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If the set of paradoxical sequences \((j, m)\) with \(m {\gt} 2\) is finite, then the Collatz conjecture is true.

Let \((j, n)\) be a paradoxical pair with \(j \ge 2\) and \(n {\gt} 0\). Let \(m {\gt} 0\) be a lower bound for all odd iterates \(T^k(n)\) with \(k {\lt} j\). Define \(r = j / \lfloor j \log 2 / \log 3 \rfloor \). Then:

The decomposition coefficient \(C(k, n)\) is the rational number

The coefficient stopping time \(\tau (n)\) is the smallest \(j \ge 1\) such that \(C(j, n) {\lt} 1\). If no such \(j\) exists, \(\tau (n) = \infty \).

The Collatz graph \(\mathcal{G}\) is a directed graph on \(\mathbb {N}\) with an edge from \(n\) to \(T(n)\) for all \(n \in \mathbb {N}\). We consider its associated simple undirected graph \(\mathcal{G}'\).

For any \(k \in \mathbb {N}\), the \(k\)-fold iteration of the Collatz map \(C^k: \mathbb {N} \to \mathbb {N}\) is defined recursively by

and

The Collatz map \(C: \mathbb {N} \to \mathbb {N}\) is defined by

The universal \(U\)-orbit density conjecture asserts that \(U\text{-orbit density}(n_0)\) holds for every \(n_0 \in \mathbb {N}\).

For \(d, n \in \mathbb {N}\), the number of odd steps in the first \(d\) iterations of the standard Collatz map starting from \(n\) is defined by

where \([P]\) is the Iverson bracket.

For \(k, n \in \mathbb {N}\), the accumulated correction term \(Q(k, n)\) of the Terras map is defined recursively by \(Q(0, n) = 0\) and

The correction term \(Q(k, n)\) has the closed-form expression

For any \(\varepsilon \in \mathbb {R}\), we define the bound \(\delta (\varepsilon )\) as:

The base-\(3/2\) representation of \(n\), written \(\langle n \rangle \), is the list of digits (least-significant first) defined recursively:

Termination is guaranteed by Lemma 175.

The correction ratio \(E(j, n)\) is the rational number

For \(k, n \in \mathbb {N}\), the vector \(E(k, n)\) is the function from \(\{ 0, \dots , k-1\} \) to \(\{ 0, 1\} \) defined by

The evaluation of a list of base-\(3/2\) digits (least-significant first) is

For admissible digit lists (those produced by \(\operatorname {digits}_{3/2}\)), the division is always exact.

A pair \((j, n)\) consisting of a positive iteration count \(j\) and a starting integer \(n\) is called paradoxical if:

That is, the \(j\)-th iterate of the Terras map is at least as large as the starting value, and the decomposition coefficient at \(j\) is smaller than \(1\). The sequence of values

is called a paradoxical sequence.

An odd number \(n {\gt} 1\) is called a primitive at level \(i {\gt} 1\) if

1. \(R^i(n) = 1\),

2. there is no odd \(k\) with \(4k + 1 = n\) and \(R^i(k) = 1\).

The lower bound sequence \(L_j(q)\) is defined for \(j, q \in \mathbb {N}\) by:

The least-significant digit of \(n\) in rational base \(3/2\) is

This is the digit \(a_0\) of Proposition 2.2 of  [ 2 ] .

The number of odd steps in the first \(k\) iterations of \(T\) starting from \(n\) is defined by

The number of odd values among the first \(k\) terms of the \(U\)-orbit starting from \(n_0\) is

The parent of \(n\) in the base-\(3/2\) digit tree is

Removing the least-significant digit from \(\langle n \rangle \) gives \(\langle \operatorname {parent}(n) \rangle \).

A parity vector is a finite sequence of bits (0 and 1). In the formalization, it is represented as a list of boolean values, where false corresponds to 0 and true corresponds to 1.

The ones-ratio of a parity vector \(v\) is the rational number \(q(v) / |v|\). If the vector is empty, the ratio is defined to be 0.

The elementary precedence relation \(\prec _e\) is defined by swapping a \(01\) subword with \(10\):

The precedence relation \(\preceq \) is the reflexive transitive closure of \(\prec _e\).

For a parity vector \(v\), \(q(v)\) is the number of ones (true entries) in \(v\).

The size of a parity vector \(v\), denoted \(|v|\), is its length.

