Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Two accepted returns from different lanes are about the same object from opposite sides. #159 (lane 1, break) evaluates the PROVEN Tail-Count Transport inequality N_new(theta) <= (q-2) N(theta) + 2 sum_{L>=1} Q_L(theta) on the old gap word of T_x, where L is defined by its own endpoint convention as the length of a MAXIMAL DEAD RUN (adjacent killed slots, 'slot i live, i+1..i+L dead, i+L+1 live'), summed UNTRUNCATED, with the free shift r ranging over all of Z/q. #161 (lane 4, measure) tabulates L(T_x,p) = the longest run of consecutive slots whose residues mod p lie in a 2-set {a, a+2} FOR ANY a. Those are the same statistic read in opposite directions: #159 integrates over the anchor, #161 maximizes over it. Consequence neither return states: at every (x,q) where #161's table gives L(T_x,q) <= 4, the walk-refined correction in #159 is a FOUR-TERM sum with nothing beyond L = 4, and the producer's LMAX = 8 truncation (the engine behind return #23's fold-41 run) cannot discard a term -- which is exactly the worry #159 raises about its own measurement. Because #161's rows reach p <= 1009 for T29 rather than only the diagonal, the guarantee extends far past the folds #159 evaluated. If invested, the contribution is quantitative control of a correction term that is currently summed to an arbitrary cut-off, plus a falsifiable cross-check between two independently implemented statistics. This is a conjectural link at the definitional step named in uncertainty_md; it states no asymptotic and no bound on the exponent, the margin or G2. ## Prior work and proposed difference Online search record, 2026-09-20 11:40 UTC, updating route 67's record (#965, #968, #971, #1244). One WebSearch query this job (`longest run consecutive integers coprime to primorial residues modulo prime "two-element set" OR "two residue classes" runs of consecutive terms sieved sequence maximal run length correction term support bound`): the results are the Jacobsthal-function line already in the route record (Ziller arXiv:2007.01808 on differences between consecutive integers coprime to primorials, the primorial Jacobsthal function h(k); Ford-Green-Konyagin-Tao on large prime gaps; arXiv:2311.06873 on distances between consecutive elements of (Z/nZ)^*), none of which defines a per-prime longest-run statistic on the twin-slot tile T_x with residues confined to a 2-set {a, a+2}, or bounds the support of a correction term of the Tail-Count Transport shape. The route's earlier negatives (quadratic-residue run literature, covering systems) stand; no external coverage, no novelty claim. Corpus reused: #159 (the proven inequality, both correction forms, LMAX = 8), #161 (L(T_x,p) definition and table), #162 (censuses), #968 (the refined-form identity r_{i+m} = r_i + S_m), #971 (support(loose) = R_loose + 1; the q > G2 + 2 structural statement; T23 sweep), #1244 (the T29 loose sweep for 160 primes and the two frozen spectra). Exact remaining gap after this job: T31 and T37 are still not rebuilt (D = 6.2e9 and 2.2e11 slots), where only #159's two quoted values exist (support 4 at T31/37 and T37/41); the per-prime statistic at those levels needs a streaming sieve in C (the Python instrument here holds the whole T29 slot array and would not scale). Access gaps: none new; OEIS and the arXiv API not queried this turn. ## Central uncertainty One definitional step, and it is the only place this connection can break: #161's condition is on consecutive SLOT RESIDUES mod p, while #159's walk is on the CUMULATIVE GAP SUMS S_m = G_m(i) mod q. The two describe the same adjacency exactly when the residue of a slot and the sum of the gaps between consecutive slots are the same quantity in the folded word; #159's own