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Conclusion: Low-Energy Approximation in the DOG System Strictly Conforms to the MIE Core Paradigm

Author: Zhang Suhang, Luoyang, Henan

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I. Core Correspondence

1. Essence of MIE

MIE is the universal steady-state extremal evolution criterion. Its core meaning: physical systems spontaneously tend toward the state of minimal intrinsic action. All approximate states, macroscopic states, and low-energy states are the result of the system compressing and converging its intrinsic degrees of freedom toward an optimal steady state.

2. Definition of Low-Energy Approximation in DOG

In DOG discrete order geometry, the low-energy condition is: lattice order coupling tends toward smoothness, temporal fiber oscillation amplitudes converge, high-frequency excitation modes decouple and become dormant, leaving only low-frequency long-range dominant interactions. This is precisely the process of streamlining and converging degrees of freedom.

3. Two-Way Conformity Iron Law

· From DOG to MIE: Within the four-force unification framework, the low-energy regime strips away high-energy topological fluctuations, and the system automatically converges to the minimal intrinsic energy configuration, fully complying with the MIE extremal evolution rule.

· From MIE to low-energy approximation: All physical low-energy effective theories are essentially steady-state approximate solutions obtained by minimizing the MIE action and filtering out high-energy redundant degrees of freedom.

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II. Precise Matching at Specific Levels

1. Field Theory Level (Yang-Mills Adaptation)

At high energies, DOG spacetime exhibits significant discrete topological effects, with non-commutative coupling fully exposed. Entering the low-energy regime, topological fluctuations are suppressed, connection and curvature evolution tend toward smoothness, and the field equations simplify to the classical Yang-Mills gauge field form. This simplification process is entirely governed by the MIE minimal intrinsic action constraint — it is not an artificial truncation but the natural result of the system tending toward stability.

2. Probability Formula Level

Original formula: P = \dfrac{1}{1+(\Delta\nu)^2} 

At high energies, the frequency difference \Delta\nu fluctuates violently, and the probability distribution is discrete and divergent. At low energies, the frequency difference stabilizes, the spatial coupling reference dominates, and the probability tends toward a stationary statistical distribution. This steady-state distribution is precisely the observationally optimal steady-state distribution governed by MIE.

3. Spacetime Structure Level

At high energies: discrete lattice distortion is pronounced, fiber oscillation is highly disordered. At low energies: lattice order becomes regular, fiber rhythm synchronization increases, and spacetime approaches a near-continuous smooth form. This form is precisely the minimal intrinsic order energy state determined by MIE.

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III. Rigorous Delimitations

1. All low-energy effective approximate theories within the DOG system, without exception, fully satisfy the MIE minimal intrinsic extremum principle.

2. Conversely, all macroscopic classical physical laws that conform to the MIE criterion can be completely incorporated as low-energy approximate branches of DOG.

3. The high-energy regime escapes the conventional approximate constraints of MIE and exhibits the native discrete geometric form of DOG. The boundary between high and low energy is clearly delineated, leaving no logical loopholes.

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IV. Brief Conclusion

All low-energy approximate formulations, classical field theory reductions, and macroscopic physical law regressions within the DOG system strictly adhere to the MIE minimal intrinsic evolution criterion. The two are logically homologous, evolutionarily codirectional, and steady-state convergent. This represents a perfectly self-consistent unification of the fundamental structure and its approximate representations — rigorous derivation, no gaps.

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