Preprint · Version 3.2.3 · 17 August 2026
In ordinary natural density, almost every positive integer reaches a Collatz value of polylogarithmic size within \(O(\log n)\) steps — and never climbs above \(n^{1+\beta}\) before it gets there.
New to Collatz?
Choose a positive integer. At each step, halve it when it is even; when it is odd, apply (3n+1). The chain below reveals the actual arithmetic one step at a time.
Standard map: odd (nmapsto3n+1), even (nmapsto n/2).
The example (7) will run when this panel enters view.
This is a finite computation for the chosen input. If the chain does not reach (1) before the safety cap, the simulation stops without drawing a mathematical conclusion. The paper proves an almost-all descent theorem, not the pointwise Collatz conjecture.
The public theorem
The Collatz conjecture predicts that every orbit reaches \(1\). This paper does not prove that. It proves a quantitative almost-all statement: outside a set of density zero, orbits come down to a polylogarithmic value quickly, and the descent is witnessed by an explicit clock.
Throughout, \(T\) is the shortcut Collatz map, \(T(n)=n/2\) for even \(n\) and \(T(n)=(3n+1)/2\) for odd \(n\). Write \(H_2\) for binary entropy and put \[ \kappa_*=1-H_2(\log_3 2), \qquad A_{\rm FP}=\frac{1}{2\kappa_*}, \qquad c_*=\frac{2}{\log(4/3)}. \]
For every fixed \(A>A_{\rm FP}\), \(c>c_*\), \(\beta>0\), and \(0<\gamma<\kappa_*(A-A_{\rm FP})\), there is \(C_{\rm tar}>0\) such that, as \(X\to\infty\), all but \(O_{A,c,\beta,\gamma}\!\left(X/(\log X)^{\gamma}\right)\) integers \(n\le X\) admit an integer \(k<c\log n\) with \[ T^k(n)\le C_{\rm tar}(\log n)^A, \qquad \max_{0\le j\le k}T^j(n)\le n^{1+\beta}. \]
The same witness \(k\) carries all three conclusions at once: the landing, the clock, and the ceiling on every earlier iterate. A moving form allows the exponent to vary with the dyadic shell and reaches the critical exponent itself, at the cost of explicit \(\log\log\) factors. A stretched-logarithmic companion covers the weaker target \(\exp((\log n)^{1-\delta})\) for every fixed \(0<\delta<1\).
Vocabulary
The argument is elementary in its ingredients but precise in its language. These six terms are used in a fixed technical sense throughout; everything later is assembled from them.
A threshold is a dyadic value \(2^q\) the orbit tries to cross from above. Its first passage is the first iterate at or below it, and the value reached there is the landing.
Used to turn a dynamical question into a counting question about where orbits first arrive.
Successive landings that start new passages are checkpoints; together they form a decreasing threshold chain. The key lemma says every later landing is also a direct first passage from the original shell.
Collapses long multi-landing histories into single passages.
The first \(M\) parity bits of an orbit depend only on \(n \bmod 2^M\), and every word of length \(M\) occurs exactly once on the dyadic shell \(I_M=[2^M,2^{M+1})\).
The only probabilistic input: exact counting on a complete Boolean cube, not a stochastic model of an orbit.
A certificate is a condition on parity data guaranteeing the stated orbit bounds. At high shell ranks it controls every prefix. At low ranks the proof instead uses a timeout: failure to cross the next threshold within the allotted steps.
The switch between these two regimes is what makes the critical exponent finite.
Transport means counting sources in the original shell whose first-passage landing lies in a specified target set — reversing the iteration at the crossing while retaining the accumulated effect of the additive \(+1\) terms.
Converts a sparse set of bad landings back into a bound on bad starting values.
A witnessing time is an iterate where the asserted descent has occurred. The same-witness orbit ceiling bounds every earlier iterate through that same time — it says nothing about the orbit afterwards.
This is the conclusion absent from the headline statements of comparable results.
Where this sits
Almost-all Collatz results differ along several axes at once: which density, which target, and whether a clock or a rate is supplied. Collapsing them into a single ranking would be misleading, so the table compares only the coordinates each work actually states.
| Work | Density | Target for almost all starts | Clock or quantitative feature |
|---|---|---|---|
| Korec (1994) | natural | \(n^\theta\), every fixed \(\theta>\log_4 3\) | no clock or quantitative exception used here |
| Inselmann (2024) | natural | \(n^\varepsilon\), every fixed \(\varepsilon>0\) | \(2\log n/\log(4/3)\) shortcut steps |
| Tao (2022) | logarithmic | every \(f(n)\to\infty\) | no single global clock in the headline theorem |
| Mazur (2026) | natural, bridged from Tao | every \(f(n)\to\infty\) | \(<436\log n\) unaccelerated steps |
| Allikvere (2026) | natural, bridged from Tao | every \(f(n)\to\infty\) | \(<12\log n\) unaccelerated steps |
| This paper | natural | \(C_{\rm tar}(\log n)^A\), every fixed \(A>A_{\rm FP}\); moving critical endpoint | every shortcut constant \(>c_*\); target-dependent rates; orbit-height bound through the witness |
The arbitrary-threshold results of Tao, Mazur and Allikvere reach smaller targets than this paper does. What is different here is the combination: ordinary natural density, an explicit polylogarithmic scale, a quantitative exceptional rate, a logarithmic clock, and a bound on every iterate up to the witness. Those axes are deliberately not collapsed into one ordering, and no priority or strongest-result claim is made.
