QTT Velocity: How Motion Emerges from Ticks, World–Cells, and the Reality Dimension

Reference: https://doi.org/10.5281/zenodo.17594186

In school mechanics, velocity is introduced in a very simple way:

$v_{\rm avg} = \frac{\Delta s}{\Delta t}, \qquad v = \frac{ds}{dt}.$

It works incredibly well, but the underlying picture is left vague: time is continuous, space is continuous, and the speed limit $|v|\le c$ is later imposed as a relativistic postulate.

In Quantum Traction Theory (QTT), this story is rebuilt from the ground up. Instead of starting from smooth space and time, QTT starts from:

Planck-scale world–cells in space and in the Reality Dimension $(x,w)$ ,
a ticked Absolute Background Clock $T$ ,
and motion as discrete steps across this combined $(x,w)$ lattice.

In this framework, the familiar velocity formulas become the continuum projection of a more fundamental, capacity–bounded, ticked velocity.

1. World–cells, the Absolute Clock, and the Reality Dimension

QTT discretises both ordinary space and the Reality Dimension into world–cells of size $\tilde\ell$ :

$(x,w) = \tilde\ell\,(n_x, n_w), \qquad n_x, n_w \in \mathbb{Z}.$

The Absolute Background Clock ticks in steps of

$\tilde t = \frac{\tilde\ell}{c},$

so that the maximal signal speed on the ledger is naturally

$c = \frac{\tilde\ell}{\tilde t}.$

Axiom A1 relates the proper time $\tau$ of a bundle and the Absolute Clock $T$ via a two–clock factor $\mathcal{N}$ . In the Einstein gauge (flat background) this becomes:

$d\tau = \mathcal{N}(x^\mu, v)\, dT \approx \sqrt{1 - \frac{v^2}{c^2}}\, dT.$

The full motion of a bundle between ticks $T_n = n\tilde t$ and $T_{n+1}$ is a step in the combined $(x,w)$ space:

$\Delta x_n = x_{n+1} - x_n, \qquad \Delta w_n = w_{n+1} - w_n,$

with both $\Delta x_n$ and $\Delta w_n$ measured in steps of $\tilde\ell$ . The spatial part $\Delta x_n$ sets the visible motion in 3–space; the Reality Dimension part $\Delta w_n$ controls how the internal carrier and proper time respond.

2. QTT velocity per tick: motion across world–cells

On the tick lattice, QTT defines the velocity per tick in the Einstein frame as:

$\mathbf{v}_n := \frac{\Delta \mathbf{x}_n}{\tilde t} = \frac{\mathbf{x}_{n+1} - \mathbf{x}_n}{\tilde t}.$

Capacity and causality on the world–cell lattice impose a maximal spatial step per tick:

$|\Delta \mathbf{x}_n| \le \tilde\ell,$

so the QTT velocity per tick is automatically bounded:

$\boxed{ \mathbf{v}_n = \frac{\Delta \mathbf{x}_n}{\tilde t}, \qquad |\Delta \mathbf{x}_n| \le \tilde\ell, \qquad |\mathbf{v}_n| \le c. }$

In other words:

velocity is literally “how many spatial world–cells you cross per tick,”
and the speed bound $|\mathbf{v}_n| \le c$ is a of the lattice and tick, not an extra postulate.

Motion in the Reality Dimension (nonzero $\Delta w_n$ ) is encoded in the internal dial phase and affects the proper-time factor $\mathcal{N}$ , but the visible 3–velocity $\mathbf v_n$ is the spatial projection of the full $(x,w)$ step.

3. Average and instantaneous velocity from ticks

Over $N$ ticks, from $T_0$ to $T_N$ , the total spatial displacement is

$\Delta \mathbf{s} = \mathbf{x}_N - \mathbf{x}_0 = \sum_{n=0}^{N-1} \Delta \mathbf{x}_n,$

and the elapsed laboratory time is

$\Delta t = N \tilde t.$

The QTT average velocity is then

$\mathbf{v}_{\rm avg} = \frac{\Delta \mathbf{s}}{\Delta t} = \frac{1}{N\tilde t} \sum_{n=0}^{N-1} \Delta \mathbf{x}_n = \frac{1}{N} \sum_{n=0}^{N-1} \mathbf{v}_n.$

Since each $\mathbf v_n$ is bounded by $c$ , the average velocity automatically satisfies $|\mathbf{v}_{\rm avg}| \le c$ as well.

