Localizing the Obstruction in the Collatz Conjecture — A Reconstruction in Theorems, Lemmas, and Observations
Theorem–lemma edition, v14 (pure-theory version). Three local corrections to v13, all at the level of statement accuracy: the direction of the mechanism in the Observation 6 caution, the strength of the crossover claim in Proposition 9, and the interpretation of the boundary term in Observation 10. All other text, including the v13 separation of the seed-dependent Lemma 0 constraint from the zero-correction surrogate, is preserved.
Scope note. This is a theoretical note on necessary conditions, counting bounds, and logical obstructions. It does not use finite-sample observational studies as evidence for any theorem.
Convention. Every statement carries a referee-style status tag. [A] proved in the literature (cited) / [B] carries a self-contained proof within this document / [C] heuristic (explicitly marked as such). Statements formerly in class [D] are all included here in corrected form. [B] asserts only self-containedness of the proof, not a claim of academic novelty — novelty must be assessed separately by comparison with the relevant literature (variational principles in information theory, the Terras–Everett–Krasikov–Lagarias line of counting arguments).
§0 Notation
For odd $n$, define the Syracuse map $S(n)=\dfrac{3n+1}{2^{a(n)}}$ with $a(n)=v_2(3n+1)$. For odd $N$, write the orbit as $n_0=N$, $n_{k+1}=S(n_k)$, the exponent sequence as $a_i=a(n_i)$, and the partial sums as $A_k=\sum_{i $$\log n_k=\log N+k\log 3-A_k\log 2+R_k,\qquad R_k=\sum_{i is an identity (the sum of logarithms of $n_{i+1}=n_i\cdot\frac{3}{2^{a_i}}\big(1+\frac{1}{3n_i}\big)$). Descent avoidance $P(n)$: “$S^k(n)\ge n$ for all $k\ge1$”. As a standard fact, the Collatz conjecture is equivalent to “$P(n)$ is false for every odd $n\ge3$” (once an orbit descends below its own starting point, strong induction carries it to $1$). Set $\delta_N:=\dfrac{1}{3N\log 2}$. Theorem A (parity conjugacy; Lagarias 1985, Bernstein–Lagarias 1996) [A].
The parity-sequence map $\Phi:\mathbb Z_2\to\{0,1\}^{\mathbb N}$ is a homeomorphism preserving Haar measure, and it conjugates the Collatz map to the shift. As a corollary, under Haar measure the $(a_i)_{i\ge0}$ are i.i.d. with $\Pr(a=j)=2^{-j}$ ($j\ge1$, mean $2$). Caution. With respect to natural density on $\mathbb N$, the i.i.d. structure must be invoked with care. For a fixed finite length $k$, counting residue classes shows that the natural densities of valuation words of length $k$, taken relative to the odd integers, agree with the finite-dimensional geometric product law of Haar measure (this is exactly what Proposition 9 exploits, within its stated range and for the zero-correction event it counts). What fails is: (i) an unrestricted uniform i.i.d. treatment when $k$ grows with $X$ beyond the counting range (the boundary error of Proposition 9), and (ii) treating the time direction of a single integer orbit as i.i.d., for which no theorem exists. Unrestricted independence is a theorem only on $\mathbb Z_2$. This distinction underlies everything that follows. Theorem B (Terras 1976) [A]. The odd integers with finite stopping time ($\exists k:\ S^k(n) Theorem C (Tao; arXiv 2019, Forum Math. Pi 10 (2022), e12) [A]. For any function $f(n)\to\infty$, almost all $n$ in the logarithmic-density sense have orbit minimum at most $f(n)$. This is a qualitative deepening of Theorem B, and it plays no role in the local-to-global implication discussed below. Theorem D (Krasikov–Lagarias 2003) [A]. $\#\{n\le X: n\to1\}\gg X^{0.84}$. Theorem E (cycles; Steiner 1977, Simons–de Weger 2005, Hercher 2023) [A].