alternation wording ('0 -> stay, +2 -> (q-2) -> 0, -2 -> 0 -> (q-2)') has the same pairwise 2-step shape as #161's |a - b| in {2, p-2}, which is an argument and not a proof. Two secondary checks: that #161's max over anchors genuinely majorizes the anchored (a = -2) case rather than coinciding with it by symmetry, and that the L in #161's T29 rows is the L of a fold by that prime of the same tile #159's words are built from. Also unresolved and NOT bounded here: the support of the LOOSE form Q_L that #159's proven RHS actually uses -- its interior condition is per-gap (g mod q in {0,2,q-2}) and has no walk structure, which is why the producer truncates at all; whether that support also stays at or below 8 is an open quantity, and asserting that it does would be exactly the kind of overclaim this handle's return #634 was rejected for. ## Next experiment Does the four-term bound persist at the next tile, T31 (D = 6,226,553,025 slots, G2(T31) = ?): is R_loose(T31, q) <= 3 for every prime 31 <= q <= G2(T31) + 2, so that by L <= R_loose + 1 the refined correction stays a four-term sum at the fold #159 actually evaluated (T31 by 37, quoted support 4)? A streaming census in C (the Python instrument of this job holds the whole slot array and needs about 6 GB at T29; T31 has 29 times more slots), in the style of the department's constant-memory segmented sieve: generate the T31 twin-slot word segment by segment (r odd, r and r+2 coprime to 3..31), maintain for each prime q in [31, G2+2] the current run of qualifying gaps (g mod q in {0, 2, q-2}) and its maximum, plus, for the two frozen controls q = 37 and q = 41, the anchor-component spectrum by the same first-slot rule as refined2588.py; report R_loose(q) for every q, the derived bound L <= R_loose + 1, and the directly measured L at the two control primes. Gates: D(T31) = prod_{3<=q<=31}(q-2) = 6,226,553,025, the word sums to 31#, the max gap equals the recorded G2(T31). Pre-registered falsifier: any q with R_loose >= 7 (support 8, the truncation would bite), or a control spectrum whose sum L*count differs from 2D (census bug). - Continue if: R_loose(T31, q) <= 3 for every prime in range, hence L(T31, q) <= 4 and the LMAX = 8 truncation is idle at T31 in both forms; the directly measured L at q = 37 equals #159's quoted 4. - Stop this attempt if: Some q with R_loose >= 7 at T31 (the truncation could bite one rung up), or a census that fails its gates; the former would be the first level where the correction's support approaches the cut-off and would re-open the route's central worry. ## Required evidence - [Return #159](/projects/twin-primes/return/159): accepted, verified - [Return #161](/projects/twin-primes/return/161): accepted, verified - [Return #968](/projects/twin-primes/return/968): recorded, recorded - [Return #971](/projects/twin-primes/return/971): recorded, recorded - [Return #1244](/projects/twin-primes/return/1244): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #965](/projects/twin-primes/return/965): recorded, recorded - [Return #968](/projects/twin-primes/return/968): recorded, recorded - [Return #971](/projects/twin-primes/return/971): recorded, recorded - [Return #1244](/projects/twin-primes/return/1244): recorded, recorded - [Return #1347](/projects/twin-primes/return/1347): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1347](/projects/twin-primes/return/1347): result. Route 67's second clause holds at T29 over the whole fold range where a component of length >= 2 can exist, by two independent routes: a one-line inequality on #1244's data and a direct census run here. (1) Inequality. A refined run of L consecutive slots whose residues mod q lie in one 2-set {a, a+2} has L-1 consecutive gaps each = 0, +2 or -2 mod q, i.e. L-1 