Fibonadic Numbers & Zeckendorf Dynamics

• Fibonadic numbers are defined through Zeckendorf expansions, ensuring a unique representation of integers as sums of non-consecutive Fibonacci numbers.
• They form an inverse limit structure with discrete dynamical properties and an ultrametric topology analogous to p-adic systems.
• Their formal power series operations are normalized to yield a commutative rig, linking advanced arithmetic, fractal geometry, and operadic decompositions.

Fibonadic numbers, in contemporary mathematical research, refer to arithmetical and combinatorial structures arising from the unique representation of integers as sums of non-consecutive Fibonacci numbers and the associated algebraic, metric, and dynamical structures obtained from this representation. Their definition and properties are intimately tied to the Zeckendorf expansion, the Zeckendorf shift, and connections to discrete dynamics, fractal geometry, and the golden ratio. The concept generalizes the classical idea of “Fibonacci-based number systems” and equips them with a rich algebraic and topological structure, leading to a commutative rig and ultrametric space with additional normalization and operadic interpretations (Haran, 13 Sep 2025).

1. Definition via Zeckendorf Expansion and Shift

A Fibonadic number is represented as a bi-infinite sequence $z = (\ldots, z_{-2}, z_{-1}, z_0, z_1, \ldots)$ with digits $z_n \in \{0,1\}$, indexed over the integers, subject to two constraints:

• Finiteness: only finitely many $z_n$ are nonzero for $n \gg 0$, ensuring the associated “φ-value” is finite.
• No consecutive ones: $z_n z_{n-1} = 0$ for all $n$, corresponding to the Zeckendorf expansion's uniqueness condition.

Given Fibonacci numbers $f_n$ (with suitable indexing), each such sequence encodes the value

$R = \sum_{n=1}^{\infty} z_n f_n.$

The Zeckendorf shift $\pi$ is defined by

$\pi(R) = \sum_{n=1}^{\infty} z_n f_{n-1}.$

The set of Fibonadic numbers $\mathcal{F}$ is then constructed as the inverse limit

$$\mathcal{F} := \varprojlim \left\{ \mathbb{N} \xleftarrow{\pi} \mathbb{N} \xleftarrow{\pi} \cdots \right\}.$$

This definition encodes not just an individual number but the entire projective sequence of representations under repeated application of the Zeckendorf shift.

2. Inverse Limit Structure and Dynamical Interpretation

Each Fibonadic number, in this setting, can be viewed as a “tower” $\{R_n\}$ with

$R_n = \sum_j z_j f_{n + j},$

with the coherence condition $\pi(R_{n+1}) = R_n$ for all $n$. This constructs $\mathcal{F}$ as an arithmetic and topological configuration space analogous to p-adic integers but built upon Fibonacci expansions and the associated shift map.

The Zeckendorf shift $\pi$ plays a role similar to the restriction maps in inverse limits, acting as an endomorphism both arithmetically and dynamically. This discrete dynamical viewpoint aligns Fibonadic numbers with ergodic theory and symbolic dynamics, with the golden ratio $\varphi$ appearing as the scaling constant in the induced metric (see below).

3. Metric, Order, and Commutative Rig Structure

Metric and Order

An ultrametric is defined on $\mathcal{F}$ by

$d(z, z') = \varphi^{\delta(z, z')},$

where $\delta(z, z')$ denotes the greatest index $n$ at which $z_n \neq z'_n$ (with $\varphi = \frac{1+\sqrt{5}}{2}$), ensuring $\mathcal{F}$ is complete and totally disconnected.

A lexicographic order is imposed: $z < z'$ if for the maximal $n_0$ with $z_{n_0} \neq z'_{n_0}$, one has $z_{n_0} = 0$, $z'_{n_0} = 1$.

Commutative Rig

Interpreting each Fibonadic number as a formal power series

$z(t) = \sum_n z_n t^n,$

with the φ-value (realization) $V_\varphi(z) = \sum_n z_n \varphi^n$, one can define addition and multiplication via power series operations, followed by a canonical “normalization” so that all coefficients remain in $\{0,1\}$ and the no-consecutive-ones constraint is restored. This is achieved via the normalization lemma, which allows a sequence of local “moves” (analogous to carry/borrow in positional notation but subject to Zeckendorf constraints) to yield a unique representative. As a consequence, $\mathcal{F}$ forms a commutative rig, i.e., a ring without additive inverses.