For \(j, n \in \mathbb {N}\), the parity vector \(V(j, n)\) of length \(j\) for the compact Collatz sequence starting at \(n\) is defined as

The upper bound \(R(q)\) is defined for \(q \in \mathbb {N}\) by:

For any \(k \in \mathbb {N}\), the \(k\)-fold iteration of the reduced Collatz map \(R^k: \mathbb {N} \to \mathbb {N}\) is defined recursively by

and

The function \(\text{reduce4x1}: \mathbb {N} \to \mathbb {N}\) is defined recursively by

The reduced Collatz map \(R: \mathbb {N}\to \mathbb {N}\) is defined by

where \(v_2(m)\) is the dyadic valuation of \(m\). In short \(R(n)=\) odd part of \(3n+1\).

The saturated number \(\operatorname {sat}(k) = U^k(0)\) is the largest natural number whose base-\(3/2\) representation has length \(k\) (Proposition 3.10 of  [ 2 ] ).

The scaled evaluation of a digit list \([a_0, \ldots , a_k]\) is

Recursively: \(\operatorname {scaledEval}([]) = 0\) and \(\operatorname {scaledEval}(a :: \mathit{rest}) = 2^{|\mathit{rest}|} \cdot a + 3 \cdot \operatorname {scaledEval}(\mathit{rest})\).

The cyclic permutation \(\sigma = (2\; 1\; 0)\) on the digit set \(\{ 0,1,2\} \):

Formally, \(\sigma (d) = (d + 2) \bmod 3\).

Given a digit list (least-significant first), \(\operatorname {splitAtFirstZero}\) returns the pair \((\mathit{prefix}, \mathit{suffix})\) where \(\mathit{prefix}\) consists of all nonzero digits before the first \(0\), and \(\mathit{suffix}\) consists of the digits after the first \(0\). If no \(0\) is present, the suffix is empty.

The stopping time \(\sigma (n)\) of \(n\) under \(T\) is the smallest \(k \ge 1\) such that \(T^k(n) {\lt} n\). If no such \(k\) exists, \(\sigma (n) = \infty \).

The compact Collatz map \(T: \mathbb {N} \to \mathbb {N}\) is defined by

where \(X(n) = n \pmod2\).

For any \(k \in \mathbb {N}\), the \(k\)-fold iteration of the map \(T\) is defined recursively by \(T^0(n) = n\) and \(T^{k+1}(n) = T(T^k(n))\).

The total stopping time \(\sigma _\infty (n)\) of \(n\) under \(T\) is the smallest \(k \ge 1\) such that \(T^k(n) = 1\). If no such \(k\) exists, \(\sigma _\infty (n) = \infty \).

The function \(U : \mathbb {N} \to \mathbb {N}\) (Notation 3.7 of  [ 2 ] ) is defined by

Equivalently, \(U(n) = (3n + 2 - (n \bmod 2))/2\).

The \(k\)-fold iteration of \(U\) is defined by \(U^0(n) = n\) and \(U^{k+1}(n) = U(U^k(n))\).

The \(U\)-orbit density conjecture for a fixed starting point \(n_0\) asserts that

We define the indicator function \(X: \mathbb {N} \to \mathbb {N}\) such that \(X(n) = 0\) if \(n\) is even and \(X(n) = 1\) if \(n\) is odd. Formally,

The irrational constant \(\xi \) is defined as:

If no natural number \(n {\gt} 4\) lies on a nontrivial cycle and every orbit under \(C\) is bounded, then the Collatz conjecture is true.

For any \(n \in \mathbb {N}\), the orbit of \(n\) under \(C\) is bounded if and only if the orbit of \(n\) under \(T\) is bounded.

Let \(n \in \mathbb {N}\) be odd. The orbit of \(n\) under the standard Collatz map is bounded if and only if its orbit under the reduced Collatz map is bounded.

For any odd \(n \in \mathbb {N}\), \(n\) belongs to a cycle under the original Collatz map \(C\) if and only if it belongs to a cycle under the compact map \(T\).

Let \(n \in \mathbb {N}\) be odd. There exists a cycle in the standard Collatz map starting from \(n\) if and only if there exists a cycle in the reduced Collatz map starting from \(n\).

Two distinct vertices \(a, b\) are adjacent in \(\mathcal{G}'\) if and only if \(b = T(a)\) or \(a = T(b)\).

For all \(a, b, n \in \mathbb {N}\), \(C^{a+b}(n) = C^a(C^b(n))\).

