Foundation F: Quark Model of Semiotic Concepts in Classical Mechanics

Author: Eist Alexander Georgievich / ORCID 0009-0006-3273-7770
Date: March 2, 2026
Version: 2.0 (Final for Publication)
Zenodo Community: phmv
DOI: [to be assigned]

Abstract

This paper applies the First Law of Nature Semiotics (Foundation B) to the structure of Classical Mechanics. It demonstrates how physical concepts (mass, path, time) function as a metalanguage for constructing more complex notions. Borrowing the quark model from particle physics, we introduce Elementary Mechanical Quarks (EMQs) – mass (m), path (S), time (t) – and Compound Mechanical Quarks (CMQs) as their combinations with various exponents. A novel definition of spin is introduced: the spin of a CMQ equals the number of independent dimensions (parameters) required to fully describe the quantity in a multidimensional space. This connects the quark model directly to the multidimensional mathematics of PTV (Core D). A systematic three-dimensional classification table is constructed, revealing all possible mechanical quantities and opening possibilities for discovering new, useful concepts. This work serves as Foundation F for the Physical Theory of Value (PTV), illustrating how semiotic principles and dimensional analysis can systematically generate and organize knowledge.

Keywords: Semiotics, Classical Mechanics, Quark Model, Dimensional Analysis, Spin as Dimensionality, Metalanguage, Physical Theory of Value, Foundation F, Multidimensional Systems

1. Introduction: Semiotics and the Language of Physics

1.1 The First Law of Nature Semiotics

According to the First Law of Nature Semiotics (Foundation B), a thinking system (human, animal, or machine) operates not with real objects but with their models – simplified representations encoded in an internal language. In physics, this language consists of physical quantities and their units of measurement, which together form a metalanguage – a system where signs (e.g., "meter," "kilogram," "second") have a direct physical connection to reality via standards and etalons.

1.2 The Fundamental Triad of Classical Mechanics

Throughout the history of Classical Mechanics, scientists have chosen concepts most convenient for solving specific problems. Three fundamental concepts emerged as the elementary basis:

- Mass (m) – a scalar quantity representing the amount of matter
- Path (S) – a vector quantity representing spatial displacement
- Time (t) – a vector quantity representing temporal duration

All other mechanical concepts – velocity, acceleration, force, energy, momentum, and countless others – are combinations of these three basic elements in various powers and products.

1.3 The Quark Model Analogy

In the early 20th century, particle physics introduced the quark model to understand the structure of elementary particles. Hadrons (protons, neutrons, etc.) were shown to be combinations of more fundamental quarks. This paper borrows this powerful conceptual tool and applies it to the structure of mechanical concepts themselves.

Just as quarks combine to form particles, Elementary Mechanical Quarks (EMQs) combine to form Compound Mechanical Quarks (CMQs) – the familiar and unfamiliar quantities of classical mechanics.

2. The Quark Model of Mechanical Concepts

2.1 Elementary Mechanical Quarks (EMQs)

2.2 Compound Mechanical Quarks (CMQs)

A Compound Mechanical Quark is any product of EMQs raised to arbitrary powers:

\[
\text{CMQ} = m^\alpha \cdot S^\beta \cdot t^\gamma
\]

where:
- \(\alpha, \beta, \gamma\) are the spins (exponents) of the respective EMQs
- Spins can be any rational number: integers (positive, negative, or zero), fractions, or irrational numbers (in principle)

Examples:
- Velocity: \(S^1 t^{-1}\) → \(\alpha=0, \beta=1, \gamma=-1\)
- Force: \(m^1 S^1 t^{-2}\) → \(\alpha=1, \beta=1, \gamma=-2\)
- Energy: \(m^1 S^2 t^{-2}\) → \(\alpha=1, \beta=2, \gamma=-2\)

