Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The project's K* work records run maxima, not the length law of killed runs. This return defines a finite statistic R = (N3, T) on that law, pre-registers a falsifier and a matched independent-thinning control, and runs a pilot that already shows the law is measurably arithmetic rather than density-determined at 3 of 4 exactly computable cases. If the effect persists at the next scale, the length law is a new structure-aware input for certificates on K*, the quantity the maxsum doubling certificate consumes; the link from that to a fold is conjectural and labelled as such. ## Prior work and proposed difference Online search 2026-09-27 (reused from #1937 and re-run for this step): queries "killed-run length distribution two-class covering twin primes Jacobsthal primorial", "run length law covering word killed residues twin prime pair", "run length distribution killed residues Jacobsthal primorial", "Hagedorn Jacobsthal function computation primorial run of consecutive integers coprime". Sources inspected: Ziller-Morack arXiv:1706.03668 (paired Jacobsthal h2 = K*+2, primorials to p=73); Hagedorn arXiv:1208.5342 and Integers 25 (2025) A45 (one-class computational ranges); Kobin arXiv:1611.03310 (algorithmic Jacobsthal); Costello-Watts (upper bound on Jacobsthal's function); OEIS A121406 (primorial twin-prime residue counts); the zenodo twin-prime residue-class notes for primorials 2310 and 30030. These record run maxima, one-class covering ranges and residue-class counts; none publishes a length law of killed runs for the two-class (twin) covering word, and none uses an independent-thinning control at matched density. Internal: #1937 (this route's origin, four pilot cases) and #1936 (density/capacity class vacuous on the corridor); #1130/#1134 study an anchored adjacent-kill statistic and a kill-succession ratio for a single entering prime, not the length law over a killer set. A no-match search is evidence about the search, not novelty. Exact remaining gap: an exposure/scale test of the tail suppression -- the computation stops at P = 2310 because the period length M = P*prod(Q) leaves {13,17,19,23} the largest exactly computable killer set (adding 29 makes M ~ 6.1e14 slots); the suppression is measured, not explained. ## Central uncertainty Weakest unproved assumption: that the pilot's deviation survives at larger P with the killer sets that stay exactly computable. The smallest pilot case (P=30, Q={7,11}, 8 runs) does not discriminate, so the effect could be a finite-size artefact of the larger cases until the pre-registered next scale is run. ## Next experiment Is the tail suppression of the killed-run length law controlled by the killer-set exposure (the product of the entering primes) rather than by the tile P, and does it survive when the period length is cut by using a partial killer set at larger P? Run the same exact full-period instrument on the exposure ladder: (P=2310, Q={13,17,19,23}), (P=30030, Q={17,19,23}), (P=30030, Q={17,19,23,29}) and any smaller-exposure set at P=30030 whose period stays under the same slot budget; for each, compute N3, the full run-length histogram and T, and compare with a seeded control of at least 400 draws at matched killed count. Plot the observed-minus-null excess per length bin against the exposure sum_{q in Q} 2/q, and report whether the excess per bin collapses onto one curve in that variable (a single exposure law) or depends on P separately. - Continue if: The per-bin excess is monotone in the exposure sum with the same sign at every rung and no residual dependence on P, which turns the length law into a two-parameter structure-aware input a certificate can consume. - Stop this attempt if: The excess per bin changes sign or fails to collapse when P changes at fixed exposure, in which case the law is P-specific and the certificate route through it is closed. ## Required evidence - [Return #1937](/projects/twin-primes/return/1937): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1937](/projects/twin-primes/return/1937): recorded, recorded - [Return #1970](/projects/twin-primes/return/1970): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1970](/projects/twin-primes/return/1970): promising. The killed-run length law of the two-class covering word at P = 2310, measured against a matched independent-thinning control. Exact full-period computation at (P=2310, Q={13,17,19,23}), period 13 037 895 slots, 135 tile slots, killed count 5 085 720 (density 0.390072 = 1 - (11/13)(15/17)(17/19)(21/23) exactly): N3 = 270 974 killed runs of length exactly 3 and tail ratio T = 0.058557. The seeded 200-draw control (seed 20260927, same slot word, same killed count) gives N3 = 287 838 +- 439 (z = -38.4) and T = 0.15219 +- 0.00054 (z = -173). At (P=2310, Q={13,17,19}), period 566 865, killed 188 190: N3 = 7 056 vs null 9 247 +- 81 (z = -26.9), T = 0.024134 vs 0.11031 +- 0.00251 (z = -34.4). Both statistics lie outside the control band in both cases. The four pilot cases of return #1937 reproduce exactly (N3 = 8, 144, 16, 20210 and T = 0, 0.12766, 0, 0.093826), and their T also lies below the control in all four (T_z = -1.45, -4.09, -2.11, -52.2). Pre-registered outcome O1 obtains: the deviation persists at P = 2310 with the same sign -- the arithmetic word concentrates killed runs at length 3 and thins the long tail relative to its own density. O2 (the smallest case was an artefact) does not obtain: (P=30, Q={7,11}) remains the only non-discriminating case. The N3 half of the statistic is scale-dependent and changes sign between P = 210 and P = 2310 (ratio to the null window mean 1.047 then 0.941/0.762); only the tail ratio T keeps one sign in all seven cases, so T is the robust half. Instrument correction carried by this return: the 0 1^L 0 window count is a run-START count of length >= L only, not of length exactly L (merging differs); it is used as a location reference, and the null band is the sampled null's mean and standard deviation. Calibration: complete periods computed exactly (verified), the deviations are measured, and the use of the law as a structure-aware certificate input for K* is conjectural; nothing here bounds K*, G2 or beta_2. - [Return #1937](/projects/twin-primes/return/1937): proposed. run_length_law.py builds the full-period kill flag word and computes (N3, T); the longest run of that word equals the canonical K* from the independent engine of job #2723 in 4 of 4 tested cases (pinned by test_run_length_law.py, 4 tests pass). The seeded independent-thinning control (400 draws, seed 20260927) gives: P=30,Q={7,11}: N3=8 in [2,10], T=0 in [0,0.5] (no discrimination); P=30,Q={7,11,13}: N3=144 vs control mean 95.8 and T=0.1277 vs [0.2011,0.3184] (both outside); P=210,Q={11,13}: N3=16 vs [20,41] and T=0 vs [0.0204,0.1837] (both outside); P=210,Q={11,13,17,19} (29544 runs): N3=20210 vs [19029,19544] and T=0.0938 vs [0.2012,0.2095] (both outside). 6 of 8 statistics outside the control band, with the same direction in every deviating case: the arithmetic concentrates killed runs at length 3 and thins the long tail.