The dependency spine
The proof splits sharply in two. Steps 1–2 are exact probabilistic certification; steps 3–6 are deterministic Collatz transport and assembly. Read the six nodes left to right, then open any one in the lab below.
The key design choice: steps 3 and 4 are shared by both main theorems. Applying them with one fixed barrier gives the stretched-logarithmic rate; applying them with shrinking barriers and the low-rank timeout gives the polylogarithmic target.
Guided proof lab
Open a stage to inspect its input, its mathematical burden, and its output. The output of each panel is exactly the input consumed by the next one.
Input: a dyadic shell \(I_M\)→Output: a complete Boolean cube of parity words
The first \(M\) parity bits of the orbit of \(n\) depend only on \(n \bmod 2^M\), and the induced map is a bijection onto \(\{0,1\}^M\). So the shell \([2^M,2^{M+1})\) contains exactly one representative of every parity word. The exact affine iterate formula then writes \(T^k(n)\) explicitly in terms of that word.
Input: the parity cube→Output: a dense set with a deterministic orbit envelope
Words whose partial odd-step counts deviate too far from the mean are counted directly by a binary-entropy tail bound. Discarding them removes an exponentially small proportion of the shell and leaves a set on which the orbit obeys a two-sided envelope at every time in the block — a pathwise bound, obtained by exact counting rather than by any independence assumption.
Input: a set of bad landing values→Output: a bound on the sources that reach them
For a fixed passage time and landing value, reversing the map bounds how many sources can arrive there. The reverse product retains the accumulated effect of the additive \(+1\) terms as an explicit loss term, so the estimate stays exact rather than heuristic. This is what lets a sparse bad set at the target be transported back to a sparse set of bad starts.
Input: a chain of successive landings→Output: one direct first passage from the source shell
If an orbit first drops below \(2^{30}\), and from that landing later first drops below \(2^{27}\), then no earlier iterate was ever below \(2^{27}\) — so the second landing is the original orbit's own first passage below \(2^{27}\). Iterating this observation along the whole decreasing chain means multi-stage histories never have to be counted as separate events, and their reverse losses telescope.
Input: a linear horizon of candidate times→Output: a set of size \(O(\sqrt{M\log M})\)
This is the quantitative pivot. Aggregating over every time up to the horizon costs a factor linear in \(M\). Rank-dependent barriers instead confine every time at which certification can first fail to one interval of size \(O(\sqrt{M\log M})\). Replacing a linear counting loss by a square-root loss is precisely what moves the reachable target from a fixed power down to a polylogarithm.
Input: blocks below the switch rank→Output: the exponent margin \(\Delta_M\), hence \(A_{\rm FP}\)
Below the switch, positive certification is dropped. A block is declared failed only if it misses its next threshold within the allotted time — and such a block must have an unusually large terminal odd-step count, whose exact binomial upper tail has rate \(\kappa_*\log 2\). Balancing that tail against the square-root time cost leaves the margin \(\kappa_*L-\tfrac12\log_2 M-\log_2\log M\); its divergence is the endpoint condition, and the leading power changes sign at \(A\kappa_*=1/2\).
Graduate study lab
The theorem has an order. First fix a target exponent \(A\) above the critical value, a clock \(c\) above \(c_*\), and a ceiling \(\beta>0\); only then is an exceptional exponent \(\gamma\) admissible, and only below \(\kappa_*(A-A_{\rm FP})\). Move the controls to see which conclusions travel together.
Model a very large input as \(n=10^{D}\). The slider moves \(D\), so each step changes the number of digits of the starting value.
The digit ratio tends to zero: the certified value is polylogarithmic in \(n\), so it has \(O(\log\log n)\) digits against the input's \(\asymp\log n\). A fixed power \(n^{\alpha}\) only ever shrinks the digit count by the constant factor \(\alpha\).
Boundary discipline. The controls stop at \(A_{\rm FP}=9.9911133419\ldots\) and \(c_*=6.9521189935\ldots\), because equality is not claimed at either. The paper proves existence of \(C_{\rm tar}\) and an asymptotic onset; this calculator does not invent numerical values for either one.
