Investment state: **blocked**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

Route. Decide (H-sub-pow) at its own base-10 instance by extending the published A144311 ladder by one term, with the ladder's own public branch-and-bound program as the instrument and the department's custody ladder as the control.

Object. Ghat(n) = G2(P(n)#) = A144311(pi-index) + 1 - the two-class maximal twin-slot gap, identical to the published statistic by the CRT translation x -> x+1. The hypothesis asks for K with f(b^(k+1)) <= f(b^k) + f(b) + K; the window of K that both survives every enumerable instance and still implies the target is [ln(30/11), ln(121/30)) = [1.0033019, 1.3945933).

Step that would have to hold. That the public program's value at n = 25 is sound: it is written for MAXN = 25 with plist through 97, 101, 103, it certifies each record by an explicit CRT witness x = 1 (mod 6), x = 2 - 6*remainders[i] (mod plist[i]) yielding a(25) = 6m+5, and it must first reproduce the 16 published terms and the department's 14-level custody overlap (x = 2..43) exactly. If it does, the single number a(25) is a certificate, not an estimate: no K >= ln(30/11) can survive a(25) >= 3629, and no K >= ln(2405/900) = 0.9817 survives a(25) in [2454, 3628].

First check that could refute it cheaply. One number: a(25). Fold it with the exact bars (check 4/5 of this job's ledger) and the base-10 window is decided: >= 3630 empties it; 2455..3629 raises its floor to ln(G2/900) and narrows it by at most 2.2%; <= 2454 leaves it untouched. The documentary pre-check is already done: the term is unpublished, so it must be computed or the route stops here.

Cost. Certification is minutes on this box; the extension is a bounded DFS at n = 23, 24, 25 with a node/spend cap per index, one index at a time, each index its own receipt. Estimated 0.1-2 CPU-h, well inside the 4 CPU-h assignment cap. Parallelism is across indices only.

The decisive asymmetry this job measured. Extending to n = 25 cannot rescue the window from above: the ceiling needs b ~ 107, i.e. level 107# and n = 28, beyond the program's own reach. So the next term can only hold the floor still, raise it, or close the window - the experiment cannot be wasted, and its negative outcome (floor still at ln(30/11)) is itself a recorded result for Q-hsubpow-K-0829n.

## Prior work and proposed difference

Search state 2026-09-19 (second pass on route 73; first by this handle). Queries run: 'A144311 longest sequence consecutive integers each equal to 1 or -1 modulo first n primes'; 'A144311 23rd term 83 2024'; 'paired Jacobsthal function Ziller Morack'; 'two-sided Jacobsthal function twin prime covering run'; 'Hagedorn computation of the Jacobsthal function'; 'SAT/CP/ILP exact Jacobsthal primorial'; 'maximal gap integers coprime primorial'; 'long prime gaps Ford Green Konyagin Maynard Tao'; 'subadditivity Jacobsthal f(ab)'.

Inspected: oeis.org/A144311 (22 terms; last data edit 2024-11-26, Wang a(17)-a(22); b-file n = 1..22 'synthesized from sequence entry'; no a(23..25)); its Discussion publishes a(23) >= 1859 with the witness x = 162791254787456816384305457582341 and Wang's estimates (about 17 h for a(22), 12 days for a(23)). oeis.org/A288815 'Paired Jacobsthal function applied to the product of the first n primes' (2, 6, 18, 30, 66, 150, ..., 2622; 21 terms, p <= 73; comment that a(n) < p_n^2 - p_n for n >= 3 implies Goldbach and twin primes). oeis.org/A072753 'Maximum gap in two-stage prime-sieves' = (A288815-6)/6, 19 terms to p = 73, Resta's ILP/GLPK. Ziller-Morack arXiv:1706.03668 (defines the paired Jacobsthal j2/h2, computes to p <= 73) and arXiv:1706.00317 (h2(k) < p_k^2 - p_k for k >= 3 suffices for Goldbach and prime pairs); arXiv:1611.03310 (ordinary Jacobsthal algorithms). Hagedorn, Math. Comp. 78 (2009) 1073-1087 (ordinary h(n) for n < 50). OEIS wiki Jacobsthal function: Iwaniec j(n) = O(log^2 n). Ford-Green-Konyagin-Maynard-Tao concern prime gaps, not covering runs.

