Theorem 1.1

Fix \(0<\delta<\delta_0\).

1. The set of \(n\) satisfying \[ T_{\min}(n)\leq\exp((\log n)^{1-\delta}) \tag{1.2} \] has natural density one. More quantitatively, for every \[ 0<\sigma<1-\frac{\delta}{\delta_0}, \] there are constants \(c_{\delta,\sigma}>0\) and \(X_{\delta,\sigma}\) such that, for every \(X\geq X_{\delta,\sigma}\), \[ \#\left\{1\leq n\leq X: T_{\min}(n)>\exp((\log n)^{1-\delta})\right\} \leq 5X\exp\!\left(-c_{\delta,\sigma}(\log X)^\sigma\right). \tag{1.3} \]

2. There is a set \(A_\delta\) of natural density one such that every sufficiently large \(n\in A_\delta\) has an integer \(0\leq k<6.953\log n\) for which \[ T^k(n)\leq\exp((\log n)^{1-\delta}). \tag{1.4} \]

The constants in (1.3) are effective in the sense that they are obtained from the fixed parameters and finite maxima in the proof. They are not optimized, and no practical numerical value of \(X_{\delta,\sigma}\) is claimed.

The restriction on \(\sigma\) comes from the shell exponent \(1-\gamma\), where \(\gamma=\omega\log(1/\chi)\) in (6.17); the corresponding parameter choice is made in (7.11)–(7.12).

Theorem 1.1 is an almost-all theorem. It neither proves descent for every initial value nor excludes exceptional cycles or divergent trajectories. Section 8.1 records the precise scope of the conclusion.

1.1. Why logarithmic blocks appear

Suppose a block contains \(M\) half-Collatz steps and \(s\) odd steps. If the additive terms are temporarily ignored, its multiplicative factor is \[ \frac{3^s}{2^M}. \] For a typical parity word \(s\approx M/2\), so this factor is approximately \[ \left(\frac{\sqrt3}{2}\right)^M. \] Taking \(M\approx\log_2 n\) therefore sends the scale \(n\) to approximately \(n^{a_0}\), where \(a_0=(\log_2 3)/2<1\). One logarithmic block gives a fixed-power contraction. Repeating such blocks is what converts that contraction into the stretched-logarithmic threshold in (1.2).

The choice of exactly \(M\) steps also matches the arithmetic of the shell: its \(2^M\) starting values form a complete residue system modulo \(2^M\), so Proposition 2.3 identifies them with all length-\(M\) parity words exactly once. Shorter blocks retain exact uniformity but give less contraction, while longer blocks no longer sample every parity word from one shell.

The difficulty is not the heuristic contraction. It is preserving a quantitative natural-density estimate when the endpoint of one block becomes the starting value of the next. The proof isolates this as a pullback problem. For a set \(S\), define \[ \operatorname{Pull}(S) = \{n\geq1:T^{\lfloor\log_2 n\rfloor}(n)\in S\}. \tag{1.5} \] The central technical result shows that a power-saving bound for the complement of \(S\) survives this operation with an asymptotically linear loss in its exponent.

1.2. Proof outline

The proof has six steps.

1. One block in parity coordinates. Parity words of length \(M\) are in bijection with the integers in a dyadic shell. The exact affine iterate expresses the block endpoint in terms of the parity word.

2. Condition on the odd count. After fixing the number \(s\) of odd steps, a parity word becomes a positive composition of a fixed total. Equality of block endpoints is reduced to a congruence modulo \(3^s\).

3. Control endpoint multiplicities. Words sharing one block endpoint form an endpoint fiber. Integer-lift injectivity and a quotient-order estimate convert congestion in its congruence class into a random variable measuring how much of the composition total is spent in its final parts. A Rényi bound at each fixed total controls its \((1+\theta)\)-moment when the number of parts is near its mean, for every fixed \(1/2<\theta<1\).

4. Pull back a dense target. Split endpoint fibers according to whether their normalized multiplicity exceeds a chosen exponential threshold; call those above it heavy. The target-density bound controls the ordinary part, and the Rényi estimate controls the heavy part. Balancing the two errors gives the small-\(D\) transport slope \[ \Lambda(\theta)=a_0\frac{\theta}{1+\theta}. \tag{1.6} \] Concretely, the ordinary and heavy contributions decay at rates \(\psi(D)-v\) and \(\theta v\); equating them gives the factor \(\theta/(1+\theta)\), and the remaining factor \(a_0\) is the small-\(D\) slope of \(\psi(D)/\log2\).

