Fundamental Axioms of Scale Physics

I am writing this as a draft of foundational axioms for Scale Physics. My goal is to first solidify the conceptual backbone,and then build a structure from which transformation laws, dynamic equations, and testable consequences can be derived from these axioms.

Introduction

In standard physical description, the state of a system is defined by physical quantities such as position, time, momentum,energy, and field. Scale Physics additionally posits that the scale of observation is an integral variable of the physical description.

Therefore, a general physical state is considered not only as

$$X=X(x,t)$$

but in the form of

$$X=X(x,t,s)$$

Here, $s$ represents the physical scale at which the system is examined.

From this starting point, I propose the following axioms.

Axiom 1: Scale is a Fundamental Variable of the Physical State

A complete description of a physical system cannot be finalized merely by providing spatial and temporal coordinates.The scale at which the system is defined must also be specified.

Thus:

$$\mathrm{\Psi }=\mathrm{\Psi }(x,t,s)$$

Here:

• $x$: spatial state,
• $t$: temporal state,
• $s$: observation or interaction scale.

Conclusion

The same system can exhibit different effective physical behaviors at scales $s_1\mathrm{\ne }{s}_2$.

Therefore, in general, it must be allowed that:

$$\mathrm{\Psi }(x,t,s_1)\mathrm{\ne }\mathrm{\Psi }(x,t,s_2)$$

This is the fundamental axiom of Scale Physics.

Axiom 2: Physical Quantities Can Be Scale-Dependent

The value of a measured physical quantity does not necessarily have to be independent of scale in all cases.

A general physical quantity can be expressed as:

$$Q=Q(s)$$

There is a transformation between two scales:

$$Q(s_1)\to Q(s_2)$$

In the simple self-similar case, this transformation can be of the form:

$$Q(\lambda s)={\lambda }^{{\alpha }_Q}Q(s)$$

Here, ${\alpha }_Q$ is the scale exponent of the relevant quantity.

This specific form is not an axiom, but a model that can be derived from the axiom or tested experimentally. We must specifically preserve this distinction.

Axiom 3: Scale Change is a Physical Transformation

The transition from one scale to another is mathematically represented by a transformation operator:

$$S_{\lambda }:\mathrm{\Psi }(x,t,s)\to {\mathrm{\Psi }}^{\mathrm{\prime }}(x^{\mathrm{\prime }},t^{\mathrm{\prime }},\lambda s)$$

Here, $\lambda$ is the scale transformation ratio.

Therefore, one of the fundamental mathematical problems of Scale Physics is:

$$S_{\lambda }=?$$

The advanced form of the theory must explicitly determine how this operator transforms which quantities.

For instance, it must be additionally derived which of the transformations:

• $x\to x^{\mathrm{\prime }}$
• $t\to t^{\mathrm{\prime }}$
• $E\to E^{\mathrm{\prime }}$
• $p\to p^{\mathrm{\prime }}$
• $m\to m^{\mathrm{\prime }}$ are physical.

Axiom 4: The Form of Physical Laws Must Be Separated from the Observed Effective Behavior

The observed physical behavior can change when the scale changes. However, this does not necessarily mean that the fundamental law of nature has changed.

Therefore, a distinction is made:Fundamental law $\mathrm{\ne }$ scale-specific effective law

In the strong form of Scale Physics, the aim is to define the effective laws observed at different scales within a common superstructure.

Symbolically, it can be considered as:

$${\mathcal{L}}_s={\mathcal{T}}_s({\mathcal{L}}_0)$$

Here, ${\mathcal{L}}_0$ represents the fundamental structure, and ${\mathcal{L}}_s$ represents the effective dynamics observed at a specific scale.

Axiom 5: Physical Structures Can Conserve Motifs Across Scales

A system does not have to be exactly identical at different scales. However, certain structural relations can be conserved.

We can call this conserved structure a scale motif.

Let a motif be denoted by:

$$M(s)$$

In a self-similar system, it can be:

$$M(s)\sim M(\lambda s)$$

However, “similarity” does not have to be geometric equality.

The conserved property could be;

• topological structure,
• interaction network,
• correlation,
• ratio,
• dynamic relation,
• statistical distribution.