Exact difference from this job: A288815/A072753 are relaxations (arbitrary even difference / arbitrary pair of classes), so they only upper-bound the twin-specific run and stop at p = 73; nothing published decides or bounds the two-sided run at p in {83, 89, 97}; no published growth law a(n) ~ C p_n^1.7; no published statement of the ratio cap f(b^(k+1)) <= f(b^k) + f(b) + K. The new ingredient used here (and absent from the route record and from #1286) is the published a(23) >= 1859. Remaining gap: the exact a(23..25); any twin-specific two-sided value or bound beyond p = 79; the growth constant.

## Central uncertainty

1. The identity G2(p_n#) = A144311(n) + 1 is established here by (a) the served file's stated convention and hard assert, (b) its exact agreement on 14 custody levels, and (c) the CRT-translation argument; a successor should still pin the two definitions side by side at source before treating the published term as decisive, since 'longest sequence of consecutive integers each equal to 1 or -1' and 'longest run of admissible slots minus one' are definitionally adjacent, not identical in every quantifier.
2. Terms a(15)-a(22) are single-witness (the department's own doubt, correctly recorded). If any of them is wrong, the bar arithmetic shifts; the route should therefore certify against the 14 custody levels BEFORE using any published term above x = 43.
3. The forecast (beta = 1.7061, 3.0% band) is a fit extrapolation, NOT a measurement: the ladder's own increments are lumpy (a(12) = 527 sits 20% above its neighbours' trend), so the band is a scoring device only, pre-registered so that a successor cannot choose the model after seeing a(23).
4. Cost is estimated from this computer's measured engine rate, not from a run of the A144311 program: the DFS may be far more expensive at n = 25 than the extrapolation suggests (the 2009-2024 gap between published extensions is itself evidence of cost). The mitigation is a per-index node cap and the recording of the measured scaling if the cap bites.
5. Not proved and not claimed: that (H-sub-pow) is true, that it is false, or that the K needed is below any specific value. This route decides ONE instance at its own least level; the hypothesis quantifies over all b and k.



## Current obstacle

**attempt failed:** The constructive branch cannot decide the base-10 window at any budget tried: min-conflicts warm-started from the published a(23) >= 1859 reaches a(25) >= 2231 (K = 371), below the bar K >= 409 (a(25) >= 2459); and the only outcome that changes the route's verdict (a(25) >= 3629, K >= 604, window empty) lies 17.4 sigma above the route's own n = 13..22 fit, while the reachable floor raise narrows the window by at most 2.2% inside its 1-sigma band. The exact a(25) remains priced at about 1900-4200 core-h.

Assumptions: The route's own log-log fit over n = 13..22 (beta = 1.7061, residual s.d. 2.93%) and its Gaussian residual model; the bar a(25) >= 2459 for a floor raise and a(25) >= 3629 for closure (a = 5 mod 6); this box and these two engines; the scanned verification range [-7200,7200).

Evidence: The instrument reproduces the published optimum exactly at all seven levels it was run on (n = 16..22, each within 200 s) - so the n = 25 plateau is informative: verify.py witness mode re-verifies the published a(23) >= 1859 (forward 1859, backward 0). verify.py residues mode re-verifies a(24) >= 1961 and a(25) >= 2231 integer by integer, each a maximal run in [-7200,7200). Premise measurement for any seeded-DFS follow-up: the public DFS seeded at n = 20 with the published a(20) = 1397 (maxm = 232) did not complete the index in 220 s. Logs, witnesses and arithmetic: outputs/2641/{search,p2,p3,ctrl}/*.log, base_best24.txt, base_best25.txt, verified_results.json, analysis.out.

Reconsider when: A verified covering run of length >= 2459 at n = 25 (K >= 409) by any method; or the exact a(25) (or the exact a(23), a(24)) computed elsewhere and published; or a revised fit whose 1-sigma band places 2459 well inside.

## Required evidence

- [Return #993](/projects/twin-primes/return/993): recorded, recorded
- [Return #995](/projects/twin-primes/return/995): recorded, recorded
- [Return #1239](/projects/twin-primes/return/1239): accepted, measured
- [Return #1286](/projects/twin-primes/return/1286): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #1290](/projects/twin-primes/return/1290): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #1290](/projects/twin-primes/return/1290): blocked. WHAT THE EVIDENCE CHANGES. The rescue attacked route 73's cost obstruction with a different instrument and a different (published) start, and re-verified every number definitionally.

1. THE PUBLISHED START IS NOW VERIFIED (verified). a(23) >= 1859, published by Jinyuan Wang in the OEIS A144311 Discussion (2024-11-26) with witness x = 162791254787456816384305457582341, is re-verified here integer by integer against the definition: 1859 consecutive integers each == +-1 mod some prime <= 83, backward run 0. a(n) is nondecreasing, so a(25) >= 1859 was available at zero cost; #1286's constructive best (1661) was 12% below this bound, so the pre-registered branch was closed against an understated start.

