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The attack log

You asked for the paths, walked, logged, deeper. Here they are — the four serious lines of attack on the conjecture, actually executed: ranking-function candidates killed at computed counterexamples, the cycle gauntlet run through exact convergents of log₂3, the covering induction pushed to modulus 2¹⁰²⁴, the divergence bias measured across real orbits. Each log ends at the same place the field's best mathematicians ended: a terminal wall, precisely characterized. What no honest log can end with is a proof — the conjecture is open, and these instruments show exactly why.

PATH A · Direct descent — hunt for a ranking function

The textbook way to prove termination: find a potential φ that provably decreases every step (or every odd-block). If any candidate below had survived, the conjecture would be settled. Watch each one die at a specific, computed integer.

›DEPTH 1 — testing 10 candidate potentials for strict descent, n ≤ 16,384
››φ(n) = n (raw magnitude, per step) — DIES at n = 3: φ(3) = 3 → φ(10) = 10 (no decrease)
››φ(n) = bitlength(n), per step — DIES at n = 3: φ(3) = 2 → φ(10) = 4 (no decrease)
››φ(n) = bitlength(n), per odd-block — DIES at n = 3: φ(3) = 2 → φ(5) = 3 (no decrease)
››φ(n) = popcount(n), per odd-block — DIES at n = 3: φ(3) = 2 → φ(5) = 2 (no decrease)
››φ(n) = 1·bitlength + 1·popcount, per odd-block — DIES at n = 3: φ(3) = 4 → φ(5) = 5 (no decrease)
››φ(n) = 2·bitlength + 1·popcount, per odd-block — DIES at n = 3: φ(3) = 6 → φ(5) = 8 (no decrease)
››φ(n) = 1·bitlength + 2·popcount, per odd-block — DIES at n = 3: φ(3) = 6 → φ(5) = 7 (no decrease)
››φ(n) = 3·bitlength + -1·popcount, per odd-block — DIES at n = 3: φ(3) = 4 → φ(5) = 7 (no decrease)
››φ(n) = 2·bitlength + 3·popcount, per odd-block — DIES at n = 3: φ(3) = 10 → φ(5) = 12 (no decrease)
››φ(n) = 4·bitlength + 1·popcount, per odd-block — DIES at n = 3: φ(3) = 10 → φ(5) = 14 (no decrease)
›DEPTH 2 — retreat to a weaker demand: “every n descends below itself within m steps, for some uniform m.” Computing stopping-time records for n ≤ 16,384:
››σ(2)=1 · σ(3)=6 · σ(7)=11 · σ(27)=96 · σ(703)=132 · σ(10,087)=171
››The record σ keeps growing with the range — no uniform m survives the search. (Terras proved the records are unbounded: stopping time exceeds any m on a positive-density-zero but infinite set.)
›TERMINAL STATE — every locally-checkable potential dies because ×3 rewrites the whole binary string (global carries) while ÷2 trims one bit; a working φ must tolerate the 27-style excursions, i.e. it must already know the orbit comes back. This wall is why no ranking-function proof exists. PATH A: EXHAUSTED, NOT CLOSED.

PATH B · Cycle exclusion — the Diophantine gauntlet

A nontrivial cycle with a odd and b even steps forces 0 < b − a·log₂3 < a/(2V), where every cycle element exceeds the verification bound V = 2^60. So b/a must approximate log₂3 from above absurdly well — a condition only continued-fraction convergents can meet. This engine computes log₂3 to 320 fixed-point bits, derives its convergents exactly, and tests each against the inequality.

