collatz – drilling even deeper with gpt 5.5

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I treated the seven stages this way:

Gateway = first step where the trajectory falls below its starting input.
That is the cleanest “loss of sovereignty” point, because from there the number has entered the known path of a smaller number.

Terminal descent = the final power-of-2 chain to 1, usually 16 → 8 → 4 → 2 → 1.

For inputs 2 through 10,000, all 9,999 trajectories reached 1.

Profile statistics: all inputs 2–10,000

Stage / Metric	Median	Mean	90th %ile	99th %ile	Maximum
Private ascent steps	0	2.50	7	32	94, input 7527
Private turbulence reversals	0	2.95	9	39	101, input 703
Private peak value	8,080	31,691	40,566	319,397	27,114,424, input 9663
Private peak ÷ start	1.00×	6.13×	6.76×	64.88×	2806×, input 9663
Post-peak wandering steps	1	2.68	5	23	76, input 7279
Gateway step	1	5.18	13	52	132, input 703
Inherited river steps	63	75.49	140	177	252, input 9257
Terminal descent steps	4	4.31	4	10	14, input 5461
Total trajectory steps	73	84.98	150	189	261, input 6171

Odd inputs only: the nontrivial private biographies

The all-input table is distorted by even numbers, because every even number immediately drops below itself in one step. So the odd-only profile is more revealing.

Stage / Metric	Median	Mean	90th %ile	99th %ile	Maximum
Private ascent steps	3	5.01	12	43	94, input 7527
Private turbulence reversals	3	5.90	15	53	101, input 703
Private peak value	20,764	58,386	70,468	599,056	27,114,424, input 9663
Private peak ÷ start	4.50×	11.27×	15.19×	116.96×	2806×, input 9663
Post-peak wandering steps	3	4.36	8	32	76, input 7279
Gateway step	6	9.37	21	70	132, input 703
Inherited river steps	66	77.69	143	178	252, input 9257
Terminal descent steps	4	4.31	4	10	14, input 5461
Total trajectory steps	81	91.36	158	197	261, input 6171

The big hidden pattern

The private part is usually short. Across all inputs, the median gateway step is only 1. Even among odd inputs, the median gateway step is only 6.

So most numbers do not remain sovereign very long.

The typical trajectory is:

short private biography → long inherited river → tiny terminal descent

The average all-input decomposition is roughly:

5 private steps + 75 inherited river steps + 4 terminal descent steps

That means most of the Collatz journey is not “new story.” It is inherited structure.

Key extreme cases

Role	Input	What it does
Longest private ascent	7527	Climbs privately for 94 steps before reaching its private peak
Most turbulent private path	703	Has 101 direction reversals before gateway
Highest private peak	9663	Reaches 27,114,424, about 2806× its start
Longest post-peak wandering	7279	Wanders 76 steps after peak before gateway
Latest gateway	703	Does not fall below itself until step 132
Longest inherited river	9257	Spends 252 steps after gateway before terminal descent
Longest total trajectory	6171	Takes 261 total steps

Where 27 sits

For input 27:

Stage	Value
Total steps	111
Private ascent steps	77
Private peak	9,232
Peak multiplier	341.93×
Private turbulence reversals	73
Gateway step	96
Gateway value	23
Post-peak wandering	19 steps
Inherited river	11 steps
Terminal descent	4 steps: 16 → 8 → 4 → 2 → 1

So 27 is genuinely dramatic, but it is not the champion in this range. It is a clean showcase number because its story is legible: long climb, wild turbulence, clear peak, then entry into the inherited river.

The strongest new insight

More than half of inputs reach their global maximum after they have already crossed below their starting value.

That means their tallest mountain is not privately earned. It is inherited from the river of a smaller number.

So the Collatz landscape is not simply:

every number climbs, peaks, and falls.

It is more like:

many numbers surrender early, then inherit someone else’s mountain.