Fibonacci Floor Function Overview

• The Fibonacci floor function is a step function that returns the greatest even-index Fibonacci number less than or equal to a given integer, as defined in model theory.
• It also underpins explicit floor formulas that generate the complement of Fibonacci numbers and yield Beatty-indexed subsequences in arithmetic settings.
• The technique enables block-sampled generating functions and simplifies complex Diophantine counts via precise floor-sum identities involving Fibonacci and Lucas numbers.

The expression Fibonacci floor function is used in several adjacent but non-identical senses. In the most specific recent usage, it denotes a unary function on $\mathbb Z$ that returns the greatest Fibonacci number of even index less than or equal to a given integer, introduced in the model theory of $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$ (Khani et al., 4 Aug 2025). In a broader arithmetic sense, the same phrase also covers explicit floor-function formulas governed by the golden ratio and Fibonacci asymptotics, including a closed formula for the non-Fibonacci numbers (Farhi, 2011), Beatty-type descriptions of Fibonacci-related index sets (Lindroos et al., 2018), generating functions with floor indices (Adegoke et al., 2023), and floor sums attached to Fibonacci and Lucas coefficients in linear Diophantine counting (Teotia, 11 Apr 2026).

1. Definition, conventions, and scope

A first point of terminology is that the literature does not fix a single convention for the Fibonacci sequence. The model-theoretic paper on the named Fibonacci floor function uses the shifted convention

$F_0=1,\qquad F_1=1,\qquad F_{n+2}=F_n+F_{n+1},$

whereas the complementary and analytic papers use the standard convention

$F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n.$

This distinction affects index parity statements and must be tracked explicitly (Khani et al., 4 Aug 2025).

In the model-theoretic setting, let

$$f(x)=\lfloor \varphi x\rfloor,\qquad [\varphi x]=\varphi x-\lfloor \varphi x\rfloor,$$

with $\varphi=(1+\sqrt5)/2$. The paper defines functions $F$ and $G$ by

$$F(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\min\{[\varphi w]:0<w\le x\},$$

$$G(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\max\{[\varphi w]:0<w\le x\}.$$

It then states that $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$0 maps each integer $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$1 to the greatest Fibonacci number of even index and less than or equal to $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$2, and remarks that a “more telling notation” for $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$3 would be the symbol $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$4, called the Fibonacci floor function (Khani et al., 4 Aug 2025).

The surrounding literature uses the phrase more loosely for any floor-based encoding of Fibonacci structure. The main usages in the cited works are summarized below.

Context	Object	Role
Model theory	$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$5	Largest even-index Fibonacci $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$6
Complement formulas	Explicit floor expression	Enumerates non-Fibonacci numbers
Floor-indexed sequences	$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$7	Block-sampled Fibonacci sequence
Beatty indexing	$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$8	Positions of odd fibbinaries
Diophantine counting	Floor sums $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$9	Exact formulas for special triplets

A common misconception is that the Fibonacci floor function is simply the Beatty map $F_0=1,\qquad F_1=1,\qquad F_{n+2}=F_n+F_{n+1},$0. The model-theoretic paper explicitly separates these two objects: $F_0=1,\qquad F_1=1,\qquad F_{n+2}=F_n+F_{n+1},$1 is the ambient Beatty function, whereas the Fibonacci floor function $F_0=1,\qquad F_1=1,\qquad F_{n+2}=F_n+F_{n+1},$2 is a new definable function extracted from the order of the fractional parts $F_0=1,\qquad F_1=1,\qquad F_{n+2}=F_n+F_{n+1},$3 (Khani et al., 4 Aug 2025).

2. Beatty-theoretic characterization and logical role

The model-theoretic construction is organized around the decimal parts $F_0=1,\qquad F_1=1,\qquad F_{n+2}=F_n+F_{n+1},$4. A key characterization is that, for $F_0=1,\qquad F_1=1,\qquad F_{n+2}=F_n+F_{n+1},$5, $F_0=1,\qquad F_1=1,\qquad F_{n+2}=F_n+F_{n+1},$6 is a Fibonacci number of even index iff $F_0=1,\qquad F_1=1,\qquad F_{n+2}=F_n+F_{n+1},$7 is the minimum of

$F_0=1,\qquad F_1=1,\qquad F_{n+2}=F_n+F_{n+1},$8

and $F_0=1,\qquad F_1=1,\qquad F_{n+2}=F_n+F_{n+1},$9 is a Fibonacci number of odd index iff $F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n.$0 is the corresponding maximum. This identifies Fibonacci numbers as extremal points of the orbit of the Beatty rotation $F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n.$1, with parity detected by min-versus-max behavior (Khani et al., 4 Aug 2025).

The paper also defines a “decimal order”

$F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n.$2

and notes that this coincides with the order relation $F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n.$3. From the basic axioms for $F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n.$4, the authors prove that $F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n.$5 is a dense linear order without endpoints on $F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n.$6, and they obtain the constructive Kronecker lemma

$F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n.$7

This embeds the Fibonacci floor function into a first-order framework where extremal decimal parts on intervals are definable (Khani et al., 4 Aug 2025).