›DEPTH 1 — log₂3 = 1.584962500721156… (computed to 320 bits by exact squaring; convergents below are exact integers)
›DEPTH 2 — testing convergents b/a against the cycle inequality ‖a·log₂3‖ < a·2^−61:
››b/a = 19/12 (below) — gap ≈ 2^-5.7 vs required < 2^-57.4 → wrong side, excluded
››b/a = 65/41 (above) — gap ≈ 2^-5.9 vs required < 2^-55.6 → EXCLUDED
››b/a = 84/53 (below) — gap ≈ 2^-8.4 vs required < 2^-55.3 → wrong side, excluded
››b/a = 485/306 (above) — gap ≈ 2^-9.4 vs required < 2^-52.7 → EXCLUDED
››b/a = 1,054/665 (below) — gap ≈ 2^-14.0 vs required < 2^-51.6 → wrong side, excluded
››b/a = 24,727/15,601 (above) — gap ≈ 2^-15.2 vs required < 2^-47.1 → EXCLUDED
››b/a = 50,508/31,867 (below) — gap ≈ 2^-16.5 vs required < 2^-46.0 → wrong side, excluded
››b/a = 125,743/79,335 (above) — gap ≈ 2^-17.5 vs required < 2^-44.7 → EXCLUDED
››b/a = 176,251/111,202 (below) — gap ≈ 2^-17.6 vs required < 2^-44.2 → wrong side, excluded
››b/a = 301,994/190,537 (above) — gap ≈ 2^-23.4 vs required < 2^-43.5 → EXCLUDED
››b/a = 16,785,921/10,590,737 (below) — gap ≈ 2^-23.7 vs required < 2^-37.7 → wrong side, excluded
››b/a = 17,087,915/10,781,274 (above) — gap ≈ 2^-25.8 vs required < 2^-37.6 → EXCLUDED
››b/a = 85,137,581/53,715,833 (below) — gap ≈ 2^-27.6 vs required < 2^-35.3 → wrong side, excluded
››b/a = 272,500,658/171,928,773 (above) — gap ≈ 2^-28.5 vs required < 2^-33.6 → EXCLUDED
››b/a = 357,638,239/225,644,606 (below) — gap ≈ 2^-28.6 vs required < 2^-33.3 → wrong side, excluded
››b/a = 630,138,897/397,573,379 (above) — gap ≈ 2^-32.6 vs required < 2^-32.4 → SURVIVES: first admissible cycle shape
››b/a = 9,809,721,694/6,189,245,291 (below) — gap ≈ 2^-32.8 vs required < 2^-28.5 → wrong side, excluded
››b/a = 10,439,860,591/6,586,818,670 (above) — gap ≈ 2^-36.0 vs required < 2^-28.4 → SURVIVES: first admissible cycle shape
››b/a = 103,768,467,013/65,470,613,321 (below) — gap ≈ 2^-37.1 vs required < 2^-25.1 → wrong side, excluded
››b/a = 217,976,794,617/137,528,045,312 (above) — gap ≈ 2^-39.5 vs required < 2^-24.0 → SURVIVES: first admissible cycle shape
››b/a = 1,193,652,440,098/753,110,839,881 (below) — gap ≈ 2^-42.4 vs required < 2^-21.5 → wrong side, excluded
››b/a = 8,573,543,875,303/5,409,303,924,479 (above) — gap ≈ 2^-43.3 vs required < 2^-18.7 → SURVIVES: first admissible cycle shape
››b/a = 9,767,196,315,401/6,162,414,764,360 (below) — gap ≈ 2^-43.6 vs required < 2^-18.5 → wrong side, excluded
››b/a = 18,340,740,190,704/11,571,718,688,839 (above) — gap ≈ 2^-45.6 vs required < 2^-17.6 → SURVIVES: first admissible cycle shape
››b/a = 83,130,157,078,217/52,449,289,519,716 (below) — gap ≈ 2^-48.8 vs required < 2^-15.4 → wrong side, excluded
››b/a = 683,381,996,816,440/431,166,034,846,567 (above) — gap ≈ 2^-48.9 vs required < 2^-12.4 → SURVIVES: first admissible cycle shape
›DEPTH 3 — first surviving shape: a ≥ 397,573,379 odd steps, b ≥ 630,138,897 even steps ⇒ any nontrivial cycle has length ≥ 1,027,712,276 — with every element above 2^60.
›TERMINAL STATE — the bound grows as V grows (press “go deeper”), but for every V infinitely many admissible shapes remain: the convergents of log₂3 never run out. Baker-type transcendence bounds sharpen the gauntlet; nothing known closes it for ALL periods. PATH B: BOUNDS ENORMOUS, EXCLUSION INCOMPLETE.

PATH C · Covering induction — Terras' engine, run to exhaustion

Work mod 2^k: a residue class whose k-step parity vector has a odd steps carries multiplier 3^a/2^k, and provably descends when 3^a < 2^k. The uncovered density is the exact binomial tail u(k) = 2^−k · Σ C(k,a) over a ≥ k·log₃2 — computed below in exact integer arithmetic. If u(k) ever hit zero, the conjecture would follow by induction.

›DEPTH 1 — exact uncovered density by modulus:
››k = 8: descent needs a < 6 of 8 odd steps; uncovered u(k) = 1.445e-1 (≈ 2^-3) — NONZERO
››k = 16: descent needs a < 11 of 16 odd steps; uncovered u(k) = 1.051e-1 (≈ 2^-4) — NONZERO
››k = 32: descent needs a < 21 of 32 odd steps; uncovered u(k) = 5.509e-2 (≈ 2^-5) — NONZERO
››k = 64: descent needs a < 41 of 64 odd steps; uncovered u(k) = 1.638e-2 (≈ 2^-6) — NONZERO
›DEPTH 2 — decay rate check: u(k) shrinks geometrically (≈ 2^-0.031 per step of k) — this is exactly Terras' density-1 theorem materializing in the numbers.
››But u(k) > 0 for every k — provably forever, since the all-odd parity vector always has 3^k > 2^k. The uncovered classes always contain infinitely many integers.
›TERMINAL STATE — the covering argument proves “almost all n descend” and can be iterated (Korec, Tao push it to almost-bounded values), but the exceptional set shrinks and never provably empties. Density-1 is structurally unable to reach ALL. PATH C: CONVERGES TO 100%, NEVER ARRIVES.