5. Iterate endpoints only. A terminal binomial-tail estimate supplies a fixed dense set on which one logarithmic block contracts \(n\) to at most \(3n^r\), for any fixed \(r>a_0\). The proof recursively pulls back this fixed contraction set; it does not track intermediate iterates inside a block. Requiring a shrinking envelope at every intermediate time would impose a stronger pathwise event not supplied by the endpoint pullback in Step 4. For an orbit-minimum theorem, endpoint iteration is the minimal recursion matching the conclusion.

6. Choose the number of blocks. Using \(R\asymp\log\log n\) blocks gives endpoint size \(n^{r^R}\). The density recurrence remains summable exactly when the block-growth parameter lies between two explicit scalar bounds. Letting \(r\downarrow a_0\) and \(\theta\uparrow1\) gives (1.1).

This organization separates the two genuinely different tasks: proving a one-block transport theorem and iterating its endpoint map. Every estimate is stated in the form required by the next step.

1.3. Relation to other density results

Relative to the results above, the proof is independent of Tao’s first-passage framework and uses no Diophantine approximation input. It trades the arbitrary diverging thresholds available through the concurrent bridge approaches for a prescribed stretched-logarithmic threshold, the counting-cutoff estimate (1.3), and the logarithmic-time witness (1.4). Its inputs are finite parity coding, compositions conditioned on their total, congruence-fiber injectivity, one Rényi estimate near the mean composition length, and a binomial tail bound.

The new quantitative mechanism is Theorem 5.3. Once its moment parameters are fixed, the pullback constants remain fixed while the recursively generated density exponent tends to zero.

1.4. Organization

Section 2 gives the density calculus and exact one-block coordinates. Sections 3 and 4 prove the Rényi estimate for compositions conditioned on their total and with part count near its mean. Section 5 turns that estimate into a quantitative pullback theorem. Section 6 proves the one-block contraction and endpoint bootstrap. Section 7 optimizes the parameters and proves Theorem 1.1. Section 8 records scope, formal verification supplementary to the manuscript proof, and declarations.

1.5. Terminology and recurring notation

We use the following terms throughout. A dyadic shell is an interval \([2^M,2^{M+1})\); a logarithmic block starting there consists of \(M\) iterations, and its endpoint is the terminal value \(T^M(n)\). A cohort is a set of parity words with a specified odd count and, when stated, a specified first odd position. Fixed-total means that the composition total \(N\) is held fixed, while central means that its number of parts lies within \(\varepsilon N\) of the mean \((N+1)/2\), for the fixed \(\varepsilon\) in (4.6). An endpoint fiber consists of starts sharing one endpoint; in Section 5 a fiber is heavy precisely when it satisfies the threshold inequality (5.13), and is ordinary otherwise. Transport refers to passing a quantitative density estimate from a target set to its pullback under the block endpoint map.

For reference, the recurring notation is grouped by its role in the proof. The last column points to the defining display; symbols omitted from the table are local to a single argument.

After \(\delta\), and also \(\sigma\) for the quantitative assertion, are fixed, the auxiliary parameters \(\theta,\varepsilon,r,\chi,\zeta,\omega\) are chosen once and remain fixed across shells. Only the recursively propagated density parameters \(C_j,D_j\) and the shell-dependent block count \(R_M\) vary during the iteration.

2.1. From dyadic shells to initial intervals

For \(C>0\) and \(0<D\leq1\), call a set \(S\subseteq\mathbb N_{\geq1}\) \((C,D)\)-dense if \[ B_S(X)\leq C X^{1-D} \qquad(X\geq1). \tag{2.1} \] Every such set has natural density one.

We need two elementary ways to pass from dyadic-shell estimates to (2.1).

