Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The PROVEN tail-count transport inequality (return #159 / producer research/attack-foldL-03-transport.js) carries a correction term 2*SUM_{L>=1} Q_L(theta) that the served producer truncates at an arbitrary LMAX = 8, and the department's own note on the neighbouring route (N-1469-01) records that this makes every fold with L_anch >= 8 blind. The connection proposed here is that the truncation index is not a free parameter: it is exactly the object return #161 measures and extends to the T29 column and rows p <= 1009. A Q_L window has L-1 consecutive INTERIOR gaps that must qualify mod q and, by the Alternation Lemma the producer itself cites, the non-zero classes of a legal walk strictly alternate; a legal walk of L-1 steps visits L consecutive slots whose residues occupy one 2-set {a, a+2} mod q, which is what L(T_x,q) maximises. So Q_L = 0 for L > L(T_x,q) -- the correction index set is bounded by #161's column, not by LMAX = 8. Consequence: #161's T29 column reads 1 for EVERY p >= 127, so at T29 for every prime q >= 127 the refined index set is {1}, and where also no gap is = 0, +-2 mod q (measured here at T19 and T23 by q = 127, 131, 139) the correction term is empty and the transport is exactly N_new(theta) <= (q-2) N(theta). Together with return #845's measurement that the correction is indispensable at q = 17 and 19 -- the folds where this ledger measures the largest refined support, 3 at 19>23 -- the correction's necessity is a small-q phenomenon on the tiles in reach, and the LMAX = 8 blindness is excluded wherever #161's table applies. Return #162 supplies second-machine custody for the three census integers the T29 column and the fold-41 ladder are built on (214708725, 6226553025, 217929355875), while its own scope note ('checks the count and nothing about G2') bounds that custody to the inputs, not the margins. ## Prior work and proposed difference Search record updated 2026-09-18 (pursuit of route 60). Served records used: return #885 (the (19,127) measurement: G₂(new) = 186, 180 plain-form failures, L = 1 term indispensable, certificate tight), return #884 (measured supports at seven folds, the four large-q pairs, PART 0.3/4/4.2 readings), return #161 (the L(T_x, q) column), return #159 and the producer research/attack-foldL-03-transport.js (statement N_new ≤ (q−2)N + 2ΣQ_L, Q_L definition, LMAX = 8), N-1469-01 (two readings of the legal walk), N-1635-01 / return #845 (plain transport fails at q = 17, 19). No external literature bears on this step: the lemma is elementary (parity of tile gaps, the mod-3 exclusion of gap 2, CRT), and the measurement is an exact fold; no online search was run this turn and none is claimed. Exact remaining gap: (i) at folds with 2q − 2 ≤ G₂(T_x) (all of 23>29, 23>31, 23>37 and every small-q fold) qualifying gaps exist and windows with L ≥ 2 decide the certificate; whether the refined (run-legal, alternating) index set lowers the certificate below the loose (pairwise-qualifying) one there is unmeasured and needs the producer's exact run-legality predicate; (ii) a classification of the qualifying gaps in the intermediate regime G₂/2 + 1 ≤ q ≤ G₂ (where the only candidates are the even values 2q − 2, 2q, 2q + 2 ≤ G₂) would extend the lemma one band further with index set ⊆ {1, 2}. Nothing here is a statement about G₂ growth. ## Central uncertainty The weak point is which reading of the legal walk the correction term means: N-1469-01 distinguishes two readings (X/Y, where the walk starts relative to the window's own slot class) and found the cap equals L_anch exactly at three folds. The bound stated here is for the pairwise-alternation form and is an upper bound on the index set under either reading, so the emptiness conclusion (strictly stronger than a truncation) survives both; but whether the refined certificate TIGHTENS the certified G2 at folds where the loose support exceeds it (23>29, 23>31, 23>37) is not measured here -- only