to redistribute acceleration between the internal and observable levels of the system. However, the formulas given do not yet imply the ability to eliminate inertial loads, rotate a body without interaction, or create reactionless motion.
To move on to physical application, it is necessary to experimentally establish:
\[\tag{50} \begin{gathered} \text{Do real analogs of linear channels exist?}\\ \text{What process creates their multiplicative relationship?}\\ \text{Is the zero proper acceleration of each channel preserved?}\\ \text{Are finite corrections }\rho_N\text{ and }\phi_N \text{ observed?} \end{gathered} \] Today, the design provides a mathematical algorithm, a digital or analog model, and a specific law of finite deviations. Potential future applications will begin only after the discovery of physical degrees of freedom corresponding to elementary splittings and a way to control their projection.
Conclusions
The complete operator of internal motion
\[\tag{51} J(t)=\j^{\varpi t}=\ep+\em e^{i\omega t} \] is presented as the limit of the product of identical linear splitting states:
\[\tag{52} \boxed{ \j^{\varpi t} =\lim_{m\to\infty} \prod_{k=1}^{2^m} \left[ \ep+\em\left(1+i\frac{\omega t}{2^m}\right) \right]. } \] Each elementary state moves uniformly:
\[\tag{53} \boxed{ \dot L_{m,k} =i\em\frac{\omega}{2^m} =\operatorname{const}, \qquad \ddot L_{m,k}=0. } \] Nevertheless, their multiplicative projection has an acceleration that, in the limit, exactly coincides with the acceleration of the rotating component of the original operator:
\[\tag{54} \boxed{ \lim_{m\to\infty}\ddot J_m(t) =-\omega^2\em e^{i\omega t} =\ddot{\j^{\varpi t}}. } \] Unlike the decomposition of the previous part, the new scheme immediately restores the entire operator, does not use conjugation, and does not require opposite branches. The sine and cosine coordinates appear later as the real and imaginary components of a single complex phase.
The immediate practical application of this result is the construction of mathematical, digital, and analog models of collective nonlinearity. A possible future application is the control of the observed curvature and acceleration through the constant velocities of internal channels, if the physical existence of such channels and the law of their multiplicative projection are confirmed experimentally.
The main result can be formulated briefly: exact accelerated rotation can be represented as the limit of a multiplicative assembly of states, each of which itself moves rectilinearly, uniformly, and has no acceleration. The observed circular motionand does not arise after the coordinate projection of the assembled internal operator.