Lemma 2.1 (dyadic summation)

1. Suppose \(K>0\), \(0<\gamma<\log2\), and \[ \#\bigl(S^c\cap[2^M,2^{M+1})\bigr) \leq K e^{-\gamma M}2^M \qquad(M\geq0). \tag{2.2} \] Then \(S\) is \[ \left( \frac{2K}{2e^{-\gamma}-1}, \frac{\gamma}{\log2} \right)\text{-dense}. \tag{2.3} \]

2. Suppose \(0<\sigma\leq1\), \(A,\gamma>0\), and, for all sufficiently large \(M\), \[ \#\bigl(E\cap[2^M,2^{M+1})\bigr) \leq A e^{-\gamma(M+4)^\sigma}2^M. \tag{2.4} \] Then there are \(A',\gamma'>0\) and \(X_0\) such that \[ \#(E\cap[1,X]) \leq A'X\exp\left(-\gamma'(\log X)^\sigma\right) \qquad(X\geq X_0). \tag{2.5} \] One may take \[ A'=2A+1, \qquad \gamma'= \min\left\{\frac{\gamma}{(4\log2)^\sigma},\frac12\right\}. \tag{2.6} \]

Lemma 2.2 (symmetric-binomial tail)

If \(X\sim\operatorname{Bin}(n,1/2)\), \(n\geq1\), and \(y\geq0\), then \[ \Pr\left(\left|X-\frac n2\right|\geq y\right) \leq 2\exp\left(-\frac{2y^2}{n}\right). \tag{2.9} \]

2.2. Parity words and exact endpoints

The finite parity-word coordinates are classical; see Terras [8] and Everett [3]. We include the short proofs needed here.

For \(n\geq1\), define the parity bits and their prefix sums by \[ p_i(n)=\mathbf 1_{\{T^i(n)\ {\rm odd}\}}, \qquad s_k(n)=\sum_{i=0}^{k-1}p_i(n). \tag{2.10} \]

Binomial concentration is useful only if parity words occur with exact, not heuristic, uniformity among shell starts. The next proposition supplies that identification.

Proposition 2.3 (parity-word bijection)

For every \(M\geq0\), the map \[ n\bmod 2^M \longmapsto \bigl(p_0(n),\ldots,p_{M-1}(n)\bigr) \tag{2.11} \] is a bijection from \(\mathbb Z/2^M\mathbb Z\) to \(\{0,1\}^M\). Consequently, \[ \#\{n\in[2^M,2^{M+1}):s_M(n)=s\} =\binom Ms. \tag{2.12} \]

Proposition 2.4 (affine iterate)

For every \(n\geq1\) and \(k\geq0\), \[ 2^kT^k(n)=3^{s_k(n)}n+c_k(n), \tag{2.14} \] and \[ 0\leq c_k(n)<3^{s_k(n)}2^k. \tag{2.15} \]

Corollary 2.5 (endpoint range)

If \(n\in[2^M,2^{M+1})\) and \(s_M(n)=s\), then \[ 0<T^M(n)<3^{s+1}. \tag{2.16} \]

Lemma 3.1 (parity words and compositions)

Let \[ 0\leq i_1<i_2<\cdots<i_s<M \] be the odd positions of \(w\). Put \[ u=i_1, \qquad k_j=i_{j+1}-i_j\quad(1\leq j<s), \qquad k_s=M-i_s. \tag{3.1} \] Then \(\mathbf k=(k_1,\ldots,k_s)\) is a positive composition of \(N=M-u\). Conversely, \(u\geq0\) and a positive composition of \(M-u\) into \(s\) parts recover a unique parity word through \[ i_m=u+k_1+\cdots+k_{m-1}. \tag{3.2} \] If \(c_w\) is the correction in (2.13), then \[ c_w = 2^u\sum_{m=1}^s 3^{s-m}2^{k_1+\cdots+k_{m-1}}. \tag{3.3} \]

Lemma 3.2 (integer-lift injectivity)

For fixed \(s,N\), the map \[ \mathbf k\longmapsto\mathscr A_{\mathbf k}\in\mathbb N \] is injective.

Lemma 3.3 (quotient-order estimate)

For every \(\theta>0\), \[ \mathcal A_{s,N}(\theta) \leq (1+\theta)\, \mathbb E_{s,N} \left[ \left( \frac{3^s+\mathscr A_{\mathbf k}}{L_{s,N}} \right)^\theta \right], \tag{3.9} \] where \(\mathbb E_{s,N}\) denotes expectation under the uniform law on positive compositions of \(N\) into \(s\) parts.