2.3 Spin of an Elementary Mechanical Quark

The spin of an EMQ is simply its exponent in the CMQ expression:
- Integer spin: e.g., \(S^2\) has spin 2 for the path quark
- Negative spin: e.g., \(t^{-1}\) has spin 1 for the time quark
- Zero spin: indicates the quark is not present in the CMQ
- Fractional spin: e.g., \(S^{1/2}\) has spin 1+2=3 for the path quark

2.4 Spin of a Compound Mechanical Quark – Definition

The spin of a CMQ is defined as the total number of independent dimensions (parameters) required to fully describe that quantity in a multidimensional space. It is calculated as:

\[
\text{Spin}_{\text{CMQ}} = \sum_{\text{integer } n} |n| + \sum_{\text{fractional } p/r} (|p| + |r|)
\]

where the sum is taken over all EMQs present in the CMQ.

2.4.1 Rationale for This Definition

This definition follows from the multidimensional approach of PTV (Core D):
- Each occurrence of an EMQ represents one dimension in the parameter space
- Negative exponents still represent a dimension – just one that contributes inversely
- Fractional exponents indicate compound dimensions: the numerator and denominator represent separate contributions to the total dimensionality
- Taking absolute values ensures that negative exponents do not reduce the dimensionality count

2.4.2 Examples of Spin Calculation

| Mass | \(m^1\) | m: +1 | \|+1\| = 1 | 1 | One dimension (mass space) |
| Path | \(S^1\) | S: +1 | \|+1\| = 1 | 1 | One dimension (space) |
| Time | \(t^1\) | t: +1 | \|+1\| = 1 | 1 | One dimension (time) |
| Velocity | \(S^1 t^{-1}\) | S: +1, t: -1 | \|+1\| + \|-1\| = 1+1 | 2 | Two dimensions: space + time |
| Acceleration | \(S^1 t^{-2}\) | S: +1, t: -2 | \|+1\| + \|-2\| = 1+2 | 3 | Space + time counted twice |
| Force | \(m^1 S^1 t^{-2}\) | m:+1, S:+1, t:-2 | 1+1+2 | 4 | Mass + space + time² |
| Momentum | \(m^1 S^1 t^{-1}\) | m:+1, S:+1, t:-1 | 1+1+1 | 3 | Mass + space + time |
| Energy | \(m^1 S^2 t^{-2}\) | m:+1, S:+2, t:-2 | 1+2+2 | 5 | Mass + space² + time² |
| Area | \(S^2\) | S: +2 | \|+2\| = 2 | 2 | Two spatial dimensions |
| Volume | \(S^3\) | S: +3 | \|+3\| = 3 | 3 | Three spatial dimensions |
| Frequency | \(t^{-1}\) | t: -1 | \|-1\| = 1 | 1 | One time dimension |
| Angular velocity | \(t^{-1}\) (same) | t: -1 | \|-1\| = 1 | 1 | One time dimension |

2.4.3 Handling Fractional Spins

| Fractional path | \(S^{1/2}\) | p=1, r=2 | \|1\| + \|2\| = 1+2 | 3 |
| Fractional time | \(t^{2/3}\) | p=2, r=3 | \|2\| + \|3\| = 2+3 | 5 |
| Mixed fractional | \(m^{1/2} S^{1} t^{-2/3}\) | m:1/2 (1+2), S:1 (1), t:-2/3 (2+3) | (1+2) + 1 + (2+3) = 3+1+5 | 9 |

This reflects the idea that fractional exponents represent fractal dimensions or scale-dependent properties, requiring more parameters for complete description.

2.5 Key Insight: Spin as Dimensionality

> The spin of a Compound Mechanical Quark equals the number of independent dimensions (parameters) of that quantity when considered as a multidimensional system.

This insight creates a direct bridge between:
- Foundation F (Quark Model of Classical Mechanics)
- Core D (Mathematical Modeling Framework of PTV)

In Core D, any system is described by \(d\) parameters, with volume \(V = \prod (c_j k_j)\). The spin defined here is precisely that \(d\) – the dimensionality of the parameter space.

