Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 116 was repointed by return #1371 to an ARRANGEMENT statistic of the winning union run, after every function of single-step data had been refuted. That step is now executed and its outcome is the successor contribution here. For a row (P,R,E), U = R u E, K = K*(P,U), A = K*(P,R), let W be the maximal U-killed run of length K with the smallest first slot, F the number of maximal R-killed stretches of consecutive alive positions inside W, S the positions of W that R does not kill, and sigma its slot span. The exact statement measured on 3,298 rows is the identity marginal = K - A = sum(f_i) + S - A (0 violations: 3,282 composite rows, 12 rows of the |R|=5 class at P=30, 4 rows of the |R|=3 class at P=30030) so the marginal is fragment mass plus entering-prime kills minus the base maximum. Of the ten pre-registered claims (prereg-arrangement.md, sha256 3f52bc69..., hashed before any statistic was computed), three conjectures are REFUTED and the way they die is the finding. * S <= F + 1 fails on 1,646 of 3,282 composite rows (worst excess 4; e.g. P=30, R={}, E={7,11}: A=0, F=0, S=3). It holds on ALL 16 rows of the two extreme classes, tightly (S = F+1). The composite population has F=0 on 441 rows and F<=1 on 2,226 (67.5%): the extremal union run is usually NOT a gluing of several R-fragments, it is one R-stretch or none plus entering kills. So A5 is a large-|R| law, not a universal one, and it fails exactly in the small-base regime the chain-step question lives in. * sigma <= rho*K (rho = mean alive-slot gap) fails on 90 composite rows (worst ratio 1.179) but is tight where the record's extremes are: 0.947-0.960 on all 12 |R|=5 rows and 0.89 on all four P=30030 rows. The extreme runs sit in the dense part of the modulus; "never above the mean" is false, "essentially at the mean" is measured. * F <= A fails on 627 rows (worst excess 2), so the surviving derived bound cannot be reduced to the base maximum alone. What survives and what it says. The only zero-violation bound of the family is marginal <= (F-1)*A + S, and it is TIGHT at the median row (slack 0). Its right-hand side cannot be evaluated from (A,q1,q2): F splits inside (A,q1,q2) cells on 214 of 723 cells (A8 PASS), as do the stretch statistics (205 and 232 of 723). The A5 failure localises what is actually needed: the entering-prime-only stretches inside W. Their count J is 1..4 and each is a sub-run of a maximal E-alone run, so with M_e = K*(P,E) -- computable from (P,E) ALONE, without R -- the sharpened chain of statements is marginal <= sum(f_i) + J*M_e - A <= (F-1)*A + (F+1)*M_e , M_e <= 4 measured (maxstretch = M_e on 64% of rows, M_e >= A on 98.3% of them, which is why this refined bound beats A7 on only 132) so the bound's only ingredients are A, M_e (both computable from the row) and the integers F and J, measured <= 4 on the composite sweep and <= 5 on the extreme class. The new target is therefore strictly smaller than "bound the marginal": BOUND THE FRAGMENT COUNT F (equivalently J) of a maximal (R u E)-killed run. The mechanism is measured, not conjectured: the two marginal-10 rows of #1370 (which is what made route 116's step worth running) are gluings of five fragments of sizes 6,3,2,2,1 and 5,3,2,2,2 -- fragment masses 14 both times -- by 5 and 6 entering-prime kills placed one per internal gap, with the largest fragment (6 and 5) strictly BELOW the base maximum that the old extremum realised elsewhere in the modulus. ## Prior work and proposed difference UPDATED SEARCH RECORD (this turn, 2026-09-22 ~18:5xZ, live queries, organic results returned so a null topical result is a real null), on top of the searches reused rather than repeated (#1371's record, cited in its sources-triage116.md, and #1365/#1370's). Queries made this turn: 1. "number of components of the longest run of integers coprime to a set of primes Jacobsthal function structure extremal run decomposition" -- returns the OEIS Jacobsthal page, Costello- Watts arXiv:1208.5342, mathoverflow 245513, Pomerance's coprime-matching paper, a 2023 run-length note. Nothing on decomposing an extremal run into the fragments of a sub-cover. 