PATH D · Divergence forcing — what an escaping orbit must do

In the accelerated map T (odd n → (3n+1)/2), t steps with o odd scale the value like n·3^o/2^t, so a divergent orbit must sustain odd-fraction o/t ≥ log2/log3 = 0.630930… at every horizon, forever — while Terras equidistribution says typical orbits run at 1/2. This engine measures the longest any real orbit sustains the required bias.

›DEPTH 1 — required permanent bias: o/t ≥ 0.630930 (exactly log 2 / log 3). Sweeping odd seeds n ≤ 50,000 for the longest sustained-bias prefix:
››n=3 holds it for 3 steps · n=7 holds it for 6 steps · n=27 holds it for 58 steps · n=703 holds it for 80 steps · n=10,087 holds it for 104 steps · n=35,655 holds it for 134 steps
››Every orbit in range breaks the bias within ~134 steps and then falls. The bias records grow only logarithmically with the range — exactly the random-walk prediction, and exactly what makes a real divergent orbit a statistical monster.
›DEPTH 2 — the honest asymmetry: no finite sweep can rule the monster out. It is not even proven that any orbit of 5n+1 diverges — the tool “prove this specific orbit escapes” does not exist in mathematics yet, and its dual “prove no orbit escapes” is this conjecture.
›TERMINAL STATE — divergence is heuristically impossible (the bias must beat equidistribution forever) and rigorously unexcluded. This is the hard core of the problem. PATH D: MEASURED, NOT CLOSED.

PATH E · Joint research, iteration 1 — the modular Lyapunov family, DECIDED

The natural complete proof: a potential φ(n) = log₂n + w(n mod 2^16) that strictly decreases every accelerated step. Unlike most proof shapes this family is decidable — w exists iff every cycle of the congruence graph mod 2^16 has negative weight. This engine decides it, live, by verifying explicit positive-weight cycles edge by edge in exact arithmetic. Full write-up: research/RESEARCH.md in the repo.

›DEPTH 1 — hunting nonnegative cycles in the congruence graph mod 2^16 (any one kills the whole family):
››shadow of x = −1 (the T-fixed point; all-odd loop) — closes: true, all 1 edges lawful: true, mean +0.58496 bits/step, exact positivity 3^1 > 2^1: true → CERTIFICATE-KILLING CYCLE
››shadow of the −5 cycle (parity 1,1,0) — closes: true, all 3 edges lawful: true, mean +0.05664 bits/step, exact positivity 3^2 > 2^3: true → CERTIFICATE-KILLING CYCLE
››shadow of the −17 cycle (7 odd / 4 even in 11 steps) — closes: true, all 11 edges lawful: true, mean +0.00861 bits/step, exact positivity 3^7 > 2^11: true → CERTIFICATE-KILLING CYCLE
›DEPTH 2 — verdict at modulus 2^16: family PROVEN EMPTY — and the x = −1 self-loop exists at every modulus (−1 is a genuine fixed point of T on ℤ), so the emptiness holds for ALL k, not just those tested.
›TERMINAL STATE — the three killing cycles are precisely the known cycles of Collatz on negative integers: finite-modulus arithmetic cannot see the sign of the domain, so any modular certificate must also certify orbits that genuinely cycle. The §6 no-go filter, upgraded from warning to machine-checked emptiness proof. PATH E: FAMILY DEAD, OBSTRUCTION NAMED. Iteration 2 candidates are listed in research/RESEARCH.md.

PATH F · Joint research, iteration 2 — certified density bounds, live

The Applegate–Lagarias tree-search method: censuses of the inverse Collatz tree rooted at 8 become certified statements 'at least T distinct integers ≤ x reach 1', i.e. π₁(x) ≥ x^γ at that x. The DP over classes mod 3^j is always conservative (min over 3-adic lifts) and was validated against exact brute-force enumeration. Full-depth runs (γ ≥ 0.7244 at modulus 3^11, depth 128) in research/iteration2.ts; this panel re-derives the trajectory live at browser scale.

›DEPTH 1 — certified census bounds, computed now in this browser:
››modulus 3^4, depth 40: ≥ 1.19e+3 distinct integers ≤ 2^19.7 reach 1 (via b=9 branch steps) → γ ≥ 0.5174
›DEPTH 2 — the exponent climbs with compute (0.5174 here; 0.7244 at full depth in the repo runner) — past the tree-search method's published-era numbers (≈0.654, 1995) though still below the 0.84 record, which uses the stronger difference-inequality machinery (our iteration 3 target). TERMINAL STATE of this method: it certifies density exponents, and density can never reach the conjecture itself (§8). PATH F: BOUNDS REAL AND GROWING, TOTALITY OUT OF REACH BY DESIGN.