Thus, fractality evolves into a more general concept than purely geometric self-similarity.

Axiom 6: Exact Self-Similarity Across Scales is Not Mandatory

Real physical systems do not need to behave identically across an infinite number of scales like ideal mathematical fractals.

Therefore, the relationship:

$$M(s_1)\simeq M(s_2)$$

might be valid within a specific scale range.

That is: While fractal or scale-invariant behavior is present in the range $s_{min}<s<s_{max}$​, different physical regimes can emerge outside of this.

This axiom is important because it prevents Scale Physics from turning into an unnecessarily strong claim like everything is fractal at every scale.

Axiom 7: There Are Regime Transitions Between Scales

The behavior of a system may change continuously with scale, but it can also exhibit qualitative changes at certain critical scales.

For a critical scale $s=s_c$, different effective physical regimes can emerge in the regions $s<s_c$​ and $s>s_c$.

Therefore, the scale space of physical reality can be divided into regimes as:

$$S=\{R_1,R_2,\mathrm{.}\mathrm{.}\mathrm{.},R_n\}$$

This idea can later form the foundation of a critical scale points theory.

Axiom 8: The Combined Effect of Components Does Not Have to Be Reduced to the Sum of Individual Effects

In multi-component systems:

$$E_{system}$$

generally does not have to be identical to

$$\sum _i{E}_i$$

The cooperative term can be defined as:

$$C=E_{system}-\sum _i{E}_i$$

Thus, we obtain:

$$E_{system}=\sum _i{E}_i+C$$

Here $C$ could be the most fundamental mathematical starting point of the Cooperative Fractal Effect.

In the more general case, since:

$$C=C(s)$$

the cooperative effect is also scale-dependent.

Axiom 9: Cooperative Effect Can Saturate

The cooperative effect does not necessarily have to increase limitlessly as the system grows.

In certain systems, a saturation can occur in the form:

$$C(N)\to C_{max}(N\to \mathrm{\infty })$$

For such a system, it can be defined as:

$$\underset{N\to \mathrm{\infty }}{\lim }C(N)=C_{max}$$

However, one must be careful here again: Saturation should not be made a universal physical axiom.

The safer fundamental principle is this: The functional form of the cooperative effect dependent on scale and system size is determined by the physical system; in some systems, this effect can approach a finite saturation value.

Thus, Saturated CFE becomes a specific model and is not confused with the foundational theory.

Axiom 10: Observation Scale and Physical Scale Must Be Separated

This distinction is quite important for the future of the theory.

Let us define the actual characteristic scale of a system as:

$$s_p$$​

and the observation/measurement resolution as:

$$s_o$$

In the general case, it can be:

$$s_p\mathrm{\ne }{s}_o$$

The observed quantity emerges as:

$$Q_{obs}=M(Q,s_p,s_o)$$

Therefore, the measured behavior can be influenced not only by the system itself but also by the relationship between the system scale and the observation scale.

This point can form the mathematical basis of the “scale blindness” concept.

Axiom 11: Scale Blindness Can Produce an Incomplete Effective Description

If an observation method cannot resolve the structure at certain scales, the observed system might not represent the entire scale structure of the physical system.

Symbolically:

$${\mathrm{\Psi }}_{obs}={\mathcal{P}}_{s_0}[\mathrm{\Psi }]$$

Here, ${\mathcal{P}}_{s_0}$​ can be thought of as a projection that selects the information allowed by the observation scale.

Thus, it could be that:

$${\mathrm{\Psi }}_{obs}\mathrm{\ne }{\mathrm{\Psi }}_{full}$$

However, the conclusion that dark matter or dark energy automatically stems from scale blindness should not be drawn from this axiom. These are hypotheses that need to be tested separately.

Axiom 12: Physical Scale Structure Must Be Measurable

This is the fundamental condition that removes Scale Physics from being merely a philosophical interpretation.

The scale dependency of a physical system must correspond to an observationally measurable quantity.

For example, if:

$$Q(s)$$

is measurable,

$$\frac{dQ}{d\ln s}$$

could be one of the fundamental quantities showing the variation against scale.