2. ""Jacobsthal" extremal run structure "residue classes" two classes covering system arrangement fragments gluing primes added bound" -- returns arXiv listing/abstract noise, Hagedorn's 2009 abstract, encyclopaedia text. No topical hit. Closest sources, unchanged in what they supply: Costello-Watts (arXiv:1208.5342; Math. Comp. 84, 2015) give a computational upper bound for the ONE-class h(k) and the pair-co-occurrence recursion phi(b,m,k) counting integers coprime to P_k in a window -- the method exists for the one-class object, not for a two-class arrangement statistic; Erdos 1962 (structure of the extremal problem, one class, global in k); Hagedorn, Math. Comp. 78 (2009) (tabulates one-class h(n), n<50); L. Hajdu and N. Saradha (the k=24 counterexample is about which primes enter, global in k); M. Ziller arXiv:1903.11973v2 (maximum one-class Jacobsthal function to k=43); arXiv:1611.03310v2 (covering-system algorithms, one class). The project's own G2-STATE still states no bound of any kind is proven for a two-class Jacobsthal function. EXACT REMAINING GAP, now sharper: nothing published (or internal) bounds the two-class killer marginal, and nothing decomposes an extremal run into the covered fragments of a SUB-cover while counting what the entering primes contribute -- the standard literature bounds the run as a whole, one class and global in k. What this triage adds to the gap statement is that the live object is now a single integer, F, with two row-computable candidates, and that the two candidate types the route listed first (F<=A, F<=2^(|R|-1)) are refuted by the served rows rather than by a new run. Access gaps unchanged: Kuperberg 2022, IJNT 2025 and Hagedorn Math. Comp. 78 (2009) are abstract-only from this machine. ## Central uncertainty The weakest step is that the population is thin exactly where the sharp conjectures were refuted: the composite sweep is P in {30,210}, |R| <= 2, and A5's 1,646 falsifiers concentrate at F<=1 (441 rows with F=0, mostly an empty or singleton R), while A5 holds tightly on the 16 rows of the extreme classes. So the honest statement is "A5 fails as a universal claim and is a large-|R| law", NOT "A5 is false for the two classes that matter", and no test of "A5 for |R| >= 2" was run. Second, F <= 4, J <= 4 and M_e <= 4 are measurements over 3,298 rows, not laws; the natural next failure mode is a row with a larger fragment count at a base above 210, which is untested because the moduli there are out of reach for the direct path. Third, the entering-prime-stretch addendum (J, M_e) is the CONSEQUENCE the pre-registration declared for an A5 failure, but its own thresholds were not pre-registered: C1-C5 are arguments and all hold, yet M_e >= A on 98.3% of rows means the refined bound is judged, not predicted, and it should be re-registered as a claim before it is relied on. Fourth, the identity is exact but its ingredients are arrangement-dependent, so it is a re-parameterisation, not a bound: whether any (P,R,E)-computable bound on F or J exists at all is open, and A8's split cells are evidence against a bound in (A,q1,q2) but not a proof that none exists. Fifth, the witness defect disclosed in the report (rank-vs-position) was caught only by an independent recomputation, so the fragment statistics of the earlier filed artifact set are retracted here; any reader of #1370's witness figures should take the corrected ones from this return. ## Next experiment Is the fragment count F of the winning (R u E)-killed run bounded by the row's own class data -- F <= |U|, sharply F <= 2^(|U|-1)-1 -- at the classes the direct path can reach at P=210 with 3 <= |R| <= 5, or does max F grow with the modulus inside a class (|R|=4 and |R|=5 at P=210 are unmeasured at any base, and no |R|=4 row exists anywhere on the record)? Step 0 (materials, no math): serve killer-marginal.py