4. Canonical Representatives and the Normalization Lemma

The normalization lemma guarantees the existence of a unique canonical representative for any element of $\mathcal{F}$ (i.e., any Laurent series with the prescribed constraints). The normalization consists of two moves:

• (A) Reducing pairs of adjacent ones by propagating a ‘carry’ to higher indices;
• (B) Splitting digits $z_n \geq 2$ appropriately.

After repeated application, every Laurent series reduces uniquely to a binary sequence with no consecutive ones, matching the uniqueness from Zeckendorf’s theorem. This normalization is essential for making addition and multiplication operations on $\mathcal{F}$ well-defined and compatible with the φ-value function: $V_\varphi(z) = V_\varphi(P(z)),$ where $P$ is the normalization projection.

5. Principal Units, Quotients under Shifts, and the Circle Structure

The multiplicative action by $\Phi$ (where $\Phi$ is the image of $t$ in $\mathcal{F}$) corresponds to digit shifts. Considering the quotient

$$S = (\mathcal{F} \setminus \{0\})/ \{ \Phi^n : n \in \mathbb{Z} \}$$

gives shift-invariant classes of Fibonadic numbers. A fundamental domain for these classes is the set of “Principal Units”

$$\mathcal{D} = \{ z \in \mathcal{F} : z_0 = 1,\ z_n = 0\ \forall n > 0 \}.$$

The φ-value restricts to $\mathcal{D}$ as a surjection onto $[1, \varphi]$, and modulo scaling by powers of $\varphi$, this establishes a covering of the circle by φ-values, realizing a “fractally” self-similar quotient structure.

6. Connections to Number Theory, Combinatorics, Dynamics, Operads, and Fractals

The structure of Fibonadic numbers synthesizes advanced themes:

• Number Theory: $\mathcal{F}$ reflects the unique Zeckendorf representation, reminiscent of numeration systems and p-adic number theory, but grounded in the Fibonacci recurrence and the arithmetic properties of the golden ratio.
• Combinatorics and Trees: The Farey-type graphs and associated binary trees mirror the arithmetic layering in $\mathcal{F}$. The combinatorial “arithmetic layers” correspond to subtrees and operadic compositions, linking Fibonadic numbers to algebraic operads and the combinatorics of partitions.
• Discrete Dynamics: The Zeckendorf shift is a discrete dynamical map with an action on digit expansions, analogous to β-transformations and substitution dynamical systems.
• Fractal and Geometric Aspects: The underlying Farey fractal informs the self-similar, recursive, and ultrametric structure of $\mathcal{F}$, further manifested in the quotient circle and the scaling properties induced by $\varphi$.
• Operads and Arithmetic Layering: The “levels with associated functions encoding the decomposition of X into arithmetic layers” (as referenced in the original) highlight a hierarchical, operadic decomposition within $\mathcal{F}$, compatible with the inductive and fractal architecture of the Farey graph and its finite subtrees (Haran, 13 Sep 2025).

7. Significance and Emerging Applications

Fibonadic numbers provide a bridge between arithmetic, symbolic dynamics, fractal topology, and modern algebraic structures:

• They offer a rigorous framework for arithmetic operations compatible with Zeckendorf representations and normalization.
• The metric, order, and rig structure give rise to new avenues of paper in non-Archimedean geometry and combinatorial arithmetic.
• The quotient by shifts and connection to the circle via φ-values point toward deeper symmetries and possible connections to dynamical systems, ergodic theory, and even coding of geometric objects like tilings and quasicrystals.
• The embedding in operadic and fractal settings suggests new research on hierarchical, recursive structures in both number theory and discrete geometry.

The phrase “Fibonadic numbers” thus no longer refers solely to integers representable as sums of Fibonacci numbers but rather encodes a multi-layered, dynamically rich, and canonically normalized structure at the intersection of classical sequences and contemporary arithmetic geometry (Haran, 13 Sep 2025).