2.6 Interactive 3D Visualization

An interactive web-based model of the quark classification space is available at:
[http://md.c-europe.eu/index30.php](http://md.c-europe.eu/index30.php)

This tool allows researchers to:
- Select spins for mass (m), path (S), and time (t) using sliders
- Visualize the corresponding CMQ in real time
- Explore the three-dimensional space of mechanical quantities
- Discover unnamed combinations for potential new concepts

The interface currently supports spins from -3 to +3 for each EMQ, covering most commonly used mechanical quantities and many unexplored regions.

3. The Three-Dimensional Classification Space

Since we have three EMQs (m, S, t), the space of all possible CMQs is three-dimensional, with axes corresponding to the spins of each EMQ.

3.1 Coordinate System

- X-axis: Spin of mass (\(\alpha\))
- Y-axis: Spin of path (\(\beta\))
- Z-axis: Spin of time (\(\gamma\))

Each point \((\alpha, \beta, \gamma)\) in this integer lattice (extended to rationals) corresponds to a unique mechanical quantity.

3.2 Visualization: Slices at Constant Path Spin

The original Russian document presents detailed tables for various slices. Below are the key slices translated and interpreted with the new spin definition.

Figure 3: Slice at Path Spin \(\beta = 1\)

This slice contains all quantities where path appears to the first power – the most familiar mechanical concepts.

| \(t \backslash m\) | -2 | -1 | 0 | 1 | 2 |
|

| 2 | \(m^{-2}S^1 t^2\) | \(m^{-1}S^1 t^2\) | \(S^1 t^2\) | \(m^1 S^1 t^2\) | \(m^2 S^1 t^2\) |
| 1 | \(m^{-2}S^1 t^1\) | \(m^{-1}S^1 t^1\) | \(S^1 t^1\) | \(m^1 S^1 t^1\) | \(m^2 S^1 t^1\) |
| 0 | \(m^{-2}S^1\) | \(m^{-1}S^1\) | \(S^1\) (Path) | \(m^1 S^1\) | \(m^2 S^1\) |
| -1 | \(m^{-2}S^1 t^{-1}\) | \(m^{-1}S^1 t^{-1}\) | \(S^1 t^{-1}\) (Velocity) | \(m^1 S^1 t^{-1}\) (Momentum) | \(m^2 S^1 t^{-1}\) |
| -2 | \(m^{-2}S^1 t^{-2}\) | \(m^{-1}S^1 t^{-2}\) | \(S^1 t^{-2}\) (Acceleration) | \(m^1 S^1 t^{-2}\) (Force) | \(m^2 S^1 t^{-2}\) |

Spin values (dimensionality) for selected cells:
- Path (\(S^1\)): spin = |1| = 1
- Velocity (\(S^1 t^{-1}\)): spin = |1| + |-1| = 2
- Momentum (\(m^1 S^1 t^{-1}\)): spin = 1 + 1 + 1 = 3
- Force (\(m^1 S^1 t^{-2}\)): spin = 1 + 1 + 2 = 4

Figure 4: Slice at Path Spin \(\beta = 0\)

This slice contains quantities where path does not appear – pure mass-time combinations.

| \(t \backslash m\) | -2 | -1 | 0 | 1 | 2 |
|

| 2 | \(m^{-2} t^2\) | \(m^{-1} t^2\) | \(t^2\) | \(m^1 t^2\) | \(m^2 t^2\) |
| 1 | \(m^{-2} t^1\) | \(m^{-1} t^1\) | \(t^1\) | \(m^1 t^1\) | \(m^2 t^1\) |
| 0 | \(m^{-2}\) | \(m^{-1}\) | 1 | \(m^1\) | \(m^2\) |
| -1 | \(m^{-2} t^{-1}\) | \(m^{-1} t^{-1}\) | \(t^{-1}\) (Frequency) | \(m^1 t^{-1}\) | \(m^2 t^{-1}\) |
| -2 | \(m^{-2} t^{-2}\) | \(m^{-1} t^{-2}\) | \(t^{-2}\) | \(m^1 t^{-2}\) | \(m^2 t^{-2}\) |

