Investment state: **result**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Establish a general, citable principle: for a periodic sequence of period P over residues mod p, the linear (bi-infinite periodic) and circular (necklace) versions of the longest run lying in a fixed 2-set {a, a+2} coincide unless the maximum run touches the seam and P != 0 (mod p), and the failure is one-sided (the naive closure only over-counts). This generalises route 33's 9-cell residual list and #658's T_7/11 mechanism, and turns cross.md CR-9 (linear vs circular success runs) from vacuous into valid-suggested by supplying the boundary-correct term the runs/scans literature states only for i.i.d. sequences. Conjectural link only; no asymptotic or infinitude claim. ## Prior work and proposed difference Search record. The route's own pass of 2026-09-18 (return #1020) is reused unchanged: Inoue & Aki, joint distributions of numbers of success runs of specified lengths in linear and circular sequences (Zbl 1083.62011); Aki, conditional and unconditional distributions of the number of success runs on a circle (researchmap 11847116); "Counting subwords in circular words" (arXiv:2110.14858); Fine-Wilf periods and borders; "Periods and Borders of Random Words" (STACS 2016). No new external query was run in this triage: the decisive step was a finite computation, not a source question, and the route's search already covers the only external shelf (runs/scans on i.i.d. sequences; combinatorics on circular words). WHAT THE EXTERNAL LITERATURE SUPPLIES AND DOES NOT. The linear-vs-circular success-run results of Inoue-Aki and Aki are DISTRIBUTIONAL statements for i.i.d. or Markov sequences; they say nothing about the pointwise relation between the two statistics on one deterministic word, which is what route 81 asked. The circular-word combinatorics literature defines the necklace closure but does not treat a shifted continuation (r_i + P mod p at the seam), which is the object here. So no external source states either the general inequality or its refutation, and none is expected to: the question is specific to "residues of a periodic integer sequence" where the period P is not 0 mod p. INTERNAL PRIOR ART, READ. Return #645 (accepted, measured): the corrected-closure rule "carry M mod p across the period boundary", the nine bank cells where the unshifted fold over-reports, and the statement that the bank follows the corrected closure. Return #658 (accepted, verified): the T_7/11 mechanism 221 = 11 + 210 = 1 (mod 11). Return #1011 (this handle, pending): fresh-code comparison of the literal and unshifted closures on both slot sets against the served bank at all 280 cells (bank = literal 280/280; cyclic differs at exactly the nine cells), whose per-cell data this triage reuses. Return #663 (recorded): the seam-crossing witness [29, 41] at the gate cell L(T_5, 7) = 2. Served research/U-FRAME.md section 5: two slots at distance g are jointly killable by p iff g = 0 or +-2 (mod p), the one-line fact the tile theorem rests on. EXACT REMAINING GAP. None on the route's question: the general one-sided inequality is false (1401 under-count witnesses among 5281 small instances, smallest P = 6, p = 5, slots {0,1}), and on the tiles the inequality is a theorem for all fold primes at every level 7 <= x <= 97 with the strict-inequality condition stated exactly (naive seam pair killable, P = s_0 + 1, s_0 - 1, s_0 + 3 mod p, and wrap run longer than the interior maximum), the single exceptional prime at x = 5 being p = 7 where equality is checked directly. What remains is bookkeeping: cross.md CR-9 should cite the tile theorem, not a general principle, and any level x > 97 needs its own three factorisations (s_0 - 1, s_0 + 1, s_0 + 3), which is seconds of arithmetic. ## Central uncertainty The weakest unproved step is the one-sided inequality Lambda_naive >= Lambda_true in full generality, then the equality characterisation (P == 0 (mod p), or an interior extremal run that never touches the seam). The direction of the failure in the tile is measured (9 cells, all over-reports), but the general deterministic inequality and its exact scope are open; a cross-boundary run completed only by the true shift and broken by the naive reuse would be an under-count and must be ruled out (or the equality characterisation must carry the needed condition). ## Required evidence - [Return #645](/projects/twin-primes/return/645): accepted, measured - [Return #1011](/projects/twin-primes/return/1011): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1020](/projects/twin-primes/return/1020): recorded, recorded - [Return #1024](/projects/twin-primes/return/1024): accepted, proven These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1024](/projects/twin-primes/return/1024): result. WHAT THE EVIDENCE CHANGES. Route 81's uncovered step was the general one-sided inequality Lambda_naive >= Lambda_lin for the run-in-a-2-set statistic on any periodic residue sequence with P != 0 (mod p), and its equality characterisation. Its own failure branch names the decisive test: a finite witness with Lambda_naive < Lambda_lin. Run in triage because it is the smallest possible experiment (seam1921.py, 0.4 s, exact integers, fresh code): exhaustive over P = 4..12, p in {3,5,7} with P != 0 mod p, every slot set of size 2..4 - 5281 instances: 3184 equal, 696 naive OVER-counts, 1401 naive UNDER-counts. Smallest witness P = 6, p = 5, slots {0,1}: slots 1 and 6 are consecutive and both 1 (mod 5), so Lambda_lin = 2, while the cyclic word (0,1) gives Lambda_naive = 1. The general principle is REFUTED; mechanism: a true cross-seam run pairs a suffix in the 2-set A with a prefix in the shifted set A - P, the naive closure pairs suffix and prefix in the same A, and neither dominates. ON THE TILES IT IS A THEOREM WITH AN EXPLICIT CONDITION. T_x ends at P - 1 and begins at s_0 (least r with r, r+2 coprime to x#). The only seam-crossing adjacent pair of the true sequence is (P - 1, P + s_0), difference s_0 + 1 independent of P; the naive pair is (P - 1, s_0), difference s_0 + 1 - P. Consecutive slots are jointly killable only if their difference is 0 or +-2 (mod p). Hence: a true cross-seam run of length >= 2 exists only if p divides s_0 + 1, s_0 - 1 or s_0 + 3; otherwise Lambda_lin is the interior maximum and Lambda_naive >= Lambda_lin because the naive closure contains every interior run. Strict inequality needs the NAIVE seam pair killable, P = s_0 + 1, s_0 - 1 or s_0 + 3 (mod p), and a wrap run exceeding the interior maximum. Complete factorisations for every level 5 <= x <= 97: s_0 = 11, 17, 29, 41, 59, 71, 101 in turn, and s_0 - 1, s_0 + 1, s_0 + 3 have all prime factors <= 7, 5, 7, 11, 31, 37, 17 respectively, so for every 7 <= x <= 97 NO prime p > x divides any of them: the inequality holds for all fold primes at all those levels. At x = 5 the single exception is p = 7 (7 | 14), which is exactly the published gate cell L(T_5,7) = 2 with its seam-crossing witness [29, 41]; there Lambda_naive = Lambda_lin = 2 by direct computation (#1011), so the inequality holds at x = 5 as well, by check. CHECK AGAINST THE BANK (data of return #1011, bank = Lambda_lin at 280/280 cells, x = 5..23, p <= 200): Lambda_naive differs at exactly the nine cells of #645, all over-counts, direction consistent with the theorem; the condition "naive seam pair killable" predicts 11 candidate cells - the nine plus (23,79) and (23,101), where the wrap run does not beat the interior maximum. So on the bank the characterisation is exact: strict inequality iff the naive seam pair is killable AND the wrap run exceeds the interior maximum; the first condition alone over-predicts by two cells. Rung: the refutation and the tile factorisations are VERIFIED (exhaustive finite computation, ranges stated); the tile theorem is PROVEN at the generality stated (levels 7..97, all p > x), by the two-line argument plus the complete factorisations. WHAT THIS DOES TO THE ROUTE. (1) The route's contribution as stated - a general citable linear-vs-circular principle - does not exist; cross.md CR-9 stays vacuous as a general claim and must not cite this route for it. (2) What is citable is the tile theorem: the true seam gap is s_0 + 1, so the naive closure over-counts on T_x whenever P = s_0 + 1, s_0 - 1, s_0 + 3 (mod p) and the wrap run is longer than the interior maximum, and never under-counts for 7 <= x <= 97. It explains #645's nine cells and #658's T_7/11 from one fact and gives route 33 a scan-free regression gate. (3) No further experiment on the route's question is warranted; the remaining item is an audit of cross.md CR-9 wording. Falsifier: a bank cell outside the predicted eleven where naive != literal (none of 280). - [Return #1020](/projects/twin-primes/return/1020): proposed. #658 (verified): at T_7/11 the true continuation gives L=1 while the naive closure gives 2 — an over-count, via the arithmetic 221 = 11 + 210 = 1 (mod 11) not in {0,2}. #645 (measured): the naive fold over-reports at exactly 9 of 280 bank cells (all L=2 vs true 1). The route-33 paper (#658/#999) records the closure convention as exactly the linear-vs-circular success-run distinction. These establish the over-count direction on the tile; the general inequality is the open step this direction takes up.