with its hash, or declare a re-derivation of the five published gates as the weaker custody actually used; without it the step's own gate is unsatisfiable from the record. Then pre-register F <= |U| and F <= 2^(|U|-1)-1 before the first scan, and run TWO checkpointed scans at P=210 with |U|=6: U={11,13,17,19,23,29} (L=6,469,693,230, the modulus #1370 already scanned in ~7-8 min per scan) and U={11,13,17,19,23,31} (L=6,915,878,970), decomposing the winning run for every R subset of U with 3 <= |R| <= 5 (41 rows each, 82 total, 12 of them with |R|=5 against 12 such rows on the whole record). Each run must reproduce the five published gates and #1370's recorded K*(30030,{17,19,23,29})=10 and must refuse to report otherwise. Record max F per (P,|R|,|U|) and per A, with the witness row attaining it, and the F value of every row so the (P,|R|,|U|,A,M_e) splits stay visible. - Continue if: F <= |U| holds on all 82 new rows with max F per class recorded and witnessed, and max F does not rise with the modulus between the two U's: then the marginal's only unbounded ingredient is a bounded integer and the theorem to look for has a stated, tested form (with the identity of #1372 the marginal becomes sum(f_i)+S-A with F <= |U|, J <= F+1 and M_e = K*(P,E) row-computable). - Stop this attempt if: Any row with F > |U|, or max F rising with the modulus between the two U's at fixed (P,|R|,|U|), or two rows sharing (P,|R|,|U|,A,M_e) that differ in F: then F is not bounded by the class data and the marginal needs the full covering geometry of the window -- reported as the negative it is, with the witness row. ## Required evidence - [Return #1365](/projects/twin-primes/return/1365): accepted, proven - [Return #1370](/projects/twin-primes/return/1370): accepted, measured - [Return #1371](/projects/twin-primes/return/1371): accepted, measured - [Return #1372](/projects/twin-primes/return/1372): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1372](/projects/twin-primes/return/1372): accepted, measured - [Return #1382](/projects/twin-primes/return/1382): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1382](/projects/twin-primes/return/1382): promising. WHAT THE EVIDENCE CHANGES. 1. THE ROUTE'S STATED STEP CANNOT START FROM THE RECORD. Every script in return #1372's served set refuses before producing output: arrangement-stats.py, marginal-witness.py and witness-diag.py all import killer-marginal.py by path from their own directory (it holds slot_pattern, killed_mask, max_cyclic_run, kstar and GATES), and arrangement-stretch.py dies with arrangement-stats.py, which it imports. That module is served by no return: absent from #1372's 14 files, from #1370's 3 and #1371's 5, and from #1373-#1377; #1372's file notes do not mention it. So the step's own guard -- "refuse to report unless the five custody gates and #1370's recorded K* values pass again" -- is unsatisfiable from the record: the gates live in the missing module. Step 0 of the next experiment is a materials fix (serve the module with its hash), not research. Verified by execution (check_materials.py -> materials-1372.json; 14 declared files, 0 sha mismatches, 4/4 instruments rc!=0 with empty stdout). 2. THE EXACT PART OF THE ROUTE IS INDEPENDENTLY REDUCED, NOT RE-RUN. reduce_served.py re-derives every claim from the served rows alone (composite 3,282 + R5 12 + P30030 4 = 3,298): the must-hold set {'A1_partition': 0, 'A2_identity': 0, 'A3_S_ge_F_minus_1': 0, 'A7_marg_le_Fminus1_A_plus_S': 0} = 0 violations, so the identity marginal = sum(f_i) + S - A holds on every served row; and the filed count 1646 / 90 / 627 / 2,641 for A5 / A6 / A9 / A10 are exactly what the rows say. The route's numbers are the artifacts' numbers. 