PATH G · Iteration 4A — cycle-exclusion push at the live frontier

Direction (A) of the research menu: aim the exact-convergent gauntlet at the current verification frontier and state coverage honestly — by the best-approximation theorem the exclusion covers every odd-step count below the surviving convergent, not only convergents themselves.

›DEPTH 1 — gauntlet at V = 2^71 (all cycle elements exceed V):
››any nontrivial cycle: ≥ 137,528,045,312 odd steps, total length ≥ 355,504,839,929 — and the exclusion covers ALL a below 137,528,045,312, since ‖a·log₂3‖ ≥ ‖q_n·log₂3‖ for every a < q_(n+1).
›DEPTH 2 — LITERATURE UNBLOCKED (iteration 6.1): Hercher 2023 proved no m-cycles with m ≤ 91 and that verification up to 3·2⁶⁹ suffices for the next bound K ≥ 1.375×10¹¹ odd members; Barina 2025 published verification to 2⁷¹ > 3·2⁶⁹. The corollary therefore stands on published results — and its constant is EXACTLY the convergent this engine derived independently: 137,528,045,312. Two machineries, one integer.
›DEPTH 3 — beating m ≤ 91 itself still needs the papers' internal linear-form constants; full texts return 403 under this environment's network policy (search allowed, fetch blocked). Logged precisely as far as the sources reach.
›STATUS: NEXT CYCLE BOUND ESTABLISHED FROM PUBLISHED RESULTS + INDEPENDENT EXACT AGREEMENT · FULL METHOD REPLICATION BLOCKED BY NETWORK POLICY.

PATH H · Iteration 4B — Krasikov–Lagarias difference inequalities, attempted

Direction (B): the 0.84-record machinery. We derived the counting system from scratch rather than pretending to remember the paper — and the derivation itself produced the finding below.

›DEPTH 1 — derived exactly: f_r(x) = f_(r/2)(x/2) + (odd-class term with argument 3x+1). The odd term's argument INCREASES, so downward induction can keep only the doubling term.
››FINDING — the downward-inductive fragment of the difference system telescopes into exactly the inverse-tree census of iteration 2 (Path F). The two attacks are the same attack; all additional strength of KL 2003 lives in upward-induction lemmas.
›DEPTH 2 — rebuilding those lemmas from memory risks a silently weaker system presented as the real one — the exact failure mode this log forbids. BLOCKED pending the 2003 paper. STATUS: CLARIFIED, NOT REBUILT.

PATH I · Iteration 4C — the universal modular kill, live

Direction (C), automated invariant search, first hypothesis class: certificates whose state is n mod m for ANY finite modulus m (not just 2-powers), any bounded lookahead. Decided wholesale: positive integers ≡ −1 (mod m·2^B) ride the ×3/2 growth pattern inside the class −1 mod m, so iteration 3's telescoping kill applies at every modulus. This panel machine-verifies the pattern for every m up to the slider, in exact BigInt arithmetic, right now.

›DEPTH 1 — sweeping m = 2 … 5,000: 30-step persistence of class −1 with v = 1 growth:
››ALL 4,999 moduli verified — no modulus escapes the −1 obstruction.
›THEOREM (sweep + the fixed-point argument for all m): no finite-modulus certificate — any m, any bounded lookahead — can prove the conjecture. Descent proofs must consume unbounded 2-adic information. This closes the entire hypothesis class that iterations 1 and 3 sampled. STATUS: CLASS CLOSED.

PATH J · Iteration 4D — mining beyond the Terras horizon, live

Direction (D): inside the first ~log₂n accelerated steps, parity vectors are exactly equidistributed (Terras' theorem). Beyond that horizon nothing is proven — so measure what real orbits actually do there. This panel runs the sweep in your browser.

›DEPTH 1 — 100,000 orbits from n = 2^20, T-steps 1–40:
››odd-fraction inside horizon (steps 1–20): 0.49960 · beyond horizon (steps 21–40): 0.50030 · binomial σ ≈ 0.00035
››naive deviation beyond the horizon: 0.8σ (binomial, assumes independent orbits)
››CAVEAT, discovered by this very panel: consecutive-seed orbits merge and share tails, so they are NOT independent and the naive σ understates the error — this window and the 500k script window disagree wildly on significance (0.8σ vs 8.8σ). Try “go deeper” and watch the number move.
›STATUS: SUGGESTIVE DEFICIT, SELF-CORRECTED — not a certified anomaly until a block-bootstrap over merge-clusters replaces the naive error bar. The overstatement and its correction are both logged (research/RESEARCH.md): the instruments are allowed to catch each other.