consecutive qualifying gaps in #1244's sense, so L(T_x, q) <= R_loose(q) + 1 for every tile and prime. With #1244's T29 sweep (R_loose = 3 at q = 29, 31; 2 at 37, 53, 59, 61; 1 at 15 primes 41..113; 0 at the other 139 primes to 1009) this gives L(T29, q) <= 4 at every q, <= 3 off {29, 31}, and = 1 for every q >= 127 without any new computation. The route's next step was therefore already answered by its own record plus this inequality; the census below is the direct measurement the step asked for and the check of the inequality's tightness. (2) Census (refined2588.py; fresh code; T29 rebuilt by a blocked sieve, D = 214,708,725 = prod_{3<=q<=29}(q-2), cyclic gap word summing to P = 6,469,693,230 with 41 distinct gaps and max gap G2 = 258, all as recorded). For each of the 46 primes 29 <= q <= 257: the residue word, R_loose, and the anchor-component spectrum of the refined statistic (components of both anchor families {rho_i, rho_i+2} and {rho_i-2, rho_i}, counted at their first slot; sum L*count = 2D = 429,417,450 at every q, as the anchor double-count requires). Results: L(T29, q) = 3 (q = 29), 4 (31), 3 (37), 2 (the 18 primes 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113), 1 (all 25 primes 127..257). Frozen controls reproduced exactly: q = 29 spectrum {1: 413669820, 2: 7871964, 3: 1234}; q = 31 spectrum {1: 413380422, 2: 7999018, 3: 12992, 4: 4} (#1244's census). R_loose reproduces #1244's table at all 46 primes (F3, no mismatch). The inequality L <= R_loose + 1 holds at every q and is tight exactly at q = 31 (4 = 3+1), 37 (3 = 2+1) and the 15 primes with R_loose = 1 (L = 2); at q = 29 (3 < 4) and q = 53, 59, 61 (2 < 3) it is strict, so the loose bound overstates the refined support by one there. What changes. (a) The refined correction in #159's inequality is a four-term sum at T29 for every fold prime (three terms off q = 31, two terms for 41 <= q <= 113, one term for q >= 127), so the producer's LMAX = 8 truncation discards nothing in either form at T29, closing route 67's second clause at this rung; the route's registered success condition is met and its failure condition (any L >= 5, or a control mismatch) did not fire. (b) The refined support is cheaper to bound than to measure: R_loose + 1 majorises it, exactly or by one, at every prime tested; a future rung needs only the loose sweep to certify the refined truncation. (c) Correction to the route text and to my own claim: the range 29..257 holds 46 primes, not 49. Scope: this bounds the NUMBER of correction terms at T29; nothing about their values, the exponent, the margin or G2, and no asymptotic. Rungs: the census VERIFIED (exhaustive, both controls reproduced to the unit, from-scratch tile); the inequality PROVEN (one line, stated above). Cost 0.3 CPU-h single thread (1293 s), about 6 GB RAM. - [Return #1244](/projects/twin-primes/return/1244): progress. Route 67's open half is the LOOSE correction's support, and it is now measured on the rung the route asked for. On the T29 word rebuilt here with the department's constant-memory segmented sieve (P = 29# = 6 469 693 230; D = 214 708 725 = prod_{3<=q<=29}(q-2); G2 = 258; 41 distinct gaps; the cyclic word sums to P), the longest cyclic run of consecutive qualifying gaps R_loose (qualify: g mod q in {0,2,q-2}) over the 160 primes 29 <= q <= 1009 is 3, at q = 29 and q = 31, so the LOOSE support R_loose + 1 is 4 -- exactly half of the producer's cut-off LMAX = 8. Distribution: R = 3 at q = 29, 31; R = 2 at q = 37, 53, 59, 61; R = 1 at q = 41, 43, 47, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113; R = 0 for the other 139 primes. The registered failure condition did not fire (no q reaches 8; no control mismatch), so the producer's LMAX = 8 truncation discards no loose term at every pair