A sandbox for the 3n+1 problem — run the experiments that show why it's hard, in your browser, at any size (BigInt throughout).
Odd (Syracuse) steps: n → (qn+1)/2^v₂. Enter any integer — plain digits or 2^1000-1 form. The previous run stays as a gray ghost for comparison.
Where do the divisions come from? From carries. In 3n+1 = 2n + n + 1 the carry propagates exactly through the trailing bits of n that agree with the 2-adic expansion of −1/3 = …0101010101₂.
Highlighted suffix = the match with …0101. Its length is exactly the number of halvings the next step performs. Watch the orbit consume its trailing bits and manufacture new ones out of carries.
Pick any division pattern you want — say v₂ = 2,1,1,3,1,2,1,4 for the first eight odd steps. There is exactly one residue class mod 2^(Σkᵢ+1) whose orbits do it (Terras 1976). Trailing bits are freely programmable, so no theorem can force carries to lengthen within any fixed number of steps.
Start from 2¹⁰⁰⁰−1 (a thousand 1-bits, compresses to ~96 bits). Carries inject entropy: the compressed size (dashed) climbs to meet the bit length exactly at the peak, and from then on the number is incompressible and rides the average drift down. A random seed starts incompressible and just descends. Under 5n+1 the same local statistics produce divergence.
Half of the complexity hypothesis is free — C(n_k) ≤ C(n₀) + O(log k) holds for any computable map — which is exactly why it buys nothing: the 5n+1 orbit of 7 is a bounded-complexity sequence that apparently diverges.
A seed 2^k−1 dictates its first k−1 divisions (all v₂=1, the climb). Past that horizon the orbit must manufacture its own bits out of carries — deterministically. Are the manufactured divisions statistically random? Run it and see: the histogram is geometric(1/2) to within noise.
Aggregate offline run (seeds 2^k−1, k=200…1000): 38,299 post-horizon steps, mean v₂ = 2.007 vs 2.0 ideal, χ² = 16.5 on 10 df (not rejected), lag-1 autocorrelation 0.007. Same seeds under 5n+1: mean v₂ = 1.990 — identical local statistics, opposite fate.
The hoped-for bound v₂(3n+1) ≥ f(carry chain) is an identity: v₂(3n+1) equals the length of the trailing match between n and the 2-adic integer −1/3. The carry chain in 2n+n+1 propagates exactly through that region. No sharper local statement is possible — the microscope above is the whole story of a single step.
"Carry chains can't stay short forever" is false pointwise: the pattern programmer builds an n for any division schedule, and 2^k−1 keeps carries minimal for k−1 straight steps while growing by (3/2)^(k−1). Bad patterns of length m occupy exactly one residue class mod 2^m — density 2^(−m) — which is how Terras proved almost every n descends. Density is provable; every is the open problem.
The map extends to the 2-adic integers, where it has other cycles: −1 → −1 and {−5, −7, −10}. The worst climbers are 2-adic impostors — 2^k−1 → −1 as k → ∞ — so any digit-pattern metric that contracts every step would pass to the limit and give two fixed points, which contractions can't have. Worse, on ℤ₂ the map is conjugate to the full 2-shift (entropy log 2; Lagarias 1985, Bernstein–Lagarias 1996): maximally chaotic in exactly those coordinates. The salvage is contraction on average against an invariant measure — which is precisely the architecture of Tao's 2019 result.
Every structural mechanism here — carries, complexity, digit statistics — holds verbatim for 5n+1 (replace −1/3 with −1/5). Same mean v₂ = 2, same thermalization. Only the drift flips: log₂5 − 2 = +0.32, and orbits like 7 climb without apparent end. So no argument built from these mechanisms alone can prove descent: a working proof must use log₂3 < 2 quantitatively. The margin is 0.415 bits per odd step; the conjecture says no orbit keeps its long-run mean v₂ below 1.585 forever, while short-run means can provably be pushed to 1.
