The Coherence-Point Principle
Geometry of Number-Theoretic Structures

GhostDrift Mathematical Institute

This note proposes the "Coherence-Point Principle" as a unified framework for understanding multiplicative number-theoretic inequalities, particularly those sharing the structure of the ABC conjecture. When three distinct number-theoretic lenses—expected value, typical value, and arithmetic weight—align in the same direction for "almost all $n$", we define their intersection as a coherence point. From this perspective, the validity of complex inequalities emerges as a form of statistical necessity.

The "Coherence-Point Principle" discussed here serves as the number-theoretic model for the "Principle of Coherence" employed throughout the GhostDrift theory. It describes the mechanism by which the synchronization of different phases (expectation, typicality, arithmetic structure) at a single point generates strong constraints (inequalities) on the entire system.

1. The Coherence Template and Definitions

To structure multiplicative inequalities, we introduce the following "Coherence Template".

Definition 1.1 (Coherence Template).
Let $\Total(n)$ be a quantity dependent on an integer $n$. If $\Total(n)$ can be decomposed into a main term $\MainMass(n)$ and an excess term $\ExcessMass(n)$ such that $$ \Total(n) = \MainMass(n) + \ExcessMass(n), $$ we call this structure a Coherence Template.

Example in the context of the ABC inequality: $$ \begin{align} \Total(n) &= \log n \\ \MainMass(n) &= R(n) = \log \rad(n) \\ \ExcessMass(n) &= \delta(n) = \sum_{p|n} (\nu_p(n)-1)_+ \log p \end{align} $$ Here, $\delta(n)$ represents the excess logarithmic weight derived from the squarefull part (where the exponent is 2 or more).

Such formulations based on $\rad(n)$ and $abc$-triples are well-organized in classical texts like Hardy–Wright [HW08, Ch. II], standard introductions to analytic and probabilistic number theory [Ten15, §I.1], and surveys on the ABC conjecture [Wal15]. In this decomposition, we define a point where the balance between the main term and the excess term is statistically "coherent".

Definition 1.2 (Coherence Point).
For a parameter $X$, an increasing function $f(X)$ with respect to $X$, and constants $c, \varepsilon > 0$, an integer $n \le X$ is a Coherence Point if it simultaneously satisfies the following two conditions: $$ \begin{cases} \ExcessMass(n) \le \frac{\varepsilon}{2} \cdot f(X) & (\text{Suppression of Excess}) \\ \MainMass(n) \ge c \cdot f(X) & (\text{Dominance of Main Mass}) \end{cases} $$ In the case of ABC, it is natural to adopt $f(X) = \log \log X$ as the scaling function.
Convention 1.3 (Notation for Mean and Density).
In this note, we use the uniform distribution on the finite set $\{1,\dots,X\}$ for $X \ge 1$. For a bounded function $F:\mathbb{N}\to\mathbb{R}$, we write $$ \E_{n\le X}[F(n)] := \frac{1}{X}\sum_{1\le n\le X} F(n). $$
The (natural) density of a subset $A\subset\mathbb{N}$ is defined by $$ d(A) := \lim_{X\to\infty}\frac{1}{X}\#\{n\le X : n\in A\}, $$ provided the limit exists. Expressions like "density 1" or "density 0" in the text refer to this density $d(\cdot)$.

The treatment of such natural densities appears in classical papers [Dro89] and, more recently, in studies combining model theory/SMT with natural density [ToZ25], as well as recent research precisely evaluating the natural density of various arithmetic sets [LuoRen23, Tro21].
Note 1.4 (Scale Conventions for $X$ and $n$).
We consider the situation where the upper bound parameter $X$ is fixed and $1\le n\le X$ varies. Regarding the scaling function $f(X)$, we adopt the following conventions: If necessary, one can fully formalize the above notation by choosing $X=X(n)$ at a comparable scale to $n$ for each $n$ and reading $f(n)$ as $f\bigl(X(n)\bigr)$. This type of "shorthand for $X\asymp n$" appears as standard notation in probabilistic number theory literature [Kow21, §2].

2. The Coherence-Point Principle

Under this setting, the following general principle holds. This abstracts number-theoretic phenomena from the viewpoint of "coherence".

Theorem 2.1 (Coherence-Point Principle).
Consider the asymptotic limit $X \to \infty$. Let $f(X)$ be an increasing function of $X$ such that $f(X)\to\infty$ as $X\to\infty$. Assume that $\ExcessMass, \MainMass:\mathbb{N}\to[0,\infty)$ constitute a coherence template $$ \Total(n)=\MainMass(n)+\ExcessMass(n). $$
Suppose the following two conditions hold: Then, for any $\varepsilon>0$, the density of the set of $n\le X$ that are not "coherence points" is 0. That is, there exists a set $S_\varepsilon\subset\mathbb{N}$ of density 1 such that for all $n\in S_\varepsilon$, $$ \ExcessMass(n) \le \frac{\varepsilon}{2}\,f(n),\qquad \MainMass(n) \ge c\,f(n) $$ hold simultaneously.