PATH K · Iteration 4E — Berg–Meinardus functional equation: refused, deliberately

Direction (E): validated numerics on the analytic reformulation.

›Faithful work requires the exact functional equation from the paper; running rigorous numerics against a half-remembered equation would manufacture confident nonsense — the one failure mode this log forbids. BLOCKED pending literature access. An honest refusal is logged as a result: it marks where the map ends.

PATH L · Iteration 5L — thermodynamics: divergence as a second-law violation

Nature framing with teeth: treat log₂n as energy — odd steps inject 0.585 bits, halvings dissipate 1. A divergent orbit must beat the drift forever, and large-deviation theory prices that at 1 − H(θ) = 0.05004 bits of improbability per step (θ = log2/log3). The prediction is testable against exact combinatorics: Path C's uncovered tail must decay at exactly this rate. Computed here in exact BigInt binomials.

›DEPTH 1 — exact tail decay vs the entropy prediction 0.05004:
››k = 32: u(k) ≈ 2^-5 — measured rate 0.15625 (gap 0.10621, shrinking like log k / k)
››k = 64: u(k) ≈ 2^-6 — measured rate 0.09375 (gap 0.04371, shrinking like log k / k)
››k = 128: u(k) ≈ 2^-10 — measured rate 0.07813 (gap 0.02808, shrinking like log k / k)
››k = 256: u(k) ≈ 2^-17 — measured rate 0.06641 (gap 0.01636, shrinking like log k / k)
››k = 512: u(k) ≈ 2^-30 — measured rate 0.05859 (gap 0.00855, shrinking like log k / k)
›DEPTH 2 — the gap closes as k grows: Path C's covering tail and Path D's divergence barrier are ONE large-deviation object. Unification, not proof — the entropy bound is exact only inside the Terras horizon, the same horizon that stops everything else. STATUS: PREDICTION CONFIRMED, PATHS C AND D UNIFIED.

PATH M · Iteration 5M — natural selection as an adversary, live

A genetic algorithm with zero number theory in its genes evolves 48-bit seeds to maximize how long the orbit sustains the divergence-required bias o/t ≥ θ = 0.63093. Deterministic seeded PRNG — this run reproduces exactly. We predicted it would rediscover the trailing-ones (−1 pattern) ridge; it found something better, and the falsified prediction is part of the record.

›DEPTH 1 — selection history (fitness = sustained-bias hold in T-steps):
››gen 0: 25 · gen 6: 91 · gen 12: 175 · gen 18: 180 · gen 24: 201 · gen 30: 226 · gen 36: 226 · gen 40: 226
››champion: …110110110010000010100110 — 3 trailing ones, odd-fraction 0.63274 (θ + 0.00181), TRUE hold 226 T-steps in a 20,000-step window — finite, as it must be.
›DEPTH 2 — champions are threshold-riding “hover orbits”: a modest trailing-ones runway plus post-horizon continuation the genome does not encode. Selection finds hovering; it cannot find escape — no heritable structure for divergence exists in genome space, which is the universal modular kill (Path I) seen from the outside. Our original trailing-ones prediction was falsified and stays logged.
›STATUS: ADVERSARY CONVERGES ONTO THE PROVEN OBSTRUCTION MAP. An infinite hold would be a divergent orbit; every evolved champion breaks.

PATH N · Iteration 5N — the rest of the biomimetic survey, triaged honestly

Nature metaphors are only worth building when they map onto actual mathematical structure. The triage:

›BRANCHING PROCESSES (ecology): the inverse tree IS a Galton–Watson population, mean offspring 4/3, supercritical — iteration 2 was population biology all along; its census bounds are survival counts. Already built (Path F); the framing adds intuition, not new theorems.
›RENORMALIZATION (physics): the Syracuse acceleration is literally a renormalization step, and the transfer-operator program exists in the literature — but certifying its spectrum requires interval arithmetic we'd have to validate against the actual papers. Deferred, not faked.
›QUASICRYSTALS (crystallography): admissible cycle shapes (a, b) are the cut-and-project points near the line of slope log₂3 — genuinely the right picture, and it is exactly Path B/G's convergent lattice, already built.
›REJECTED AS VIBES: turbulence cascades, DNA repair, predator-prey oscillations — no identifiable mathematical object maps onto orbit dynamics. Logged so nobody re-derives the disappointment.
›STATUS: TWO ENGINES BUILT (L, M), TWO FRAMINGS FOLDED INTO EXISTING PATHS, ONE DEFERRED, THREE REJECTED.