In the smooth limit where many ticks fit into any laboratory time interval ( $\Delta t \gg \tilde t$ ), we can treat time as a continuous parameter $t \simeq n\tilde t$ and define the instantaneous velocity by:

$\mathbf v(t) := \lim_{\Delta t \to 0} \frac{\Delta \mathbf s}{\Delta t} = \frac{d\mathbf s}{dt}, \qquad |\mathbf v(t)| \le c.$

So the familiar derivative definition $\mathbf v = d\mathbf s / dt$ is not fundamental. It is the continuum projection of the ticked motion of a bundle across world–cells, with the speed bound inherited from the QTT lattice.

4. The Reality Dimension and four–velocity

The full motion of a bundle in QTT lives in the combined $(x,w)$ space: the visible 3–space $\mathbf x$ plus the Reality Dimension coordinate $w$ . The Absolute Clock $T$ and proper time $\tau$ are related through the internal dynamics in the Reality Dimension:

$d\tau = \mathcal{N}(x^\mu, v)\, dT.$

Defining the four–position $X^\mu = (ct, \mathbf x)$ and proper time $\tau$ , the QTT four–velocity reads:

$U^\mu = \frac{dX^\mu}{d\tau} = \gamma(v)(c,\mathbf v), \qquad \gamma(v) = \frac{1}{\sqrt{1 - v^2/c^2}}.$

Here:

the Reality Dimension motion (steps in $w$ ) shapes how fast proper time $\tau$ accumulates for each tick $dT$ ,
the visible three–velocity $\mathbf v$ is the spatial projection of the full motion in $(x,w)$ ,
and the usual relativistic structure $U^\mu = \gamma(v)(c,\mathbf v)$ appears as a geometric consequence of the world–cell motion and the two–clock relation.

In other words, the standard four–velocity of special relativity is not assumed; it is what you get when you look at how bundles traverse the $(x,w)$ ledger from the perspective of proper time.

5. Unified QTT velocity law (boxed)

QTT’s description of velocity can be summarised in one unified “velocity ledger”:

$\boxed{ \begin{aligned} \mathbf v_n &= \dfrac{\mathbf x_{n+1} - \mathbf x_n}{\tilde t}, & |\mathbf x_{n+1} - \mathbf x_n| &\le \tilde\ell, & |\mathbf v_n| &\le c, \\[0.4em] \mathbf v_{\rm avg} &= \dfrac{\Delta \mathbf s}{\Delta t} = \dfrac{1}{N}\sum_{n=0}^{N-1} \mathbf v_n, & \Delta t &= N\tilde t, \\[0.4em] \mathbf v(t) &= \dfrac{d\mathbf s}{dt}, & |\mathbf v(t)| &\le c, & U^\mu &= \gamma(v)(c,\mathbf v). \end{aligned} }$

6. What velocity really means in QTT

From the QTT perspective, velocity is no longer a primitive derivative. It is:

the ticked displacement across spatial world–cells per Planck tick $\tilde t$ ,
bounded by the lattice light speed $c = \tilde\ell/\tilde t$ ,
and the visible projection of a deeper motion in the combined $(x,w)$ space that includes the Reality Dimension.

The textbook formulas

$v_{\rm avg} = \frac{\Delta s}{\Delta t}, \qquad v = \frac{ds}{dt}, \qquad |v|\le c$

are still correct — they are just revealed to be the macroscopic shadow of a discrete, capacity–regulated velocity living on the QTT world–cell ledger.

From this point of view, special relativity’s speed limit, four–velocity, and time dilation all arise because of how Reality–Dimension motion and world–cell steps conspire behind the scenes to produce the smooth velocities we measure in the lab.

QTT Velocity: How Motion Emerges from Ticks, World–Cells, and the Reality Dimension

1. World–cells, the Absolute Clock, and the Reality Dimension

2. QTT velocity per tick: motion across world–cells

3. Average and instantaneous velocity from ticks

4. The Reality Dimension and four–velocity

5. Unified QTT velocity law (boxed)

6. What velocity really means in QTT

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1. World–cells, the Absolute Clock, and the Reality Dimension

2. QTT velocity per tick: motion across world–cells

3. Average and instantaneous velocity from ticks

4. The Reality Dimension and four–velocity

5. Unified QTT velocity law (boxed)

6. What velocity really means in QTT

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