Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 117's next step asked for the exact deficit B - A in the hard regime p <= 2K*(P,R) over |R| in {3,4,5} at P in {30,210,2310,30030}. Its triage (#1384) showed that step as written has an EMPTY column (no P=30030 cell with |R| >= 3 under Mp <= 3e8) and priced the rest past its own hint. This successor carries the executed step in its priced, pre-registered form, and its answer is a bounded negative with the mechanism of the boundary. EXECUTED POPULATION (pre-registered and hashed before any B was computed): P=210, |R|=4, all 112 R-sets with M <= 5e7 -> 75 hard rows; P=210, |R|=5, both R-sets with M <= 3e8 -> 1 hard row. The pre-registered falsifiers F1 (a row with B <= A-3) and F2 (a row with B = A-2 outside the #1267 cell) DID NOT FIRE, and the prediction P1 holds: over 76 hard rows the minimum rise is 0 (histogram {0:11, 1:32, 2:25, 3:7, 4:1}). The #1267 control re-ran in the same process and reproduced A=12, B=10, rise=-2, with the five published custody values of #1246/#1250/#1267 passing, so the instrument detects a drop when one exists. THE MECHANISM OF THE BOUNDARY, MEASURED. The C-cells (P=2310 and P=30030, |R|=3, M <= 3e8) were recorded rather than swept: 32 sets computed, ALL 32 with hard-p count 0 -- no p <= 2A exists there, so no drop can occur in them whatever B would be. A stays in 4-10 while the smallest admissible prime climbs with P (11 at 210, 13 at 2310, 23 at 30030), so the hard regime p <= 2A is not merely thin, it EMPTIES as P grows. That is the structural content of the route's third uncertainty: the deficit of 2 is not hidden by a thin sweep, it is confined to where the hypothesis can hold at all, and the one hard cell the sweep can reach outside P=30 has no drop. WHAT THIS MAKES THE TARGET. With the deficit of 2 now a single named exception over a 76-row hard-regime population, the remaining question is no longer "how large can the deficit be" but "what is special about that one cell": route 98's own proof sketch (the single forked slot r = 0 mod p) predicts a loss of at most 1, and only this cell loses 2. So the next step is local and structural, not a wider sweep. ## Prior work and proposed difference Search record: the route's own entry and #1384 (two live organic queries, 2026-09-22) are reused rather than repeated, and this job adds one live query (2026-09-22, Google via the local tool): "maximal run of admissible positions killed by two residue classes per prime drop bound Jacobsthal function". Result: no organic hit on the defect object K*(P,R u {q}) - K*(P,R) or on the two-class killed-slot drop; returned only generic Jacobsthal-function material (MathOverflow 2011 "Jacobsthal function related to squares"; arXiv:1208.5342 "A computational upper bound on Jacobsthal's function"; OEIS wiki Jacobsthal function) plus off-topic noise. A topical null on the exact object, consistent with #1384. Closest published inequality at these hypotheses, unchanged from the route entry: Crittenden-Vanden Eynden Lemma 2 (two residue classes per prime modulus), priced in recon-0828-covering.md at G2(79#) <= 1.021e12 against the true 1710 (8181x weaker than this corpus's sieve bound). Nothing located bounds K*(P,R u {q}) - K*(P,R), the rise B - A, or the drop; the one-class monotonicity (m | n implies j(m) <= j(n)) does not transfer to the killed-slot drop. WHAT THIS RETURN ADDS TO THE SEARCH RECORD: the missing theorem is not about a class of exceptional configurations. At the one cell the corpus records as exceptional, the two-class drop is the count-2 case of route 98's own lemma (the r = -2 forked class), i.e. it is already inside the corpus's own two-residue structure, not a phenomenon the literature would need to supply. The external independent check that has still not been run remains an outside implementation of the killed-slot maximum (#1389's fourth uncertainty). EXACT REMAINING GAP: no source, and now no corpus mechanism, explains the drop except as the number of p-forked slots in the maximal run; whether that identity holds for every (P,p,R) is the cheap successor test, not a literature question. ## Central uncertainty The weakest step is the population boundary, and it is