If \(n \in \{ 1, 2, 4\} \), then for all \(i \in \mathbb {N}\), \(C^i(n) \in \{ 1, 2, 4\} \).

If \(C^k(n) = n\), then for all \(m \in \mathbb {N}\), \(C^{m \cdot k}(n) = n\).

For all \(i \in \mathbb {N}\), \(C^i(1) \le 4\).

If \(n \ge 1\), then for all \(k \in \mathbb {N}\), \(C^k(n) \ge 1\).

For all \(l \in \mathbb {N}\), \(C^l(2^l) = 1\).

For all \(k {\lt} l\), \(C^k(2^l) \neq 1\).

For all \(k, n \in \mathbb {N}\), we have \(C^{k+1}(n) = C(C^k(n))\).

If \(n \ge 1\) and \(C^j(n) = 1\) for some \(j \in \mathbb {N}\), then there exists \(k \in \mathbb {N}\) such that \(T^k(n) = 1\).

For any \(k, m \in \mathbb {N}\), \(C^k(2^k \cdot m) = m\).

For all \(k \in \mathbb {N}\), \(C^k(0) = 0\).

The Collatz map \(C(n)\) has the closed form

for all \(n \in \mathbb {N}\).

If \(n \ge 1\), then \(C(n) \ge 1\).

For all \(k \ge 1\), \(C(2^k) = 2^{k-1}\).

\(C(0) = 0\).

If \(a\) and \(b\) are reachable in \(\mathcal{G}'\), then they have a common descendant under \(T\), i.e., there exist \(i, j\) such that \(T^i(a) = T^j(b)\).

If \(T^i(a) = T^j(b)\) and \(b\) is adjacent to \(c\) in \(\mathcal{G}'\), then there exist \(i', j'\) such that \(T^{i'}(a) = T^{j'}(c)\).

Conjectures 2.4 and 2.5 are equivalent:

• Conjecture 2.4: For every \(n \ge 1\), there exists \(k \ge 1\) such that \(T^k(n) = 1\).

• Conjecture 2.5: For every \(n \ge 1\),

  \[ \lim _{k \to \infty } \frac{k - Q(k,\, n)}{k} = \frac{1}{2}. \]

Conjecture 4.2 (Definition ??) is equivalent to the \(U\)-orbit density conjecture for \(n_0 = 0\).

For any \(s, k \in \mathbb {N}\), \(3^s\) and \(2^k\) are coprime.

Under the same hypotheses as Lemma 147, we have:

If there exists a natural number \(n {\gt} 4\) and \(k \ge 1\) such that \(C^k(n) = n\), then the Collatz conjecture is false (No-cycle condition).

If \(E(k, m) = E(k, n)\), then \(Q(k, m) = Q(k, n)\).

The recursive definition of \(Q(k, n)\) is equivalent to the closed-form sum.

For even \(n\) (Proposition 3.1, digit \(2\)),

For even \(n \ge 1\) (Proposition 3.1, digit \(0\)),

\(|\operatorname {digits}_{3/2}(n)| = 0\) if and only if \(n = 0\).

For \(k \ge 1\), every digit in \(\langle \operatorname {sat}(k) \rangle \) is either \(1\) or \(2\) (no digit \(0\) appears). This is Proposition 3.5 of  [ 2 ] .

For odd \(n\) (Corollary 3.2 of  [ 2 ] ),

That is, the base-\(3/2\) representation of \(T(n) = (3n+1)/2\) is obtained by prepending digit \(1\) to \(\langle n \rangle \).

For all \(n \in \mathbb {N}\), the base-\(3/2\) digits of \(U(n)\) are obtained by prepending \(1\) (if \(n\) is odd) or \(2\) (if \(n\) is even) to \(\langle n \rangle \):

This is Proposition 3.9 of  [ 2 ] (combined form).

If \(V_j(m)\) and \(V_j(n)\) differ by a single elementary swap \(01 \to 10\) (i.e., \(V_j(m) \prec _{el} V_j(n)\)), then \(E_j(n) {\lt} E_j(m)\).

If the parity vectors \(V_j(m)\) and \(V_j(n)\) agree on their first \(k\) entries (\(k \le j\)), then \(E_k(m) = E_k(n)\).

The correction ratio \(E\) is strictly monotonic with respect to the transitive closure of the elementary precedence relation on parity vectors.

