Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Record the exact form of the lower leg, which is what route 98's own uncertainty asks for when a counterexample appears, and hand the pursuit a question it can actually settle. The measured picture over 15747 exact rows: in the easy regime p > 2K*(P,R) the leg B >= A holds (15061 rows, minimum rise 0), so route 98(ii)'s conditional theorem is confirmed and its hypothesis is tight; in the hard regime p <= 2K*(P,R) every drop occurs (6 rows), the sharpest being the record's own #1267 pair with B = A - 2. Success in this direction is an exact statement of the deficit: either a bound B >= A - d with d explicit over the hard regime, or the exact finite list of configurations where d = 2, together with the demonstration that outside that list d = 1. Either form is consumable by the boundary sandwich, because the sandwich's upper side (route 98's C2, the -1 bound) is untouched by the lower side and is separately equivalent to the strict statement that the tile dial's rise equals the killer marginal minus one -- 0 counterexamples in the same 15747 rows (#1365). ## Prior work and proposed difference The route's own search record (same session, 2026-09-21, on the killer marginal and the base extension h(k) vs h(k-1)) is reused rather than repeated, per the research protocol. Refreshed today, 2026-09-22, with two live queries whose organic results make a topical null a real null: (1) "Jacobsthal function monotonicity adding a prime to the modulus j(np) versus j(n) covering runs of integers coprime" -- MathOverflow 70307; arXiv:1611.03310v2 (algorithms for j(n), covering-system formulation); P. Erdos 1962, "On the integers relatively prime to n and on a conjecture of Jacobsthal"; the OEIS wiki entry on the Jacobsthal function; Costello-Watts's explicit bound h(k) < 2^k * ... . All of it is ONE class and global in k: the one-class analogue of the rise is trivially non-negative (m | n implies j(m) <= j(n), immediate from the definitions), so no one-class monotonicity statement transfers to a drop on the killed-SLOT two-class object. (2) "covering system maximum gap two residue classes Jacobsthal function killed slots maximal run" -- the only topical hit is the project's own recon-0828-covering.md, which prices Crittenden-Vanden Eynden's Lemma 2 at k_i = 2 (its hypotheses are exactly two residue classes per prime modulus): G2(79#) <= 1.021e12 against the true 1710, 8181x weaker than this corpus's own 4.2665 sieve bound at the same level, and it records that no covering-systems result bounds a finite interval coverable by two classes per prime at polynomial scale, and that the free-shape two-class relaxation is a published ladder (A288815 / A144311 family). EXACT REMAINING GAP, unchanged and now sharper: no published statement, and none in the project's own G2-STATE, bounds K*(P, R u {q}) - K*(P,R) or the rise B - A; the closest published inequality is a covering threshold at the wrong scale, and it bounds no deficit. Access gaps unchanged: Kuperberg 2022, IJNT 2025 and Hagedorn, Math. Comp. 78 (2009) are abstract-only from this machine. ## Central uncertainty The weakest unproved step is whether the deficit of 2 is isolated. It is measured at exactly one row, whose A is the maximal run length the record has at P=30, |R|=4, and the other five drops sit at A in {9,12} in the same hard regime; but the sweep's other |R|=4 rows number only 100 and its |R|=5 rows only 12, so "isolated" is a statement about a thin region, not a law. Second, no mechanism is offered for the deficit at all: route 98's own proof sketch (the single forked slot r = 0 mod p) predicts loss at most 1 and does not explain a loss of 2, so the repair needs either a second forked-slot mechanism or an explicit exception. Third, the deficit could be a small-P artefact: all six drops are at P=30, and whether P=210 or 2310 has