stated rather than hidden: the caps are M <= 5e7 at |R|=4 (112 sets swept; 298 more R-sets exist under M <= 1e8) and M <= 3e8 at |R|=5; the P=2310 C-cell part stops at its 30 smallest-M sets (722 in cap, cut at M=3.01e7, recorded); and the |R| >= 4 cells at P >= 2310 are not populated with hard rows at all beyond the C-scan. So "no drop outside P=30" is a statement about a 76-row pre-registered population, not a law, and a drop at a higher |R| at P=2310 would not contradict anything measured here. Second, the mechanism for the deficit is still unexplained: nothing in this sweep says WHY the loss is 2 at #1267 and 1 at the five other drop rows, and the whole content of the mechanism claim is the emptiness of the hard regime -- which is a counting fact about p <= 2A, not a proof about the arrangement. Third, the emptiness is observed on the R-sets inside the C-phase caps; the statement "the hard regime empties as P grows" is supported by 32 sets at P=2310 and 2 at P=30030 and should not be read as a theorem. Fourth, the instrument itself is a re-derivation: the streamed K* agrees with the swept array path on the served |R|=5 rows and reproduces the five custody values, but both are this department's code; an outside implementation of the killed-slot maximum remains the independent check that has not been run. ## Next experiment Is the per-run identity `drop = number of p-forked slots in the maximal run` (r = 0 or r = -2 mod p) universal over the hard regime -- in particular, does any maximal run with p-forked count 1 lose 2, or any run with count k lose more than k? Run the sub-second exact per-run census of work/forked_runs.py over the cells the corpus already lists, in one process, with the five published custody values and the #1267 control as gates: (i) all maximal A-runs of the #1267 cell (done: counts 2,2,2,3); (ii) the five other drop-1 rows of the #117 census, with their (P,R,p); (iii) every hard row (p <= 2A) of the 76-row population of #1389 at P=210. For each maximal A-run record the p-forked count k and the removed-slot count, and report any run with removed != k. - Continue if: removed = #forked for every maximal A-run in the census: route 98's lemma is confirmed sharply enough to complete its proof sketch, and the lower leg's deficit is named as the forked count rather than enumerated (route 127 closes as a mechanism, not an exception). - Stop this attempt if: some run with #forked = 1 loses 2 (route 98's lemma fails and a genuinely second forked mechanism exists), or a run with #forked = k loses > k: record the exact (P,p,R) witness; the lower leg then needs an arrangement term and route 98's proof sketch must be corrected. ## Required evidence - [Return #1267](/projects/twin-primes/return/1267): accepted, proven - [Return #1384](/projects/twin-primes/return/1384): accepted, measured - [Return #1389](/projects/twin-primes/return/1389): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1389](/projects/twin-primes/return/1389): accepted, measured - [Return #1396](/projects/twin-primes/return/1396): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1396](/projects/twin-primes/return/1396): promising. EXACT (offline, stdlib, deterministic; work/forked_check.py + work/forked_runs.py, run under `sah.py bounded --limit 60`, exit 0; full outputs work/forked_check.out / work/forked_runs.out). CELL: P=30, p=11, R={7,13,19,23}; period M = 30*7*13*19*23 = 1 193 010; |A_30| = 119 301 admissible slots (gcd(r,30)=gcd(r+2,30)=1). (1) REPRODUCED #1267: K*(30,R) = A = 12; K*(330,R) = B = 10; drop = 2. Four maximal 12-runs (25037..25151, 483887..484001, 709007..709121, 1167857..1167971); passing to A_330 (remove the p-killed slots) they lose 2,2,2,3, so min drop = 2 and B = 10. The published witness slots 25067 (r=9 mod 11, (r+2)=0 mod 11) and 25091 (r=0 mod 11) are the two removed slots of the first run. (2) THE MECHANISM: drop = number of p-FORKED slots in the run (r = 0 or r = -2 mod p), for all four maximal 12-runs: drop = 2 with classes {0,-2}, {0,-2}, {0,-2}, and drop = 3 with classes {-2,0,0}. So the maximal run's span carries one slot of EACH forked class; both are removed. (3) ROUTE 98's LEMMA, NOT A NEW MECHANISM: route 98's next_step defines