In particular, on the same set $S_\varepsilon$, $$ \Total(n) \le \Bigl(1+\varepsilon'\Bigr)\MainMass(n) $$ holds, where $\varepsilon' = \varepsilon/(2c)$.

The structure of extracting "typical points" from uniform boundedness of expectation and a density-1 lower bound follows a pattern very similar to standard frameworks in probabilistic number theory [Ten15, Kow21, Sch07], such as Erdős–Kac type results and Turán–Kubilius type inequalities.

Proof.

Fix $\varepsilon>0$. For each $X\ge 1$, define the set of points violating the "excess term condition" as $$ B_\varepsilon^{(\mathrm{ex})}(X) := \bigl\{1\le n\le X : \ExcessMass(n) > (\varepsilon/2)\,f(X)\bigr\}. $$ Applying Markov's inequality under the notation of Convention 1.3 yields $$ \frac{\#B_\varepsilon^{(\mathrm{ex})}(X)}{X} \le \frac{2}{\varepsilon f(X)}\, \E_{n\le X}\bigl[\ExcessMass(n)\bigr] \le \frac{2C_0}{\varepsilon f(X)}. $$ By the assumption $f(X)\to\infty$ and (A), the right-hand side converges to 0 as $X\to\infty$. Thus, the density of $B_\varepsilon^{(\mathrm{ex})}(X)$ is 0. This procedure of deriving density 0 from uniform boundedness plus Markov's inequality is analogous to typical "almost all" type arguments based on mean value theorems [Ten15, §II.1].

Similarly, let the set of points violating the main term condition be $$ B^{(\mathrm{main})}(X) := \bigl\{1\le n\le X : \MainMass(n) < c\,f(X)\bigr\}. $$ Assumption (B) can be restated as $$ \lim_{X\to\infty} \frac{\#B^{(\mathrm{main})}(X)}{X} = 0. $$ That is, the "points violating the main term condition" also have density 0.

Now, define the set of "non-coherence points" at scale $X$ as $$ \Bad_\varepsilon(X) := B_\varepsilon^{(\mathrm{ex})}(X)\ \cup\ B^{(\mathrm{main})}(X). $$ Then, $$ \frac{\#\Bad_\varepsilon(X)}{X} \le \frac{\#B_\varepsilon^{(\mathrm{ex})}(X)}{X} + \frac{\#B^{(\mathrm{main})}(X)}{X} $$ always holds. Since both terms on the right-hand side converge to 0 as $X\to\infty$, $$ \lim_{X\to\infty}\frac{\#\Bad_\varepsilon(X)}{X} = 0. $$ This is exactly a restatement of the first claim of Theorem 2.1: "for any $\varepsilon>0$, the density of the set of $n\le X$ that are not coherence points is 0."

The latter statement, "That is, there exists a set $S_\varepsilon\subset\mathbb{N}$ of density 1 such that...", is a shorthand interpretation of this fact at the scale $X\asymp n$, restated in the language of "coherence points" from Definition 1.2. (Strictly speaking, this can be formalized by choosing $X=X(n)$ at a comparable scale for each $n$. See also Note 1.4 regarding this point.) This completes the proof of the theorem.

3. Three Axes of Coherence

This principle is supported by the intersection of three independent number-theoretic phenomena (axes). We unravel this structure using the ABC inequality as an example.