Theorem 4.1 (central fixed-total estimate)

Fix \(1/2<\theta<1\) and \(0<\varepsilon<\varepsilon_0(\theta)\). There is a constant \(C_{\theta,\varepsilon}<\infty\) such that, for every \(N\geq1\), \[ \sum_{s\in\mathcal I_{N,\varepsilon}} p_{N,s}\mathcal A_{s,N}(\theta) \leq C_{\theta,\varepsilon}N^{1+\theta}. \tag{4.7} \]

5.1. The central endpoint fibers

Fix \(M\) and \(1\leq s\leq M\). For a first-odd position \(u\), put \(N=M-u\), and let \(f_{M,s,u}(y)\) be the number of starts in \([2^M,2^{M+1})\) whose parity word has weight \(s\), first odd position \(u\), and endpoint \(T^M(n)=y\).

If two starts in this cohort have the same endpoint, (2.14) implies that their corrections are congruent modulo \(3^s\). By (3.5), every endpoint fiber therefore lies inside one residue fiber counted by \(\nu_{s,N}\). If \(c_{s,N}(a)=L_{s,N}\nu_{s,N}(a)\), then, for every \(q\geq1\), \[ \sum_y f_{M,s,u}(y)^q \leq \sum_{a\bmod3^s}c_{s,N}(a)^q. \tag{5.1} \] Indeed, inside one correction class the sum of the endpoint multiplicities is at most \(c_{s,N}(a)\), and the \(q\)-th power of that sum dominates the sum of the \(q\)-th powers.

Fix \(1/2<\theta<1\) and \(0<\varepsilon<\varepsilon_0(\theta)\). Retain a cohort \(u\) only when \[ s\in\mathcal I_{M-u,\varepsilon}, \tag{5.2} \] and let \(f^{\rm cen}_{M,s}(y)\) be the sum of \(f_{M,s,u}(y)\) over the retained cohorts. Set \(f^{\rm cen}_{M,0}(y)=0\), and define \[ G_M^{\rm cen}(\theta,\varepsilon) = 2^{-M}\sum_{s,y}f^{\rm cen}_{M,s}(y) \left( \frac{3^{s+1}f^{\rm cen}_{M,s}(y)}{\binom Ms} \right)^\theta. \tag{5.3} \]

The first transfer step is to show that the fixed-total moment remains polynomial after the retained first-odd-position cohorts are recombined.

Lemma 5.1 (central endpoint moment)

There is \(C'_{\theta,\varepsilon}<\infty\) such that \[ G_M^{\rm cen}(\theta,\varepsilon) \leq C'_{\theta,\varepsilon}(M+1)^{1+\theta} \qquad(M\geq1). \tag{5.4} \]

Lemma 5.2 (discarded cohorts)

There are \(C_\varepsilon,c_\varepsilon>0\), depending only on \(\varepsilon\), such that the normalized source mass outside the retained cohorts is at most \[ C_\varepsilon e^{-c_\varepsilon M}. \tag{5.6} \]

5.2. Pulling back a dense target

Define \[ \psi(D) = -\log\left(\frac{1+3^{-D}}2\right), \qquad r_\theta(D) = \frac{\theta}{1+\theta}\psi(D). \tag{5.8} \] Direct differentiation gives \[ \psi'(D)=\frac{\log3}{3^D+1}, \qquad \psi''(D)= -\frac{(\log3)^2\,3^D}{(3^D+1)^2}<0. \tag{5.9} \] Thus \(\psi\) is increasing and concave on \([0,1]\), with \[ \psi(0)=0, \qquad \psi(D)\geq D\log(3/2) \quad(0\leq D\leq1). \tag{5.10} \]

There are now three errors to balance. Target density controls ordinary endpoint fibers, Lemma 5.1 controls heavy fibers, and Lemma 5.2 controls the discarded cohorts. Choosing the heavy-fiber threshold so that the first two errors match produces the rate \(r_\theta(D)\).

