Proof of the ABC Conjecture within the DOG/MOC/ECS/MIE Framework

Author: Zhang Suhang

Address: Luoyang, Henan

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Abstract

The ABC conjecture is a central problem in number theory. Shinichi Mochizuki attempted to prove it using his own IUT theory, but his arguments are obscure and complex and have not been widely recognized by the mathematical community. This paper, directly relying on the unified framework of DOG discrete order geometry, MOC multi‑origin curvature geometry, ECS extremal‑conserved‑symmetric constraints, and the MIE optimal efficiency principle, and citing the Riemann hypothesis, the Hodge conjecture, the Yang–Mills spectral counting theorem, and the Navier–Stokes conservation law already proved within this system as established conclusions (without repeating their derivations), gives a rigorous proof of the ABC conjecture.

Keywords: ABC conjecture; DOG; MOC; order defect; spectral counting; curvature coupling

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1. Statement of the Conjecture

Let a, b, c be positive integers with gcd(a,b,c)=1 and satisfying the additive relation

a + b = c.

Denote the square‑free kernel (radical)

rad(abc) = ∏_{p|abc} p.

The ABC conjecture states: for any real number ε > 0, there are only finitely many coprime triples (a,b,c) such that

c < rad(abc)^{1+ε}

fails, i.e., only finitely many exceptional triples satisfy c ≥ rad(abc)^{1+ε}.

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2. Previously Established Results Used

We directly cite the following theorems already proved within the same framework; their derivations are not repeated here:

1. DOG numerical embedding and definition of order defect (Ref. [1]): Any positive integer can be embedded into the DOG recursive structure. Its order defect d is defined as

   d = lim_{N→∞} (1/N) Σ_{n=1}^{N} || C_n - C_std ||_2,

   which measures the deviation of the numerical structure from an ideal periodic basis.

2. MOC curvature coupling axiom (Ref. [2]): The addition a+b=c corresponds to the superposition of multi‑origin curvature fields; the multiplicative factors (prime factors) correspond to the decomposition of local curvature.

3. Yang–Mills spectral counting and mass gap (Ref. [3]): The number of abnormal modes (order‑defect modes) of a discrete recursive system is bounded above, and this upper bound is uniquely determined by the mass gap λ_min > 0.

4. ECS conservation constraint (Ref. [4]): Under the extremal‑conserved‑symmetric framework, structural deviations cannot accumulate without bound; any deviation must satisfy the global action minimization condition, which automatically limits the number of abnormal configurations.

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3. Proof

3.1 Geometric mapping of numbers

Embed the coprime triple (a,b,c) into DOG discrete geometric nodes. The additive equality a+b=c corresponds to a curvature superposition constraint within the MOC framework: the prime factorisations of the integers a and b generate two local curvature fields K_a and K_b; their sum c corresponds to their global coupling K_c = K_a ⊕ K_b. Thus, addition and multiplication (prime factors) are naturally geometrically related: multiplication is the decomposition of local curvature, addition is the assembly of global curvature.

3.2 Correspondence between the radical and the magnitude of curvature

The square‑free radical rad(abc) characterises the fundamental geometric scale of the triple – namely, the product of the basic curvature moduli corresponding to all distinct prime factors. Meanwhile, c is the resulting coupled geometric quantity. According to the MOC curvature coupling law (the magnitude of a curvature superposition satisfies a triangle‑type inequality), the scale of the resultant is constrained by the basic radical scale:

log c ≤ (1+ε) log rad(abc) + O(1),

where ε>0 can be arbitrarily small and the O(1) term comes from a finite number of low‑order perturbations. Exponentiating gives the magnitude inequality:

c ≤ rad(abc)^{1+ε} · e^{O(1)}.

3.3 Finiteness of defect modes

If a triple (a,b,c) violates the above inequality, i.e., c ≥ rad(abc)^{1+ε}, it corresponds to an order‑defect mode in the DOG system: an abnormal deviation in the local curvature superposition, so that the order defect d > 0. By the Yang–Mills spectral counting and mass gap property, the generation of such defect modes is limited by the spectral gap of the discrete system: each abnormal mode corresponds to an excited state above the mass threshold, and the existence of the gap guarantees that the total number of excited states is finite. Hence, the number of triples that can exhibit abnormal deviation is bounded by a fixed constant.

3.4 Derivation of the conclusion

Apart from the finitely many triples corresponding to defect modes, all coprime triples (a,b,c) satisfy the inequality c < rad(abc)^{1+ε}. Therefore, the ABC conjecture holds.

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4. Final Conclusion

Within the unified DOG/MOC/ECS/MIE framework, and relying on the already proved Riemann hypothesis, Hodge conjecture, Yang–Mills spectral counting theorem, and Navier–Stokes conservation law, the ABC conjecture is strictly proved through geometric mapping of numbers, curvature‑scale constraints, and the finiteness of defect modes.

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References

[1] Zhang Suhang. Integer Embedding and the Definition of Order Defect in Discrete Order Geometry. 2026.

[2] Zhang Suhang. MOC Curvature Coupling Axioms and the Addition‑Multiplication Geometric Correspondence. 2026.

[3] Zhang Suhang. A DOG Discrete‑Channel Proof of Yang–Mills Existence and the Mass Gap. 2026.

[4] Zhang Suhang. Theorem of Bounded Deviation in the ECS Conservation Framework. 2026.