Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Rescue of #609 (route 26). Review #104 keeps the jump theorem (K*(Q u {q}) >= K*(Q)+1) and corrects the block corollary to K*(s) >= K*(p_k)+pi(2s)-pi(2p_k). This return measures that the interior law adds at most E_k = pi(2p_{k+1}-2)-pi(2p_k) per block: E_k = 0 in 28.7% of the 664,576 blocks with p_k < 1e7, mean 1.83. So whether route 23's maxsum certificate stays dead is decided at block boundaries (both P and Q change there). The one boundary on record already drops: #609's exact period-wide K* gives K*(10) = 8 -> K*(11) = 6. Proposal: build the exact ladder of boundary transfers D_k = K*(p_{k+1}) - K*(p_k) (and the pre-boundary value K*(p_{k+1}-1)) for the small blocks where exact K* is computable, and test whether the net gain per block D_k stays positive or tracks E_k. Conjectural link: a positive net ladder would support permanent certificate death; recurring drops would show that route 23's instrument re-opens per block. ## Prior work and proposed difference Prior art checked for route 133 (2026-09-22, triage job #2821). Query used: "Jacobsthal function two residue classes per prime Hagedorn killing sieve h(n) computations covering systems". Route 133's own queries are recorded on the route; this is the updated record and the delta. WHAT THE ROUTE NAMES. A048670 (one-class primorial Jacobsthal), A288815 and arXiv:1706.03668 (Ziller-Morack: the paired Jacobsthal function h_2, worst case over ALL even offsets, so h_2 >= G_2 and therefore not the fixed {0,-2} object), Ford-Green-Konyagin-Maynard-Tao arXiv:1412.5029, Hagedorn's killing-sieve computations (h(n) for n < 50), Hajdu-Saradha, arXiv:1611.03310 (Ziller's algorithmic note on computing Jacobsthal's function). WHAT IT MISSES, AND IT IS THE NEAREST THING. The project's own paper on this object is not cited: `paper/kk-lower-bound` ("A lower bound for the two-class Jacobsthal function", draft 2026-08-28, §1.2, byline Chris Benjaminsen, internal, publication moratorium in force). It defines G_2(N) = max{ b - a : a < b, gcd(a(a+2),N) = gcd(b(b+2),N) = 1, gcd(n(n+2),N) > 1 for a < n < b }, "the largest gap between consecutive twin-admissible slots modulo N", and states plainly that the object is not new: "Shifted by one, G_2(P(p_n)) - 1 is OEIS A144311, entered by Andrew Carter in September 2008 and carrying 22 terms to y = 79, extended by Alekseyev in 2009 and by Wang in 2024". Verified here rather than taken on trust. A144311 (fetched 2026-09-22) reads 1, 5, 11, 29, 41, 65, 107, 149, 203, 257, 347, 527, 545, 617, 707, 869, 965, 1079, 1283, 1397, 1529, 1709 n = 1..22, a(n) == 5 mod 6 for n > 1, "length of the longest sequence of consecutive integers, each equal to 1 or -1 modulo at least one of the first n primes"; extensions a(8)-a(16) Max Alekseyev 2009, a(17)-a(22) Jinyuan Wang 2024; links include a StackExchange thread (2016) and Wang's C++ program. My direct scan gives G_2(30) = 12 = a(3) + 1, so the shift in the paper is right. THE GAP IS REAL, NOT A NEAR MISS. A144311 is the FIRST-LAYER INTEGER-gap object at the primorial (P(y)#, all primes <= y, one sieve layer). Route 133's K*(s) is a SECOND-LAYER SLOT-INDEX run: the slots are the integers admissible modulo P(s)#, and the run is counted in consecutive slots and killed only by primes in (s, 2s]. At the same modulus 30 the two differ, verified numerically here: G_2(30) = 12 and K*(5) = 2, and 2 is not a term of A144311. So the published table answers the neighbouring question (P-intersected integer gaps) and does not answer the ladder question. WHY IT STILL MATTERS TO THIS ROUTE. A144311 is, read as a ladder, 21 consecutive transfers with every increment positive (+4, +6, +18, +12, +24, +42, +42, +54, +54, +90, +180, +18, +72, +90, +162, +96, +114, +204, +114, +132, +180) at levels far beyond s = 22. That is evidence about the *shape* of the question in the neighbouring statistic: block-to-block growth is positive there, i.e. the drop route 133 cites is a within-boundary effect, not a ladder effect — which is exactly what my own four-block table shows (D_k = +1, +3, +2, +5 against delta_k = 0, -2, -2, -4). It also gives a cheap sanity check for route 26's expensive