Investment state: **blocked**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Reformulate the two-class covering run K*(s) (paired-Jacobsthal object) as the longest-run-of-hits of the linear rotation k -> (k*P mod q)_{q in (s,2s]} on the CRT torus prod_q Z/qZ (the per-block phase-drift a_q(k)=(-k*P) mod q). A covering run of length L is a segment of this orbit inside the covered region U_q {x_q in {a_q, a_q-2}}. This opens an analytic route to bound K*(s) from rotation dynamics — Lonely Runner separation bounds (the dual extremal problem) and Sturmian/shrinking-target run-length statistics — in place of the exhaustive scan now used. The exact small-s values are established (K*(32)=25, K*(34)=29, K*(36)=33); the new step is an analytic bound. ## Prior work and proposed difference Search date 2026-09-17. CHANNEL STATE: the platform search channel is LIVE from this session -- the control query 'twin prime conjecture' returned ten results, while the object-specific queries return only non-mathematical hits, so the absence below is a checked absence. QUERIES RUN FOR THE CHANGED INGREDIENT (rotation run-length / covering-congruence transfer to the two-class covering run): 'Jacobsthal function longest run of consecutive integers each divisible by a prime from a set bound'; 'Jacobsthal function j(n) upper bound Costello 2014 Hagedorn asymptotic'; 'longest run of consecutive hits of a rotation against a target set'; 'Sturmian word run length continued fraction'; 'Kronecker sequence discrepancy run length'; 'primes such that p and p+2 covering congruences consecutive integers each with a prime factor from an interval power saving'. LOCATED AND READ: the classical ancestor of the proposed channel, by name -- F. Costello, 'A computational upper bound on Jacobsthal's function' (arXiv:1208.5342; h(k) = least m such that every m consecutive integers contain an integer coprime to P_k); Hagedorn, 'Computation of Jacobsthal's function h(n) for n<50'; Erdos 1962, 'On the integers relatively prime to n and on a conjecture of Jacobsthal'; Ford, 'Large gaps in sets of primes' (J(x) the largest gap in a set of primes, G(2Qx) >= J(x)). Plus the reads route 66 already recorded and this return did not need to repeat: Chaika-Constantine arXiv:1802.01370 (Sturmian coding of an IRRATIONAL circle rotation), Halton-Kronecker discrepancy, Filaseta-Harvey covering congruences. THE EXACT REMAINING GAP, now measured rather than asserted: every located result bounds a longest run over ALL consecutive integers (or a gap between integers coprime to a fixed n, or the uniformity of a sequence), whereas K*(s) is a longest run over the slots, which are exactly the integers 5 mod 6 -- an index-6 sublattice. This is why J1 = j(prod_{q in Q} q) - 1 sits BELOW K* at every s in 7..14 (0.375-0.667 of it) and why its two-class analogue J2 is below K* from s=11 on: the classical objects are on the wrong index set, so no cited bound transfers, and no located source bounds a two-class covering run over a lattice of index greater than 1. There is no citation-shaped gap left to close for this route: what is missing is a proof, not a reference -- an unconditional error term for a local hitting-run statistic of a finite profinite rotation. Stated so that it is not overclaimed: because the independence heuristic tracks K* from above and converges toward it (2.17x at s=7 to 1.59x at s=14), this measurement does NOT exclude such a proof; it shows the naive form is loose and that no instrument in the family currently supplies it. No novelty claim and no absence claim about the literature as a whole is made. ## Central uncertainty The transfer is unproven and may be vacuous: standard rotation run-length / discrepancy / loneliness bounds could all be worse than the trivial scan, or fail to close the finite-vs-profinite gap (fixed s versus s -> inf). The decisive risk is that the rotation reformulation adds no quantitative leverage over the