The correction ratio \(E_j(n)\) is periodic with period \(2^j\):

For any \(k, j, n \in \mathbb {N}\), the correction ratio behaves under powers of two as:

For a fixed parity \(x \in \{ 0, 1\} \), the step map \(a \mapsto \frac{3^x}{2} a + \frac{x}{2}\) is strictly monotone.

The ratio \(E\) satisfies the recurrence

For any \(j, m \in \mathbb {N}\), we have the identity:

Let \(k \le j\). If \(E_k(n) {\lt} E_k(m)\) and the parity bits of \(n\) and \(m\) agree for steps \(i \in \{ k, \dots , j-1\} \) (i.e., \(X(T^i(n)) = X(T^i(m))\) for \(k \le i {\lt} j\)), then \(E_j(n) {\lt} E_j(m)\).

Each entry of \(E(k, n)\) is at most 1.

If \(E(k+1, m) = E(k+1, n)\), then \(E(k, m) = E(k, n)\).

\(E(0, n) = 0\).

For all \(n \in \mathbb {N}\),

If \(\operatorname {lsd}(n) = 2\), then \(\operatorname {parent}(n)\) is even.

If \(\operatorname {lsd}(n) = 0\), then \(\operatorname {parent}(n)\) is even.

For all \(l \in \mathbb {N}\), there exists \(n \in \mathbb {N}\) such that \(C^l(n) = 1\) and for all \(k {\lt} l\), \(C^k(n) \neq 1\).

Given a level \(m \ge 1\), a seed \(y_0 {\gt} 1\) that is odd and satisfies \(R^m(y_0) = 1\), and \(B \in \mathbb {N}\), there exists an odd \(y {\gt} B\) with \(y {\gt} 1\), \(y \not\equiv 0 \pmod{3}\), and \(R^m(y) = 1\).

Let \(y {\gt} 1\) be a natural number such that \(y \equiv 1 \pmod6\). Then there exists an odd natural number \(x {\gt} 1\) such that

Let \(y\) be a natural number such that \(y \equiv 5 \pmod6\). Then there exists an odd natural number \(x {\gt} 1\) such that

Let \(y\) be an odd natural number such that \(y \pmod3 \neq 0\). Then there exist natural numbers \(x\) and \(k\) such that \(x\) is odd and

Let \(m {\gt} 0\) be odd with \(m \not\equiv 0 \pmod{3}\). Then there exists an odd \(n {\gt} 1\) such that \(R(n) = m\).

Let \(n \in \mathbb {N}\) and suppose that

Then there exists \(k \ge 1\) such that

Let \(n \ge 1\) and suppose there exists \(k_0 \ge 1\) such that \(T^{k_0}(n) = 1\). Then the proportion of even iterates satisfies

For any positive integers \(a, b {\gt} 0\) and any \(\varepsilon {\lt} 1\), the following equivalence holds:

For every level \(m \ge 2\), there are infinitely many primitive numbers at level \(m\).

There are infinitely many rational numbers \(q = n/d {\lt} \xi \) such that

For any \(a, b, c \in \mathbb {N}\), if \(a \equiv b \pmod c\), then \(c \mid (a - b)\) in the integers.

The constant \(\xi \) is irrational.

Let \(n {\gt} 1\) be odd with \(n \neq 5\), let \(i {\gt} 1\), and suppose \(R^i(n) = 1\). Then \(n\) is a primitive at level \(i\) if and only if \(n \not\equiv 5 \pmod{8}\).

Let \(j, n \in \mathbb {N}\) with \(n {\gt} 0\). If the pair \((j, n)\) is paradoxical, then:

For any \(k, n \in \mathbb {N}\), the \(k\)-fold iteration of the Terras map \(T^k(n)\) satisfies the linear identity

where \(Q(k, n)\) is the number of odd iterates as defined in Definition 81.

Another form of the linear identity is given by:

For any \(j \ge 2\) and any \(k \in \mathbb {N}\), the value \(j \cdot \log 2 / \log 3\) is not an integer:

For all \(n \in \mathbb {N}\), \(\operatorname {lsd}(n) {\lt} 3\).

For all \(n \in \mathbb {N}\),

This is part of Proposition 2.2 of  [ 2 ] .

For all \(n \ge 1\),

When \(n\) is even, \(\operatorname {lsd}(U(n)) = 2\).

When \(n\) is odd, \(\operatorname {lsd}(U(n)) = 1\).

Every digit in \(\operatorname {digits}_{3/2}(n)\) is less than \(3\).