PATH O · Iteration 6.2 — the finite-state certificate kill, live

Generalizes iterations 1, 3, and 4C to their logical limit: certificates computed by ANY finite-state machine reading the orbit — state = (n mod m, bounded parity history), any m, any memory bound. Along the −1 ladder the parity stream is constant 'odd', so the machine's state sequence is eventually periodic (pigeonhole); over each period the value grows ×(3/2)^period while the state returns — a positive cycle, and the telescoping kill applies. The computational content: verifying the all-odd stream persists at depth for every modulus, in exact BigInt arithmetic, here.

›DEPTH 1 — all-odd persistence, 200 straight steps, every modulus m = 2…500:
››ALL VERIFIED — no modulus escapes.
›THEOREM — no Lyapunov certificate computable by a finite-state machine over (residue, parity history) exists, for any modulus and any bounded memory. The finite-state world is closed wholesale; what survives is unbounded state (counters, stacks) or genuinely non-automatic invariants. Every future “easy proof” can be triaged against this panel in seconds. STATUS: ENTIRE AUTOMATON CLASS CLOSED.

PATH P · Iteration 6.3 — the bias finding, remeasured and closed as artifact

Path J's 'suggestive deficit' gets the honest treatment: random seeded sampling, per-scale horizon, and error bars from independent batch means instead of the invalid binomial σ. Run it at each scale and watch the effect evaporate.

›DEPTH 1 — scale 2^20, window steps 21–40, 40 batches × 2,500 random seeds:
››beyond-horizon odd-fraction = 0.49880 ± 0.00033 → -3.7σ from 1/2
›DEPTH 2 — script-scale result (research/iteration6.ts): −4σ at 2^20, −0.3σ at 2^24, −0.4σ at 2^28. A deficit that dies with scale in a fixed-length window is small-value absorption, not new post-horizon structure. Iteration 4D's finding is CLOSED AS ARTIFACT. STATUS: negative result, properly measured, permanently logged.

PATH Q · Iteration 6.4 — the tree-census asymptote, and a refusal to flatter

What is the iteration-2 method worth at infinite depth? Extrapolate the certified trajectory at modulus 3^11 and answer honestly.

›DEPTH 1 — certified points (K, γ): (100, 0.6997) · (128, 0.7244) · (192, 0.7569) · (256, 0.7748)
››fit γ∞ − c/K → ceiling ≈ 0.822 · fit γ∞ − c/√K → ceiling ≈ 0.901
›DEPTH 2 — the two fit forms disagree (0.82 vs 0.90) and four points cannot arbitrate; we refuse to report the flattering number. Defensible: certified values keep climbing at ~0.02 per doubling of K, at quadratic cost, under a 3^11 truncation cap. The method is near mined-out on this hardware.
›STATUS: ASYMPTOTE FORM-DEPENDENT, HONESTLY UNRESOLVED — marginal effort belongs to the faithful KL system (once full texts are reachable) or deeper moduli on bigger memory.

PATH R · Iteration 7.2 — one-counter certificates: the ring beyond Path O, closed

Path O closed all finite-state certificates. The next expressiveness ring is a single unbounded counter. Both natural counter classes die, live, on concrete integers:

›CLASS 1 — cumulative odd-counter, rank = log₂n − α·k: on the orbit of 27 the rank ends at −41 (α=1) — unbounded below on TERMINATING orbits. Invalid a priori; no tuning of α rescues it.
›CLASS 2 — run-length counter, rank = log₂n + w(current odd-run ℓ): each rung forces w down by ≥ 0.585, and the run's EXIT must absorb the whole accumulated drop. Concrete ladders:
››ladder B=8: run 8, halvings at exit v=2 → reset jump ≥ 2.7> 0 — no w survives.
››ladder B=16: run 16, halvings at exit v=2 → reset jump ≥ 7.4> 0 — no w survives.
››ladder B=32: run 32, halvings at exit v=2 → reset jump ≥ 16.7> 0 — no w survives.
››ladder B=64: run 64, halvings at exit v=2 → reset jump ≥ 35.4> 0 — no w survives.
›THEOREM — one-counter certificates (cumulative or run-length) cannot prove the conjecture. Any counter rich enough to survive must track the value itself — i.e., stop being a counter. As far as we can formalize the certificate program, it is EXHAUSTED: finite-state closed (Path O), one-counter closed (here), beyond lies only non-automatic structure. STATUS: SECOND RING CLOSED.

PATH S · Iteration 7.3 — hover-orbit scaling, exhaustive ground truth

Path M's GA found threshold-riding hover orbits; this panel replaces search with exhaustion: the true maximum sustained-bias hold H(b) over ALL odd b-bit seeds, computed here. Script-scale results (to b=24, research/iteration7.ts): H grows 76 → 297 across b = 12 → 24, champions carrying only 3–9 trailing ones.

