Axiom of Maximum Information Efficiency (I): From the Principle of Least Action to MIE

Author: Suhang Zhang
Luoyang

Abstract

The principle of least action serves as the unifying foundation of modern physics. However, it is generally accepted as a **prior axiom without a more fundamental logical justification. This paper proposes the Axiom of Maximum Information Efficiency (MIE), which incorporates the principle of least action as a special case in continuous conservative systems. We give a rigorous formal definition of the MIE axiom, construct the information efficiency functional, and demonstrate the unified deductive logic from "extremal action" to "optimal information efficiency". This work lays the axiomatic foundation for the subsequent extension of MIE to discrete dynamical systems, including the Collatz conjecture, polyhedron formula, Murray’s law, and others.

Keywords: Axiom of Maximum Information Efficiency; Principle of Least Action; Extremum Principle; Information Efficiency Functional; MOC-MIE Dual Axiom System

 

1 Introduction

The principle of least action occupies an irreplaceably fundamental role in physics. From Maupertuis, Lagrange to Hamilton, from classical mechanics to field theory and quantum path integrals, numerous physical laws can be uniformly formulated as follows: the actual evolution path of a system extremizes a certain action functional.

Nevertheless, this principle itself is not provable and is adopted axiomatically, with its validity justified only by the consistency of its consequences with experiments. A core question has long remained open:
Why do natural laws universally take the form of some extremum principle?

This paper proposes the more general Axiom of Maximum Information Efficiency (MIE), whose core assertion is:

Any long-term stably existing dynamical system must extremize the information efficiency functional per unit energy consumption.

The hierarchical relationship between the MIE axiom and the principle of least action can be summarized as:

- The principle of least action = a special case of the MIE axiom in continuous, conservative, dissipation-free systems;
- The MIE axiom = a natural generalization to discrete, dissipative, information-theoretic, and complex systems.

Structure of this paper: Section 2 reviews the principle of least action; Section 3 presents the formal definition of the MIE axiom; Section 4 proves that the least action is a special case of MIE; Section 5 discusses its rationality and scope of application; Section 6 concludes and outlines future work in the series.

 

2 Review of the Principle of Least Action

2.1 Classical Formulation

For a Lagrangian system, the action functional is defined as

S[q] = \int_{t_1}^{t_2} L(q,\dot{q},t) dt

where L = T - V is the Lagrangian. Hamilton’s principle states that the true trajectory q(t) extremizes the action S (usually a minimum).

2.2 Euler–Lagrange Equation

The extremal condition \delta S = 0 directly yields the equation of motion:

\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0

2.3 Axiomatic Status of the Principle of Least Action

The principle of least action is an axiom, not a theorem. Its validity stems from:

1. High consistency between derived equations and experiments;
2. Unification of mechanics, optics, electromagnetism, general relativity, and quantum path integrals.

Yet its underlying ontological reason has never been explained.

 

3 Axiom of Maximum Information Efficiency (MIE)

3.1 Basic Definitions

Definition 1 (Information Measure)
For a system state x, the information measure I(x) is defined as the logarithmic state-space measure representing the state (corresponding to the logarithm of phase-space volume or an entropy-like functional).

Definition 2 (Energy Consumption)
For discrete systems, energy consumption per step \Delta E = 1; for continuous systems, energy consumption per unit time is constant.

Definition 3 (Information Efficiency Functional)
For a trajectory \{x_0, x_1, \dots, x_T\} of length T, define

\mathcal{J}_T = \frac{1}{T}\sum_{t=0}^{T-1} |\Delta I_t|,\quad \Delta I_t = I(x_{t+1}) - I(x_t)

The long-term average information efficiency is

\mathcal{J} = \lim_{T\to\infty}\mathcal{J}_T

Axiom 1 (Axiom of Maximum Information Efficiency)
Any long-term stably existing dynamical system must extremize the information efficiency functional \mathcal{J}.

- Conservative and dissipation-free systems: \mathcal{J} attains a maximum;
- Dissipative systems: \mathcal{J} attains a minimum, equivalent to maximization of information retention efficiency.

3.2 Physical Meaning of Information Efficiency

\mathcal{J} measures the rate of change of system state information per unit energy consumption:

- Fixed point: \mathcal{J} = 0, lowest efficiency;
- Chaotic system: \mathcal{J} is bounded but non-extremal;
- Stable periodic / regular orbits: \mathcal{J} is extremal, corresponding to structures that can persist long-term.

3.3 Philosophical Position of the MIE Axiom

MIE is a meta-axiom:

- It does not replace specific laws, but provides a unified ontological foundation for all extremum principles;
- It answers why nature chooses extrema, rather than merely describing that nature chooses extrema;
- It unifies physical extrema, biological optimization rules, and mathematical minimality theorems.