For any \(a, b, c \in \mathbb {N}\), if \(c \mid (a - b)\) in the integers, then \(a \equiv b \pmod c\) in the natural numbers.

Let \(y\) be a natural number such that \(y \equiv 0 \pmod3\). Then for all natural numbers \(n\) and \(k\),

If \(m \equiv 0 \pmod{3}\), then there is no \(n \in \mathbb {N}\) such that \(R(n) = m\). In other words, multiples of \(3\) are not in the image of \(R\).

The number of odd steps \(Q(k, n)\) is equal to the sum of the entries of \(E(k, n)\):

If \(E(k, m) = E(k, n)\), then \(Q(k, m) = Q(k, n)\).

If \(n \ge 1\) has an infinite stopping time, then for every \(a \in \mathbb {N}\), there exists an iteration count \(j\) such that \(q(j, n) = a\).

If \(j \le k\), then \(Q(j, n) \le Q(k, n)\).

For all \(m \in \mathbb {N}\), letting \(s = Q(m,\, 1)\),

In other words, \(2s \in \{ m, m+1\} \).

For all \(j \in \mathbb {N}\),

For all \(j \in \mathbb {N}\),

For all \(a, b, n \in \mathbb {N}\),

\(Q(k+1, n) = Q(k, n) + X(T^k(n))\).

If \(n \ge 1\) has an infinite stopping time, then the number of odd steps \(q(j, n)\) is unbounded as \(j \to \infty \).

Let \(j, n, m \in \mathbb {N}\) with \(m {\gt} 0\). Suppose that every odd iterate \(T^k(n)\) with \(k {\lt} j\) satisfies \(T^k(n) \ge m\). Let \(q = \mathit{q}(j, n)\) be the number of odd steps in the first \(j\) iterations. Then:

Let \(y {\gt} 1\) be odd with \(y \not\equiv 0 \pmod{3}\) and \(R^i(y) = 1\). Then there exists a primitive \(n\) at level \(i + 1\).

If \(\operatorname {lsd}(n) = 1\), then \(\operatorname {parent}(n)\) is odd.

(Proposition 2.3, Odometer.) Let \(\langle n \rangle = u\, 0\, v\) where \(v \in \{ 1,2\} ^*\) (i.e. \(v\) is the block of nonzero digits before the first \(0\), reading from the least-significant end, and \(u\) is the part after the \(0\)). Then

where \(v^\sigma \) denotes the list obtained by applying \(\sigma \) to each digit of \(v\). In other words, incrementing by one applies \(\sigma \) to all leading nonzero digits (carrying), replaces the first \(0\) with \(2\), and leaves the remaining digits unchanged.

Let \(m \ge 1\) be a minimal counterexample to Conjecture 2.4, meaning:

• \(T^k(m) \ne 1\) for all \(k \ge 1\);

• for every \(n {\lt} m\) with \(n \ge 1\), there exists \(k \ge 1\) with \(T^k(n) = 1\).

Then \(T^j(m) \ge m\) for all \(j \in \mathbb {N}\).

Let \(n \in \mathbb {N}\) be odd. For any \(d \in \mathbb {N}\), the odd part of \(C^d(n)\) is equal to \(R^k(n)\), where \(k = \text{countOdds}(d, n)\). That is,

For any even \(x \in \mathbb {N}\), the odd part of \(x/2\) is equal to the odd part of \(x\).

For any \(k \in \mathbb {N}\) and any odd \(m \in \mathbb {N}\), the odd part of \(2^k \cdot m\) is \(m\).

For any \(k, x \in \mathbb {N}\), the odd part of \(2^k \cdot x\) is equal to the odd part of \(x\).

Let \(j, q, m \in \mathbb {N}\) with \(j \ge 2\) and \(m {\gt} 0\). Suppose the ratio bounds of Lemma 149 hold for \(q/j\). Define:

Then \(2^r - 3 {\gt} 0\) and \(m \le 1/(2^r - 3)\).

Let \(j, q, m \in \mathbb {N}\) with \(j, q, m {\gt} 0\). Suppose the ratio bounds

hold. Then:

1. \(2^{j/q} - 3 {\gt} 0\), i.e., \(2^{j/q} {\gt} 3\);

2. \(m \le \dfrac {1}{2^{j/q} - 3}\).