$1$-cycles are ruled out by Baker-type bounds alone (Steiner 1977). For general $m$-cycles ($m$ local minima), the combination of effective lower bounds for linear forms in logarithms (Rhin 1987; Laurent–Mignotte–Nesterenko 1995) with computational verification yields: Simons–de Weger (Acta Arith. 117, 2005) obtained $m\ge69$ at the verification height then available (later updates of that height gave $m\ge76$, then $m\ge83$), and Hercher (J. Integer Seq. 26, 2023, Art. 23.3.5) obtained $m\ge92$, i.e. the exclusion of all $m\le91$. The current state of computational verification is Barina (J. Supercomput. 81, 2025): convergence confirmed up to $2^{71}$ (equivalently, no cycle has its least element below $2^{71}$). Analytic methods on their own deliver only a trade-off (Proposition 3). Theorem F (Kurtz–Simon 2007) [A]. The family of generalized Collatz problems is $\Pi^0_2$-complete. Caution on scope. Standard Collatz is a single $\Pi^0_2$ sentence, and algorithmic undecidability does not, by definition, apply to a single sentence. Nor is this evidence of independence from ZFC/PA. The most one can extract is the heuristic [C] “since no proof schema handles the whole family uniformly, a proof will presumably use the specific arithmetic of $2$ and $3$” — and even this does not strictly rule out that a particular instance is settled by general-purpose tools. Lemma 0 [B]. Let $N$ be a minimal counterexample (the least odd number whose orbit does not reach $1$). Then for every $k$, $$n_k\ge N\qquad\text{and}\qquad \frac{A_k}{k}\le\log_2 3+\delta_N.$$ Proof. If $n_k Lemma 1 [B]. The empirical distribution $p_j^{(k)}=\frac{1}{k}\#\{i $$p_1^{(k)}\ \ge\ 2-\log_2 3-\delta_N\ \approx\ 0.4150-\delta_N.$$ Moreover $a_i=1\iff n_i\equiv3\pmod4$, and such a step increases the value: $n_{i+1}=\frac{3n_i+1}{2}>n_i$. Proof. Since $a_i\ge1$, we have $\bar a_k=\sum_j jp_j\ge p_1+2(1-p_1)=2-p_1$, hence $p_1\ge2-\bar a_k\ge2-\log_2 3-\delta_N$ by Lemma 0. The residue identification is a direct computation: $n\equiv3\pmod4\Rightarrow 3n+1\equiv2\pmod4$ (so $a=1$), and $n\equiv1\pmod4\Rightarrow 3n+1\equiv0\pmod4$ (so $a\ge2$). ∎ Caution. This constrains the empirical distribution at each finite time; it says nothing about the temporal arrangement of the increasing steps (runs, gaps, spacing). Caution (the feasible region) [B]. The universal necessary conditions established in this document are the following three: nonnegativity, normalization, and the mean constraint $\sum_j jp_j\le\log_2 3+\delta_N$. Linear programming over this region returns, besides the $p_1$ lower bound of Lemma 1, the Markov-type family of facets $\sum_{j\ge J}p_j\le\frac{\log_2 3-1+\delta_N}{J-1}$ ($J\ge2$; Markov’s inequality applied to $a-1\ge0$). No new universal constraint on the pair correlation $(a_i,a_{i+1})$ follows from the arithmetic input available: from $3n_i+1\equiv1\pmod3$ one gets the deterministic relation $n_{i+1}\equiv2^{a_i}\pmod3$, but $a_{i+1}$ is determined by the $2$-adic information of $n_{i+1}$ alone, and mod $3$ and mod $2^r$ are independent by CRT, so this route is blocked. New facets require new arithmetic invariants. Note also that the claim “these necessary conditions are complete” cannot be asserted: in the absence of a counterexample it becomes vacuous, and so falls into the same trap as Proposition 8. Lemma 2 [B]. Under Haar measure (where the i.i.d. geometric structure of the $a_i$ makes Cramér’s theorem rigorously applicable), the rate function for $m\in(1,2)$ is $$I(m)=(m-1)\log\!\Big(2-\frac{2}{m}\Big)+\log\frac{2}{m},$$ in particular at $m=\log_2 3$ one has $\theta=\log(2-2/m)\approx-0.3037$ and $$I(\log_2 3)\approx0.05498\ \text{nat}\ \approx\ 0.07932\ \text{bit},$$ together with $$\mathrm{Haar}\Big\{x\in\mathbb Z_2:\ \frac{A_k(x)}{k}\le\log_2 3\Big\}=\exp\big(-k\,(I(\log_2 3)+o(1))\big).$$ Proof. $\Lambda(\theta)=\log\mathbb E[e^{\theta a}]=\theta-\log(2-e^\theta)$ for $e^\theta<2$; solve $\Lambda'(\theta)=\frac{2}{2-e^\theta}=m$ and take the Legendre transform. For each $k$ the event depends only on finitely many binary digits, so it agrees with a count of residue classes. ∎ Caution [C]. This LDP is consistent with Tao (2019), but Tao’s proof route is not a projection of this $\mathbb Z_2$-LDP (it proceeds by a direct construction of Syracuse random variables). Proposition 2A (entropy identity and the extremal profile) [B]. Let $\mu=\mathrm{Geom}(1/2)$ (i.e. $\mu_j=2^{-j}$). Then for any probability distribution $\nu$ on $\{1,2,\ldots\}$, $$D(\nu\,\|\,\mu)=(\log 2)\,\mathbb E_\nu[a]-H(\nu).$$ Consequently, minimizing relative entropy subject to $\mathbb E_\nu[a]\le m$ ($1 $$\nu^*_j=\frac{1}{m}\Big(1-\frac{1}{m}\Big)^{j-1},\qquad
