QTT Force: When Newton’s Second Law Becomes a Planck-Bounded Tick Law

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

In every physics course, we meet force as a slogan:

$\sum \mathbf{F} = \frac{d\mathbf{p}}{dt}, \qquad \sum \mathbf{F} = m\mathbf{a}$

It works brilliantly, but in the classical story it’s really just a definition: “force is whatever changes momentum,” and “inertia is just there.”

In Quantum Traction Theory (QTT), this picture is sharpened. Force is no longer a primitive concept. Instead, it becomes:

the capacity flux that rephases a fixed internal Planck-scale carrier, changing the visible momentum per tick.

From this tick-based viewpoint, Newton’s law reappears as the smooth, low-velocity shadow of a deeper, discrete and bounded update rule.

1. The Planck carrier: a built-in curvature and a maximal acceleration

QTT starts from the idea that each massive bundle is carried by an internal circular motion in the Reality Dimension, with a fundamental step length and tick:

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

This internal carrier has radius $\tilde\ell$ and speed $c$ , so its hidden centripetal curvature (a “built-in acceleration”) is

$a_\ast = \frac{c^2}{\tilde\ell} = \frac{c}{\tilde t}.$

QTT interprets $a_\ast$ as the maximal logical acceleration: it is the acceleration required to change a visible speed from $0$ to $c$ in a single substrate tick $\tilde t$ . Visible accelerations are never bigger than this. On the tick lattice

$T_n = n\tilde t, \qquad n \in \mathbb{Z}_{\ge 0},$

we define tick-wise velocity and acceleration in the Einstein gauge:

$v_n := \frac{x_{n+1} - x_n}{\tilde t}, \qquad |v_n| \le c,$ $a_n := \frac{v_{n+1} - v_n}{\tilde t}.$

QTT then writes this as a projection of the Planck curvature:

$a_n = \alpha_n a_\ast, \qquad \alpha_n \in [-1,1], \qquad |a_n| \le a_\ast.$

There is no such thing as “infinite acceleration” here: all visible motion is built from finite, bounded changes per tick.

2. Momentum from the dial action

Momentum in QTT is not defined by $m\mathbf{v}$ ; it is derived from the dial action. The free action of a mass $m$ is

$S_{\rm free} = - \int mc^2\, d\tau = - \int mc^2 \sqrt{1 - \frac{v^2}{c^2}}\; dT.$

This yields the familiar relativistic Lagrangian

$L(v) = -mc^2 \sqrt{1 - \frac{v^2}{c^2}},$

and the canonical momentum

$\mathbf{p} = \frac{\partial L}{\partial \mathbf{v}} = \gamma m \mathbf{v}, \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$

Evaluated at tick $T_n = n\tilde t$ :

$\mathbf{p}_n = \gamma_n m \mathbf{v}_n.$

In the low-velocity regime $v \ll c$ , this reduces to the Newtonian approximation $\mathbf{p}_n \simeq m\mathbf{v}_n$ . Momentum is thus the coarse-grained record of how the internal carrier has been rephased by capacity flows.

3. QTT force = tick-wise change of momentum

QTT now defines the force on a bundle at tick $n$ as the tick-wise momentum update:

$\mathbf{F}_n := \frac{\mathbf{p}_{n+1} - \mathbf{p}_n}{\tilde t}.$

Using $\mathbf{p}_n = \gamma_n m \mathbf{v}_n$ and the definition of acceleration, we obtain

$\mathbf{F}_n = m\frac{\mathbf{v}_{n+1} - \mathbf{v}_n}{\tilde t} + \mathcal{O}\!\left(\frac{v^2}{c^2}\right) = m\mathbf{a}_n + \mathcal{O}\!\left(\frac{v^2}{c^2}\right).$

Because $|\mathbf{a}_n| \le a_\ast$ , we get an absolute local bound on force:

$|\mathbf{F}_n| \le m a_\ast = m\frac{c^2}{\tilde\ell}.$

For a Planck mass bundle

$m_P = \frac{\hbar}{c\tilde\ell},$

this becomes the Planck force scale

$F_P = m_P a_\ast = \frac{c^4}{G}.$

In QTT language, $F_P$ is not just a dimensional curiosity; it is the maximal mechanical capacity flux per world–cell.

4. The continuum limit: recovering Newton’s law

On laboratory scales we average over many ticks, so that $t \simeq n\tilde t$ becomes effectively continuous and

$\mathbf{F}(t) = \lim_{\Delta t \to 0} \frac{\Delta \mathbf{p}}{\Delta t} = \frac{d\mathbf{p}}{dt}, \qquad \mathbf{p} = \gamma m \mathbf{v}.$

For $v \ll c$ and constant $m$ this reduces to the familiar Newtonian form:

$\mathbf{F}(t) \simeq m\frac{d\mathbf{v}}{dt} = m\mathbf{a}(t).$

But QTT insists that even in this smooth limit the hidden bounds remain:

$|\mathbf{a}(t)| \le a_\ast = \frac{c^2}{\tilde\ell}, \qquad |\mathbf{F}(t)| \le m a_\ast.$

So the textbook law $\mathbf{F} = m\mathbf{a}$ is now seen as:

the low-velocity, continuum shadow of a discrete, capacity-bounded momentum update at the Planck tick.

5. The unified QTT force law (boxed)

We can summarise the QTT force law in a single “ledger-friendly” set of equations:

$\boxed{ \begin{aligned} \mathbf{F}_n &= \dfrac{\mathbf{p}_{n+1} - \mathbf{p}_n}{\tilde t}, & \mathbf{p}_n &= \gamma_n m \mathbf{v}_n, & \mathbf{a}_n &= \dfrac{\mathbf{v}_{n+1} - \mathbf{v}_n}{\tilde t}, \\[0.4em] |\mathbf{v}_n| &\le c, & |\mathbf{a}_n| &\le a_\ast = \dfrac{c^2}{\tilde\ell}, & |\mathbf{F}_n| &\le m a_\ast. \end{aligned} }$

In the continuum limit:

$\mathbf{F} = \frac{d\mathbf{p}}{dt}, \qquad \mathbf{p} = \gamma m\mathbf{v}, \qquad \mathbf{F} \simeq m\mathbf{a}\quad(v\ll c).$

6. What “force” really means in QTT

In QTT language, force is not a mysterious “push” or “pull” that we add by hand. It is:

the capacity flux that rephases the internal S $^1$ carrier,
the tick-wise update of momentum on the Planck lattice,
and a quantity that is bounded by the fixed internal curvature $a_\ast$ .

The classical law $\mathbf{F} = m\mathbf{a}$ is still there — it’s just not the deepest layer anymore. Underneath it, QTT reveals a more fundamental picture:

force is a Planck-bounded, quantised change of momentum per tick, caused by redirecting a fixed internal curvature into visible motion in space.

From this perspective, Newton’s Second Law looks less like a postulate and more like a large-scale approximation

QTT Force: When Newton’s Second Law Becomes a Planck-Bounded Tick Law