Let us define:

$${\beta }_Q=\frac{dQ}{d\ln s}$$

Thus:

• ${\beta }_Q=0$: scale-independent region,
• ${\beta }_Q\mathrm{\ne }0$: scale-dependent region,
• Prominent changes in ${\beta }_Q$: can be investigated as regime transition candidates.

This section is more of a direct mathematical consequence/research tool than an axiom.

Axiom 13: Scale Physics Must Reproduce Standard Physics at the Appropriate Limit

The new structure must not arbitrarily eliminate existing physics.

For a specific classical limit $S_0$:

$$\underset{s\to s_0}{\lim }{F}_{scale}\to F_{standard}$$

is expected.

For instance, the following must be investigated separately in the future:

• Scale Mechanics $\to$ Newton’s Mechanics
• Fractal Wave Dynamics $\to$ Quantum Mechanics
• Fractal Space-Time $\to$ General Relativity

These transitions should not be accepted as true from the beginning; they must be demonstrated mathematically.

Axiom 14: The Physical Value of the Theory Depends on Its Ability to Produce New Cross-Scale Predictions

The strong scientific version of Scale Physics should not merely reinterpret existing phenomena.

Starting from an initial state:

$$\mathrm{\Psi }(s_1)$$

it must be able to predict the state at another scale:

$$\mathrm{\Psi }(s_2)=S_{s_2\mathrm{/}{s}_1}\mathrm{\Psi }(s_1)$$

And this prediction must be compared with experiment:

$${\mathrm{\Psi }}_{theory}(s_2)\leftrightarrow {\mathrm{\Psi }}_{experiment}(s_2)$$

This is the actual scientific test.

Axiom 15: Scale Physics Must Be Falsifiable

I believe it is necessary to establish the last axiom particularly strongly:Every fundamental scale law must have a definable condition of falsifiability.

For instance, if the theory predicts for a specific physical quantity:

$$Q(s)\propto s^{\alpha }$$

experimental data systematically falling outside the error margin allowed by this relationship should mandate the rejection of the model.

Therefore, the theory should not only say:“I can explain this.”

At the same time, it must be able to say: “If this measurement occurs, my model is wrong.”

Let Us Simplify the Axiomatic Core

I do not actually propose leaving these 15 items with equal weight as “fundamental axioms”. A scientifically cleaner structure would be to reduce these to 5 foundational axioms:

I. Scale Axiom

$$\mathrm{\Psi }=\mathrm{\Psi }(x,t,s)$$

The physical state is defined by the scale parameter.

II. Scale Transformation Axiom

$$\mathrm{\Psi }(s_2)=S_{s_2\mathrm{/}{s}_1}\mathrm{\Psi }(s_1)$$

The physical relationship between scales can be represented by a transformation law.

III. Scaled Structure Axiom

Physical systems can conserve motifs, correlations, or interaction structures within specific scale ranges; they can transition to different regimes at critical scales.

IV. Cooperativity Axiom

$$E_{system}=\sum _i{E}_i+C(s)$$

The effective behavior of a multi-component system does not have to be reduced to the independent sum of its components.

V. Physical Testability Axiom

Scale dependence must produce measurable results, return the limits where existing physics is valid, and create falsifiable predictions.

In My Opinion, the Core of the Theory

If we compress this even further, we can summarize Scale Physics in a single thought:

Physical state = space + time + scale + relationship between scales

In other words, if the center of gravity in Newtonian mechanics is motion, in relativity space-time and the observer, and in quantum mechanics state and measurement, the fundamental new axis in the structure I am trying to build becomes:

scale and scale transformation.

The Next Critical Step

I believe we should not immediately move on to new physical applications from here on out. We must derive the mathematics from these axioms. Especially, if we establish the ${\delta }_{\lambda }$​ operator in Axiom II, the work will progress seriously. Because then we can ask whether

$${\delta }_{{\lambda }_1}{\delta }_{{\lambda }_2}={\delta }_{{\lambda }_1{\lambda }_2}$$

and we can start building the Scale Transformation Algebra.

This would be the point where Scale Physics transitions from axioms to mathematical theory.