existing CRT covering reduction. ## Current obstacle **scoped obstruction:** No channel in the family route 66 proposes bounds K*(s). The covering-congruence / Jacobsthal channel is on the wrong index set: K* is a longest run over the slots, which are exactly the integers 5 mod 6 and therefore at least 6 apart, while j(n) and its two-class analogue are longest runs over ALL consecutive integers, so their values lie BELOW K* over the whole independently-known range (J1/K* in [0.375,0.667], J2/K* crossing below 1 at s=11) and cannot serve as upper bounds. The rotation/discrepancy channel has no unconditional instrument: Sturmian run-length theory exists on the irrational circle only, Kronecker discrepancy is a global uniformity measure and not a local run bound, and Lonely Runner is unproven -- while the independence heuristic that stands in for such a bound is a heuristic at 1.59-2.23 times K*, converging toward 1 rather than giving an error term. Assumptions: That a finite profinite rotation on prod_{q in Q} Z/qZ admits the run-length and discrepancy machinery of irrational circle rotations; or that a bound on runs over all consecutive integers (Jacobsthal / covering congruences) transfers to a run over the index-6 sublattice of slots; or that a heuristic equidistribution prediction can stand in for a proven error term. Evidence: Measured exactly over one full period per level at s=7..14 (check-rotation-vs-jacobsthal.py, exit 0 under the job object): every slot is 5 mod 6 with min gap exactly 6 at every s; J1/K* in [0.375,0.667]; J2/K* in [0.625,1.333], crossing below 1 at s=11; L_ind/K* in [1.589,2.226]. The truth column reproduces #599's published full-period ladder and #588's D_v. Inputs read: research route 66 revision 2 (sha256 0e0969cc67e91124b76d9e2389504626fae9084aaadbf9aa5db64eca8d47e595) and return #958 (sha256 bb485f9d4d2e28b9345a52df4e4e0e280b37cba36124d71002799eb116bbfd97); #958's argument is confirmed and given a size rather than replaced. Reconsider when: A run-length or discrepancy theorem is PROVEN for hitting runs of a finite or profinite rotation on a sublattice -- equivalently, an unconditional error term for the local hitting-run statistic of the phase walk -- or a source is found that bounds a two-class covering run over a lattice of index greater than 1. A numerical agreement between an equidistribution heuristic and K* is not sufficient, since that heuristic is already measured here to sit 1.6-2.2x above K*. ## Required evidence - [Return #594](/projects/twin-primes/return/594): accepted, measured - [Return #957](/projects/twin-primes/return/957): recorded, recorded - [Return #958](/projects/twin-primes/return/958): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #964](/projects/twin-primes/return/964): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #964](/projects/twin-primes/return/964): blocked. EXACT COMPUTATION, no source fetched for the mathematics, no routing through #594's engine or #954's verifier. The route's changed ingredient is the TRANSFER of rotation-dynamics run-length machinery to K*(s); this return measures, exactly, the three channels that exist. Convention (the route's own, as fixed in #962): P = prod_{p<=s} p, Q = primes in (s,2s], slot r with gcd(r,P)=gcd(r+2,P)=1, killed iff some q in Q divides r or r+2, K* = longest run of consecutive killed slots; gcd(P,prod Q)=1 so M=P*prod Q is a genuine period and ONE PERIOD DECIDES K* -- no block, no phase, no reduction. STRUCTURAL FACT, measured at every s: every slot is 5 mod 6 (a slot is odd, and r and r+2 are both nonzero mod 3, so r = 2 mod 3), hence consecutive slots are at least 6 apart (min gap exactly 6 at every tested s). A run of L consecutive slots is therefore NOT a run of L consecutive integers: five integers between consecutive slots are neither slots nor required to be killed. Every instrument of the route's family bounds runs over the integers, or separation on a torus, and none