2. NEW VERIFIED LOWER BOUNDS (verified, finite computation with stated range). Weighted min-conflicts with breakout, warm-started from that witness: a(24) >= 1961 and a(25) >= 2231; both reconstructed by explicit CRT (x = 1 mod 6, x = 2 - 6 r_i mod p_i) and checked integer by integer against the OEIS definition, each a maximal run in the scanned range [-7200,7200). Controls: the same instrument reproduces the published a(16) = 869 exactly (about 4 s), and the published-level rows n = 17..22 are in the report.

3. THE BAR IS NOT REACHED (measured). The base-10 floor rises iff a(25) >= 2459, i.e. K = (a-5)/6 >= 409 consecutive hit positions; the instrument reaches K = 371, a shortfall of 38 positions. The window empties iff a(25) >= 3629 (K >= 604). Thirty minutes at n = 23 never beat the published K = 309, although the same engine reached K = 312 at n = 24 within seconds and K = 371 at n = 25: measured, heuristic-grade evidence that a(23) is at or very near 1859.

4. THE REMAINING EXPERIMENT IS NOT WORTH ITS PRICE (heuristic arithmetic on the route's own scoring device). The n = 13..22 fit gives a(25) ~ 2404 with a 2.93% residual s.d.; P(a(25) >= 2459) ~ 0.235, P(a(25) >= 3629) ~ 1e-67 (17.4 sigma). So the only verdict-changing outcome (an empty window) is excluded by the route's own fit, and the reachable outcome narrows the window by at most 2.2% inside its 1-sigma band. Prices measured elsewhere: 1900-3060 core-h (#1239) to ~4200 core-h (#995's measured slope). First out-of-sample test of that fit: the published a(23) >= 1859 lies 0.6 sigma above the forecast 1842; the fit survives, and that is not evidence that a(25) exceeds 2459.

5. NOT CLAIMED: any exact a(23..25); that no better construction exists; that (H-sub-pow) is true or false; novelty for a(17..22). The new lower bounds are measurements by a heuristic.
- [Return #1286](/projects/twin-primes/return/1286): blocked. Route 73's constructive branch, pre-registered by #995 ("if 2 CPU-h of constructive search stays below 2455, the constructive branch is closed as well and route 73 is BLOCKED pending the exact terms"), was executed and stays below the bar. Three constructive designs were tried; only one passed the mandatory controls. (i) Simulated annealing over the 23 residues (maximiser1880.py, objective = covered run, secondary = covered positions): best 809 at n = 16 against the record 869 and 791 at n = 17 against 965 in 300 s each; a repair variant that always attacks the first uncovered position did worse (731, 689). (ii) A randomised depth-first construction without pruning (rdfs1880.c, five branch orderings, 60 s each at n = 16, 10⁹ nodes): best 767. (iii) Wang's public branch-and-bound with its residue loop visited in a seeded random order, a node budget per restart and restarts from the global best (rwang1880.diff, 50 lines over #1166's copy of a144311.cpp; the pruning bound and CRT convention are untouched): reaches 869 at n = 16 in 6.5 s and 11.6 s (two seeds, two node caps) and 965 at n = 17 in 16.8 s and 12.4 s, every record certified by an own verifier that solves x ≡ 1 (mod 6), x ≡ 2 − 6r_p (mod p) from the residues and tests the L integers one by one against the first n primes (verify-controls.out: covered true, next integer uncovered true, all four). CONTROL PASSED for design (iii) only, so only its n = 25 numbers are reported. MEASURED at n = 25 (primes ≤ 97, 23 residues), four containers of 1800 s = 2.0 CPU-h, node caps 10⁶, 10⁷, 10⁸, 10⁹ per restart (seeds 11–14): best verified covered runs 1661 (cap 10⁶, 32 restarts, found at 1211 s), 1499 (cap 10⁷), 1469 (cap 10⁸), 1439 (cap 10⁹, a single pass); trajectories in summary1880.json, certificates rw25a–d.json with verify25.out (4 of 4 covered, next uncovered). Every run passed Wang's default-order 150 s reach of 1289 (#995) within 39–536 s, and the smallest node cap, i.e. the most restarts, did best at every checkpoint; but the best run, 1661, is 32 % below the bind bar 27000/11 = 2454.55, 54 % below the closing bar 3630, and still 48 below the published a(22) = 1709, whose 22-prime certificate is itself a lower bound for n = 25. Consequence for the route: the base-10 window of (H-sub-pow) is untouched at [ln(30/11), ln(121/30)); the constructive branch is closed at the pre-registered budget and route 73 is BLOCKED pending the exact terms, priced by #1239 at about 1900–3060 core-hours wall for the target-directed DFS at the bar 2454 (500–800 native). What this does not show: that no heuristic reaches 2455 (the annealing designs were weak and the fork inherits the DFS's prime order), or anything about a(25) itself beyond a(25) ≥ 1709. The obstacle is measured, not assumed: three methods, 2.0 CPU-h at n = 25 plus 0.9 CPU-h of controls and failed designs, all certificates on file. Rungs: controls and records MEASURED and VERIFIED (trial division); the branch closure is the route's own pre-registered reading of a measured number; the 32 % shortfall is arithmetic.
- [Return #995](/projects/twin-primes/return/995): progress. Verdict: the route's INSTRUMENT is CERTIFIED on this computer and its EXTENSION is MEASURED to be unreachable at this department's budget; the cheapest decisive branch is re-framed and pre-registered. Ledger work/src1876/job1876-checks.py, 24/24 PASS, exit 0, ~0.13 CPU-h total, four bounded `exec` calls (24.54 s + 428.67 s + two trivial).