Figure 5: Slice at Path Spin \(\beta = 2\)

This slice contains quantities with path squared – areas and related concepts.

| \(t \backslash m\) | -2 | -1 | 0 | 1 | 2 |
|

| 2 | \(m^{-2}S^2 t^2\) | \(m^{-1}S^2 t^2\) | \(S^2 t^2\) | \(m^1 S^2 t^2\) | \(m^2 S^2 t^2\) |
| 1 | \(m^{-2}S^2 t^1\) | \(m^{-1}S^2 t^1\) | \(S^2 t^1\) | \(m^1 S^2 t^1\) | \(m^2 S^2 t^1\) |
| 0 | \(m^{-2}S^2\) | \(m^{-1}S^2\) | \(S^2\) (Area) | \(m^1 S^2\) | \(m^2 S^2\) |
| -1 | \(m^{-2}S^2 t^{-1}\) | \(m^{-1}S^2 t^{-1}\) | \(S^2 t^{-1}\) | \(m^1 S^2 t^{-1}\) | \(m^2 S^2 t^{-1}\) |
| -2 | \(m^{-2}S^2 t^{-2}\) | \(m^{-1}S^2 t^{-2}\) | \(S^2 t^{-2}\) | \(m^1 S^2 t^{-2}\) (Energy) | \(m^2 S^2 t^{-2}\) |

Energy (\(m^1 S^2 t^{-2}\)): spin = 1 + 2 + 2 = 5 dimensions

Figures 6-7: Negative Path Spins (\(\beta = -1, -2\))

These slices contain quantities with inverse path – spatial densities and gradients.

Example from \(\beta = -1\):
- Inverse length: \(S^{-1}\) – spin = |-1| = 1
- Velocity gradient: \(S^{-1} t^{-1}\) – spin = 1 + 1 = 2
- Mass per length: \(m^1 S^{-1}\) – spin = 1 + 1 = 2
- Mass flow per length: \(m^1 S^{-1} t^{-1}\) – spin = 1 + 1 + 1 = 3

4. Purpose and Applications of the Quark Model

4.1 Systematization of Knowledge

Just as Mendeleev's periodic table predicted unknown chemical elements, this quark model predicts possible mechanical quantities that may have been implicitly used but never formally named. The three-dimensional space contains many empty cells – combinations of spins that correspond to potentially useful concepts.

4.2 Discovery of New Concepts

By examining cells with low spin values that are rarely or never used, researchers can identify new compound concepts that might simplify calculations in specific domains:

| Cell \((\alpha,\beta,\gamma)\) | Expression | Spin | Potential Application |
|

| (1, -1, 0) | \(m S^{-1}\) | 2 | Mass per unit length – linear density |
| (1, -2, 0) | \(m S^{-2}\) | 3 | Mass per unit area – surface density |
| (1, -3, 0) | \(m S^{-3}\) | 4 | Mass per unit volume – density (already named) |
| (0, 1, 1) | \(S t\) | 2 | Space-time interval (non-relativistic) |
| (1, 1, 1) | \(m S t\) | 3 | "Mass-action" – useful in statistical mechanics? |
| (0, 2, -1) | \(S^2 t^{-1}\) | 3 | Rate of area change |
| (0, 2, -3) | \(S^2 t^{-3}\) | 5 | Related to power in rotational systems? |

4.3 Educational Value

The quark model provides a visual and systematic way to teach dimensional analysis and the structure of physical theories. Students can see how all complex concepts emerge from three basic building blocks, and how the spin as dimensionality concept connects to multivariable calculus and multidimensional geometry.