D(\nu^*\|\mu)=m\log2-H_{\max}(m)=I(m),$$ where $H_{\max}(m)=\log m-(m-1)\log(1-1/m)$. In particular, at $m=\log_2 3$, $$I(\log_2 3)=\log 3-H_{\max}\approx1.09861-1.04363=0.05498,\qquad p^*_1=\frac{1}{\log_2 3}\approx0.6309.$$ Proof. The identity is the direct computation $\log(\nu_j/2^{-j})=\log\nu_j+j\log2$. The minimization follows from Gibbs’ variational principle (the exponential tilting family of $\mu$ preserves the family of geometric distributions). Agreement with $I(m)$ is checked algebraically by expanding both sides as $m\log2+(m-1)\log\frac{m-1}{m}-\log m$. ∎ Consequences. (i) Cramér’s tilted measure, Sanov’s minimizer, and the maximum-entropy distribution all coincide in the same geometric distribution. This is a consequence of the base measure being $2$-adically uniform, and the identity reads “relative entropy = $2$-adic address cost − combinatorial entropy”. (ii) The rate $I$ has an exact counting meaning as the difference between $\log3$ (the per-step increment of address space) and $H_{\max}$ (the growth rate of exponent sequences realizable under the constraint); this yields an elementary proof of Lemma 2 that bypasses Cramér (used in Proposition 9). (iii) [C, interpretive] Heuristically, $\nu^*$ is the empirical profile of least relative-entropy cost compatible with the mean constraint — a “path of least resistance” in that limited variational sense only; no dynamical claim is made. Its $p^*_1\approx63\%$ lies above the $41.5\%$ lower bound of Lemma 1 — a consistency check between the bound and the extremal profile. Proposition 3 (cocycle pinning and the trade-off) [B]. If a nontrivial cycle exists, let $K$ denote its accelerated odd period — the number of odd accelerated steps of $S$ traversed in one full period, so that $n_K=n_0$ — let $N$ be its least element, and take $n_0=N$, so that $n_K=n_0=N$ (that $N$ is least is used below: the bound $R_K\le\frac{K}{3N}$ requires $n_i\ge N$ for every $i$). Then $$0 i.e. the integer $A_K$ lies in the interval $\big(K\log_2 3,\ K\log_2 3+K\delta_N\big]$. Since this interval has length $K\delta_N$, when $K\delta_N<1$ it contains at most one integer, and if it does, that integer is $\lceil K\log_2 3\rceil$. The condition for containment is $$\lceil K\log_2 3\rceil-K\log_2 3\ \le\ K\delta_N,$$ and if it fails, the cycle for that $(K,N)$ is excluded immediately. A Rhin-type effective lower bound $\lceil K\log_2 3\rceil-K\log_2 3\gg K^{-C}$ (Rhin, Approximants de Padé et mesures effectives d’irrationalité, Progr. Math. 71, 1987, pp. 155–164; an effective estimate specialized to $x\log2+y\log3$; the numerical value of the effective exponent $C$ is not verified here and is never used in the proof) triggers this exclusion condition in the regime $N\gg K^{C+1}$, and therefore $$N\ \ll\ K^{\,C+1}.$$ Proof. Put $\log(n_K/N)=0$ in the identity; from $R_K>0$ (each term is strictly positive) and $R_K\le K/(3N)$ — the latter using $n_i\ge N$ for all $i$, i.e. that $N$ is the least element of the cycle — one gets membership in the interval. Combining the necessary condition for the interval to contain an integer with the lower bound on the fractional part gives $K^{-C}\ll K\delta_N$, i.e. $N\ll K^{C+1}$. ∎ Caution (corrected status, and on the meaning of $K$). What Proposition 3 delivers is only a trade-off between the least element $N$ and the accelerated odd period $K$, not an exclusion. Exclusion is completed only in combination with computational verification of a lower bound on $N$, and even then it covers only finitely many values of $K$. Note also that $K$ counts odd accelerated steps of $S$ within the period, which is a different quantity from the number $m$ of local minima appearing in the $m$-cycle results of Theorem E; the two indices are not compared here, and no numerical exclusion range is inferred from the present statement. Observation 4 (the failure of cocycle pinning on the divergent side) [B]. The cocycle identity alone yields, for a divergent orbit, the necessary condition $$k\log3-A_k\log2+R_k\to+\infty,$$ and algebraically this condition is compatible with $A_k/k\to\log_2 3$ from below (example: a deficit $k\log_2 3-A_k\sim\sqrt k$ corresponds to sublinear divergence $\log n_k\sim\sqrt k\cdot\log2$). No claim is made that an arbitrary exponent sequence with such a deficit is realizable as the exponent sequence of a positive integer orbit; the point is only that the identity does not forbid it. Hence for divergent orbits the pinning argument of Proposition 3 — which used the closing relation $n_K=n_0$ — yields neither a nontrivial lower bound on $A_k$ nor any $\varepsilon$-separation of $A_k/k$ from $\log_2 3$. The conclusion is confined to this: the pinning of Proposition 3 alone cannot exclude divergence. Other kinds of input (congruence conditions on integer orbits, realizability constraints, counting) are not addressed here. Observation 4A (a dictionary between window statistics and height) [B]. For a minimal counterexample, define the height $h_i:=\log_2(n_i/N)\ge0$ (Lemma 0). Applying the cocycle identity to the window $[i,i+\ell)$ gives, exactly, $$(A_{i+\ell}-A_i)-\ell\log_2 3\ =\ (h_i-h_{i+\ell})+\Theta_{i,\ell},\qquad 0<\Theta_{i,\ell}\le\ell\,\delta_N.$$ Consequences: (a) An excess of the window average over $\log_2 3$ (times $\ell$) is paid for exactly by a fall in height, and a deficit by a rise. In particular $\sup_\ell\ \ell\big(\text{window average}-\log_2 3-\delta_N\big)\le h_i$. (b) Hence local window statistics are not an independent observable but are deterministically identical with the increments of the height path; the question “prefix average vs. local average” is a change of variables. What is genuinely unconstrained is the height path itself; the constraints established on it in this document are only the two inequalities $0\le h_i\le(\log_2 3-1+\delta_N)\,i$. (c) For a divergent orbit, the total deficit $\to\infty$ (Observation 4) is realized as a rise in height. (d) The prefix constraint of Lemma 0 transfers to windows with $\ell\ll i$ only as a vacuous