A hypothetical cycle with C odd steps and A halvings forces 0 < A·log2 − C·log3 < C/(3·n_min): A/C must approximate log₂3 absurdly well. Continued fractions plus Baker's theorem on linear forms in logarithms, combined with the 2⁷¹ verification, push any cycle past ~10¹¹ steps (Eliahou's method; Hercher 2023 excludes all k-cycles, k ≤ 91). That is the mixed-radix idea working — but only for periodic orbits, where the coordinate transformation closes into a single Diophantine equation.
Trailing bits drive the dynamics and are consumed every step; a positive integer has only log₂n₀ of them before the orbit runs on manufactured carries alone. The seed-horizon lab shows the manufactured stream is statistically perfect randomness — mean 2.007, geometric histogram, no autocorrelation. Divergence would require this deterministic stream to stay biased below mean 1.585 forever. Equivalent to the conjecture, but it names the exact difficulty: prove the orbit can't conspire against its own carries.
Checklist for any proposed proof mechanism:
Three more candidate mechanisms tested, three exact autopsies. "Negative curvature" (long bad runs arithmetically suppressed): across 3,000 random 256-bit seeds, upward excursions match the iid coin-flip model exactly — P(orbit ever climbs ≥ x bits) ≈ 2^(−x), and the Cramér equation t^(log₂3) = 2t−1 has root t = 2 only for multiplier 3 (measured tail slope −1.14 vs −1 predicted; mean longest bad run 8.53 vs 8.57 iid). No conspiracy in either direction. A residue-graph approach mod 2^k·3^m collapses to an identity: the next odd number is ≡ (−1)^v₂ (mod 3) — the entire 3-adic side is deterministically slaved to the division sequence (0 violations in 30,053 steps) and carries no independent constraint.
The headroom autopsy became a theorem. A proposed "binary headroom" Lyapunov functional h = 1 − n/2^bits (distance to the next power of two — an MSB quantity) was predicted to drift down for 3n+1 and up for 5n+1. Measured over 1.27M odd steps: E[Δ] = +0.0001 for both maps, flat in v₂ — dead by its proposer's own criterion. Reason: the shell coordinate x = frac(log₂ n) evolves as x′ = x + log₂3 + log₂(1+1/(3n)) (mod 1), and the division count, being an integer, drops out of the fractional part entirely. The top of the number is a rigid irrational rotation, independent of the carries (verified to 1.5×10⁻¹³ over a million steps).
The factorization. To O(1/n), the Syracuse map splits into two independent exactly-solvable factors: a zero-entropy rotation by log₂3 on the mantissa circle (the Benford factor — Kontorovich–Miller 2005, Lagarias–Soundararajan) and a maximal-entropy full shift on trailing bits (Terras, Tao), coupled only by ε = log₂(1+1/(3n)). Around a cycle, Σε = a − c·log₂3 — so the Baker/continued-fraction cycle bounds are exactly lower bounds on accumulated coupling over closed orbits. Along open orbits the accumulated coupling converges (every term is < 2⁻⁷¹ for any surviving counterexample) while the bit-walk fluctuates like √k, so no Baker-style bound can exist for escaping trajectories.
The boundary in one sentence: the constants 2 and 3 can currently be exploited only where an orbit closes — and the conjecture lives entirely in orbits that don't.
Attribution post-scriptum (2026-07-15): a closely related result — an explicit near-conjugacy of the Collatz map to a circle rotation (base-6 formulation, error O(1/x)) — appeared in arXiv:2601.04289 (Jan 2026). The rotation-factor observation is therefore concurrent literature, not unique to this session; our Syracuse-time variant and its consequences for MSB Lyapunov functionals stand as verified here.
The shrinking-target collapse. A proposed reformulation asked whether positivity constraints on the mantissa rotation become empty for below-critical division scripts. Answer: positivity constrains nothing. Over the reals, the p-step Syracuse composite x ↦ (3^p·x + D)/2^A is an increasing affine map with D > 0, so every division script is realizable from every real x₀ > 0 with all iterates positive. Over ℤ₂ every script is realizable (Terras). Jointly, a residue class mod 2^N contains integers in every real window of length 2^N, so the nested constraints are nonempty at every finite stage. The obstruction to bad orbits is integrality of the infinite limit — a global, adelic condition invisible to both completions separately — not the inequality n > 0.