drops at larger |R| is untested because the row files' windows reach only |R|=4 at those bases. ## Next experiment In the cells that still contain hard rows -- P=210 with |R| in {4,5} -- is B >= A-1 on every row except the single #1267 cell, or does B = A-2 (or worse) occur elsewhere in the hard regime, and does any row attain B <= A-3? Streaming K* (job2753/stream-run.py: gated on the five published custody values and cross-checked against the swept array instrument) over the HARD rows only (p <= 2K*(P,R), computed per row from A) of the cells that still have them: P=210, |R|=4 for every R-set with M <= 3e8 and every hard p; P=210, |R|=5 likewise; and the P=2310 / P=30030, |R|=3 cells, which currently contain ZERO hard rows and are to be recorded as empty rather than swept. The reach-cells/hard-window instruments fix the cell list first; the sweep refuses to report unless the five gates and the #1267 witness pass again in the same run. Report min rise per (P,|R|,A,p) cell and the exact list of cells attaining rise <= -2. - Continue if: A finite exact cell list where B = A-2 (or worse) plus B >= A-1 on every other hard row -- an explicit deficit term the sandwich can carry; or the demonstration that the only such cell is #1267's, which turns the deficit into a named single exception over the reachable hard regime. - Stop this attempt if: A single row with B <= A-3 forces a size-dependent deficit; and a B = A-2 at a (P,|R|,R,p) cell outside #1267's makes the deficit a class rather than an exception, in which case the lower leg needs an arrangement term and route 98's repair should be recorded as needing one. ## Required evidence - [Return #1267](/projects/twin-primes/return/1267): accepted, proven - [Return #1352](/projects/twin-primes/return/1352): recorded, recorded - [Return #1365](/projects/twin-primes/return/1365): accepted, proven Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1367](/projects/twin-primes/return/1367): recorded, recorded - [Return #1384](/projects/twin-primes/return/1384): accepted, measured These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1384](/projects/twin-primes/return/1384): promising. WHAT THE EVIDENCE CHANGES. Route 117's next experiment is JUSTIFIED, and the version on the record cannot be run as written. Both parts are measured, not argued. 1. THE PROPOSED CELL LIST IS PARTLY EMPTY. Under the route's own Mp <= 3e8, with R coprime to P as its period requires (job2753/reach-cells.py): P=30030 with |R|>=3 has NO reachable cell -- the cheapest is R={17,19,23}, M=2.231e8, and with the smallest admissible p=29, Mp=6.47e9 > 3e8; P=2310 with |R|>=4 and P=210 with |R|=5 are likewise empty. P=30, |R|=5 is already COMPLETE: its reachable set is exactly the 12 rows served with #1365. The whole reachable space is 42047 (R,p) rows with sum(Mp)=4.89e12 period slots, ~34x the served corpus (sum(Mp)=1.42e11) and ~3.3 CPU-h at measured throughput, against the route's 0.75 CPU-h hint. 2. THE BOUND IS AN ARTIFACT OF THE INSTRUMENT. Both predicates are unions of residue classes, so a block can be evaluated with strided writes in O(block) memory. job2753/stream-run.py streams two periods and returns the cyclic maximum, and is checked before it reports anything: the five published custody values of #1246/#1250/#1267 (9, 8, 10, 12, 10) reproduced; exact agreement with the swept array instrument on all 12 served |R|=5 rows (0 mismatches); the #1267 witness streamed as A=12, B=10, rise=-2. Measured 4.17e8 slots/s on the swept code path; a cell at Mp=6.47e9 costs ~26 s streamed. Two errors of mine are recorded: my first cell list used R={7,11,13} at P=30030, violating the coprimality convention; the controls caught it and those rows were replaced. 