The Coherence-Point Principle: Deciphering the ABC Conjecture via the Geometry of Number-Theoretic Structures

This article proposes the "Coherence-Point Principle" as a unified framework for understanding multiplicative number-theoretic inequalities, most notably the ABC conjecture.

When three distinct number-theoretic lenses—Expected Value, Typical Value, and Arithmetic Weight—synchronize at a single point, a Coherence Point emerges. From this vantage point, the validity of seemingly complex inequalities arises as a form of "statistical necessity."

The Coherence-Point Principle serves as the number-theoretic model for the "Principle of Coherence" within GhostDrift theory, elucidating the mechanism by which the synchronization of distinct phases generates robust structural constraints (inequalities).

1. Mathematical Modeling of the ABC Conjecture: Coherence Template and Definitions

To structure multiplicative number-theoretic inequalities, we introduce the concept of the "Coherence Template."

Consider a quantity $\mathrm{Total}(n)$ dependent on an integer $n$. We posit that this quantity can be decomposed into a main term $\mathrm{MainMass}(n)$ and an excess term $\mathrm{ExcessMass}(n)$, structured as follows:

$$\mathrm{Total}(n) = \mathrm{MainMass}(n) + \mathrm{ExcessMass}(n)$$

Example in the context of the ABC Conjecture:

• Total Quantity: $\mathrm{Total}(n) = \log n$

• Main Term: $\mathrm{MainMass}(n) = R(n) = \log \mathrm{rad}(n)$

• Excess Term: $\mathrm{ExcessMass}(n) = \delta(n) = \sum_{p|n} (\nu_p(n)-1)_+ \log p$

Here, the excess term $\delta(n)$ represents the "excess weight" derived from the squarefull part of $n$ (where the exponent of prime factors is 2 or greater).

Within this decomposition, we define a state where the balance between the main term and the excess term is statistically "coherent."

Definition: Coherence Point

For a parameter $X$, let $f(X)$ be an increasing function (naturally, $f(X) = \log \log X$ in the context of the ABC conjecture), and let $c, \varepsilon > 0$ be constants. An integer $n \le X$ is designated as a Coherence Point if it simultaneously satisfies the following two conditions:

1. Suppression of Excess:

   $$\mathrm{ExcessMass}(n) \le \frac{\varepsilon}{2} \cdot f(X)$$

2. Dominance of Main Mass:

   $$\mathrm{MainMass}(n) \ge c \cdot f(X)$$

2. The Coherence-Point Principle

Under these settings, the following general principle holds, abstracting number-theoretic phenomena through the lens of "coherence."

$$Statement of the Principle$$

Assume that $\mathrm{ExcessMass}$ and $\mathrm{MainMass}$ satisfy the following two conditions:

1. ExcessMass Axis (Uniform Boundedness of Expectation): The expected value of the excess term, $\mathbb{E}[\mathrm{ExcessMass}(n)]$, is uniformly bounded by a constant $C_0$.

2. MainMass Axis (Lower Bound with Density 1): The main term $\mathrm{MainMass}(n)$ possesses a lower bound $c \cdot f(X)$ on a set of density 1 (i.e., for almost all numbers).

Then, the density of numbers that are NOT coherence points is 0. In other words, almost all $n$ are coherence points, where the following inequality holds:

$$\mathrm{Total}(n) \le (1+\varepsilon') \mathrm{MainMass}(n)$$

Logic of Validity

The intuitive logic underpinning this principle is as follows:

• Regarding the Excess Term: Since the average (expected value) is small, Markov's inequality implies that points assuming extremely large values are statistically negligible (density 0).

• Regarding the Main Term: The theory of Normal Order guarantees that the term takes sufficiently large values at almost all points.

• Conclusion: Since "Coherence Points," where both conditions behave favorably, dominate the set (density 1), the inequality necessarily holds almost everywhere.

3. Three Axes Supporting Coherence

This principle is sustained by the intersection of three independent number-theoretic phenomena (axes).

Axis I: Finite Expectation of ExcessMass (Probabilistic Axis)

The excess term $\delta(n)$ assumes an extremely small value on average.

$$\mathbb{E}[\delta(n)] < \infty$$

This suggests that numbers containing squarefull parts are statistically rare.

Axis II: Typical Lower Bound of MainMass (Normal Order Axis)

The main term $R(n)$ becomes sufficiently large for almost all $n$. By the Hardy-Ramanujan theorem, the typical behavior is given by:

$$R(n) \ge \frac{1}{2} \log \log n$$

Axis III: Arithmetic Lifting (Arithmetic Axis)

If coherence for a single variable $c$ holds with probability 1, this property is "lifted" to the entire space of triples $(a,b,c)$ satisfying the additive relation $a+b=c$.

4. Discussion: Incoherent Regions as Exceptional Sets

From the perspective of this principle, numbers that could potentially serve as "counterexamples" to number-theoretic conjectures like the ABC conjecture are redefined as numbers residing in "Incoherent Regions," where coherence conditions break down.

$$E(X) \subseteq \{ n : \delta(n) \text{ is excessively large} \} \cup \{ n : R(n) \text{ is abnormally small} \}$$

The path to a complete theoretical construction reduces to the problem of shrinking the size of this incoherent region from merely "density 0" to a truly exceptional scale (a finite number or an extremely sparse set).

"How few points lie outside the coherence sphere?" This geometric intuition, suggested by GhostDrift theory, can serve as an effective compass even in the depths of number theory.

© 2025 GhostDrift Mathematical Institute