4.4 Connection to the Physical Theory of Value (PTV)

For the Physical Theory of Value, this Foundation F demonstrates several key principles:

1. Metalanguage: The EMQs (m, S, t) form the metalanguage of mechanics – just as PTV's fundamental units will form the metalanguage for value measurement.

2. Artificial Dimensionality: The spin values are assigned by the researcher based on the needs of modeling – exactly as in Core D, where dimensionality is a choice, not an inherent property of reality.

3. Systematic Classification: The quark model shows how to systematically generate all possible quantities in a domain – a methodology directly applicable to generating new Cores of value measurement.

4. Spin as Dimensionality: The definition of spin as the number of independent dimensions creates a direct bridge to Core D's volume formula:
\[
V = \prod_{j=1}^{d} (c_j k_j)
\]
where \(d\) is precisely the spin of the CMQ representing the system.

5. Fractional Dimensions: The handling of fractional spins opens the door to fractal systems and scale-dependent properties – relevant for complex economic and ecological systems.

5. Extensions and Future Work

5.1 Vector vs. Scalar Nature

The current model treats all EMQs as either scalar (mass) or vector (path, time). A complete treatment must distinguish between:
- Inner products (scalar quantities) – e.g., \(S \cdot S = S^2\) (area as scalar)
- Outer products (vector/tensor quantities) – e.g., \(S \otimes S\) (area as tensor)

This distinction affects the dimensionality count: a tensor product may introduce multiple dimensions beyond the simple spin count.

5.2 Application to Other Physical Domains

The quark model can be extended to:
- Thermodynamics: Add temperature (T) as a fourth EMQ
- Electromagnetism: Add charge (Q) as a fourth EMQ
- Relativity: Treat space and time as unified (Minkowski space) with different metric signature

5.3 Integration with PTV Cores

| Core | Connection to Quark Model |
|

| Core D (Mathematical) | Direct: spin = dimensionality \(d\) |
| Core I (Technological) | Express homogeneity in terms of m,S,t combinations |
| Core II (Teleological) | Model goal vectors in m,S,t space |
| Core VI (Negentropic) | Express entropy in m,S,t units (bits = action? \(m S^2 t^{-1}\)) |

5.4 Computational Implementation and Interactive Tools

An interactive prototype of the quark model is already available at:
[http://md.c-europe.eu/index30.php](http://md.c-europe.eu/index30.php)

This tool demonstrates the feasibility of computational exploration of the spin space. Future developments could include:
- Automatic generation of all CMQs up to a maximum spin value
- Identification of unnamed but potentially useful quantities
- Integration with dimensional analysis software
- Export of discovered quantities to mathematical software (Mathematica, MATLAB)

Researchers are encouraged to use and extend this tool as part of the open-source PTV initiative.

6. Conclusion

The quark model of semiotic concepts in Classical Mechanics provides a powerful visualization and classification tool for understanding the structure of physical knowledge. By introducing Elementary Mechanical Quarks (m, S, t) and defining Compound Mechanical Quark spin as dimensionality, we have shown that:

1. All mechanical quantities are combinations of just three fundamental concepts
2. The space of all possible quantities is three-dimensional, organized by the spins of each EMQ
3. Spin, redefined as the total number of independent dimensions, connects directly to the multidimensional mathematics of PTV (Core D)
4. Fractional spins introduce the possibility of fractal dimensions and scale-dependent properties
5. Many cells in the classification space remain unexplored – potentially hiding useful new concepts

As Foundation F of the Physical Theory of Value, this work illustrates how semiotic principles – the use of metalanguage, the artificial assignment of dimensions, and systematic classification – can generate new knowledge and provide a rigorous basis for measuring value in multidimensional systems.

The quark model is not just a pedagogical tool;

it is a generative framework for discovery, applicable far beyond classical mechanics to any domain where fundamental concepts combine to form complex quantities.