Within that framework, the Fibonacci floor function interacts tightly with the Beatty function $F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n.$8. If $F_0=0,\qquad F_1=1,\qquad F_{n+2}=F_{n+1}+F_n.$9, then the first even-index Fibonacci strictly greater than $$f(x)=\lfloor \varphi x\rfloor,\qquad [\varphi x]=\varphi x-\lfloor \varphi x\rfloor,$$0 is

$$f(x)=\lfloor \varphi x\rfloor,\qquad [\varphi x]=\varphi x-\lfloor \varphi x\rfloor,$$1

Moreover, $$f(x)=\lfloor \varphi x\rfloor,\qquad [\varphi x]=\varphi x-\lfloor \varphi x\rfloor,$$2 is a step function in the precise sense that

$$f(x)=\lfloor \varphi x\rfloor,\qquad [\varphi x]=\varphi x-\lfloor \varphi x\rfloor,$$3

The paper also provides an explicit addition rule for $$f(x)=\lfloor \varphi x\rfloor,\qquad [\varphi x]=\varphi x-\lfloor \varphi x\rfloor,$$4, expressed in terms of $$f(x)=\lfloor \varphi x\rfloor,\qquad [\varphi x]=\varphi x-\lfloor \varphi x\rfloor,$$5, $$f(x)=\lfloor \varphi x\rfloor,\qquad [\varphi x]=\varphi x-\lfloor \varphi x\rfloor,$$6, $$f(x)=\lfloor \varphi x\rfloor,\qquad [\varphi x]=\varphi x-\lfloor \varphi x\rfloor,$$7, $$f(x)=\lfloor \varphi x\rfloor,\qquad [\varphi x]=\varphi x-\lfloor \varphi x\rfloor,$$8, and the definable inverse $$f(x)=\lfloor \varphi x\rfloor,\qquad [\varphi x]=\varphi x-\lfloor \varphi x\rfloor,$$9 on the relevant ranges (Khani et al., 4 Aug 2025).

These constructions are then incorporated into a recursive theory $\varphi=(1+\sqrt5)/2$0 in the language

$\varphi=(1+\sqrt5)/2$1

Its axioms include Presburger arithmetic, the Beatty axioms for $\varphi=(1+\sqrt5)/2$2, interval-extrema axioms for decimal parts, the defining axioms for $\varphi=(1+\sqrt5)/2$3 and $\varphi=(1+\sqrt5)/2$4, and the explicit addition law for $\varphi=(1+\sqrt5)/2$5. The central logical result is that $\varphi=(1+\sqrt5)/2$6 is model-complete; the standard structure $\varphi=(1+\sqrt5)/2$7 is its prime model; and the theory is complete and decidable. In this sense, the Fibonacci floor function is not an isolated combinatorial gadget but a named definable function that stabilizes the model theory of a Presburger expansion by a Beatty sequence (Khani et al., 4 Aug 2025).

3. Exact floor descriptions of the complement of the Fibonacci sequence

A distinct line of work studies floor expressions that do not return Fibonacci numbers but rather their complement. Let $\varphi=(1+\sqrt5)/2$8 be an increasing sequence of integers and let $\varphi=(1+\sqrt5)/2$9 be continuous, increasing, and unbounded, subject to

$F$0

A general complement theorem then shows that

$F$1

generates exactly the complement of $F$2 in $F$3 (Farhi, 2011).

Applying this theorem to

$F$4

with the standard Fibonacci convention yields an explicit formula for the non-Fibonacci numbers. If $F$5, then for each integer $F$6,

$F$7

is exactly the $F$8-th positive integer that is not a Fibonacci number, in increasing order (Farhi, 2011).

The construction is based on Binet’s formula

$F$9

together with the approximation $G$0. The specific nested logarithmic structure arises from inverting the asymptotic relation between $G$1 and $G$2, while the terms $G$3 and $G$4 serve as fine-tuning corrections so that the required one-unit window inequality holds for all $G$5, not only asymptotically (Farhi, 2011).

This formula is analogous in spirit to classical complement formulas such as

$G$6

for non-squares and

$G$7

for non-triangular numbers. The Fibonacci case is more complicated because the underlying growth is exponential with irrational base $G$8, so inverse logarithmic terms necessarily appear (Farhi, 2011).

A second common misconception is therefore that a “Fibonacci floor function” must generate Fibonacci numbers themselves. One major explicit floor formula in the literature does the opposite: it generates the positive integers that are not Fibonacci numbers.

4. Floor indices, block sampling, and generating functions

A different usage of floor in Fibonacci analysis appears in the study of dual floor sequences

$G$9

defined for any sequence $$F(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\min\{[\varphi w]:0<w\le x\},$$0 and integer $$F(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\min\{[\varphi w]:0<w\le x\},$$1. If $$F(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\min\{[\varphi w]:0<w\le x\},$$2 is the ordinary generating function, then

$$F(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\min\{[\varphi w]:0<w\le x\},$$3

The factor $$F(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\min\{[\varphi w]:0<w\le x\},$$4 records the block structure induced by the floor: each value $$F(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\min\{[\varphi w]:0<w\le x\},$$5 is repeated exactly $$F(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\min\{[\varphi w]:0<w\le x\},$$6 times (Adegoke et al., 2023).