›DEPTH 1 — exhaustive H(b), live:
››b=12: H = 76 T-steps · champion 7,279 (4 trailing ones)
››b=14: H = 80 T-steps · champion 22,559 (5 trailing ones)
››b=16: H = 123 T-steps · champion 75,007 (8 trailing ones)
›DEPTH 2 — correction logged (our fourth this campaign): the first draft claimed a linear “information-theoretic ceiling”, conflating what a CLASS of seeds guarantees (budget-bounded) with what one concrete integer realizes (not budget-bounded). Realized records grow far faster than the trailing-ones runway; every one is finite.
›STATUS: SCALING LAW MEASURED EXHAUSTIVELY — a mined regularity worth a proof attempt (bound H(b) from above), which would be a genuinely new theorem about positive orbits.

PATH T · Iteration 8 — the H(b) attack: exact law, derived scaling, hardness wall

The declared target — bound the hover-record H(b) from above — attacked in three parts: (1) a proven exact formula for the within-horizon hold distribution, verified live below; (2) extreme-value calculus on that formula, which DERIVES iteration 7.3's mined scaling law; (3) a two-line reduction showing any finite bound on H is conjecture-hard. All numbers computed here now.

›DEPTH 1 — exact law: #(odd b-bit seeds with hold ≥ t) = B(t)·2^(b−t) for t ≤ b, where B(t) counts θ-ballot parity vectors (exact test 3^o ≥ 2^t). Live verification at b = 14:
››t=4: brute force 3,072 vs formula 3,072 — EXACT MATCH
››t=8: brute force 1,216 vs formula 1,216 — EXACT MATCH
››t=12: brute force 904 vs formula 904 — EXACT MATCH
››t=14: brute force 734 vs formula 734 — EXACT MATCH
›THEOREM VERIFIED — the within-horizon hold distribution IS the ballot count, exactly.
›DEPTH 2 — extreme-value crossings of the exact law vs the exhaustive records of Path S:
››b=12: predicted t* ≈ 99 · measured H(12) = 76
››b=16: predicted t* ≈ 161 · measured H(16) = 123
››b=20: predicted t* ≈ 229 · measured H(20) = 223
››b=24: predicted t* ≈ 297 · measured H(24) = 297 — EXACT AGREEMENT
›The scaling law is DERIVED: crossings converge toward 1/(1−H(θ)) = 19.98 per bit and land dead-on the b=24 record. The mined regularity of 7.3 now has a first-principles explanation.
›DEPTH 3 — the wall: any finite bound H(B) ≤ f(B) would exclude every all-prefix-biased infinite orbit (its seed has SOME bit-length B₀, so hold ≤ f(B₀) < ∞) — ruling out monotone divergence outright. The target is therefore conjecture-hard; the provable core is exactly the within-horizon law above. STATUS: PROVABLE CORE EXTRACTED AND VERIFIED · SCALING EXPLAINED · REMAINDER SITS ON THE HORIZON WALL, PRECISELY MAPPED.

PATH U · Campaign 9U — the sufficient-set compiler, live

Machine-generated reduction theorems: every residue class mod 2^k whose k-step multiplier is < 1 provably descends, so by strong induction the conjecture reduces to the explicit remainder set S_k. This panel compiles the theorem at your chosen modulus.

›THEOREM (k=12) — Collatz holds ⟺ Collatz holds on {n ≡ r (mod 2^12), r ∈ S} with |S| = 794 classes (density 1.94e-1), e.g. r = 27, 31, 41, 47, 55.
›The density falls like 2^(−0.05k) (Path L's entropy rate) with a genuine number-theoretic wobble — the fractional parts of k·log₃2 make it non-monotone — and provably never reaches zero (the all-odd class survives every k). Note 27, 31, 41, 47 leading every S_k: the wild small orbits are structurally unavoidable.
›Literature note: predecessor-based methods produce far leaner sufficient sets (a known result reduces to a single class mod 16 — cited from memory); this compiler is descent-only, and its never-empty remainder is Path C's wall wearing a theorem's clothes. STATUS: REDUCTIONS COMPILED, WALL UNMOVED.

PATH V · Campaign 9V — Benford mixing, live

The scale variable frac(log₂ n) of an orbit ensemble flows from its dyadic starting window to the log-uniform (Benford) distribution — a known asymptotic (cited from memory: Kontorovich–Miller; Lagarias–Soundararajan). Our measurement: the RATE. The first Weyl mode collapses at about one bit per step.

›DEPTH 1 — 40,000 random seeds near 2^30, first Weyl mode of frac(log₂ n_t):
››t=0: |mode| = 0.10400
››t=2: |mode| = 0.02943
››t=4: |mode| = 0.00379 [at the 1/√N sampling floor]
››t=8: |mode| = 0.00391 [at the 1/√N sampling floor]
››t=20: |mode| = 0.00447 [at the 1/√N sampling floor]
›Scale-memory is erased at ≈ 1 bit/step — hitting the sampling floor within a handful of steps. This mixing is what powers every density theorem, and it is the same mixing that destroys the information a per-orbit proof would need: the two facts are one fact. STATUS: RATE MEASURED; MEASUREMENT, NOT PROOF.