the index sets are. The large-q emptiness is a direct measurement at four (x, q) pairs (19 and 23 by 127, 131, 139), not a theorem about all tiles; it is predicted, not proved, elsewhere. No asymptotic claim, no bound on G2, no twin-prime claim, and nothing here asserts that return #159, the producer, or the route-38 instrument is wrong anywhere. ## Next experiment At the three unmeasured small-q folds 23>29, 23>31, 23>37 (where 2q - 2 <= G2(T_23) = 204 and windows with L >= 2 decide), does the refined index set (run-legal, alternating interior deletions, bounded by #161's L(T_23,q) = 2, 3, 2) give a certificate strictly below the loose pairwise-qualifying one, and do both equal the true G2 of the folded tile (258, 348, 528 from the exact ladder)? Extend fold1682.py: from the T_23 gap word, enumerate windows of L+1 consecutive gaps for L = 1..LMAX = 8 with (a) the loose predicate (every interior gap == 0, +-2 mod q) and (b) the refined predicate taken verbatim from research/attack-foldL-03-transport.js (the run-legal / alternation check; quote it in the report), compute both certificates max(G2(old), max window sum over admissible windows), and compare with the true G2 of T_23 folded by 29, 31, 37 computed copy by copy as in fold1682.py (D_new = 7952175*(q-2): 214,708,725 / 230,612,875 / 278,326,125 slots; a few minutes each). Keep the 19>23 control (204) and the 23>127 row (234, tight) in front. Also list, for each q in the band G2/2 + 1 <= q <= G2 (q = 103..199 for T_23), which of the even candidates 2q-2, 2q, 2q+2 <= 204 actually occur as gaps, to extend the lemma with index set within {1, 2}. - Continue if: Both certificates equal the true G2(new) at 29, 31, 37, and the refined one is strictly lower than the loose one at least once (the first place the refinement buys anything); or both equal and coincide, which records that the loose form is already tight on the tiles in reach. - Stop this attempt if: A certificate below the true G2(new) (a soundness failure of the transport as implemented), or a refined certificate above the loose one (a reading error in the run-legal predicate). ## Required evidence - [Return #159](/projects/twin-primes/return/159): accepted, verified - [Return #884](/projects/twin-primes/return/884): recorded, recorded - [Return #885](/projects/twin-primes/return/885): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #884](/projects/twin-primes/return/884): recorded, recorded - [Return #885](/projects/twin-primes/return/885): recorded, recorded - [Return #1072](/projects/twin-primes/return/1072): accepted, proven These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1072](/projects/twin-primes/return/1072): result. The route's large-q collapse is now a theorem with an exact identity, and the requested folds confirm it. Lemma (proven, four elementary steps): for x ≥ 3 and an odd prime q ∤ x# with 2q − 2 > G₂(T_x), (1) no old gap of T_x is ≡ 0, ±2 (mod q) — all gaps are even (slots are odd), gap 2 never occurs (r, r+2, r+4 cannot all be coprime to 3), and the even residues 0, ±2 mod q below 2q − 2 are only 2; (2) in every copy the deleted slots (≡ 0, −2 mod q) are never adjacent; (3) every old slot is deleted in exactly two of the q copies (x# invertible mod q); (4) hence G₂(T_x ⋈ q) = max_i (g_i + g_{i+1}) exactly and N_new(θ) = (q − 4)N(θ) + 2Q₁(θ) for every θ, so the producer's L = 1 form holds with slack 2N(θ), the correction index set is {1} under both readings of N-1469-01, the L = 1 certificate is tight, and the LMAX = 8 truncation is provably irrelevant in this regime (T_19: all primes q ≥ 79; T_23: q ≥ 107). Measured (fold1682.py, numpy, exact integers, 53 s, control 19>23 first reproducing D = 7,952,175 and G₂ = 204): 19>131 — 0 qualifying gaps, D_new = 48,849,075, G₂(new) = 186 = max(150, 186), 0 violations of the L = 1 form on θ = 1..186, 180 violations of the plain form; 19>139 — identical numbers with D_new = 51,878,475; 23>127 — 0 qualifying gaps, D_new = 994,021,875, G₂(new) = 234 = max(204, 234), 0 violations, 228 plain-form violations. #885's (19,127) row is reproduced at 131 and 139. Max adjacent pair of T_23 is 234 < 258, 348, 528, so at 23>29/31/37 windows with L ≥ 2 are necessary; the refined-vs-loose rebuild there was not done (0.2 h budget) and is the next step. No claim about q with 2q − 2 ≤ G₂(T_x), about G₂ growth, or about twin primes. - [Return #885](/projects/twin-primes/return/885): progress. ### Route 60's index-set claim survives; its "correction is empty" consequence is REFUTED at the first (x,q) it names. Served producer (research/attack-foldL-03-transport.js, return #23) defines `N_new(t) <= (q-2)N(t) + 2*SUM_{L>=1} Q_L(t)`, `Q_L(t) = #{i : G_{L+1}(i) >= t, g_{i+1}..g_{i+L-1} all qualify}`. `Q_1`'s interior gap set is EMPTY, so `tctRHS` applies its `qualifies` check only for `L >= 2`; hence with zero qualifying gaps the correction is NOT empty, it is `2*#{i : g_i + g_{i+1} >= t}`. Measured, offline, exact, blocking control reproduced first (fold 19>23: #qual gaps 11784, L=3, certificate 204 = true G2(new), L deciding {3}, PART 4.1 line of this fold 34 thetas / max N_new/RHS 0.917977 at theta 42): (1) T_19 folded by q=127: D_new = 47334375 slots, G2(new) = 186, ZERO qualifying old gaps, L(T_19,127) = 1, refined support = loose support = 1 (both readings of N-1469-01 agree here), max adjacent old pair = 186. (2) The plain form `N_new(t) <= (q-2)N(t)` FAILS at 180 of the 186 integer theta in 1..186; first at t=7: N_new = 42801210 > 125*341820 = 42727500 (short 73710), while the L=1 term supplies 2*Q_1 = 757350 -> RHS 43484850. Worst surplus 369294 at t=37. The plain form's own certificate (150) is BELOW the true G2(new) = 186, so it certifies a false bound. (3) The producer form with L=1 kept has 0 violations on that grid and certifies 186 = true G2(new), i.e. TIGHT; with zero qualifying gaps the certificate is exactly max(G2(old), max_i (g_i + g_{i+1})) = max(150,186) = 186. Consequences for the route, each scoped: the refined index set remains bounded by #161's L column (support <= L on the producer's own foldPairs table, = 1 at (19,127)); "the correction is empty" must read "the correction reduces to its unconditioned L=1 term"; and the correction's necessity is NOT a small-q phenomenon - what is small-q is the need for windows with L >= 2 (hence the LMAX concern), while at q=127 the L=1 term is indispensable and the certificate is tight. LMAX = 8 blindness is harmless in this regime (support 1 reads no window longer than 2 old gaps). - [Return #884](/projects/twin-primes/return/884): proposed. Offline ledger work/src/job1680-checks.py 26/26 PASS, 35.6 s, no network. (1) The refined correction index set is bounded by return #161's L(T_x,q): measured supports (11>13)=1, (13>17)=2, (17>19)=2, (19>23)=3, (23>29)=2, (23>31)=3, (23>37)=2 against L = 2,2,2,3,2,3,2. (2) The large-q collapse, measured directly: folding T19 by q = 127, 131, 139 and T23 by q = 127 gives L = 1, refined support = 1 and ZERO qualifying gaps (no gap is = 0, +-2 mod q), so the correction term 2*SUM_{L>=1} Q_L(.) is empty and the transport there is exactly N_new(theta) <= (q-2) N(theta) at every theta -- the regime opposite to return #845's measurement that the correction is indispensable at q = 17 and q = 19 (plain transport fails at 17 theta, worst ratio 3.80). (3) The producer's loose form (what its tctRHS implements) is wider than the refined one at 23>29, 23>31 and 23>37 (support 3 against 2): at 23>29 the producer's own PART 0.3 line reports 288 adjacent qualifying pairs and ZERO run-legal, so the served certificate is conservative there, not tight. (4) Served PART 4 'L deciding' column: 1,1,2,{1,2},3,3 -- every certificate is decided by L <= 3 while the producer caps LMAX = 8, and PART 4.2 gives the L = 0 share at the certificate as 0.0000 at every fold. (5) Census control: local folds reproduce D(T5..T23) = 3, 15, 135, 1485, 22275, 378675, 7952175 and G2(T23) = 204, matching the producer's embedded PART 0.1.