The theorem the collapse produced. A periodic division script (k₁,…,k_p) with A = Σkᵢ has a unique rational solution n = D/(2^A − 3^p), and D > 0 always — so sign(n) = sign(2^A − 3^p). Every subcritical script (A/p < log₂3, the only kind that can sustain growth) therefore solves to a negative rational. Verified on 20,000 random scripts: 4,079 subcritical, all negative, zero exceptions; the only integer solutions found were exactly −1 (script (1)), −5 (script (1,2)), −17/−41 (script (1,1,1,2,1,1,4) and rotations), and the trivial 1 (script (2), supercritical). The classical 2-adic impostor cycles are structurally explained: the sign flip of 2^A − 3^p exiles all subcritical periodic behavior to the negative integers — the same quantity Baker's theorem bounds is the quantity whose sign sorts good from bad.
Joint closing statement (cross-model consensus). The sign dichotomy is a consequence of exact closure, not approximate closure. No analogous sign invariant exists for aperiodic trajectories: a finite prefix determines only an affine family of completions, never a closed algebraic identity, so the distinction between positive divergent trajectories and negative periodic impostors cannot be detected by any finite-prefix invariant — it is inherently a global property of the infinite tail. That locates the remaining gap more sharply than any of the failed Lyapunov candidates it replaced.
The setup. An overnight three-way exchange (ChatGPT and Gemini proposing, Claude executing and verifying every claim in exact arithmetic) tested roughly twenty proof mechanisms end to end. Kill rate: 100%, each with a documented structural cause of death — the ledger spans pointwise carry lemmas, contraction metrics, complexity, curvature, residue graphs, MSB functionals, entropy budgets, certificates, and Gemini's formally-dressed "asymptotic decoupling bound," which was falsified constructively (the seed family 2500m+1−1 holds division density 1.0 at arbitrary altitude, defeating its own pass/fail criterion) and conceded in writing. A proposed shrinking-target reformulation collapsed too: positivity constrains nothing over ℝ₊ — the obstruction to divergent orbits is integrality, an adelic condition no single completion sees.
Machine-checked (Lean 4 + mathlib, clean build, zero sorry) — CollatzNight.lean: the unrolled orbit identity 2^(A_N)·n_N = 3^N·n₀ + D_N; the Böhm–Sontacchi cycle equation; positivity of its numerator; the sign dichotomy (subcritical cycles have negative seeds — the −1, −5, −17 impostors, certified); positive cycles force 3^p < 2^(A_p); the mod-3 slaving congruence (next odd ≡ (−1)^v₂ mod 3); and the carry theorem's arithmetic core (4^m | 3n+1 ⟺ n ends in the alternating bits 0101…01 — the truncated 2-adic −1/3).
Quantitative results verified. The divergence-candidate set B ⊂ ℤ₂ (subcritical scripts) has Hausdorff dimension H₂(1/log₂3) ≈ 0.949956 — correct via Besicovitch–Eggleston plus the Terras cylinder bijection, calibrated as routine/folklore rather than new. Rationals with odd denominator v are exactly the integer 3n+v systems (denominator invariance, verified for v = 5, 23, 181), and bounded rational orbits are provably eventually periodic. Literature settled: the night's central congruence is classical — Rozier (INTEGERS 2019) Lemma 1 calls it "a well-known expression" — so the two-completions trichotomy and the escape-constant computation are packaging and computation, not new identities.
What the night opens. Two well-posed subproblems: (i) a regularity program for the coding map between division scripts and 2-adic seeds — continuity, Hölder exponents, injectivity, images of natural subshifts — standalone symbolic dynamics, honestly not equivalent to Collatz; and (ii) eliminating bounded-gap aperiodic subcritical scripts over ℚ₊ via effective linear-independence measures for Σ 2^(A_j)/3^(j+1) — ambitious, near-equivalent to the divergence question in general, tractable-looking in the bounded-gap restriction. Closing calibration, three-model consensus: this is not a proof of the Collatz conjecture; it is a rigorously delimited structural map of why the problem resists current methods. The regularity program is now written up as a standalone problem sheet with a state-of-the-art survey: the coding-map regularity program.