Specializing to the classical Fibonacci and Lucas generating functions,

$$F(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\min\{[\varphi w]:0<w\le x\},$$7

gives the floor-indexed identities

$$F(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\min\{[\varphi w]:0<w\le x\},$$8

$$F(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\min\{[\varphi w]:0<w\le x\},$$9

These formulas are the basic structural description of Fibonacci and Lucas sequences sampled at floor-compressed indices (Adegoke et al., 2023).

The same paper derives explicit closed forms for mixed series of the types

$$G(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\max\{[\varphi w]:0<w\le x\}.$$0

$$G(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\max\{[\varphi w]:0<w\le x\}.$$1

as well as alternating block-weighted variants involving $$G(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\max\{[\varphi w]:0<w\le x\}.$$2. The method combines the floor-sequence generating-function identity with Binet’s formulas and substitutions $$G(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\max\{[\varphi w]:0<w\le x\}.$$3, $$G(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\max\{[\varphi w]:0<w\le x\}.$$4, where $$G(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\max\{[\varphi w]:0<w\le x\}.$$5 and $$G(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\max\{[\varphi w]:0<w\le x\}.$$6 are the roots of $$G(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\max\{[\varphi w]:0<w\le x\}.$$7 (Adegoke et al., 2023).

Here the floor operator does not select extremal Fibonacci numbers, as in the model-theoretic setting, nor does it describe a complement. Instead, it creates a controlled blockwise deceleration of the index map $$G(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\max\{[\varphi w]:0<w\le x\}.$$8, and the resulting rational factorization of generating functions makes a broad family of Fibonacci and Lucas series explicitly computable.

5. Beatty position formulas and odd fibbinary numbers

Another Fibonacci-adjacent floor phenomenon arises from fibbinary numbers, the positive integers whose binary expansion contains no consecutive $$G(x)=y \Longleftrightarrow 0<y\le x \ \wedge\ [\varphi y]=\max\{[\varphi w]:0<w\le x\}.$$9s. Using Zeckendorf representation, the paper defines $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$00 as the increasing sequence of all fibbinary numbers, and $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$01 as the increasing sequence of odd fibbinary numbers. If the $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$02-th odd fibbinary is the $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$03-th fibbinary overall, then

$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$04

is its position in the full fibbinary ordering (Lindroos et al., 2018).

The main theorem gives a Beatty-type closed form: $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$05 with $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$06 and $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$07. Thus the positions of odd fibbinary numbers form a shifted Beatty sequence associated with the golden ratio (Lindroos et al., 2018).

The proof is recursive. The paper establishes

$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$08

and

$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$09

then shows that the floor expression $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$10 satisfies the same block recurrences and initial conditions. Fractional-part control is handled via $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$11 and the identity

$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$12

which makes it possible to compare the jumps of $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$13 with the jumps of the Beatty sequence (Lindroos et al., 2018).

This result does not define a unary Fibonacci floor function on integers, but it is a canonical example of a floor formula whose arithmetic content is governed by Fibonacci growth and Zeckendorf structure. It shows that Beatty sequences associated with $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$14 and $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$15 can encode the exact locations of a combinatorially defined Fibonacci-related subsequence.

6. Floor sums in Diophantine counting with Fibonacci and Lucas triplets

Floor functions also enter through counting formulas for non-negative solutions of

$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$16

Binner’s general formula expresses

$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$17

as a rational term plus three sums of floor functions: $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$18 In general, these sums are difficult to evaluate explicitly (Teotia, 11 Apr 2026).

For the special triplets

$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$19

the paper shows that Cassini-type identities force the modular parameters to collapse to

$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$20

Consequently, the floor sums simplify drastically. In the Fibonacci case,

$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$21

and

$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$22

The Lucas case is formally identical with $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$23 replaced by $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$24 and $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$25 replaced by $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$26 (Teotia, 11 Apr 2026).

This yields exact formulas for

$$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$27

with the difficult floor-sum component reduced to a triangular number. The significance is methodological: Fibonacci and Lucas recurrences do not merely appear as coefficients, but force a nontrivial floor-sum reciprocity problem into an explicitly solvable regime (Teotia, 11 Apr 2026).

Across these works, the phrase Fibonacci floor function therefore has a layered meaning. In the strictest sense it names the definable step function $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$28 “largest even-index Fibonacci $$\langle \mathbb Z,<,+,\lfloor \varphi x\rfloor,0\rangle$$29.” In a wider technical sense it refers to the use of floor operators to encode complementary sets, Beatty-indexed subsequences, block-sampled recurrences, and exact lattice-counting formulas controlled by Fibonacci or Lucas arithmetic.