Finite Complexity Conjectures

• Finite Complexity Conjectures define integer complexity through a transfinite framework, linking minimal one-based representations with extremal arithmetic behavior.
• They establish that the maximal values at each complexity level are governed by three strictly decreasing, well-ordered sequences of normalized ratios.
• The conjectures explain empirical phenomena like stabilization under multiplication by 3 and isolate 'bad factorizations' to justify a definitive lower bound on integer complexity.

Finite complexity conjectures constitute a collection of deep conjectural statements regarding the structure of the integer complexity function and the extremal behavior exhibited by natural numbers under addition and multiplication. Originating with the work of Arias de Reyna, these conjectures formalize regularities observed in explicit calculations of integer complexity, encode them in precise transfinite combinatorial terms, and posit the existence of well-ordered, countable sequences governing all eventual extremal phenomena. This framework provides a conjectural resolution to the question of which natural numbers maximize size for fixed complexity, ties the study of integer complexity to transfinite recursion and well-ordering, and identifies “bad” non-additive behavior as the exclusive source of complexity regularities and anomalies.

1. Definition of Integer Complexity

The integer complexity $\|n\|$ of a natural number $n$ is the minimal number of ones required to represent $n$ using only addition, multiplication, and parentheses. Formally, let $\mathcal{E}$ denote the set of well-formed expressions built from $x$ (interpreted as $1$), $+$, $\cdot$, and parentheses. If $A \in \mathcal{E}$, define $v(A)$ recursively:

• $v(x) = 1$
• $v(A+B) = v(A) + v(B)$
• $v(A\cdot B) = v(A)\cdot v(B)$

The size (complexity) of $A$, denoted $\|A\|$, is the number of occurrences of $x$ in $A$. Then

$$\|n\| = \inf \{ \|A\| : A \in \mathcal{E}, v(A) = n \}$$

Equivalently, $\|\cdot\|$ is the maximal function $\mathbb{N}\to\mathbb{N}$ satisfying

• $\|1\|=1$
• $\|m+n\| \leq \|m\|+\|n\|$
• $\|mn\| \leq \|m\|+\|n\|$

The function is computed recursively: $$\|n\| = \min \Bigg\{ \min_{2 \leq d \mid n \leq \sqrt{n}} \|d\| + \|n/d\|,\ \min_{1 \leq j \leq n/2} \|j\| + \|n-j\| \Bigg\}$$

2. Observed Extremal Patterns and Regularities

Empirical investigation reveals two main phenomena:

• If $n$ is "free of bad 3-factors," multiplication by $3$ increments the complexity by exactly $3$.
• For each $m = 3k$, the largest $n$ such that $\|n\| = m$ belong to sharply decreasing sequences, expressible as rational numbers $a_1 > a_2 > \cdots$ of the form $n / 3^k$, and analogously for complexities $3k+1$ and $3k+2$.

These observations hint at underlying well-ordered structures controlling the maximal values at each complexity, suggesting that extremal behaviors are determined by a small number of well-structured infinite sequences of normalized ratios.

3. The Finite Complexity Conjectures (Arias de Reyna)

Notation

Let

• $$A = \{n \in \mathbb{N} : \|3^j n\| = 3j + \|n\|\ \forall j \geq 0\}$$, the set of numbers of "permanent 3-absorption."

Main Conjectures

1. Stabilization under 3-multiples: For every $n$, there exists $a \geq 0$ so that for all $j \geq a$,

$\|3^j n\| = 3(j-a) + \|3^a n\|$

Thus, for $n \in A$ one has $\|3^j n\| = 3j + \|n\|$ for all $j$.

2. Degree 1 Stability: For every $p,q \in \mathbb{N}$, there exists $a \geq 0$ such that for all $j \geq a$,

$$\|p \cdot (q \cdot 3^j + 1)\| = 3j + 1 + \|p\| + \|q\|$$

1. Enumeration via Transfinite Sequences: There exist three countable, strictly decreasing sequences of rationals $(a_\alpha)_{\alpha<\xi}$, $(b_\alpha)_{\alpha<\xi}$, $(c_\alpha)_{\alpha<\xi}$ (with $\xi$ a countable ordinal, $\omega\xi = \xi$) such that:
   • The largest $n$ of complexity $3n$ are the first terms of $3^n a_1, 3^n a_2, \ldots$
   • Analogously for $3n+1$ and $3n+2$ using $b_\alpha$ and $c_\alpha$.
2. Numerators and Denominators: All denominators in lowest terms are powers of $3$.