References

1. Eist, A.G. (2026). First Law of Nature Semiotics. Working draft (Foundation B). [Russian original available]

2. Eist, A.G. (2026). Physical Theory of Value: Multi-Core Research Program. Version 3.0. Zenodo: phmv community.

3. Eist, A.G. (2026). Mathematical Modeling of Physical Theory of Value. Working draft (Core D).

4. Bridgman, P.W. (1922). Dimensional Analysis. Yale University Press.

5. Feynman, R.P., Leighton, R.B., Sands, M. (1963). The Feynman Lectures on Physics. Addison-Wesley.

6. Landau, L.D., Lifshitz, E.M. (1976). Mechanics (3rd ed.). Butterworth-Heinemann.

7. Maxwell, J.C. (1871). Theory of Heat. Longmans, Green, and Co.

8. Schrödinger, E. (1944). What is Life? The Physical Aspect of the Living Cell. Cambridge University Press.

9. Shannon, C.E. (1948). "A Mathematical Theory of Communication." Bell System Technical Journal, 27, 379-423.

10. Landauer, R. (1961). "Irreversibility and heat generation in the computing process." IBM Journal of Research and Development, 5(3), 183-191.

Appendix: Complete Spin Classification Table

| Mass | \(m\) | (1,0,0) | 1 | Fundamental |
| Path | \(S\) | (0,1,0) | 1 | Fundamental |
| Time | \(t\) | (0,0,1) | 1 | Fundamental |
| Area | \(S^2\) | (0,2,0) | 2 | |
| Volume | \(S^3\) | (0,3,0) | 3 | |
| Velocity | \(S t^{-1}\) | (0,1,-1) | 2 | |
| Acceleration | \(S t^{-2}\) | (0,1,-2) | 3 | |
| Jerk | \(S t^{-3}\) | (0,1,-3) | 4 | |
| Momentum | \(m S t^{-1}\) | (1,1,-1) | 3 | |
| Force | \(m S t^{-2}\) | (1,1,-2) | 4 | |
| Impulse | \(m S t^{-1}\) | (1,1,-1) | 3 | Same as momentum |
| Energy (kinetic) | \(m S^2 t^{-2}\) | (1,2,-2) | 5 | |
| Energy (potential) | \(m S^2 t^{-2}\) | (1,2,-2) | 5 | Same dimensions |
| Power | \(m S^2 t^{-3}\) | (1,2,-3) | 6 | |
| Action | \(m S^2 t^{-1}\) | (1,2,-1) | 5 | Planck's constant |
| Angular momentum | \(m S^2 t^{-1}\) | (1,2,-1) | 5 | Same as action |
| Moment of inertia | \(m S^2\) | (1,2,0) | 3 | |
| Torque | \(m S^2 t^{-2}\) | (1,2,-2) | 5 | Same as energy |
| Frequency | \(t^{-1}\) | (0,0,-1) | 1 | |
| Angular velocity | \(t^{-1}\) | (0,0,-1) | 1 | |
| Density (linear) | \(m S^{-1}\) | (1,-1,0) | 2 | |
| Density (surface) | \(m S^{-2}\) | (1,-2,0) | 3 | |
| Density (volume) | \(m S^{-3}\) | (1,-3,0) | 4 | |
| Pressure | \(m S^{-1} t^{-2}\) | (1,-1,-2) | 4 | Force/area |
| Strain | dimensionless | (0,0,0) | 0 | |
| Refractive index | dimensionless | (0,0,0) | 0 | |
| Fine structure const. | dimensionless | (0,0,0) | 0 | |

Online Resources

- Interactive Quark Model: [http://md.c-europe.eu/index30.php](http://md.c-europe.eu/index30.php) – Explore the 3D space of mechanical quantities in real time
- Russian Documentation: Available at the same URL for original language reference
- Source Code: [link if available] – Open source implementation for community development

Contact: age@c-europe.eu
Telegram: +420775170171
Project Hub: https://www.researchgate.net/publication/398889201_A_Physical_Theory_of_Value_An_Axiomatic_Framework_and_Research_Program
Zenodo Community: phmv