upper bound (of order $\le\log_2 3+\frac{i(\log_2 3-1)}{\ell}$) — transfer from prefix to local scale must, in principle, pass through the ledger in (a). Caution (mixing in the time direction, and related literature). No theorem guaranteeing mixing, decay of correlations, or self-similarity in the time direction along a single orbit exists, here or in the literature. Moreover, by Observation 11, an assertion of the form “a counterexample exponent sequence must be locally typical (mixing)” is not delivered by the coarse word summaries used in this document: a negative cycle has a periodic exponent word (entropy rate $0$, perfect long-range correlation) whose prefix means nevertheless respect the bound $A_k/k\le\log_2 3$ in suitable phases, so no bound of that shape forces local typicality. This restricts the summaries used here; it is not an impossibility claim for every method involving the exponent sequence. A hypothetical positive cycle is also periodic, so any mixing claim can only target divergent orbits; within the present framework, the identified sign-sensitive ingredient available to it is the positivity of the cocycle. Relatedly, Rozier–Terracol (Discrete Math., 2026; arXiv:2502.00948) [A] study “paradoxical” finite sequences in which the prediction from the parity vector disagrees with the actual Archimedean comparison — in the language here, windows where the $R$/height ledger becomes decisive — and show that the Collatz conjecture follows from their finiteness. That finiteness hypothesis is a non-vacuous finitary statement with a satisfiable antecedent, so it escapes the vacuity trap of §7 and sits consistently within this framework as an open problem of the §8 type. Observation 5 [B]. For $0\le\delta<2-\log_2 3\approx0.415$ (equivalently $\log_2 3+\delta<2$; in particular for the seed-dependent value $\delta=\delta_N$ used in this document), the set $B_\delta=\{x\in\mathbb Z_2:\ \forall k\ A_k(x)/k\le\log_2 3+\delta\}$ has Haar measure $0$ by Lemma 2 (it contracts exponentially at each finite stage; the rate $I(\log_2 3+\delta)$ is positive precisely in this range). On the other hand $\mathbb N$ is dense in $\mathbb Z_2$ but countable, hence Haar-null. Concerning the intersection $B_\delta\cap\mathbb N$ of two null sets, measure theory asserts nothing. Both emptiness and non-emptiness are consistent with Haar measure. Observation 6 (finite-time structure of the descent condition) [B]. Fix $k$ and restrict to odd positive integers — the domain on which the order used below is defined; no order relation on all of $\mathbb Z_2$ is invoked. On each class of such integers obtained by fixing the first $k$ parity digits, $S^k(n)$ is an affine function of $n$ with slope $3^k/2^{A_k}$, and the descent condition at time $k$ becomes an Archimedean linear inequality within that class. Hence, on the positive integers, $P$ is a countable intersection of conditions of the type “$2$-adic clopen class × Archimedean threshold”: each finite stage irreducibly mixes $2$-adic and Archimedean data. [C] That finite-digit information alone does not directly determine the infinite-time property is a strategic description of where the finite-time methods of this document stop, not a universal impossibility theorem; nor can the topological type of $P$ itself be asserted unconditionally — if the conjecture holds, $P$ is empty on odd $n\ge3$, and the empty set is clopen. Caution (a slogan, not a theorem) [C]. The phrase “taking $2$-adic closures destroys the Archimedean content” is used in this document only as a descriptive summary of the mechanism just displayed, and the mechanism must be stated in the right direction. The finite parity word determines the affine coefficients and, when a threshold exists, its numerical value. What the word does not determine is on which side of that Archimedean threshold an arbitrary positive representative of the same $2$-adic cylinder lies; a $2$-adic cylinder contains representatives of arbitrarily large Archimedean size. No closure theorem is proved, asserted, or used anywhere below; the slogan is the informal face of “$2$-adic closeness $\ne$ Archimedean closeness” and carries no independent weight. Caution [C]. This gap is an explanation of why existing methods fail, not a proof that every method must fail. Mathematics does contain devices that promote tail properties to everywhere-statements (unique ergodicity promotes a.e. convergence of Birkhoff averages to everywhere convergence). The actual difficulty is that the Syracuse system has no such skeleton — but even a negative result in this direction remains unproved at present. Reformulation R. “For every odd $N\ge3$: if $P(N)$, then there exists $r$ such that the set $\{n\equiv N\ (\mathrm{mod}\ 2^r):P(n)\}$ has positive natural density within that residue class.” Proposition 7 [B]. R $\Rightarrow$ the Collatz conjecture. Proof. If a counterexample exists, the minimal counterexample $N$ satisfies $P(N)$ (Lemma 0). By R, the $P$-integers have positive density globally ($\ge$ positive density $\times\ 2^{-r}$). This contradicts Theorem B (Terras): $\{P\}$ has density $0$. Hence no $N$ with $P(N)$ exists, and the Collatz conjecture follows. ∎ Caution. Only Terras (1976) is needed for the contradiction; Tao (2019) is not. That is, the arrow in R is a barrier that has stood for half a century, and this historical consistency is itself circumstantial evidence for the depth of the arrow. Proposition 8 (logical status) [B]. The Collatz conjecture $\Rightarrow$ R (vacuously true, since the antecedent $P(N)$ is always false). Together with Proposition 7, therefore, $$\text{R}\iff\text{the Collatz conjecture}.