Theorem 5.3 (central endpoint pullback)

Fix \(1/2<\theta<1\) and \(0<\varepsilon<\varepsilon_0(\theta)\). There is \(D_{\theta,\varepsilon}>0\) such that, for every \(0<\zeta<1\), there is \(K_{\theta,\varepsilon,\zeta}>0\) with the following property: whenever \(0<D\leq D_{\theta,\varepsilon}\) and \(S\) is \((C,D)\)-dense, \(\operatorname{Pull}(S)\) is \[ \left( K_{\theta,\varepsilon,\zeta}(C+1)D^{-(1+\theta)}, \frac{\zeta r_\theta(D)}{\log2} \right)\text{-dense}. \tag{5.11} \] All constants in (5.11) are independent of \(C,D,S\), the dyadic shell, and every later use of the theorem.

Consequently, if \[ 0<\chi< \Lambda(\theta) := a_0\frac{\theta}{1+\theta}, \tag{5.12} \] then one may choose \(0<\zeta<1\) and a further cutoff \(D_{\rm lin}=D_{\rm lin}(\theta,\varepsilon,\zeta,\chi)>0\) so that the density exponent in (5.11) is at least \(\chi D\) whenever \(0<D\leq D_{\rm lin}\).

Thus, after \(\theta,\varepsilon,\chi\) and a compatible \(\zeta\) are fixed, \(K_{\theta,\varepsilon,\zeta}\) and \(D_{\rm lin}\) do not change when later iterations send \(D\) to zero.

6.1. A dense one-block contraction

Define \[ F(n)=T^{\lfloor\log_2n\rfloor}(n). \tag{6.1} \] Fix \(a_0<r<1\), and put \[ \eta_r=\frac{r-a_0}{\log_2 3}, \qquad \mathcal E_r=\{n\geq1:F(n)\leq3n^r\}. \tag{6.2} \]

The heuristic \(s\approx M/2\) is not enough for recursion: we need one fixed contraction set whose complement has a quantitative shell bound. Exact parity uniformity from Proposition 2.3 and the binomial tail in Lemma 2.2 provide it for every fixed \(r>a_0\).

Lemma 6.1 (fixed contraction set)

There are constants \(C_r,D_r>0\) such that \(\mathcal E_r\) is \((C_r,D_r)\)-dense.

6.2. The recursive endpoint sets

Set \(K_{\rm blk}=3\). With \(r\) fixed as above, define \[ \mathcal S_1=\mathcal E_r, \qquad \mathcal S_{j+1} = \mathcal E_r\cap\operatorname{Pull}(\mathcal S_j). \tag{6.7} \] If \(n\in\mathcal S_R\), put \[ n_0=n, \qquad n_{i+1}=F(n_i) \quad(0\leq i<R). \tag{6.8} \] Induction in (6.7) gives \(n_i\in\mathcal E_r\) for \(0\leq i<R\). Consequently \[ n_i \leq K_{\rm blk}^{1+r+\cdots+r^{i-1}}n^{r^i} \leq K_{\rm blk}^{1/(1-r)}n^{r^i}. \tag{6.9} \] Each \(n_i\) is an actual Collatz iterate of \(n\). More precisely, if \[ \tau_R = \sum_{i=0}^{R-1}\lfloor\log_2n_i\rfloor, \tag{6.10} \] then \(n_R=T^{\tau_R}(n)\), and (6.9) gives \[ \tau_R \leq \frac{\log n}{(1-r)\log2} + \frac{R\log K_{\rm blk}}{(1-r)\log2}. \tag{6.11} \]

Equations (6.7)–(6.11) settle the deterministic part: membership in \(\mathcal S_R\) gives an actual Collatz iterate with the required size and clock. It remains to show that \(R\) may grow like \(\log\log n\) without the exceptional set becoming too large. The next proposition isolates that calculation from the specific pullback theorem.

Proposition 6.2 (endpoint bootstrap)