The quantitative pivot
Everything in the proof turns on one exchange. A direct union over every moment at which certification could first fail costs a factor linear in the shell rank \(M\). The time-support lemmas confine those moments to a set of size \(O(\sqrt{M\log M})\) instead.
Orders only. The unspecified constant in the time-support lemmas is not modelled, so the vertical scale is indicative rather than numerical.
The ratio \(\sqrt{M\log M}/M\to0\), so the saving grows without bound as shells get larger. That is exactly the room the scalar budget needs: it is what leaves the exponent margin \(\kappa_*L-\tfrac12\log_2 M-\log_2\log M\) positive, and hence what makes a polylogarithmic target reachable at all.
One number at a time
Below is a genuine shortcut-Collatz orbit, drawn on a logarithmic vertical axis. The dashed line is the mean-drift reference with slope \(a_0-1\), not a pointwise orbit bound; green dots mark first passages, each one a landing on the next dyadic threshold in the decreasing chain.
A single orbit proves nothing. These runs are evaluated in your browser under the same map, purely to build intuition. The theorem is a statement about a set of natural density one, and no finite computation is used as a premise anywhere in the proof.
Reading the scale
Plotting the number of digits of each target against the number of digits of \(n\) makes the hierarchy visible. Fixed powers are straight lines through the origin — they always keep a constant fraction of the input's digits. The polylogarithmic target flattens out completely.
Landing constants are not modelled; the comparison is of growth rates. The stretched-logarithmic curve uses \(\delta=0.25\).
"Almost all", drawn
Each line is a real starting value sampled from the dyadic shell \([2^{26},2^{27})\), and its length is how many shortcut steps it actually needed to reach the target. The dashed line is the reference drift scale \(c_*\log n\). The theorem applies only after fixing \(c>c_*\), so crossings here are not identified with its exceptional set.
Which target these illustrations use, and why. The principal reference scale \((\log n)^{A_{\rm FP}}\) is genuinely asymptotic: it only becomes smaller than \(n\) once \(n\gtrsim3.3\times10^{15}\). Below that crossover every orbit satisfies it at \(k=0\), which would draw a descent where nothing has happened. The fixed-exponent theorem itself uses \(A>A_{\rm FP}\), while the critical endpoint carries additional log-log factors. So the finite illustrations on this page use the paper's stretched-logarithmic companion \(\exp((\log n)^{1-\delta})\) with \(\delta=0.25\) — a real descent at these scales.
A finite sample from one shell. Density one is an asymptotic statement: it permits an exceptional set at every scale, and says only that its proportion tends to zero.
| Start | Steps to target | Reference \(c_*\log n\) | Peak height |
|---|
Real values, computed from the map, against the companion target \(\exp((\log n)^{0.75})\). "Peak height" is the largest iterate before the witness, written as a power of the starting value — the quantity the same-witness ceiling bounds by \(n^{1+\beta}\).
This is an almost-all theorem, not the Collatz conjecture. It does not prove descent for every starting value, does not exclude nontrivial cycles or divergent trajectories, and says nothing about the orbit after the selected witness. No finite computation is used as a premise for any asymptotic claim.
Two parameter limits are explicitly not asserted: the pure target \(C(\log n)^{A_{\rm FP}}\) at the critical exponent with a bounded final multiplier, and the endpoint \(\delta=1\) in the stretched-logarithmic companion. \(A_{\rm FP}\) is a threshold of this argument, not a constant known to be intrinsic to the Collatz map.
Machine-checked
The written proof is self-contained. A separate Lean 4 development kernel-checks the canonical route to the principal exported theorem, including the shell exceptional ratio, the moving landing, the logarithmic clock, and the same-witness orbit ceiling.
The canonical library builds with no sorry, no
admit, no project axiom, and no missing module; the
public-root axiom reports contain only Lean's three standard axioms
propext, Classical.choice, and
Quot.sound. Some combined quantitative statements in the
manuscript are assembled from separately checked components rather than
exposed as single wrapper theorems — that declaration-level boundary is
stated explicitly on the map page.
| SSRN preprint | Paper 7290240 · DOI 10.2139/ssrn.7290240 |
| Direct full text | PDF · Version 3.2.3 |
| Versioned archive | 10.5281/zenodo.21984038 · v3.2.3 · CC BY 4.0 |
| Formalization | 10.5281/zenodo.21930432 · Apache 2.0 |
| Source | shaikidris/FirstPassageLinearTransport, tag lean-v3.2.0 |
| Commit | ef3410843bf58d69f771f5ba2c0571d54b54da59 |
| Toolchain | Lean 4.15.0 · Mathlib 9837ca9 |
The author's earlier and logically independent stretched-logarithmic endpoint-transport paper proves a restricted-range result by a different mechanism. No result from it is used as a premise here.