5-7. Explicit Set Realization: - $$\{a_\alpha\} = \{n / 3^{\|n\|/3}\ :\ \|n\| \equiv 0\!\!\pmod{3},\ n \in A\}$$, decreasing order. - $$\{b_\alpha\} = \{n / 3^{(\|n\|-1)/3}\ :\ \|n\| \equiv 1\!\!\pmod{3},\ n \in A\}$$. - $$\{c_\alpha\} = \{n / 3^{(\|n\|-2)/3}\ :\ \|n\| \equiv 2\!\!\pmod{3},\ n \in A\}$$.

8-11. Transfinite Structure, Limit Relations, and Recursion: - Limit relations connect the ordinal segments,

$$\lim_{n\to\infty} a_{\omega\beta+n} = c_\beta / 3,\quad \lim_{n\to\infty} b_{\omega\beta+n} = a_\beta,\quad \lim_{n\to\infty} c_{\omega\beta+n} = b_\beta$$

• The sequences are constructed recursively from multiplicative and additive structure, with "sporadic" (exceptional) terms inserted due to exceptionally small complexity values (i.e., "bad factorizations").

4. Empirical Regularity, Tabular Patterns, and "Bad Factorizations"

Extensive tables exhibit that the largest $n$ for $\|n\|=3k$ are, in base 3:

• $10_3=3$, $100_3=9$, $1000_3=27$, ...
• $22_3=8$, $220_3=24$, $2200_3=72$, ...
• $21_3=7$, $210_3=21$, ...

As $k$ increases, $n/3^k$ approaches limiting ratios, confirming the stabilization and recursive generation postulated by the conjectures.

Empirical illustrations include: $\begin{array}{c|cc} \text{complexity} & 3 & 6 \ \hline 10_3 & 100_3 \ 22_3 & 220_3 \ 21_3 & 210_3 \ 202_3 & 2020_3 \ 201_3 & 2010_3 \ \vdots & \vdots \end{array}$ where $n/3^k$ converges as one proceeds down each column.

"Bad factorizations"—pairs $(m, n)$ such that $\|mn\| < \|m\| + \|n\|$—are rare but crucial in breaking additivity, necessitating the transfinite corrections embodied in the conjectures.

5. Transfinite Structure and Well-Ordering

The conjectures posit that the extremal structure of integer complexity is entirely captured by transfinite, well-ordered sequences of limiting ratios up to an ordinal $\xi$ with $\omega\xi = \xi$ (the minimal such ordinal is $\omega^\omega$). Thus, all sufficiently large complexities' maximal representatives are determined by finitely many recursions plus three well-ordered master lists of ratios.

These master lists encode not only which $n$ maximize size for a given complexity mod 3, but also the precise transitions ("limit-relations") as complexity increases. The existence of only countably many recurring or sporadic anomalies underlines the "finite" character of this otherwise transfinite structure.

6. Mathematical Significance and Resolution

The finite complexity conjectures, by classifying extremal values via three decreasing sequences of rational numbers generated through well-understood operations and sporadic corrections, claim that no fundamentally new phenomenon arises in integer complexity beyond these countable master lists. The ultimate lower bound for $\|n\|$ is thus confirmed as $(3/\log 3)\log n$, as all surpassing possibilities are precluded by the conjectures’ formulas.

The resolution of Conjecture 2 on degree-1 stability, along with higher transfinite-index relations (Conjectures 8-11), has been achieved by Altman and Arias de Reyna by means of the theory of low-defect polynomials and ordinal-structured self-similarity of the set of defects $\delta(n) = \|n\| - 3\log_3 n$ (Altman et al., 2021). Here, the set of all defects is shown to be a well-ordered subset of $\mathbb{R}$ of order type $\omega^\omega$, with self-similarity $$\overline{\mathscr{D}} = \overline{\mathscr{D}} + 1$$ and explicit description via low-defect polynomials.

7. Broader Context and Open Problems

The finite complexity conjectures stand as one of the central open problems in the combinatorial theory of integer complexity. Their verification would provide a classification akin to the maximal combinatorial complexity of Boolean functions, reduce the landscape of non-additive or exceptional behaviors to a countable transfinite structure, and highlight the role of permanent 3-absorption in determining extreme cases. These conjectures tightly link arithmetic, combinatorics, and ordinals, offering a paradigm for understanding extremal phenomena in other arithmetic or algebraic structures.

A complete proof would necessitate control over all sporadic “bad factorizations” and justify the absence of new extremal ratios beyond the prescribed ordinal height, conclusively establishing that finite complexity—in this precise transfinite sense—dictates the global behavior of integer complexity for all but a controlled, well-understood family of exceptional values (Reyna, 2021, Altman et al., 2021).