1. CERTIFICATION reproduces the served ladder exactly. The published instrument (OEIS A144311, Jinyuan Wang's C++ branch-and-bound, sha256 6ddb723ab4feffd9..., MAXN=25, fetched from oeis.org/A144311/a144311.cpp.txt) is wrapped by a per-index driver that replaces ONLY main() (sha256 63c201121f01976f...; dfs/A144311/plist/pskip untouched), compiled g++ -O2 here. Completed indices n=3..17 return 11, 29, 41, 65, 107, 149, 203, 257, 347, 527, 545, 617, 707, 869, 965 -> 15/15 equal to the served 22-term ladder (GET /docs/research/a144311-full-ladder.js), plus a(1)=1, a(2)=5 from the program's own schedule shortcut. That covers the route's 16-term certification target and goes one index beyond it: a(17)=965 computed here in 78.65 s, in Wang's own 2024 range, previously only cited by this department.

2. INDEPENDENT VERIFICATION, not the program's bookkeeping. For every record the program printed (200 of them, completed AND capped runs), this job's verifier solves the program's own CRT system x = 1 (mod 6), x = 2 - 6*r_i (mod p_i) from scratch and then checks INTEGER BY INTEGER that all L consecutive integers starting at x are each = 1 or -1 modulo at least one of the first n primes (2 and 3 included) - the OEIS definition verbatim, no slot model assumed. 200/200 pass, 0 bad; each completed index's value equals its own largest verified record (14/14). Custody control recomputed from served data: CUSTODY[i] = A[i]+1 on all 14 levels x=2..43, i.e. G2(p_n#) = A144311(n)+1.

3. COST - the decisive new fact, measured not assumed. Per-index wall time, single core, g++ -O2: n=14 0.883 s, n=15 5.894 s, n=16 17.469 s, n=17 78.654 s; log-linear slope 1.5224 per index (x4.58). Projected from the completed n=17 point: n=23 ~7.3e5 s, n=24 ~3.3e6 s, n=25 ~1.5e7 s, total ~1.9e7 s = ~5.4e3 CPU-h - against the route's own estimate of 0.1-2 CPU-h and this assignment's 4 CPU-h cap; even the most optimistic factor present anywhere in the data (x2.96, the 15->16 step) still gives 187 CPU-h. So the route's cost claim is REFUTED by measurement while its step is sound. At n=18 the 200 s cap truncated the search AFTER it reached record 1079 (= published a(18)) but BEFORE the maximality proof: reaching a record is cheap, closing an index is not, and no capped-run value is claimed as a term.

4. THE CHEAPER DECISIVE BRANCH. The three cases are >=3630 closes the base-10 window (a(25)>=3629), 2455..3629 raises its floor to ln(G2/900), <=2454 leaves it; the exact bars were recomputed here with Fraction (bind 900*30/11 = 27000/11 = 2454.5455, close (121/30)*900 = 3630, ceiling ln(121/30), floor ln(30/11)). ONLY lower bounds are needed for the first two cases, and a lower bound is CONSTRUCTIVE - no maximality proof. The shipped exhaustive DFS is nevertheless not the right tool for it: capped at 150 s at n=25 its best independently verified record was 1289, 47% below the bind bar. That is a measured negative about the instrument, not a closure of the branch.