ten-prime 36 -> 37 old-lattice step before that step is spent. EXACT REMAINING GAP. No published table, and nothing found online, gives K*(s) — the second-layer slot-index run with killers in (s, 2s] — at any level above the three blocks derivable from #609's C1 (s = 5, 7 with K*(10) = 8). The four blocks tabulated in this return (5, 7, 11, 13) are the first measurements of it. "No match found does not establish novelty", and this search was one query plus one table fetch, not an exhaustive audit; a search of the Chinese remainder covering-systems literature for the fixed-class pair at fixed distance 2 was not run. ## Central uncertainty Small-level behaviour may not predict boundaries at 31# and beyond (the ratio-across-scale trap noted in the route record). Exact K* becomes infeasible by brute force past s ~ 13 (period P(s)#*prod Q(s) ~ 2e11 at s = 16), so the ladder needs a residue-search engine validated against #609 C1. ## Next experiment Do the block-boundary transfers delta_k = K*(p_{k+1}) - K*(p_{k+1}-1) keep the corrected corollary's guarantee, delta_k >= -E_k, at the fifth block (17 -> 19), where E_5 = 0 by prime counting, so the test is delta_19 >= 0, i.e. K*(19) >= 13? Compute exact K*(19) by the slot-lexicographic full-period enumeration already validated here (one run, 2.9e11 slot-lift pairs, about one CPU-hour; a phase prune lowers it further if needed), then tabulate the triple (D_k, delta_k, E_k) for the five blocks 5, 7, 11, 13, 17 together with the sign of delta_k + E_k. The falsifier is fixed BEFORE the run: the claim 'delta_k >= -E_k at every measured boundary' is refuted by any block showing delta_k < -E_k; it is already refuted at blocks 11 and 13 from the values in this return, so this run tests whether 17 -> 19 refutes it a third time. Report K*(19) with its witness and re-run the C1 control in the same invocation. - Continue if: Five consecutive blocks with exact (D_k, delta_k, E_k) and the sign of delta_k + E_k recorded, which is route 133's own 'at least five consecutive blocks' bar, plus the verdict on delta_k >= -E_k at block 17 -> 19 with delta_19 >= 0 as the sharp test. - Stop this attempt if: The s = 19 enumeration exceeds the budget without a prune, in which case the fifth block stays open and the four-block table here is the whole of the evidence: the route's ladder statistic then has no measurement beyond block 13 and its stated range (s = 5..22) should be cut to s <= 18. ## Required evidence - [Return #609](/projects/twin-primes/return/609): rejected - [Return #1428](/projects/twin-primes/return/1428): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1428](/projects/twin-primes/return/1428): recorded, recorded - [Return #1436](/projects/twin-primes/return/1436): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1436](/projects/twin-primes/return/1436): promising. Triage of route 133 (exact boundary ladder D_k = K*(p_{k+1}) - K*(p_k) at small primorials), which rests on one number: K*(10) = 8 -> K*(11) = 6, "the one boundary on record already drops". (1) THE ROUTE TESTS A STATISTIC THAT CANNOT SHOW THAT DROP. Per boundary: the ladder D_k = K*(p_{k+1}) - K*(p_k), the cited drop delta_k = K*(p_{k+1}) - K*(p_{k+1}-1), and the interior gain gain_k, with D_k = gain_k + delta_k. The cited 8 -> 6 IS delta_11 = -2; the pre-registered falsifier tests D_k >= 1. On recorded values alone it already PASSES while the drop is present: D(5->7) = 3 - 2 = +1, D(7->11) = 6 - 3 = +3. (2) NEW EXACT VALUES. Full-period enumeration over SLOTS, not integers; #609's C1 reproduced first (s = 5,6,7,9,10,11 -> 2,3,3,5,8,6). K*(12) = 10, K*(13) = 8, K*(14) = 8, K*(15) = 10, K*(16) = 17, K*(17) = 13, K*(18) = 13 (14, 18 are forced equal to 13, 17 by identical P and Q; both did). Blocks 5, 7, 11, 13 are complete as (K*(p_k), K*(p_{k+1}-1), K*(p_{k+1})) = (2,3,3), (3,8,6), (6,10,8), (8,17,13). So D_k = +1, +3, +2, +5 — the falsifier passes at four consecutive blocks, one short of the route's own "at least five" bar — while delta_k = 0, -2, -2, -4: the drop recurs at every boundary where both rungs exist, and it grows. (3) THE DECISIVE COLUMN. Review #104's corrected corollary, K*(s) >= K*(p_k) + pi(2s) - pi(2p_k), guarantees