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

Newton’s second law is usually presented as a compact rule: force is the rate at which momentum changes. In Quantum Traction Theory (QTT), that rule is not discarded. It is placed on a Planck-tick substrate, where force becomes the capacity flux that rephases a fixed internal carrier and changes visible momentum only by bounded tick-wise updates.

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

The familiar continuum statement is

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

Classically, this can read like a definition: force is whatever changes momentum, while inertia is simply assumed. QTT sharpens the statement by deriving the visible momentum update from an internal Planck-scale carrier. Newton’s law then reappears as the smooth, low-velocity limit of a discrete and bounded tick law.

1. Planck Carrier and Acceleration Bound

QTT begins with a massive bundle carried by an internal circular motion in the Reality Dimension. The carrier has a fundamental step length $\tilde\ell$ and a fundamental tick $\tilde t$ , with light speed set by their ratio:

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

Because the hidden carrier moves at speed $c$ around radius $\tilde\ell$ , its built-in curvature defines a maximal logical acceleration:

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

This is the acceleration needed to change a visible speed from $0$ to $c$ in one substrate tick. Visible acceleration is therefore finite and locally bounded. On the tick lattice, QTT writes

\begin{aligned}T_n=n\tilde t,\qquad n\in\mathbb{Z}_{\ge 0},\\[0.35em]\mathbf{v}_n=\frac{\mathbf{x}_{n+1}-\mathbf{x}_n}{\tilde t},\qquad |\mathbf{v}_n|\le c,\\[0.35em]\mathbf{a}_n=\frac{\mathbf{v}_{n+1}-\mathbf{v}_n}{\tilde t}=\alpha_n a_\ast,\qquad \alpha_n\in[-1,1].\end{aligned}

The consequence is immediate: QTT does not permit infinite visible acceleration. Motion is assembled from finite changes per Planck tick.

2. Momentum from the Dial Action

In QTT, momentum is not taken as the primitive product $m\mathbf{v}$ . It is obtained from the dial action of a mass $m$ :

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

The corresponding relativistic Lagrangian and canonical momentum are

\begin{aligned}L(\mathbf{v})=-mc^2\sqrt{1-\frac{v^2}{c^2}},\\[0.35em]\mathbf{p}=\frac{\partial L}{\partial \mathbf{v}}=\gamma m\mathbf{v},\qquad \gamma=\frac{1}{\sqrt{1-v^2/c^2}}.\end{aligned}

At tick $T_n=n\tilde t$ , the visible momentum is therefore $\mathbf{p}_n=\gamma_n m\mathbf{v}_n$ . In the low-velocity regime $v\ll c$ , this reduces to $\mathbf{p}_n\simeq m\mathbf{v}_n$ . Momentum is the coarse-grained record of how capacity flow has rephased the internal carrier.

3. QTT Force as a Tick-Wise Momentum Update

QTT defines force at tick $n$ as the finite momentum update per substrate tick:

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

Substituting $\mathbf{p}_n=\gamma_n m\mathbf{v}_n$ gives the Newtonian form as the low-velocity approximation:

\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)

Since $|\mathbf{a}_n|\le a_\ast$ , force has a local upper bound for a bundle of mass $m$ :

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

For a Planck-mass bundle,

the bound becomes the Planck force scale:

In QTT language, $F_P$ is not just a dimensional combination. It is the maximal mechanical capacity flux available to one world-cell.

4. Continuum Limit and Newton’s Law

Laboratory motion averages over enormous numbers of substrate ticks. The tick index $n$ becomes an effectively continuous time parameter, and the finite update becomes the derivative:

\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 becomes the textbook law:

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

The continuum limit smooths the ticks, but it does not erase the underlying bounds:

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

Thus $\mathbf{F}=m\mathbf{a}$ is the low-velocity, many-tick shadow of a discrete Planck-bounded momentum ledger.

5. Unified QTT Force Law

The tick law can be summarized in one aligned form:

\boxed{\begin{aligned}\mathbf{F}_n=\frac{\mathbf{p}_{n+1}-\mathbf{p}_n}{\tilde t},\qquad \mathbf{p}_n=\gamma_n m\mathbf{v}_n,\\[0.45em]\mathbf{a}_n=\frac{\mathbf{v}_{n+1}-\mathbf{v}_n}{\tilde t},\qquad |\mathbf{v}_n|\le c,\\[0.45em]|\mathbf{a}_n|\le a_\ast=\frac{c^2}{\tilde\ell},\qquad |\mathbf{F}_n|\le m a_\ast.\end{aligned}}

This compact form keeps the tick-wise definitions and their bounds in the same momentum ledger.

In the continuum approximation, the same structure reads

\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 Means in QTT

In QTT, force is not an unexplained push or pull added to motion from outside. It is:

the capacity flux that rephases the internal $S^1$ carrier;
the finite tick-wise update of visible momentum;
and a bounded projection of the fixed internal curvature $a_\ast$ .

The classical relation $\mathbf{F}=m\mathbf{a}$ remains valid where it works, but QTT gives it a deeper substrate meaning: force is a Planck-bounded, quantized change of momentum per tick, caused by redirecting fixed internal curvature into visible motion in space.

QTT therefore preserves Newton’s second law as an excellent macroscopic approximation while grounding it in a finite tick law for how momentum can change.

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