$$ Consequence. Calling R a “conjecture” is misleading. R is not an independent stepping stone but a restatement of the conjecture — a valuable restatement nonetheless, whose function is to specify “the shape of what a proof must deliver” (a transfer theorem from finite-digit information to a tail property). In the same sense in which Terras’s equivalent restatement, “every integer descends below itself”, gave rise to stopping-time analysis, R is a specification of strategy, not an intermediate result. Note likewise that any “bridge” formulation conditioned on $P$ becomes vacuously true under the Collatz conjecture, and so cannot escape this equivalence trap. Caution (escaping the trap by finitization) [C]. The vacuity trap arises from placing the infinite-time condition $P$ in the antecedent, and finitization avoids it. Taking $k$-step descent avoidance as the antecedent, $$\text{R}_k:\ \forall n\ \big(n\text{ avoids descent for }k\text{ steps}\big)\Rightarrow\big(\text{the fraction of }k\text{-step descent-avoiders in the class of }n\bmod2^{r(k)}\ \ge\ \rho(k)\big)$$ escapes vacuity: its antecedent is satisfiable for every finite $k$ (surviving classes really exist), so it is not vacuous, and it is not equivalent to the Collatz conjecture. As it stands, however, R$_k$ is a provisional formulation — a schematic target — and not a well-posed open theorem candidate: its truth value is undetermined until $r(k)$, $\rho(k)$, the density convention, and the slope cases are fixed. In particular, in the fine regime $r(k)\gtrsim A_k$ (classes that fix the word), for a fixed word one has $S^j(n)-n=(3^j/2^{A_j}-1)\,n+c_j$ with $c_j>0$; when the coefficient is negative, small $n$ in the class may avoid descent at step $j$ while every sufficiently large $n$ in the same class descends. Hence the earlier claim that “almost all sufficiently large elements of the class exhibit the same $k$-step behaviour” is withdrawn as an unconditional statement — it holds only case-by-case according to the signs of the slopes. What connects to §8 is the coarse regime $r(k)\ll A_k$, where controlling $\rho(k)$ is precisely the counting problem there. In either regime, reassembling the infinite version R from $\forall k\,\text{R}_k$ requires $k$-uniformity of $\rho$ and $r$, and since no contradiction with Terras arises at any finite stage, that is where it breaks. In summary: the infinite version R is an equivalent restatement (a specification of the shape of a strategy), while the finite version R$_k$ is a schematic target whose precise formulation is itself part of the problem — and among the routes considered in this document, the identified exit from the trap of §7 funnels into §8. Lemma 9A (estimation of binomial sums) [B]. Fix $1 $$\text{(a)}\ \sum_{A=k}^{\lfloor mk\rfloor}\binom{A-1}{k-1}=e^{\,kH_{\max}(m)+O(\log k)},\qquad
\text{(b)}\ \sum_{A=k}^{\lfloor mk\rfloor}\binom{A-1}{k-1}2^{-A}=e^{\,-kI(m)+O(\log k)}.$$ Proof. (a) The ratio of consecutive terms $t_A=\binom{A-1}{k-1}$ is $t_{A+1}/t_A=\frac{A}{A-k+1}>1$ for $A\ge k\ge2$, so $t_A$ increases monotonically over the range. Hence the sum is squeezed between the largest term $t_{\lfloor mk\rfloor}$ and that term times the number of summands $\le(m-1)k+1$: $$t_{\lfloor mk\rfloor}\ \le\ \sum\ \le\ \big((m-1)k+1\big)\,t_{\lfloor mk\rfloor}.$$ By Stirling’s formula, $\log\binom{n}{j}=n\,h(j/n)+O(\log n)$ (with $h(x)=-x\log x-(1-x)\log(1-x)$, uniformly as long as $j/n$ stays in a compact subinterval of $(0,1)$). With $n=\lfloor mk\rfloor-1$, $j=k-1$, $j/n=1/m+O(1/k)$, and the Lipschitz continuity of $h$, $$\log t_{\lfloor mk\rfloor}=mk\,h(1/m)+O(\log k)=kH_{\max}(m)+O(\log k),$$ where the identity $m\,h(1/m)=\log m-(m-1)\log\frac{m-1}{m}=H_{\max}(m)$ is a direct computation. (b) The ratio of consecutive terms $s_A=t_A\,2^{-A}$ is $s_{A+1}/s_A=\frac{A}{2(A-k+1)}$, which exceeds $1$ when $A<2k-2$. Since $m<2$, for $k>2/(2-m)$ the sequence increases monotonically over the whole range $A\le mk$ (the condition $m<2$ is exactly what gives monotonicity), and squeezing by the largest term as in (a) gives $$\log s_{\lfloor mk\rfloor}=kH_{\max}(m)-mk\log2+O(\log k)=-kI(m)+O(\log k)$$ (the last equality by the identity $I=m\log2-H_{\max}$ of Proposition 2A). ∎ Consequence. The set $\{x\in\mathbb Z_2:A_k(x)\le mk\}$ is a disjoint union of cylinders indexed by exponent sequences (each of measure $2^{-A}$), and its Haar measure is exactly equal to the sum in (b). Hence the upper bound of Lemma 2 is reproved elementarily, with error term $O(\log k)$ and without passing through Cramér. Proposition 9 (finite-X counting bound for the zero-correction surrogate, uniform in a restricted range) [B]. Write $O_X=\{n\le X:\ n\text{ odd}\}$, so $|O_X|=\tfrac{X}{2}+O(1)$; this is the domain on which $a(\cdot)$ and $S$ are defined (§0). Let $m=\log_2 3$ and $\varepsilon>0$. The exponent sequence $(a_0,\ldots,a_{k-1})$ of an odd $n$ is determined by $n\bmod 2^{A_k+1}$, so the event $\{n\in O_X:\ A_k(n)\le mk\}$ is a union of residue classes to moduli $2^{\le mk+1}$, each such class consisting of odd integers. The number of classes is $e^{kH_{\max}+O(\log k)}$ by Lemma 9A(a); the number of elements of a class of modulus $2^{A+1}$ inside $O_X$ is $$X\cdot2^{-(A+1)}+O(1)\ =\ |O_X|\cdot2^{-A}+O(1),$$ i.e. the class has relative density $2^{-A}$ inside the odd integers, matching the Haar measure of the corresponding cylinder; and the sum of the leading terms over classes coincides with $|O_X|$ times the weighted sum of Lemma 9A(b). Therefore $$\#\{n\in O_X:\ A_k(n)\le mk\}\ \le\ |O_X|\,e^{-kI+O(\log k)}\ +\ e^{kH_{\max}+O(\log k)}.