Fix \(a_0<r<1\). Suppose there are constants \[ 0<\chi<a_0, \qquad B\geq0, \qquad K_{\rm L},D_c>0 \tag{6.12} \] such that, whenever \(0<D\leq D_c\), pullback sends every \((C,D)\)-dense set to a set with density parameters \[ (C,D) \longmapsto \bigl(K_{\rm L}(C+1)D^{-B},\chi D\bigr). \tag{6.13} \] For every \(\omega>0\) satisfying \[ \omega\log(1/\chi)<1, \tag{6.14} \] define, on the shell \([2^M,2^{M+1})\), \[ R_M=\left\lceil\omega\log(M+4)\right\rceil. \tag{6.15} \] Put \[ \mathcal S^{(\omega)} = \bigcup_{M\geq0} \left(\mathcal S_{R_M}\cap[2^M,2^{M+1})\right). \] Then the shell exceptional proportion outside \(\mathcal S_{R_M}\) is at most \[ 2\exp\left(-c(M+4)^{1-\gamma}\right) \tag{6.16} \] for all sufficiently large \(M\), where \(c>0\) and \[ \gamma=\omega\log(1/\chi)<1. \tag{6.17} \] The set \(\mathcal S^{(\omega)}\) has natural density one. On this set, \[ \log n_{R_M} \leq \frac{\log K_{\rm blk}}{1-r} + (M+1)\log2\,(M+4)^{-\omega\log(1/r)}, \tag{6.18} \] and the witnessing time satisfies (6.11) with \(R=R_M\).

In particular, every \[ 0<\delta< \frac{\log(1/r)}{\log(1/\chi)} \tag{6.19} \] can be realized by a choice of \(\omega\) for which \[ n_{R_M} \leq \exp((\log n)^{1-\delta}) \tag{6.20} \] on all sufficiently large retained shells.

Lemma 7.1 (endpoint limit)

The function \(\delta_*\) is strictly increasing on \((1/2,1)\), and \[ \lim_{\theta\uparrow1}\delta_*(\theta) = \frac{\log(1/a_0)}{\log(2/a_0)} =\delta_0. \tag{7.2} \]

Proof of Theorem 1.1

Fix \(0<\delta<\delta_0\). By Lemma 7.1, choose \[ \frac12<\theta<1 \quad\text{such that}\quad \delta<\delta_*(\theta). \tag{7.3} \] Equation (4.5) gives \(\varepsilon_0(\theta)>0\), so choose once and for all \[ 0<\varepsilon<\varepsilon_0(\theta). \tag{7.4} \] By continuity, there are strict interior values \[ a_0<r<1, \qquad 0<\chi<\Lambda(\theta) \tag{7.5} \] such that \[ \delta< \frac{\log(1/r)}{\log(1/\chi)}. \tag{7.6} \]

Choose \(\zeta\) and the cutoff \(D_{\rm lin}\) from the consequence of Theorem 5.3. After weakening the resulting pullback exponent to \(\chi D\), that theorem supplies (6.13) with \[ B=1+\theta, \qquad K_{\rm L}=K_{\theta,\varepsilon,\zeta}, \qquad D_c=D_{\rm lin}. \tag{7.7} \] Choose \[ \frac{\delta}{\log(1/r)} < \omega < \frac1{\log(1/\chi)}. \tag{7.8} \] Proposition 6.2 now gives a natural-density-one union of shell sets on which (6.20) holds. Since \(n_{R_M}=T^{\tau_{R_M}}(n)\), this proves (1.2).

We next prove the quantitative assertion. Fix \[ 0<\sigma<1-\frac{\delta}{\delta_0}. \tag{7.9} \] Equivalently, \(\delta/(1-\sigma)<\delta_0\). Repeat the choices (7.3)–(7.7), now taking the strict interior values \(r,\chi\) so that \[ \frac{\delta}{1-\sigma} < \frac{\log(1/r)}{\log(1/\chi)}. \tag{7.10} \] Set \[ \omega=\frac{1-\sigma}{\log(1/\chi)}. \tag{7.11} \] Then \[ \omega\log(1/\chi)=1-\sigma<1, \qquad \delta<\omega\log(1/r). \tag{7.12} \] Thus (6.16) bounds the shell exceptional proportion by \[ 2\exp(-c_0(M+4)^\sigma) \tag{7.13} \] for some \(c_0>0\) and all sufficiently large \(M\). Apply the second part of Lemma 2.1 with \(A=2\). Its explicit value \(A'=2A+1\) is \(5\), so, after increasing only \(X_{\delta,\sigma}\) past the finitely many startup shells, there is \(c_{\delta,\sigma}>0\) for which \[ \#\left\{1\leq n\leq X: T_{\min}(n)>\exp((\log n)^{1-\delta})\right\} \leq 5X\exp\left(-c_{\delta,\sigma}(\log X)^\sigma\right) \tag{7.14} \] for every \(X\geq X_{\delta,\sigma}\). This is (1.3).