Axis I: Finite Expectation of ExcessMass (Probabilistic Axis).
The excess term $\delta(n)$ is very small on average. As $X\to\infty$, $$ \E_{n\le X}\bigl[\delta(n)\bigr] = \sum_{p}\frac{\log p}{p(p-1)}\ +\ o(1) < \infty. $$ This suggests that numbers containing squarefull parts are statistically rare. Regarding the average distribution of squarefull (powerful) numbers and their distribution along arithmetic progressions, the series of studies by Chan [Chan14 et al.] and applications to powerful numbers assuming the ABC conjecture [Cro20] provide detailed insights. Furthermore, recent results giving ergodic theorems along squarefree and squarefull numbers [LiYi25] complement the $\ExcessMass$ axis of this paper from a measure-theoretic perspective. Additionally, recent results on the distribution of $\omega(n)$ over $h$-free / $h$-full numbers [DKL24] provide a technical foundation for more precisely measuring "rare deviations" of $\ExcessMass$.
Axis II: Typical Lower Bound of MainMass (Normal Order Axis).
The main term $R(n)$ is sufficiently large for almost all $n$. By the Hardy–Ramanujan theorem, for almost all $n$, $$ R(n) \ge \frac{1}{2}\log\log n $$ holds. This is a manifestation of the "Normal Order" phenomenon on the MainMass side. The classical theorem by Hardy–Ramanujan [HR17, HW08, Ch. XXII] stating that the normal order of $\omega(n)$ is $\log\log n$ is the very skeleton of this lemma. For more modern introductions, one can refer to variants by Murty [Mur20] and recent results showing that the normal order of $\omega(n)$ remains $\log\log n$ even when restricted to $h$-free / $h$-full numbers [DKL24]. We also cite [Kow21] as a systematic introduction to probabilistic number theory as a whole.
Axis III: Arithmetic Lifting (Arithmetic Axis).
Consider a property $\mathcal{P}(c)$ concerning an integer $c$. Let $$ \mathbf{1}_{\mathcal{P}}(c) := \begin{cases} 1 & (\text{if }\mathcal{P}(c)\text{ holds})\\ 0 & (\text{otherwise}) \end{cases}. $$ On the set of coprime triples $$ \mathcal{T}(X) := \bigl\{(a,b,c)\in\mathbb{N}^3 : a+b=c,\ (a,b,c)=1,\ c\le X \bigr\}, $$ we introduce the following $\varphi$-weighted probability measure: $$ \mu_X(A) := \frac{1}{\sum_{c\le X}\varphi(c)} \sum_{\substack{c\le X\\(a,b,c)=1\\a+b=c}} \mathbf{1}_A(a,b,c), $$ where $A\subset\mathcal{T}(X)$. Then $\mu_X$ can be essentially identified with the $\varphi$-weighted average regarding $c$: $$ \nu_X(B) := \frac{1}{\sum_{c\le X}\varphi(c)} \sum_{c\le X}\varphi(c)\,\mathbf{1}_B(c). $$
If $$ \lim_{X\to\infty} \nu_X\bigl(\{c\le X : \mathcal{P}(c)\}\bigr) = 1 $$ holds (i.e., $\mathcal{P}(c)$ holds with $\varphi$-weighted density 1), then $$ \lim_{X\to\infty} \mu_X\bigl(\{(a,b,c)\in\mathcal{T}(X) : \mathcal{P}(c)\}\bigr) = 1 $$ also holds.

Therefore, if coherence for a single $c$ (e.g., coherence conditions regarding $\MainMass(c)$ or $\ExcessMass(c)$) holds with $\varphi$-weighted density 1, it naturally lifts to a set of $\varphi$-weighted density 1 of coprime triples $(a,b,c)$. The typical distribution behavior regarding $abc$-triples and $\rad(abc)$ (e.g., questions of how small/large $\rad(abc)$ is for most $c$) is organized in detail in surveys on the ABC conjecture [Wal15] and applications based on explicit versions [KNS19]. Applications to powerful numbers [Cro20] serve as examples emphasizing the role of the "excess term" derived from squarefull parts. On the other hand, viewing the framework of probability measures based on $\varphi$-weighted averages more logically/model-theoretically aligns well with recent research connecting natural density and spectral properties [ToZ25].

The proof follows by observing that the sum over $\mathcal{T}(X)$ contains $\varphi(c)$ terms for each $c$, making the two averages identically equal.

Note (Coherence Point as Intersection of Three Axes).
When the ExcessMass (Expectation) Axis, MainMass (Normal Order) Axis, and Arithmetic Lifting Axis intersect at a single point, that intersection is a "Coherence Point", where $\Total$ is constrained within $(1+\varepsilon)$ times $\MainMass$. It is this structural necessity that supports the validity of the ABC inequality at density 1.

4. Incoherent Regions as Exceptional Sets

From the perspective of this principle, the "exceptional set $E(X)$" which could potentially contain counterexamples to the conjecture is redefined as the region where coherence conditions break down.

Details: Structure of Incoherent Regions

The exceptional set $E(X)$ decomposes as follows: $$ E(X) \subseteq \{ n : \delta(n) \text{ is excessive} \} \cup \{ n : R(n) \text{ is deficient} \}. $$ Candidates for counterexamples to the conjecture are reinterpreted as "Incoherent Regions" (where coherence conditions are broken). The path to a complete theoretical construction (e.g., a full proof of the ABC conjecture) reduces to the problem of reducing the size of this incoherent region from merely "density 0" to a truly exceptional scale (e.g., $O(X/\log^A X)$ or smaller).

Future prospects include strengthening evaluations of exceptional sets for the normal order of $\log\rad(n)$ and refining sieve method evaluations for large squarefull parts. These are all attempts to quantify "how few points fall outside the coherence sphere". Attempts to precisely evaluate the "natural density of integers satisfying specific arithmetic conditions" have been actively progressing in recent years, such as density evaluations for inequalities of the form $\sigma(kn+r_1)\ge\sigma(kn+r_2)$ [LR23] and research dealing with the proportion of non-representable integers related to generalized Fibonacci sequences [Tro21].

References

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