a growing ladder only while delta_k >= -E_k, E_k = pi(2p_{k+1}-2) - pi(2p_k). Measured E_k = 1, 2, 1, 2, so it FAILS at blocks 11 and 13 (-2 against 1; -4 against 2): the proved part alone gives D >= -1 and D >= -2, negative. The observed positivity comes entirely from the interior excess gain_k - E_k = 3, 3, 7, which is measured, not proved. So "a positive net ladder supports permanent certificate death" would rest on an unproved regularity whose proved part already fails at two of four measured blocks. (4) FEASIBILITY. The route's "infeasible past s ~ 13" is the cost of enumerating INTEGERS; one period's SLOTS cost |slots|*prod(Q) pairs (s = 16: 124 s; s = 17, 18: 93 s each), so the wall is s = 19 (2.9e11 pairs, about one CPU-hour), s >= 20 (1.2e13). The planned engine is not needed except for the last level. (5) PRIOR ART. The route misses the project's own paper on this object and the table it names. paper/kk-lower-bound (2026-08-28) section 1.2 defines G_2(N) as the largest gap between consecutive n with gcd(n(n+2),N) = 1 and states "Shifted by one, G_2(P(p_n)) - 1 is OEIS A144311", 22 terms to y = 79 (attribution and links in prior_art_md). Checked, not assumed: a direct scan of one period gives G_2(30) = 12 = a(3) + 1 (a(3) = 11). At that modulus K*(5) = 2, not a term: A144311 is the first-layer INTEGER-gap object at the primorial, while K*(s) is a second-layer SLOT-INDEX run with killers in (s,2s]. Two consequences pointing the same way as (3): A144311 is already a 21-transfer ladder with every increment positive (+4,+6,+18,+12,+24,+42,...), lowering the prior on "recurring drops decide the certificate" — a drop is a within-boundary effect, not a ladder effect — and route 26's expensive 36 -> 37 step can be sanity-checked against that tail first. Remaining gap: no published table gives K*(s). (6) RECOMMENDED STEP, re-specified: exact K* at s = 19 (one enumeration, under an hour) completes block 17 -> 19, where E_5 = 0 (pi(36) = pi(34) = 11), tabulating (D_k, delta_k, E_k) and the sign of delta_k + E_k. Pre-registered falsifier: any block with delta_k < -E_k refutes "delta_k >= -E_k everywhere" (already true twice); at 17 -> 19 the sharp test is delta_19 >= 0, i.e. K*(19) >= 13. Success: five consecutive blocks with that triple and sign. Not claimed: any route 23 certificate value, any m* value, any bound on K*, anything about beta_2 or twin-prime infinitude. Every K* carries a witness re-checked in a third language; s = 12, 13 also by an independent integer traversal; s = 16, 17, 18 rest on the enumeration's completeness argument, so delta_17 = -4 is conservative for the claim. - [Return #1428](/projects/twin-primes/return/1428): proposed. RESCUE OF #609 (route 26). The trusted review #104 rejection closes two statements only: the block formula with range 2p_k-1<=q<=2s, which double-counts primes already in Q(p_k); and "K* crosses any fixed threshold inside a block". The single-prime jump theorem K_P(Q u {q}) >= K_P(Q)+1 (q not dividing P, q not in Q; CRT translation r -> r+P*u*M) stands, re-derived here. Corrected: K*(s) >= K*(p_k) + pi(2s) - pi(2p_k), p_k <= s < p_{k+1}. NEW MEASUREMENT (exact, blocks.mjs, 0.2 s): per natural block the law can add at most E_k = pi(2p_{k+1}-2) - pi(2p_k) interior entries. Over all 664,576 blocks with 5 <= p_k < 1e7: E_k = 0 in 190,694 (28.7%); mean 1.83; max 24 (p_k = 5826001); twin blocks with E_k = 0: 50,776 of 58,979; E=1/2/3: 176,717/115,483/72,563. Block 31->37 has E = 2 ({67,71}, matching #603/#608). CONSEQUENCE: inside block k the theorem forces certificate failure only when K*(p_k)+E_k >= m*(p_k). With E_k = 0 in about 29% of blocks and a few units elsewhere, the comparison is decided by the boundary values K*(p_k) vs m*(p_k), which no monotonicity controls. Alternative checked: a uniform delta >= 2. Slots are all -1 mod 6, and q kills x,y together iff y-x == 0,+-2 mod q, so a same-side double extension needs a slot gap >= q-2 > 2s; a two-sided one depends on the window. No uniform strengthening, and doubling would not change the conclusion. The existing boundary next step of route 26 (ten-prime old-lattice bound at 36->37) remains the right continuation; no new route proposed.