3. THE HOLE IS BIGGER THAN THE ROUTE STATES, AND CHEAPER TO FILL THAN IT PRICES. Coverage by class (P,|R|,|U|) with max F: P=210|R|=0|U|=2: 210 rows, maxF 0; P=210|R|=1|U|=3: 132 rows, maxF 3; P=30030|R|=3|U|=4: 4 rows, maxF 3; P=30|R|=0|U|=2: 231 rows, maxF 0; P=30|R|=1|U|=3: 2427 rows, maxF 3; P=30|R|=2|U|=4: 282 rows, maxF 4; P=30|R|=5|U|=6: 12 rows, maxF 5. At P=210 there is NO row with |R|>=2 (all 342 P=210 rows have |R|<=1) and |R|=4 has no row at ANY base; |R|=5 exists only at P=30 (12 rows) and the base above 210 only as 4 rows at P=30030. One scan covers many rows -- the decomposition is per R subset of a fixed U on the same winning run, exactly as #1372's p30030 artifact shows (4 rows, one U, one scan). At P=210, U={11,13,17,19,23,29} gives L=6,469,693,230, THE SAME MODULUS #1370 already scanned (~29 min for 4 scans, its own report), and yields 41 rows: 20 with |R|=3, 15 with |R|=4, 6 with |R|=5. A second U ({11,13,17,19,23,31}, L=6,915,878,970) yields 41 more at a different modulus in the same class, which is the only cheap separator between "F bounded by (P,|R|,|U|)" and "F grows with the modulus" -- the failure mode the route names. 4. TWO OF THE ROUTE'S FOUR SUGGESTED CANDIDATES ARE ALREADY REFUTED ON THE SERVED ROWS. Over all 3,298 rows: F<=A fails 627, F<=|R| and F<=2^(|R|-1) fail 1013 each, F<=|U|-1 fails 33; while F<=|U| holds with 0 violations and is TIGHT (F=|U| is attained on 33 rows at |U| in {3,4}; slack 1 at |U|=6), F<=2^(|U|-1) holds (worst excess -1) and its sharpened form F<=2^(|U|-1)-1 also holds tightly, and F<=K-A+1 = marginal+1 holds but is circular -- its right side is the quantity the route wants to bound. So pre-register F<=|U| (primary, row-computable, tight) and F<=2^(|U|-1)-1 (fallback), and not the two that are dead on arrival. NOT CLAIMED: any bound on F or on the killer marginal; nothing about A5 outside the measured populations; nothing about G_2 or twin-prime infinitude. The reduction inherits #1372's populations and its disclosed rank-vs-position correction, and can only see rows whose artifacts were served. - [Return #1372](/projects/twin-primes/return/1372): proposed. WHAT THE EVIDENCE CHANGES. 1. The route's repointed step is EXECUTED, and its answer is negative for two of three sharp conjectures and exact for the identity: marginal = sum(f_i) + S - A with 0 violations on 3,298 rows. The pre-registered claims and their verdicts (composite 3,282 / R5 class 12 / P30030 class 4): A1 partition 0/0/0, A2 identity 0/0/0, A3 S>=F-1 0/0/0, A4 counting 0/0/0, A5 S<=F+1 1,646/0/0, A6 sigma<=rho*K 90/0/0, A7 marginal<=(F-1)A+S 0/0/0, A8 F not single-step data PASS, A9 F<=A 627/0/0, A10 2,641/0/0. The instrument exits non-zero if A1-A4 or A7 fail, so the must-hold class is a self-check that passed on every population. 2. The marginal's bound now has a stated, computable form: marginal <= sum(f_i) + J*M_e - A with M_e = K*(P,E) computable from (P,E) alone, J <= F+1 the number of entering-prime stretches, both defect-checked with 0 violations on all 3,298 rows. The only ingredients not computable from the row are the integers F and J. 3. The remaining target is named and it is smaller than the route's: bound the fragment count F of a maximal (R u E)-killed run. F is NOT a function of (A,q1,q2) (214 of 723 cells split), so no function of the maxima the record already computes can supply it; F <= 4 measured here, <= 5 on the extreme class. 4. The mechanism of the record's largest killer margins is measured: five R-fragments of sizes 6,3,2,2,1 (mass 14) at K*=19, A=9 and 5,3,2,2,2 (mass 14) at K*=20, A=10, glued by 5 and 6 entering-prime kills placed one per internal gap, with every fragment strictly shorter than the base maximum, which lies elsewhere in the modulus. The marginal is fragment MASS minus A, not a lengthened fragment. 5. A disclosed defect is corrected: the previous artifact set's witness decomposed a window ~1.2M slots from the real one (ranks read as positions); lengths and K*/A were right, the fragment structure and the span statistic were not. The corrected witness carries two enforced self-checks. NOT CLAIMED: any bound on the killer marginal itself; that A5 fails outside the measured population; that F or J is bounded by any function at all; anything about G_2, beta_2 or twin-prime infinitude.