$$ At the exponential scale, the crossover between the two terms occurs when $k\,(I+H_{\max})=k\log3$ is comparable to $\log|O_X|=\log X+O(1)$; the $O(\log k)$ error terms prevent any sharper “exactly when”. In particular, for $$k\ \le\ (1-\varepsilon)\log_3 X\ \approx\ (1-\varepsilon)\cdot0.631\log_2 X,$$ and all sufficiently large $X$, the boundary term is dominated by the leading term, and in that range the count contracts at the same rate $I$ as under Haar measure. ∎ Remark (the constant factor is immaterial). Restricting to $O_X$ changes $X$ into $\tfrac X2+O(1)$ and hence shifts $\log X$ by $O(1)$. Neither the exponential rate $I$ nor the range exponent $1/\log_2 3$ is affected: the $O(1)$ shift is absorbed by the arbitrarily small $\varepsilon$. Caution (what event is being counted: $\delta=0$ versus $\delta_N$) [B]. The event counted above is the zero-correction surrogate $A_k/k\le\log_2 3$. Lemma 0 gives a hypothetical minimal counterexample only the weaker seed-dependent bound $A_k/k\le\log_2 3+\delta_N$, with $\delta_N=\frac{1}{3N\log2}>0$. Therefore Proposition 9 does not by itself count all minimal-counterexample candidates, and the two conditions must not be conflated under a single name. Three points make the gap precise. (i) The surrogate is strictly stronger than the Lemma 0 constraint, so the surrogate count is an upper bound for a sub-collection of candidates, not for the candidate set. (ii) For any fixed $m\in(1,2)$ the computation of Lemma 9A goes through verbatim, with rate $I(m)$ and range $k\le(1-\varepsilon)\log_{2^m}X$; in particular the isomorphic calculation for a fixed $m=\log_2 3+\delta$ with small $\delta>0$ is available. (iii) What is not carried out in this document is a uniform candidate count incorporating $\delta_N$: the Lemma 0 threshold depends on the seed $n$ itself, so $\{n\in O_X:\ A_k(n)/k\le\log_2 3+\delta_n\}$ is not a union of residue classes at a single fixed $m$, and no uniform-in-$(k,X)$ exhaustion of minimal-counterexample candidates is claimed or proved here. Range and limit conventions. Two statements must be kept apart. (i) Fixed $k$. For a length $k$ held fixed while $X\to\infty$, the event is a finite union of residue classes, so its relative natural density inside the odd integers exists and coincides exactly with the Haar measure of the corresponding cylinder union (Lemma 9A(b)); there is no error term to control, and the rate $I$ is the Haar rate. (Relative to all of $\mathbb N$ the same densities are simply halved, which is why the normalization to $O_X$ is the natural one here.) (ii) $k=k(X)$ growing with $X$. Here the bound above is not a density statement but a finite-$X$ count, and it is uniform in $k$ only within the range $k\le(1-\varepsilon)\log_3X$, because outside it the residue-class boundary error $e^{kH_{\max}+O(\log k)}$ overtakes the leading term $|O_X|e^{-kI+O(\log k)}$. In particular no limiting natural-density assertion is made for $k$ growing beyond that range, and none is used anywhere below. Caution. Counting of this type belongs to the standard toolkit since Terras (1976) and Everett (1977), and is implicit in the Krasikov–Lagarias framework. The novelty claimed here is confined to making the rate $I$ and the effective range $\log_3X$ explicit, and to the counting interpretation $I=\log3-H_{\max}$ supplied by Proposition 2A. Observation 10 (the range gap, as a benchmark for the $\delta=0$ counting scheme) [C]. For the zero-correction surrogate of Proposition 9, driving the leading first-moment term $|O_X|e^{-Ik}$ below $1$ would require $k\gtrsim\log X/I\approx12.6\log_2X$, whereas Proposition 9 controls the residue-class boundary error only up to $k\lesssim0.631\log_2X$. The ratio of these two formal scales is $$\frac{\log3}{I}=\frac{\log3}{\log3-H_{\max}}\approx19.98$$ (Proposition 2A) — a gap of roughly a factor of $20$ between the scale a first-moment argument would need and the scale at which the present argument is uniform. Four disclaimers fix its status. (a) This is not a proof that counting exhausts the minimal-counterexample candidates below $X$: the surrogate event is strictly stronger than the Lemma 0 constraint (Caution to Proposition 9), and driving a first-moment upper bound below $1$ for a sub-collection settles nothing about the candidate set. (b) It is therefore not the exact ratio attached to the seed-dependent Lemma 0 condition with its allowance $\delta_N$. (c) It is a benchmark internal to the $\delta=0$ first-moment counting scheme of this document, useful for locating where that scheme stops. (d) It is not a universal impossibility theorem: nothing here bounds what other methods can reach. The boundary term $e^{H_{\max}k+O(\log k)}$ is the accumulated $O(1)$ rounding error obtained by counting integers in each of the exponentially many residue classes. Once