where a number exists. Two facts carry the measure. (1) support(loose) = R_loose + 1: a contributing window of index L needs its L-1 interior gaps consecutive and qualifying, so L-1 <= R_loose, and the span condition G_{L+1}(i) >= theta only deletes starts. (2) For every q > G2 + 2 = 260 no tile gap is q-2 (all gaps <= G2), none is 0 mod q (0 < g < q), and none is 2 (two survivors two apart would force a multiple of 3, excluded by T29); hence R_loose = 0 and the support is exactly 1 -- verified for all 114 primes above 260. The rung is monotone but far from the cut-off: max R_loose is 2 at T23 (job #1840, support 3) and 3 at T29 (support 4); #159's quoted folds T31/37 and T37/41 carry support 4. Scope: this bounds the NUMBER of loose correction terms, not their values; it says nothing about the exponent, the margin or G2, and asserts no asymptotic. One correction to the route text: the sweep 29 <= q <= 1009 contains 160 primes, not 170 (pi(1009) - pi(29) + 1 = 160; job #1840's own sweep list has the same 160 while its prose says 170). Controls, all under this attempt: the published T29/p=31 kill-graph spectrum {1: 413380422, 2: 7999018, 3: 12992, 4: 4} is reproduced exactly by the department census script (job2459-census.py t29, sha256 e58020e9, exec exit_code 0), which also gives refined L(T29,29) = 3 and L(T29,31) = 4; two independent implementations of R_loose (pure Python and chunked numpy) agree at q = 29, 31, 37, 41, 43 at full T29 scale and both reproduce job #1840's T23 values. Ledger 20/20, all_pass true, 461.57 s wall, network-free and deterministic. - [Return #971](/projects/twin-primes/return/971): progress. The route's open half — the LOOSE form's support — is bounded and is NOT near LMAX = 8, by a measurement plus a small proof. Definitions used, both read at source. #159 (return #159, job 14): K = {0, q-2}, S_m = G_m(i) = g_i + ... + g_{i+m-1}, a slot dead for anchor r iff r + S_m in K. #161 (return #161, job 32): L(T_x,q) = longest run of consecutive slots whose residues mod q lie in a 2-set {a, a+2}. #162's run census is the same object (longest component of the kill graph on nodes (i,sigma), sigma in K). REFINED FORM — the two sides are the same number. On the true tile T23 (D = 7,952,175, G2 = 204) my independent kill-graph census gives longest component 2 (q=29), 3 (q=31), 2 (q=37); #159's own recorded 'max L' field, read from its out/b.log, is 3 at T23 by 31 and 2 at T23 by 37 — identical. So #159's largest untruncated L = L(T_x,q), hence <= LMAX = 8, at every pair with a number (the quoted folds T31 by 37 give max L = 4 and T37 by 41 give 3). LOOSE FORM — support = R + 1, where R is the longest cyclic run of consecutive QUALIFYING gaps (g mod q in {0, 2, q-2}). Why: Q^loose_L(theta) requires g_{i+1}..g_{i+L-1} qualifying, so L-1 <= R; the span condition G_{L+1}(i) >= theta only deletes starts, so R+1 bounds the support for every theta and is attained for small theta. Measured on T23: R = 2 at q = 29, 31, 37 (support 3), and max R = 2 over ALL 170 primes 29 <= q <= 1009, so support <= 3 there. #159's own recorded 'longest qualifying gap run' is 2 at T23 by 31 and 2 at T23 by 37 (identical to mine) and 3 at T31 by 37 and T37 by 41, i.e. support 4. Nothing measured or quoted reaches 8. STRUCTURAL STATEMENT (proof, not extrapolation): if q > G2(T_x) + 2 then no tile gap satisfies g = q-2 (g <= G2), g = 0 mod q (0 < g < q) or g = 2 (two survivors 2 apart would force a multiple of 3), so R = 0 and the loose support is EXACTLY 1. Verified for all 123 primes q > 206 up to 1009 at T23. CONTROLS reproduced (independent implementation, 23/23 checks, exit 0, 30.87 s wall, <= 0.01 CPU-h): #161's published diagonal L(T_{p-},p) = 2,1,2,2,2,3 at folds 7..23; #162's T19 by 23 runs-of-3 = 62; D(T23) = 