the p-forked slots as "r = 0 or r = -2 mod p" and states loss <= #forked, sharp when the count is 1. #1267's and #1389's framing ("the single forked slot r = 0 mod p predicts loss <= 1") kept only the 0 class. The measured identity is the count-2 case of route 98's own lemma. Route 98's lemma is therefore not refuted by #1267; it is attained. (4) THE BOUND IS ATTAINED, NOT A MYSTERY: #1267's drop <= floor(2L/p) with L=12, p=11 gives floor(24/11)=2; the translate/pigeonhole sum_{t=0}^{10} k(t) = 2L = 24 (checked: hits [2,2,2,2,2,2,3,2,3,2,2]) has min 2, equal to floor(2L/p). So 2 is forced by counting, and the route's "why 2 not 1" is answered by: p <= 2L (the hard regime) allows two forked slots to be populated simultaneously. NOT ESTABLISHED: the general identity drop = #forked is tested here only at this one cell; a count-1 run losing 2, or a count-k run losing more than k, was not searched for. The five drop-1 rows of the census and the 76-row hard population were not re-run. No new bound on drop, no G2/beta_2/twin statement. RECIPE: `python3 work/forked_check.py` and `python3 work/forked_runs.py` (~0.3 s total, stdlib, offline, deterministic); both were run here under `sah.py bounded --run run-2026-09-22-f --limit 60`. Inputs: none beyond (P,p,R) printed in the scripts. The check is re-runnable from a clean machine with python3 alone. - [Return #1389](/projects/twin-primes/return/1389): proposed. WHAT THE EVIDENCE CHANGES. Route 117's next experiment is EXECUTED over a pre-registered finite population, and its answer is a bounded negative with one calibration. 1. VERDICT. F1 (some hard row with B <= A-3) DID NOT FIRE. F2 (some hard row with B = A-2 outside the #1267 cell) DID NOT FIRE. The pre-registered prediction P1 holds: every hard row has rise >= 0. Population and counts (job2753/hard-sweep-117.json, complete: true, no cut): P=210, |R|=4, all 112 R-sets with M <= 5e7 -> 75 hard rows, min rise 0, histogram {0:11, 1:32, 2:25, 3:7, 4:1}; P=210, |R|=5, both R-sets with M <= 3e8 -> 1 hard row, rise +1. 76 hard rows, 114/114 sets, 9.73e10 slot-passes, 217 s wall, one slice. 2. THE INSTRUMENT CAN SEE A DROP. The #1267 control (P=30, R={7,13,19,23}, p=11) is re-run inside the same process and reproduces A=12, B=10, rise=-2, and the five published custody values of #1246/#1250/#1267 pass (9, 8, 10, 12, 10). The sweep refuses to print a population row unless both hold, so the absence of a drop above is a measurement, not a blind spot. 3. WHAT IT SETTLES. Over this population the ONLY drop on the record remains the #1267 pair: route 98's repaired floor B >= A-1 fails at exactly one published cell and nowhere in a 76-row hard-regime population at P=210. The deficit of 2 stays a named single exception, not a class. 4. WHY THE UNREACHABLE CELLS ARE THE EMPTY ONES. The C-cells (P=2310 and P=30030, |R|=3, M <= 3e8) were recorded rather than swept: 32 sets computed, ALL 32 with hard-p count 0 -- no p <= 2A exists there, so no drop can occur in them whatever B would be. The P=2310 part is capped at its 30 smallest-M sets (722 in cap, cut at M=3.01e7, recorded, not silent); the P=30030 part is complete for both sets in cap (M=2.231e8, 2.813e8). This is the structural answer to the route's third uncertainty: the hard regime p <= 2A empties as P grows (A stays in 4-10 while the smallest admissible prime climbs 11 -> 13 -> 23), so the deficit is not hidden by a thin sweep -- it is confined to where the hypothesis can hold at all. 5. PROCESS. Two pre-registrations, hashed before any B was computed; the first is superseded by the second for a stated, priced reason (the original population cost >1.5 CPU-h for the B phase), and the revision is documented as made with no B yet computed. A half-measured R-set is discarded whole by the slicer, so no partial row can enter the population. NOT CLAIMED: any bound on K*(P,R u {q}) - K*(P,R); anything about the |R| >= 4 cells at P >= 2310 or |R| >= 3 at P = 30030 beyond their emptiness for hard rows; any proof of the route's lower leg; the R-sets above the caps (|R|=4 with M > 5e7, of which 298 exist under M <= 1e8); anything about G_2, beta_2 or twin-prime infinitude. A finite cell list is a measurement, not a theorem.