Legacy field note reviewed · 2025-11-22 · upgraded 2026-06-03

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

QTT

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LexiconEndurance currentCapacity-current reading of gravitational coupling.Two UniversesGravity rowsNo infinite singularities; finite capacity per cell.Derivation AtlasG branchWhere G is treated as derived rather than primitive.Blog MapGravity routeRelated field notes on inertia, E=mc^2, and GR shadows.

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Current category: Gravity, endurance, inertia, and energy accounting

Book pages: p. 199, p. 206, p. 208, p. 697, p. 1168

DOI anchors:
10.5281/zenodo.20042843
10.5281/zenodo.20057430
10.5281/zenodo.20059779

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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: 10.5281/zenodo.17527179

The familiar continuum statement is

\displaystyle \sum F=(dp)/(dt), \sum F=ma

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 t̃, with light speed set by their ratio:

\displaystyle \tilde{\ell}, \tilde{t}, c=(\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:

\displaystyle a_{\mathrm{ast}}=(c^{2})/(\tilde{\ell})=(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

\displaystyle \begin{aligned}T_{n}=n\tilde{t}, ninZ_{\ge} 0,\\[0.35em]v_{n}=(x_{n}+1-x_{n})/(\tilde{t}), |v_{n}|\le c,\\[0.35em]a_{n}=(v_{n}+1-v_{n})/(\tilde{t})=\alpha_{n} a_{\mathrm{ast}}, \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 mv. It is obtained from the dial action of a mass m:

\displaystyle S_{free}=-\int mc^{2} d\tau=-\int mc^{2}\sqrt{1-(v^{2})/(c^{2})} dT

The corresponding relativistic Lagrangian and canonical momentum are

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

At tick $T_{n}=n\tilde{t}$ , the visible momentum is therefore $p_{n}=\gamma_{n} mv_{n}$ . In the low-velocity regime v≪ c, this reduces to $p_{n} \simeq mv_{n}$ . 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:

\displaystyle F_{n}:=(p_{n}+1-p_{n})/(\tilde{t})

Substituting $p_{n}=\gamma_{n} mv_{n}$ gives the Newtonian form as the low-velocity approximation:

\displaystyle F_{n}=m(v_{n}+1-v_{n})/(\tilde{t})+O((v^{2})/(c^{2}))=ma_{n}+O((v^{2})/(c^{2}))

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

\displaystyle |F_{n}|\le m a_{\mathrm{ast}}=m(c^{2})/(\tilde{\ell})

For a Planck-mass bundle,

\displaystyle m_{P}=(\hbar)/(c\tilde{\ell})

the bound becomes the Planck force scale:

\displaystyle F_{P}=m_{P} a_{\mathrm{ast}}=(c^{4})/(G)

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:

\displaystyle F(t)=lim_\Delta t \to 0(Deltap)/(\Delta t)=(dp)/(dt), p=\gamma mv

For v≪ c and constant m, this becomes the textbook law:

\displaystyle F(t) \simeq m(dv)/(dt)=ma(t)

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

\displaystyle |a(t)|\le a_{\mathrm{ast}}=(c^{2})/(\tilde{\ell}), |F(t)|\le m a_{\mathrm{ast}}

Thus $F=ma$ 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:

\displaystyle \begin{aligned}F_{n}=(p_{n}+1-p_{n})/(\tilde{t}), p_{n}=\gamma_{n} mv_{n}, [0.45em]a_{n}=(v_{n}+1-v_{n})/(\tilde{t}), |v_{n}|\le c, [0.45em]|a_{n}|\le a_{\mathrm{ast}}=(c^{2})/(\tilde{\ell}), |F_{n}|\le m a_{\mathrm{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

\displaystyle F=(dp)/(dt), p=\gamma mv, F \simeq ma (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 $a_{\mathrm{ast}}$ .

The classical relation $F=ma$ 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.

Book pages

Where this field note sits in the QTT Main Book (v10.01)

Use these page anchors to read the surrounding derivation in the current book version. The stable book DOI is 10.5281/zenodo.17527179.

pp. 198-216
Endurance current and gravity
Newtonian and Einstein-Hilbert shadows from the sink ledger
pp. 1181-1182
Endurance, gravity, and inertia
compact companion-paper equation ledger
p. 695
Origin of inertia
the indexed compact location for the inertia discussion
pp. 43-48
Reality Dimension and Access Law
the modern reading of early STR/reality-language posts

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QTT

Concept DOI is the citation target. The latest version under the concept family speaks. The full live index is the QTT DOI Map.

Book
Artian Geometry & Quantum Traction Theory
Main book record and ontology map; the stable citation anchor for the whole corpus.
Concept DOI: 10.5281/zenodo.17527179

Gravity
Einstein-Hilbert Coefficient from an Endurance Ledger
The endurance-ledger derivation of the Einstein-Hilbert coefficient used by gravity and curvature notes.
Concept DOI: 10.5281/zenodo.20042843

Gravity
Newton's Constant Is Not Primitive in QTT
Newton's constant current record: the six-face Artian G ledger, including the photon-edge/Fermi face and the dimensional-ruler guardrail. Cite the concept DOI for G as derived rather than primitive.
Concept DOI: 10.5281/zenodo.20057430

Gravity
Artian Inertial Mass Operator and Three-Readout Spectral Equivalence Theorem
Citable QTT source used by this field note.
Concept DOI: 10.5281/zenodo.20059779