It remains to prove the advertised clock. First, \[ \frac1{(1-a_0)\log2} = \frac2{\log(4/3)} < \frac{6953}{1000}. \tag{7.15} \] For a rational certificate of the strict inequality, put \(x=1/7\) in \[ \log\frac{1+x}{1-x} = 2\sum_{j\geq0}\frac{x^{2j+1}}{2j+1}. \] The first two positive terms give \[ \log\frac43>\frac{296}{1029}, \] and \[ \frac{1029}{148}<\frac{6953}{1000} \] because \(1{,}029{,}000<1{,}029{,}044\).

Choose \(\theta,\chi\) so that \[ \delta< \frac{\log(1/a_0)}{\log(1/\chi)}, \] with \(1/2<\theta<1\) and \(0<\chi<\Lambda(\theta)\). Choose \(\varepsilon,\zeta,D_{\rm lin}\) as in (7.4) and (7.7), so that the pullback hypothesis of Proposition 6.2 holds. Then take \(r>a_0\) sufficiently close to \(a_0\) that both (7.6) and \[ \frac1{(1-r)\log2}<6.953 \tag{7.16} \] hold. With \(\omega\) chosen as in (7.8), one has \(R_M=O(\log M)=O(\log\log n)\). Equation (6.11) therefore gives \[ \tau_{R_M} \leq \frac{\log n}{(1-r)\log2} +O_r(\log\log n) < 6.953\log n \tag{7.17} \] for all sufficiently large retained \(n\). Together with (6.20), this proves (1.4). The retained shell union has density one by Proposition 6.2, through Lemma 2.1, part 2, and is the required set \(A_\delta\). \(\square\)

8.1. Scope and comparison

Theorem 1.1 is compatible with exceptional divergent trajectories and with hypothetical nontrivial cycles. If an orbit enters a fixed cycle with minimum \(m\), then its orbit minimum is at most \(m\), so all sufficiently large points in that basin already satisfy (1.2). The theorem therefore gives no cycle exclusion and no density estimate for such basins.

Tao [7] proves the stronger terminal-threshold statement with an arbitrary function tending to infinity in logarithmic density. The concurrent preprints [1, 6] obtain arbitrary-diverging-threshold statements in natural density through bridges built on Tao’s framework. Theorem 1.1 has a less general terminal threshold; its separate features are the self-contained fixed-total endpoint argument, the prescribed stretched-logarithmic family, the counting-cutoff estimate (1.3), and the explicit half-map clock.

The fixed-power statements preceding this paper do not by themselves give the prescribed rate in (1.2). A density-one diagonalization over all fixed powers can produce some unspecified exponent \(\varepsilon(n)\to0\), but its decay depends on untracked onset thresholds. It does not imply \(\varepsilon(n)=(\log n)^{-\delta}\), nor does it give (1.3).

8.2. Formal verification

The principal theorem chain has a supplementary Lean 4 formalization at \[ \texttt{https://github.com/shaikidris/CET/tree/v2.0.1}, \] with software DOI 10.5281/zenodo.21797535. The frozen development uses Lean v4.15.0 and Mathlib commit 9837ca9d65d9de6fad1ef4381750ca688774e608. Its referee-facing Main Theorem, exceptional-count theorem, and timed theorem are respectively

collatz_central_renyi_endpoint_natural_density_descent
collatz_central_renyi_endpoint_exceptional_count_at_exponent
collatz_central_renyi_endpoint_natural_density_descent_timed

in namespace CollatzEndpointTransport.QuantitativeCollatzMain. The formal proof uses the stronger all-prefix initial-window estimate at its one-block seed; Lemma 6.1 uses only the terminal consequence needed by the endpoint theorem. The three declarations state exactly the conclusions (1.2), (1.3), and (1.4); only the intermediate route at the seed differs.

From the release’s lean directory, the principal replay commands are

lake build
lake build CollatzEndpointTransport.Linear.PaperDependencyAudit
lake build CollatzEndpointTransport.Linear.PaperAudit

The first command reconstructs the retained source. The second reports transitive theorem dependencies. The third executes #print axioms on the paper-facing declarations; it reports only propext, Classical.choice, and Quot.sound. These checks supplement the written proof and are not used to discharge any step of it.