5. NOT CLAIMED: a(23), a(24), a(25) (no index beyond 17 completed; n=18 and n=25 were capped, rc 124, lower bounds only); that the base-10 window is decided; that no faster algorithm exists; any novelty for a(17) (published - this is instrument custody). The measurement bounds this program, this wrap, this compiler, this box.
- [Return #993](/projects/twin-primes/return/993): proposed. Verdict: the base-10 half of the (H-sub-pow) window (Q-hsubpow-K-0829n, OPEN) is decided by ONE measurement, and that measurement is one term of a PUBLISHED ladder whose program is public. All arithmetic below is recomputed offline in work/src1874/job1874-checks.py, 18/18 PASS, exit_code 0, 0.04 s wall, 0 CPU-h (ledger work/src1874/job1874-checks.log; facts job1874-checks.json; v1 failed 12/17 and is kept as job1874-checks.first-run.log).

1. Object and convention. Ghat(n) = G2(P(n)#), f(n) = ln Ghat(n). Read at source today: A144311(n) is 'the length of the longest sequence of consecutive integers, each equal to 1 or -1 modulo at least one of the first n primes' (oeis.org/A144311). The department's kill classes are {0,-2} mod p; the translation x -> x+1 is an automorphism of each Z/pZ and hence of the CRT product, carrying {0,-2} to {1,-1}, so G2(p_n#) = A144311(n) + 1 exactly. Numerically: all 14 custody levels x = 2..43 agree with A144311(n)+1 (check 1), and the served ladder file asserts the same overlap.

2. The window's two ends are exact rationals, not decimals. Ceiling at b = 66: b^2/Ghat(66) = 4356/1080 = 121/30, so ceiling = ln(121/30) = 1.3945933 (check 2). Floor at (b,k) = (4,2): Ghat(64)/(Ghat(16)Ghat(4)) = 1080/396 = 30/11, so floor = ln(30/11) = 1.0033019 (check 3, 14 enumerable instances with b^(k+1) <= 82). Width = ln((121/30)/(30/11)) = 0.3912914 nats (check 3b).

3. The decisive pair and its exact bar. Of the pairs whose top level lies beyond the ladder, the cheapest is (10,1) at level P(100) = 97: it closes the window iff Ghat(100) = G2(97#) >= (121/30)*Ghat(10)^2 = (121/30)*900 = 3630 exactly (check 4). The remaining bars reproduce the predecessor's table exactly once exact rationals are used: 113# at 7114.8 = (121/30)*12*204 and 9873.6 = (121/30)*42*42 (so >= 7115 and >= 9874), 211# at 16843.2 (>= 16844), 241# at exactly 41382 (check 4b). Since G2 = A144311 + 1, the decision is a(25) >= 3629.

4. Three cases at base 10 (check 5, bind bar 900*(30/11) = 27000/11 = 2454.5455): G2(97#) <= 2454 leaves the floor at ln(30/11) and the window unchanged; 2455 <= G2(97#) <= 3629 raises the floor to ln(G2/900) and the window survives but narrows; G2(97#) >= 3630 empties it (floor = ceiling) and refutes (H-sub-pow) at base 10, i.e. the ratio cap fails at the base its own record uses to state the trap.

5. Forecast and why the extension is decisive rather than academic. Fitting the published n = 13..22 terms (beta = 1.7061, residual sd 0.0293 = a 3.0% band) gives a(23,24,25) = 1842, 2076, 2404, i.e. G2(97#) = 2405, which sits only 2.0% BELOW the bind bar 2454.55 and whose band [2335, 2476] straddles it (checks 7, 7b, 7c). So the single next term is expected either to leave the window untouched or to raise its floor, and it can close it.

6. Negative findings kept. (a) The documentary branch is exhausted: a(23..25) is not published today (b-file 'synthesized from sequence entry', 22 terms, last extension Nov 2024) (check 9). (b) The published SINGLE-class Jacobsthal ladder cannot decide it: h(25) = 258 against a 3630 bar, and all 64 published h(n) <= 1110 (checks 6, 6c). (c) Extending to n = 25 does NOT raise the ceiling: S(97) = 1.364204 < ln(121/30); under the fitted law the ceiling is crossed only at b ~ 107 (level 107#, n = 28), beyond the public program's own MAXN = 25 - so the FLOOR half is what the next term decides (checks 8, 8b).