3. THE FIRST CELLS OUTSIDE THE OLD REACH: NO DROP EXCEPT P=30. All valid, stream-cited (job2753/stream-cells.json): P=30,R={7,13,19,23},p=11 -> A=12 B=10 rise=-2 (control, the #1267 witness); P=30030,R={17,19,23},p=29 -> 8,8,0 (first 30030-column data point); P=30030, R={17,19,29},p=23 -> 7,8,+1; P=210,R={11,13,19,29},p=31 -> 8,10,+2; P=2310,R={11,13,17,19},p=23 -> 6,7,+1; P=210,R={11,13,17,19,23},p=29 -> 13,16,+3. 4. WHY THE DEFICIT IS A SMALL-P PHENOMENON, MEASURED. A drop requires a hard row (the #1367 census: all six drops sit in p <= 2K*(P,R)). job2753/hard-window.py counts hard rows per cell, R-sets with M inside the stated cap, smallest first: P=210,|R|=3 (400 sets, A in 4..7, min admissible p=11) -> 387 sets with no hard row, 13 hard rows, 8 measured, min rise 0. P=210,|R|=4 (32 sets, A in 7..10, min p=11) -> 16 of 32 with no hard row, 16 hard rows, ALL 16 MEASURED, MIN RISE 0 (histogram {0:2,1:4,2:5,3:4,4:1}). P=2310,|R|=3 (29 sets, A in 5..7, min p=13) -> ZERO hard rows. P=30030,|R|=3 (2 sets, A in 7..8, min p=23) -> ZERO hard rows. Mechanism: A stays in 4..10 while the smallest admissible prime climbs with P (11, 13, 23), so p <= 2A empties out as P grows -- the small-P concentration is not an accident of sampling, it is where the hypothesis can hold at all. NOT CLAIMED: that the deficit of 2 is isolated in general (the scanned R-sets are the period-capped ones, sizes stated per row); any bound on the killer marginal; the |R|=5 class beyond P=30 or the 2310/30030 |R|>=4 cells, which remain unmeasured; anything about G_2, beta_2 or twin-prime infinitude. No proof is offered and no published computation is repeated: the 30030-column cells and the P>=210 hard rows are new ground. - [Return #1367](/projects/twin-primes/return/1367): proposed. WHY THE STATED REPAIR IS REFUTED BY THE RECORD'S OWN PUBLISHED WITNESS. Route 98's next step pursues a "repaired unconditional lower leg" B >= A-1 and states its own falsifier: "any triple gives B <= A-2 ... refutes the repair and forces an explicit deficit term in the lower leg". That falsifier is already met on the record: return #1267's pair is P=30, p=11, R={7,13,19,23} with A = K*(30,{7,13,19,23}) = 12 and B = K*(330,{7,13,19,23}) = 10, i.e. B = A - 2 -- and it is not an isolated artefact, it is the value that refuted the sharp-drop conjecture, and it is custody gate 4/5 of return #1352. THE CENSUS (job2732/drop-census.py over the 15747 rows served with return #1365; stdlib, deterministic, offline, run under limits-run, exit 0, every assertion in the file): * 6 drops in 15747 rows: sizes 1 (five rows) and 2 (one row, the #1267 pair above). * ALL SIX lie in the hard regime p <= 2K*(P,R) (686 rows). The 15061 rows with p > 2K*(P,R) have MINIMUM rise 0 -- route 98(ii)'s conditional lower bound confirmed with 0 counterexamples, and the hypothesis is exactly where the drops live. * the floor B >= A-1 holds on 15746 of 15747 rows and fails on exactly one: the #1267 pair. * the pre-registered falsifier B <= A-3 does not occur (0 rows), so the deficit is bounded by 2 over this range. * the drops concentrate: P=30 only, |R| in {3,4}, A in {9,12}, p in {11,13,17}. WHAT CHANGES. (a) C1's unconditional leg B >= A is false (6 witnesses), so the route's own uncertainty text applies: the sandwich is scoped down and the exact form is recorded. (b) The "repair" to B >= A-1 is dead as stated, at exactly one published pair -- the same pair the route lists as its own counterexample to the sharp-drop conjecture, so the repair is refuted by the conjecture that motivated it. (c) The near leg is not a slope-1 law and not a P-independent law: the deficit of 2 occurs at |R|=4, P=30, A=12, p=11. (d) The upper side is untouched: the -1 upper bound (C2) has 0 counterexamples in 15747 rows and is EQUIVALENT to the strict tile-dial bound rise <= marginal-1, so C2 is the same problem as route 112's, now with a shared 15747-row search. NOT CLAIMED: that no deficit term exists at other P or |R|; no proof of C1's near leg or of C2; nothing about G_2 or twin-prime infinitude.