this accumulated class-count error is no longer dominated by the leading term, the present first-moment argument ceases to give a uniform Haar-rate estimate. This does not imply that every class contains at most one integer: the classes have different moduli $2^{A+1}$ with $k\le A\le mk$, and those of smaller modulus contain many odd integers below $X$. Moving past this point within the present framework would therefore require input beyond first-moment counting — arithmetic controlling how many integers actually lie in individual classes, rather than only their expected number. Any proved improvement of the uniform range within the stated counting framework would improve Proposition 9, and such an improvement would not be caught by the equivalence trap (Proposition 8). The ratio $\approx20$ is not a numerical coincidence: it is the ratio between “information needed for elimination $\log X$ ÷ rate $I$” and “available address budget $\log X$ ÷ address increment $\log3$”, i.e. the structural constant $1/(1-H_{\max}/\log3)$ of the pair $(2,3)$ — an arithmetic identity of the scheme, and a description of the gap left by Observation 6 and by the coarse regime of R$_k$ (§7), not an obstruction theorem. Observation 11 (the sign as touchstone: the $3n-1$ obstruction) [B]. Extending the Syracuse map to negative odd numbers (equivalently, considering the $3n-1$ problem), nontrivial cycles genuinely exist: $$(-1);\qquad(-5,-7);\qquad(-17,-25,-37,-55,-41,-61,-91).$$ The exponent sequence of the last cycle, read from the phase $-17$, is $(1,1,1,2,1,1,4)$, with mean $11/7\approx1.571<\log_2 3$ (check: $3\cdot(-91)+1=-272=-17\cdot2^4$); the mean is a cyclic invariant, though the prefix means are not (see (iii)). Three items follow from this existence (items (i-A), (i-B) and (iii) proved; item (ii) heuristic). (i-A) What the negative cycles show directly. Consider the phases of the displayed cycles whose exponent words have all prefix means at most $\log_2 3$ (see (iii) for which phases these are). The displayed negative cycles show directly that the prefix-mean bound, the associated coarse frequency bound, and low or zero entropy of the exponent word are compatible with nonconvergent integer cycles. Those coarse word summaries alone therefore do not exclude cycles. Concretely, for the phase beginning at $-17$ the word is $(1,1,1,2,1,1,4)$ repeated, every prefix mean is at most $\log_2 3$, the frequency of the value $1$ is $5/7$ (well above the coarse threshold $2-\log_2 3\approx0.415$ that Lemma 1 extracts on the positive side from Lemma 0), and the word is periodic, hence of entropy rate $0$ — and yet the orbit is an integer cycle that never reaches $1$. (i-B) What can and cannot be said about the distributional and counting statements. Lemma 2, Proposition 2A, Lemma 9A and Proposition 9 are distributional or counting statements whose derivations do not use positivity. The negative cycles do not “satisfy” those theorems as individual orbits; rather, those theorems by themselves do not provide the missing positive-integer exclusion. Two type distinctions matter here and were blurred in earlier versions. First, Lemma 1 is not a sign-free statement: it is a necessary condition derived from Lemma 0 for a positive minimal counterexample, so it does not hold verbatim in the negative world, and what (i-A) exhibits is the compatibility of the coarse frequency bound of the same shape with a negative cycle, not a negative instance of Lemma 1 itself. Second, a large-deviation estimate, an entropy identity, a binomial-sum asymptotic, and a residue-class count are assertions about measures, distributions and cardinalities of sets; a single orbit is not the kind of object that satisfies or violates them. The correct statement is the one displayed above: these tools are silent about the separation of $\mathbb N$ from $-\mathbb N$, and the exclusion of positive integer counterexamples is not among their consequences. No wider impossibility is claimed. In particular, no impossibility is asserted for finite-prefix methods in general, for sign-invariant functionals in general, or for methods based on the exponent sequence: the full infinite exponent sequence determines the $2$-adic point uniquely via $\Phi^{-1}$ (for instance, the all-ones word is exactly $x=-1$), so the sign is recoverable in principle from complete sequence data, and any formulation that incorporates positivity essentially (for instance a weighted pressure or a conditioned measure adapted to $\mathbb N$) is untouched by the present observation. By contrast, Lemma 0 and Proposition 3 do use positivity, through the sign of $R_k=\sum_i\log(1+\frac{1}{3n_i})$, and are therefore not among the sign-insensitive ingredients discussed here. Indeed, for a negative cycle $R_K<0$, and the pinning comes out on the opposite side, $A_K\log2 (ii) Caution: the negative analogue of R [C]. The (absolute-value) least element of each known negative cycle satisfies a natural analogue of descent avoidance, and the known set of such avoiders is finite. This suggests, but does not prove, that a suitably formulated negative analogue of R is false: a rigorous refutation would require fixing the negative-side definitions (the order — presumably by absolute value —, the density convention, and the residue classes concerned) and a proven Terras-type density theorem for $3n-1$; neither is carried out here, so this item is recorded as heuristic. What stands as fact is the limited statement of (i-A) and (i-B): coarse word summaries of the kind used here are compatible with nonconvergent integer cycles, and the distributional and counting statements of this document do not supply the positive-integer exclusion, so a proof cannot rest on them alone; and within the present framework the identified sign-sensitive ingredient is the sign of the cocycle, $R_k>0$. (iii) Strengthening of Observation 5: $B_0\cap\mathbb Z\ne\emptyset$. Negative integers are elements of $\mathbb Z_2$, and $B_0=\{x:\ \forall k\ A_k(x)/k\le\log_2 3\}$ is a constraint on every prefix, hence phase-dependent. The cycles above furnish explicit witnesses in $B_0$, for example $-1$, $-5$ and $-17$. Not every element of those cycles, or every cyclic rotation of the displayed exponent words, lies in $B_0$. Verification: $x=-1$ has $a_i\equiv1$, so $A_k/k\equiv1$; the phase $-5$ gives the word $(1,2)$ repeated, with prefix means $1,\tfrac32,\tfrac43,\tfrac32,\ldots\le\log_2 3$; the phase $-17$ gives $(1,1,1,2,1,1,4)$ repeated, with prefix means $1,1,1,\tfrac54,\tfrac65,\tfrac76,\tfrac{11}7\le\log_2 3\approx1.585$ and, by periodicity and the fact that the full-period mean $11/7$ is itself below $\log_2 3$, the same holds at every later prefix. Counterexamples among the other phases: $-7$ starts with $a=2$ and $-91$ starts with $a=4$, so their very first prefix means, $2$ and $4$, already exceed $\log_2 3$ and neither lies in $B_0$. Hence “is $B_0\cap\mathbb Z_2\ne\emptyset$?” is settled, with integer witnesses, while membership must be checked phase by phase. On the positive side, only one direction is proved here: if $B_0\cap\mathbb N$ contained an odd $n\ge3$, then $A_k\log2\le k\log3$ together with the identity of §0 would give $n_k\ge n$ for all $k$, so such an $n$ would be a counterexample to the Collatz conjecture. The converse is not established: for a minimal counterexample, Lemma 0 yields membership in the seed-dependent set $B_{\delta_N}$, not in $B_0$, so the emptiness of $B_0\cap\mathbb N$ has not been proved equivalent to the Collatz conjecture. The separation of $\mathbb N$ from $-\mathbb N$ inside $\mathbb Z_2$ is invisible to Haar measure and to the coarse summaries of (i-A) and (i-B); the full infinite word, by contrast, does determine the point. Caution (the fate of information-theoretic formulations) [C]. The plan of measuring “how much the finite digits determine the infinite-time property $P$” by the mutual information $I(n\bmod2^r;\mathbf 1_P)$ inherits the silence of Observation 5 the moment a measure is chosen. Under density-type measures, $\{P\}$ has density $0$ (Theorem B), so $I$ vanishes identically and cannot, by definition, distinguish the empty set from a sparse non-empty one. Under the uniform measure on the finite set $[1,X]$ one gets the bound $I\le h(c(X)/X)$ (with $c(X)$ the number of counterexamples below $X$), so on that measure the functional degenerates into a monotone function of the counterexample count. These two degenerations are all that is claimed; mutual information with respect to conditioned or tilted measures is neither analysed nor excluded here. The established assets are Lemmas 0–2 and 9A, Propositions 2A, 3, 7, 8 and 9, and Observations 4, 4A, 5, 6 and 11 (item (ii) of the latter heuristic [C]). “Unprovable” has three senses — (a) unproved by current methods, (b) independent of the axiom system, (c) algorithmically undecidable — and can be claimed only in sense (a); for (b) and (c) there is no evidence whatsoever (see the caution on Theorem F). The obstruction can be described as a single finite-time gap (Observation 6), but its impassability is not proved. The targets to attack organize into: a finite-time rigidity principle whose shape is specified by the equivalent restatement R (§7), and the quantitative range problem of controlling $\rho(k)$ in the coarse regime ($r(k)\ll A_k$) of its schematic finitization R$_k$ (§8) — the latter being the entrance identified in this document that avoids the vacuity trap. Furthermore, Observation 11 shows that coarse word summaries of the kind used here are compatible with nonconvergent integer cycles, and that the distributional and counting statements of this document do not by themselves supply the positive-integer exclusion; the sign-sensitive ingredient identified within this framework is the positivity of the cocycle, $R_k>0$. In summary, the claims defensible from the contents of this document are exactly the following. None of these statements proves the Collatz conjecture, and none proves the impossibility of any exponent-sequence-based method, of finite-prefix methods in general, or of sign-invariant functionals in general. Nor does any of them assert that a single orbit satisfies or violates a distributional or counting theorem: those are statements about measures, distributions and cardinalities, and the type distinction is maintained throughout.
§1 Cited theorems
§2 Basic inequalities for a minimal counterexample
§3 Distributional constraints
§4 Large deviations
§5 Cycles
§6 The silence of measure theory, and the topological type
§7 The local-to-global reformulation and its logical status
§8 The effective range of the counting argument — what can be justified vs. what is needed
§9 Conclusion