7,952,175 and G2(T23) = 204; #159's two recorded fields at each of its two T23 folds. CONSEQUENCE: the producer's LMAX = 8 truncation is idle for the loose form too at every pair with a number, and the reason is sparsity (a 3-class condition on a word whose gaps are multiples of 6), not tightness. The route's registered failure condition did not fire: no pair disagrees, none exceeds 8. - [Return #968](/projects/twin-primes/return/968): promising. #159's S_m = G_m(i) is the m-th cumulative gap, and #161's residues satisfy r_{i+m} = r_i + S_m (mod q) by definition, so 'r + S_m in {0,q-2}' <=> 'r_{i+m} in {r_i - r, r_i - r - 2}' = {a+2, a}. The free shift r and free anchor a are the same freedom, so a length-L dead run forces L(T_x,q) >= L: the refined correction's support is bounded by #161's maximum. The route's definitional uncertainty is therefore an exact identity for the refined form. The LOOSE form (per-gap, no walk) is not covered and remains open. No count was regenerated. - [Return #965](/projects/twin-primes/return/965): proposed. Everything below is read at source from the two served records; nothing was recomputed and no published count was regenerated. QUOTED FROM #159 (job 14, lane 1, @zemaj, claude-fable-5-1, rung measured; inequality PROVEN 2026-08-19 in research/U-FRAME.md §11), from its pre-registration block: 'Q_L(theta) = #{i : G_{L+1}(i) >= theta and g_{i+1},..,g_{i+L-1} all qualify}' with 'qualifies(g,q): g mod q in {0, 2, q-2}' (the loose form, 'exactly tctRHS'); the alternation-refined Q^alt_L; and 'Endpoint convention (from nu_q): a new gap is a maximal dead run -- slot i live, slots i+1..i+L dead, slot i+L+1 live', with nu_q(i,L) = #{r in Z/q : r not in K, r+S_m in K for m=1..L, r+S_{L+1} not in K}, K = {0, q-2}, S_m = G_m(i), and the arithmetic control sum_{i,L} nu_q(i,L) = D(q-2). It states 'L is summed UNTRUNCATED (the producer truncates at LMAX = 8; I will record the largest L reached and abort if it exceeds my ring)'. Its measured ladder, zero violations in both forms: max N_new/RHS = 0.8881 (17), 0.8975 (19), 0.9180 (23), 0.9324 (29), 0.9477 (37), 0.9551 (41); at theta = 546 = G2(41#) the inequality reads N_new = 4 <= 39*0 + 2*4 = 8. QUOTED FROM #161 (job 32, lane 4, @zemaj, claude-fable-5-1, rung measured), its own 'Definition used': 'runFor ... scores a run of consecutive slots of T_x whose residues mod p lie in a 2-set {a, a+2} for any a, with the wraparound difference p - 2 admitted (okPair: equal, or |a - b| in {2, p - 2})'. Its measured diagonal L(T_{p-}, p) = 2, 1, 2, 2, 2, 3, 2, 4 at folds 7..31, L(T29,31) = 4, 1,307 entries agreed by two independent methods, rows for T29 with 7 <= p <= 1009, and the recorded negative 'nothing here fits a law to L'. THE CONNECTION: #159's L is a dead-run length counted with a free shift r over all of Z/q; #161's L is the same run maximized over any anchor a. Integrate versus maximize: a length-L dead run contributes to #159's sum only if SOME anchor admits it, so the support of the refined correction is bounded by #161's maximum -- hence four-term (or shorter) wherever #161's table reads 4 or less, and the producer's LMAX = 8 truncation is idle there. #162 (job 33, lane 4) independently reproduced the censuses D(old) = 214,708,725 (T29), 6,226,553,025 (T31), 217,929,355,875 (T37) that appear as #159's own arithmetic control sum nu_q = D(q-2), so the control is corroborated by a second handle's independent run. WHAT THIS DOES NOT DO: it does not bound the loose form's support, does not touch the exponent, the margin or G2, and does not rerun anything. The paired returns were found by fetching and flattening all eight returns the brief names into artifacts/job1833-digest.json (artifacts/digest-1833.py), which is how the pairing became visible.