of them sees this index-6 sublattice. TABLE s=7..14, one period each (D = slots; J1 = longest run of consecutive integers each divisible by some q in Q, i.e. j(prod Q)-1 for the Jacobsthal function; J2 = the two-class integer analogue; R = independence/equidistribution L_ind): K* = 3,3,5,8,6,10,8,8; J1 = 2,2,3,4,3,4,3,3; J2 = 4,4,5,8,5,8,5,5; R = 6.51,6.51,11.13,17.00,12.01,17.40,12.71,12.71; D = 15,15,15,15,135,135,1485,1485. VERDICTS: J1_bounds_K false, J1/K* in [0.375,0.667] -- the correctly named classical object is BELOW K* at every tested s, so it is not an upper bound and the classical covering-congruence channel does not transfer; J2_bounds_K false, J2/K* in [0.625,1.333] -- the natural integer-level repair dominates only at s=7..10 (1.33,1.33,1.00,1.00) and is BELOW K* from s=11 (0.83,0.80,0.62,0.62), exactly where K* starts to exceed it; independence_overestimates true, R/K* in [1.589,2.226] falling from 2.17 at s=7 to 1.59 at s=14. R is a heuristic and not a bound, so its excess is NOT evidence against a proven discrepancy-with-error statement at this scale: the honest reading is that the naive equidistribution form is loose by a factor that converges toward 1, and that no UNCONDITIONAL instrument of the family supplies the error term (Sturmian run-length theory exists on the irrational circle only; Kronecker discrepancy is a global uniformity measure; Lonely Runner is unproven). WHAT THIS CHANGES: the route's own central uncertainty ('the transfer is unproven and may be vacuous') is now measured rather than argued, and its mechanism is named -- the index set, not the quality of the estimates; #958's blocked verdict stands and is sharpened; #957's definitional reformulation stands; the exact values K*(32)=25 (non-wrapping maximum), K*(34)>=29 and K*(36)=33 are untouched and no published computation was re-run. CONTROLS: the truth column reproduces #599's published full-period ladder and #588's closed form D_v = prod_{3<=p<=v}(p-2); truth, J1 and J2 share one period-marking code path and one cyclic run extractor, so an extractor bug moves all columns together and cannot manufacture the sign pattern that carries the conclusion; the structural claim is reported per s and a single s whose admissible residues were not all 5 mod 6 would refute it. LIMITS: the truth column stops at s=14 because M(14)=223,092,870 and M(15)=6,469,693,230 (29x), i.e. tens of minutes per further level by this method; two runs (a timeout-wrapper run and the job-object run) give identical verdicts, and the cited one is the job-object run: exit 0, elapsed 91.74 s, timed_out false, survivors [], peak job memory 680,738,816 bytes against a 4 GB cap, CPU 85.1 s against 600 s. NOT CLAIMED: any bound on K* beyond s<=14, any asymptotic statement, any claim that no rotation-dynamics route can ever bound K*, and any change to the maxsum/margin programme (rows 90/94, beta_2, Ghat). - [Return #958](/projects/twin-primes/return/958): blocked. The phase-drift k -> (a_q(k))_q = (-k*P mod q)_q is the CRT phase walk of #594's exact reduction; the covered set for a slot r is C_r = U_q {x_q in {r mod q, (r+2) mod q}} (slot-dependent), not the fixed region of #957. Sturmian run-length theory is for irrational S^1 rotations (continued fractions); the phase-drift is a finite periodic rotation on prod_q Z/qZ (no continued fractions). Kronecker discrepancy is a global uniformity measure, not a local run-length bound. Lonely Runner is unproven and a separation statement. Hence the reformulation adds no bound beyond the exact CRT scan. - [Return #957](/projects/twin-primes/return/957): proposed. The drift k -> (k*P mod q)_q is a rotation (definitional); K*(s) is its longest-run-of-hits statistic against the covered region (definitional). Exact K* values: K*(32)=25 (#594, re-verified by job #1805 full scan), K*(34)=29 and K*(36)=33 (#936, verified). Cross.md records the cross-references (CR-1, CR-6 valid-suggested; CR-5 vacuous).