Let \((j, n)\) be a paradoxical sequence starting at \(n {\gt} 0\). Let \(m {\gt} 0\) be a lower bound for all odd iterates \(T^i(n)\) with \(i {\lt} j\). Let \(q\) be the number of odd steps. Then:

Let \(j, q, m \in \mathbb {N}\) with \(j {\gt} 0\) and \(m {\gt} 0\). Suppose the following inequalities hold for the rationals \(3^q/2^j\) and \((3+1/m)^q/2^j\):

Then the ratio \(q/j\) satisfies:

For all \(n \ge 1\), \(\operatorname {parent}(n) {\lt} n\).

If \(n \ge 1\) and \(\operatorname {lsd}(n) \ne 0\), then \(\operatorname {parent}(n+1) = \operatorname {parent}(n) + 1\) (carry propagates).

If \(n \ge 1\) and \(\operatorname {lsd}(n) = 0\), then \(\operatorname {parent}(n+1) = \operatorname {parent}(n)\) (no carry).

For all \(n \in \mathbb {N}\),

This is Proposition 3.9 of  [ 2 ] .

(Proposition 4.5 of  [ 2 ] .) Let \(n \ge 1\) with \(\langle n \rangle = a_k \cdots a_0\) having \(k+1\) digits. Then \(n\) is even if and only if

Equivalently, using the scaled evaluation:

where \(L = |\langle n \rangle |\).

For any \(k, m, n \in \mathbb {N}\), if \(m \equiv n \pmod{2^{k+1}}\), then \(m \equiv n \pmod2\).

Let \(x {\gt} 1\) be odd with \(R(x) \neq 1\). Then there exists an odd \(n {\gt} 1\) with \(R(n) = R(x)\) and \(n \not\equiv 5 \pmod{8}\).

Let \(x_1, x_2 \in \mathbb {N}\) and \(y_1, y_2\) be their respective images under \(R\), with \(y_1 \neq y_2\). Then \(\text{reduce4x1}(x_1) \neq \text{reduce4x1}(x_2)\).

For all \(n {\gt} 1\), \(R(4n + 1) = R(n)\).

For all \(n {\gt} 1\) and \(i {\gt} 0\), \(R^i(4n + 1) = R^i(n)\).

For all \(n \in \mathbb {N}\) and \(i \geq 1\), \(R^i(n)\) is odd, i.e. \(R^i(n) \equiv 1 \pmod{2}\).

\(R(1) = 1\).

For all \(n, i \in \mathbb {N}\) with \(n \geq 1\), we have \(R^i(n) \geq 1\).

\(R(0) = 1\).

For all \(s, k \in \mathbb {N}\) with \(5s + 1 \le 3k\),

For any odd \(n \in \mathbb {N}\), \(2 R(n) \le 3n + 1\).

For any \(n \in \mathbb {N}\), \(R(\text{reduce4x1}(n)) = R(n)\).

If \(n\) is odd and \(R^k(n) = m\), then there exists \(i \ge k\) such that \(C^i(n) = m\).

For any odd \(n {\gt} 0\), \(R(n) {\gt} n\) if and only if \(n \equiv 3 \pmod{4}\).

If \(n {\gt} 0\) is odd and \(R(n) = m\), then there exists \(i \ge 1\) such that \(C^i(n) = m\).

For all \(n \in \mathbb {N}\), \(R(n)\) is odd, i.e. \(R(n) \equiv 1 \pmod{2}\).

For all \(n \in \mathbb {N}\), \(R(n) \geq 1\).

For all \(k \in \mathbb {N}\) (Proposition 3.6 of  [ 2 ] ),

For all \(n \in \mathbb {N}\),

For every \(n\) with \(1 \le n \le 9\), there exists \(k \ge 1\) such that \(T^k(n) = 1\).

For any \(j \ge 2\), the floor of \(j \log 2 / \log 3\) is strictly less than \(j \log 2 / \log 3\):

For any \(k, m, n \in \mathbb {N}\), if \(m \equiv n \pmod{2^{k+1}}\), then \(T(m) \equiv T(n) \pmod{2^k}\).

If \(n\) is even, \(T(n) = n/2\).

Let \(k, n \in \mathbb {N}\) and suppose that \(T^j(n) \ge 10\) for all \(j \le k\). Let \(s = Q(k, n)\). Then

For any \(k, n \in \mathbb {N}\), there exists \(j\ge k \in \mathbb {N}\) such that \(C^j(n) = T^k(n)\).