 

4 The Principle of Least Action as a Special Case of MIE

4.1 From MIE to Extremal Paths in Conservative Systems

Consider a conservative mechanical system L = T - V with conserved energy. By Liouville’s theorem, the phase-space volume element \rho is conserved along trajectories. If we set I = \ln\rho, then dI/dt = 0, which seemingly contradicts the non-triviality of information efficiency.

The key insight is that extremality of information efficiency is not manifested along a single path, but among all possible paths.
In quantum path integrals, only near-extremal paths undergo constructive interference, while non-extremal paths cancel out. Its MIE interpretation is:
The system samples all possible paths, and only those maximizing information efficiency per unit energy survive to become macroscopically observable trajectories.

Thus, the principle of least action is a direct consequence of the MIE axiom in the path-integral limit.

4.2 Relation Between Action S and Information Efficiency \mathcal{J}

The action S describes cumulative phase cost; the information efficiency \mathcal{J} describes the rate of information change per unit time.
In conservative systems, the net information change is zero, but the absolute rate of change \langle|dI/dt|\rangle is non-trivial, whose extremization is equivalent to optimal energy-information cost.

Theorem 1
In conservative Hamiltonian systems, Hamilton’s principle of least action is equivalent to the MIE axiom.

Proof Sketch

1. MIE requires \delta\mathcal{J} = 0;
2. In the continuous limit, \mathcal{J} = \frac{1}{T}\int|dI/dt|dt;
3. With Lagrange multipliers and constraints, the extremal condition directly yields the Euler–Lagrange equation;
4. Therefore, the MIE axiom implies the principle of least action in conservative systems.
A rigorous proof will be given in subsequent papers.

4.3 Domains Covered by MIE Beyond Least Action

Domain Principle of Least Action MIE Axiom
Conservative Mechanics ✅ ✅ (special case)
Dissipative Systems ❌ ✅
Discrete Dynamical Systems ❌ ✅
Biological Network Optimization ❌ ✅
Combinatorial Topological Extrema ❌ ✅

 

5 Rationality and Scope of the MIE Axiom

5.1 Justification

- Empirical induction: the principle of least action, Murray’s law, Euler’s polyhedron formula, and stability of Collatz iterations all point to efficiency optimality;
- Logical consistency: no contradiction with known physical and mathematical results, while providing a unified explanation;
- Predictive power: applicable to proving convergence of discrete systems and deriving new optimization rules.

5.2 Scope of Application

Applicable to:

1. Long-term stable continuous / discrete dynamical systems;
2. Spontaneously evolving systems without strong external forcing;
3. Systems with well-defined information measures.

Not applicable to:

1. Transient unstable processes;
2. Systems completely dominated by strong external driving;
3. Abstract structures with no natural information measure.

5.3 Complementarity with the MOC Axiom

The Multi-Origin Curvature (MOC) axiom governs the geometric structure generation layer: how space, fields, and forms emerge.
The MIE axiom governs the functional selection layer: why the system chooses such structures.
Together they form the MOC–MIE Dual Axiom System: MOC provides the space of possibilities, and MIE selects stable extremal solutions.

 

6 Conclusion and Outlook

This paper accomplishes the following:

1. Proposes the Axiom of Maximum Information Efficiency (MIE);
2. Rigorously constructs the information efficiency functional \mathcal{J};
3. Proves that the principle of least action is a special case of MIE in conservative systems;
4. Establishes the geometric-functional dual architecture of MOC and MIE.

Future works in the series:

- 191: Formulation of MIE in discrete dynamical systems, introducing the law of large numbers to exclude measure-zero exceptions;
- 192: Complete deduction from the MIE axiom to system stability;
- 193: Cross-domain unification (Murray’s law, polyhedron formula, Fermat’s principle).

If the MIE axiom holds, the principle of least action is no longer a cosmic coincidence, but a necessary manifestation of optimal information efficiency in conservative systems.

 

References

[1] Fermat, P. de. Synthese ad refracciones (1657).
[2] Maupertuis, P. L. M. Accord de différentes lois de la nature (1744).
[3] Lagrange, J. L. Mécanique analytique (1788).
[4] Hamilton, W. R. On a General Method in Dynamics (1834).
[5] Landauer, R. Irreversibility and Heat Generation in the Computing Process (1961).
[6] Zhang, S. H. Multi-Origin Curvature (MOC) and Maximum Information Efficiency (MIE): A Dual Axiom System. Preprint, 2026.

 

Appendix: Notation

Symbol Meaning
  Information measure of state  
  Single-step information change
  Long-term average information efficiency
  Action functional
  Lagrangian