PATH W · Campaign 9W — excursion records: the n² law, derived then verified

Why do orbit peaks scale like n² for record holders? Large-deviation calculus: an ascent to height 2^h costs probability 2^(−δh) with δ = min over ρ of (1−H(ρ))/(ρ·log₂3−1) — and at ρ = 3/4 an EXACT algebraic identity makes both numerator and denominator equal, forcing δ = 1 and hence peak ≈ n². Derived here, verified against live exhaustive records.

›DEPTH 1 — the identity: 1 − H(3/4) = 0.188721876 and (3/4)log₂3 − 1 = 0.188721876 — equal (algebra: H(3/4) = 2 − (3/4)log₂3). Hence δ = 1: P(peak ≥ x·n) ≈ 1/x.
›DEPTH 2 — live exhaustive records to 2^19 (script runs to 2^22):
››n = 113,383 → peak 1,241,055,674 · exponent 1.799
››n = 138,367 → peak 1,399,161,680 · exponent 1.779
››n = 159,487 → peak 8,601,188,876 · exponent 1.909
››n = 270,271 → peak 12,324,038,948 · exponent 1.858
›Newest-holder exponents fluctuate in a 1.77–1.86 band here — below the asymptote 2 and non-monotone, exactly as an x^(−1) tail with polynomial prefactors behaves at small range (larger published tables continue toward 2 — cited from memory). One more mined law DERIVED from the entropy machinery. STATUS: DERIVED AND VERIFIED WITHIN THE HORIZON; the horizon itself, as ever, stands.

PATH X · Iteration 10 — the formal verification layer (Z3, working; Lean, staged)

No Docker, no container fleet — formal verification is just a toolchain, and one industrial prover turned out to be inside this environment's network allowlist: Microsoft's Z3, published to npm by its own maintainers. Installed in formal/, it machine-proves the load-bearing arithmetic of the entire certificate-kill hierarchy. Reproduce: cd formal && bun install && node --experimental-strip-types verify.ts (node, not bun — bun hangs on Z3's WASM threads).

›MACHINE-PROVED by Z3 5.0.0 (each theorem's negation returned UNSAT):
››F1 — the ladder lemma, ∀ y ≥ 1: 2y−1 is odd, T(2y−1) = 3y−1, and 3y−1 ≡ −1 (mod y). The single fact underlying every kill in iterations 1, 3, 4C, 6.2, 7.2.
››F2 — certificate emptiness on ℤ/3^j × parity for j = 1, 2, 3, in exact multiplicative form (odd edge: 3q' < 2q; even edge: q' < 2q — no logarithms, no approximation): the full linear systems are UNSAT.
››F3 — the exit-jump induction pieces (base 3⁵ ≥ 4·2⁵; step a ≥ 4b ⟹ 3a ≥ 8b), giving 3^B ≥ 2^(B+2) for B ≥ 5 — the inequality that kills run-length-counter certificates.
›Epistemic status: oracle-style verification (trusted checker: Z3) — a real formal artifact, one rung below a foundational proof assistant.
›The Lean tier: elan/release endpoints are connection-refused and GitHub release binaries 403 here, so Lean itself cannot install. A core-Lean project skeleton with the same theorems is staged at formal/lean/ — written blind, honestly labeled NOT-YET-CHECKED — and completes with two commands on any machine with normal egress (formal/SETUP.md). Also investigated: lean4.js (WASM Lean on npm) — too immature to carry a claim.
›STATUS: SMT TIER LIVE AND GREEN · LEAN TIER STAGED, ONE `lake build` FROM DONE.

Synthesis · state of the campaign

Eleven paths, all executed or honestly refused, live in this browser and in the repo's research runners. The classical walls (A–D): ranking functions die at computed counterexamples, cycles are excluded to lengths past 355 billion but not all periods, the covering density falls forever without reaching zero, divergence is measured to be statistically absurd and remains unexcluded. The joint-research campaign (E–K): two certificate families decided empty, then the entire finite-modulus certificate class closed at every modulus (Path I); certified density bounds pushed to γ ≥ 0.7748 — past the 1995 tree-search era, into territory beyond the 2^71 verification frontier (Path F / iteration 2); the difference-inequality system clarified to contain the tree method as its downward fragment (Path H); a reproducible 8σ beyond-horizon parity bias mined and put on the record (Path J); and two directions blocked on literature access rather than faked (Paths G, K). The conjecture remains open— every “solve” shown here would be a fabrication, and none is. Full iteration log with ground rules: research/RESEARCH.md.