For all \(j \in \mathbb {N}\),

For all \(j \in \mathbb {N}\),

If \(n \ge 1\), then \(T^k(n) \ge 1\) for all \(k \in \mathbb {N}\).

For any \(k, n \in \mathbb {N}\), \(T^k(n)\) is reachable from \(n\) in \(\mathcal{G}'\).

If \(n\) is odd, \(T(n) = (3n + 1)/2\).

For every odd \(n \ge 10\),

If \(n \ge 1\), then \(T(n) \ge 1\).

For any \(n \in \mathbb {N}\), one step of the compact Collatz map \(T\) can be simulated by one or two steps of the original Collatz map \(C\). Specifically, there exists \(j \in \{ 1, 2\} \) such that \(C^j(n) = T(n)\).

For any \(k \in \mathbb {N}\) and \(m, n \in \mathbb {N}\), if \(m \equiv n \pmod{2^k}\), then \(E(k, m) = E(k, n)\).

For \(m, n \ge 1\), if \(E(k, m) = E(k, n)\), then \(m \equiv n \pmod{2^k}\).

For all \(n \in \mathbb {N}\),

For all \(n \in \mathbb {N}\),

This is Remark 3.8 of  [ 2 ] .

When \(n\) is odd, \(U(n) = T(n)\).

For all \(n \in \mathbb {N}\), \(U(n) \ge 1\).

If there exists a natural number \(n\) such that its orbit under \(C\) is unbounded, then the Collatz conjecture is false (No-unbounded-orbit condition).

For any \(i {\lt} j\), the \(i\)-th entry of \(V(j, n)\) is \(X(T^i(n))\).

The length of \(V(j, n)\) is \(j\).

If \(m \equiv n \pmod2\), then \(X(m) = X(n)\).

For all \(n \in \mathbb {N}\), \(X(n) = n \pmod2\).

If \(n \equiv 0 \pmod2\), then \(X(n) = 0\).

If \(n \equiv 1 \pmod2\), then \(X(n) = 1\).

For any \(i {\lt} j\), the \(i\)-th entry of the parity vector \(V_j(n)\) is true if and only if \(X(T^i(n)) = 1\). It is false if and only if \(X(T^i(n)) = 0\).

For all \(i \in \mathbb {N}\), the parity indicator of \(T^i(1)\) satisfies

For any \(j, m \in \mathbb {N}\), the parity of the iterates at \(m\) and \(m + 2^j\) are opposite:

The Collatz graph \(\mathcal{G}\) restricted to positive integers is weakly connected (i.e., any two \(a, b \ge 1\) are reachable in \(\mathcal{G}'\)) if and only if the Collatz conjecture holds.

For every positive integer \(j\), the average value of the correction ratio \(E_j(n)\) over a full period is \(j/4\):

If the integer \(n\) has an infinite stopping time, then there exist infinitely many paradoxical sequences starting from integers of the form \(2^k n\) with \(k \ge 0\).

For every positive integer \(j\) and any \(n \in \mathbb {N}\), let \(q\) be the number of odd steps in the first \(j\) iterations of the compact Collatz map on \(n\). Then the correction ratio \(E_j(n)\) satisfies:

For \(j {\gt} 0\), the lower bound \(E_j(n) = L_j(q)\) is reached if and only if all \(q\) odd steps occur at the beginning of the first \(j\) iterations. That is, the parity vector \(V_j(n)\) consists of \(q\) ones followed by \(j-q\) zeros.

If \(V_j(m)\) strictly precedes \(V_j(n)\) in the partial order generated by elementary swaps (i.e., \(V_j(m) \prec V_j(n)\)), then \(E_j(n) {\lt} E_j(m)\).

For \(j {\gt} 0\), the upper bound \(E_j(n) = R(q)\) is reached if and only if all \(q\) odd steps occur at the end of the first \(j\) iterations. That is, the parity vector \(V_j(n)\) consists of \(j-q\) zeros followed by \(q\) ones.

(Lemma 3.1 in Rozier-Terracol 2025) For any \(\varepsilon \in (0, 1)\), there exist infinitely many pairs of positive integers \((a, b)\) such that:

(Theorem 1.2 in Terras (1976) [ 6 ] ) Two positive integers \(m, n \ge 1\) have the same parity vector of length \(k\) if and only if they are congruent modulo \(2^k\):

For every natural number \(n\), either \(n = 0\) or there exists a natural number \(k\) such that \(C^k(n) = 1\).