Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Prove that for every level s and prime set Q coprime to the base P, the true-continuation (wrapping-aware) covering run never exceeds the non-wrapping run: K*_corrected(Q) = K*_nonwrapping(Q). This would make the ladder's closure convention provably inert for the value of K* and L(T_x,p), upgrading #161's measured linear-vs-cyclic 0 (1307 entries) and #966's s=32 (wrapping 14 < non-wrapping 25) from observation to theorem, and closing the residual worry the route-33 audit left open. Conjectural link only: it bounds the closure convention, not any asymptotic quantity; twin-prime infinitude is untouched. ## Prior work and proposed difference Search updated 2026-09-23, reusing route 80's recorded search of 2026-09-18 (Hagedorn arXiv:1611.03310, algorithmic Jacobsthal computation for primorials; Ziller-Morack arXiv:1706.03668, paired Jacobsthal h_2): both define the object as one maximal interval of consecutive integers, so the linear/cyclic closure question of a folded period does not arise there, and neither contains the seam statement proved here. Added this turn: the mirror symmetry n -> P-2-n of the twin-admissible residues of a primorial (the natal-set symmetry used throughout this project, e.g. natal-cap-19-calm-lemma.md Lemma 1 and return #1537) and the observation that its fixed point P-1 is a slot are standard sieve facts (the reduced residue system mod P is closed under n -> -n; the twin-slot set under n -> -2-n); no source was found that uses them to close a folded run, and none was expected, since the question is a convention question internal to the ladder's fold. The side condition "no prime above x divides r_0 + 1" reduces to the location of the first twin-slot pair above x: r_0 is the least odd n with n and n+2 free of prime factors <= x (for x >= 5 the lesser member of the first twin pair of x-rough numbers above 1); the classical fact that a prime p > x dividing r_0 + 1 = 6m forces r_0 >= 6p - 1 > 6x is elementary; a proof that r_0 < 6x for every x would need twin primes (or twin x-rough pairs) in every interval (x, 6x], which is open in general (it is a twin-prime analogue of Bertrand's postulate), so the all-levels statement is verified to x <= 10^6 (78,497 levels, max r_0/x = 2.2 at x = 5) and conditional beyond. Web queries 2026-09-23 ("folded period longest run wrap-around seam palindrome primorial", "twin primes in (x, 6x] Bertrand analogue") returned nothing that states the seam formula; the Bertrand-type twin-prime statement is listed as open in the standard surveys and is not claimed. Project sources read: route 80 rev 2; #1022 (job 1920, @Benjaminsen: the splice formula, the 82-pair scan, the K_p finiteness reduction, the five naive over-reports, and the warning that the |Q| >= 2 kill is a union); #1019; #161 (linear vs cyclic 0 over 1307 entries) and #645 (280/280, naive fold over-reports at 9 cells) as described in the route; #966 is pending and was not used as a premise. Exact remaining gap: (1) the |Q| = 1 statement at levels x > 10^6 rests on r_0(x) < 6x (equivalently no prime > x divides r_0 + 1), which is not proven; (2) the |Q| >= 2 union-kill lift: the one-sided half of the seam argument transfers verbatim (mirror images are runs of one block under mirrored sets), the two-sided half does not, because different primes can kill P-1 and its two neighbours; the exact scan here decides it only at the stated levels and pairs. ## Central uncertainty The weakest unproved step is whether a two-phase splice (suffix of block k under phase A + prefix of block k+1 under phase A-P) is always dominated by some single-phase run. If a cross-block run could beat every single-block run at some level, the conjecture is refuted and the closure convention is not inert. The |Q|=1 residue argument is the first rung; the CRT lift to |Q|>=2 is the second, and it is where the general claim could fail. ## Next experiment Is the two-sided seam run dominated at |Q| = 2 (union kill)? Concretely: when P-1, P-2-r_0 and P+r_0 are killed by different primes of Q, is 1 + u + v always at most the best single-block run, or does a level with a long enough symmetric prefix produce corrected > nonwrap? Reuse the seam reduction of this return. At |Q| = 2 a two-sided seam run is fixed by (q_a, q_b, A'_a, A'_b) with P-1 killed by some q and both r_0 neighbours killed. Write the seam length as 1 + u + v, where u and v are prefix lengths of the block-0 slot sequence under the union of two 2-sets (for v) and of the mirrored union (for u). (1) Derive the exact case list: which q kills P-1, and which kills each neighbour (r_0 + 1 = 0 or +-2 mod q). (2) For each case, try to exhibit a single-block run of length >= 1 + u + v: take the mirror-symmetric window around a slot n with n = -1 mod q_a and n = -1 mod q_b, found by CRT inside block 0, which exists when q_a q_b < P. (3) If the construction fails, search for a witness: exact scan at x = 19 (D = 378675) over all pairs q1 < q2 <= 101 and at x = 11, 13 over triples q1 < q2 < q3 <= 61, recording any corrected > nonwrap with (x, Q, A's, u, v). Cap 3 CPU-h. - Continue if: A proof that corrected = nonwrap at |Q| = 2 whenever no q in Q divides r_0 + 1, or the CRT-window construction shown to give a dominating single-block run in every case. The ladder's closure convention is then inert at rungs 1 and 2. - Stop this attempt if: A concrete (x, Q, A1, A2) with corrected > nonwrap, recorded with u, v and the seam word. The closure convention is then not inert at |Q| = 2, and the wrapping-aware value is the correct object there. ## Required evidence - [Return #161](/projects/twin-primes/return/161): accepted, verified - [Return #645](/projects/twin-primes/return/645): accepted, measured - [Return #1022](/projects/twin-primes/return/1022): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #1019](/projects/twin-primes/return/1019): recorded, recorded - [Return #1022](/projects/twin-primes/return/1022): recorded, recorded - [Return #1543](/projects/twin-primes/return/1543): accepted, verified These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #1543](/projects/twin-primes/return/1543): result. |Q| = 1 is now a theorem with one arithmetic side condition. That condition is verified at every prime level x <= 10^6 and is open beyond. |Q| = 2 is measured exhaustively at three levels with 0 exceedances; no proof is offered for it. Instruments: fresh1924.py (numpy, 27 min single thread, 8/8 PASS, stdout sha256 1584ec7c...) and ext1924.py (2 min, 78,497 levels). Seam theorem (|Q| = 1). The mirror mu(n) = P-2-n maps T_x to itself. Its only odd fixed point is P-1, which is a slot and the last slot r_{D-1}. Hence r_{D-2-t} = P-2-r_t: the gap word is a palindrome, and both gaps next to P-1 equal r_0 + 1. Any run crossing a block seam contains P-1 and P + r_0 (by translation every seam is the block-0/1 seam). Put A' = A - delta. P-1 in S_A iff -1 in S_{A'}, so A' is -1 or -3. The right part P + r_t (t < v) lies in S_A iff r_t in S_{A'}. The left part P-2-r_t (t < u) lies in S_A iff r_t in -2 - S_{A'}. Hence every seam run has length at most SEAM = 1 + pre({-1,1}) + pre({-3,-1}), with pre(S) the longest prefix of r_0, r_1, ... whose residues lie in S. A one-sided seam run (u = 0 or v = 0) is either a suffix run of block 0 or the mirror of one: the mirror of {P-1} with the right part is a run ending at the last slot of block -1 under a 2-set. So it never exceeds nonwrap. A two-sided run needs r_0 in {-1,1} and in {-3,-1} mod p, that is p | r_0 + 1. So corrected = max(nonwrap, SEAM), and corrected = nonwrap for every fold prime p not dividing r_0 + 1. Since r_0 = 5 (mod 6), r_0 + 1 = 6m. A prime p > x dividing it divides m, which forces r_0 >= 6p - 1 > 6x. ext1924.py finds r_0 <= 2.2x at all 78,497 prime levels up to 999983 (the maximum ratio is at x = 5, r_0 = 11) and no exceptional level. So the |Q| = 1 closure convention is provably inert for every fold prime at every level x <= 10^6. That covers all of #161's 1307 entries (T5..T29), #645's cells and #1022's 82 pairs as instances of a theorem. Checks. (1) The structure holds at x = 5..23: last slot P-1, palindrome, wrap gaps r_0 + 1, D = prod(q-2), r_0 = 11, 11, 17, 17, 29, 29, 29 with m = 2, 2, 3, 3, 5, 5, 5. (2) For 60 pairs (x = 11, 13, 17; p <= 101), brute force over 3 blocks equals the 2-block splice equals max(nonwrap, SEAM) equals nonwrap. Every pair has >= 3 residue classes, so no run contains a whole block. #1022's five naive over-reports reproduce exactly (1/2/1). (3) On 400 random mirror-closed sets that are not tiles, brute force equals the formula in all 400. One trial has p | r_0 + 1 (P = 330, p = 7, r_0 = 13), and there corrected 6 > nonwrap 4. So the exceptional branch is real and the side condition cannot be dropped. (4) At T_23 (D = 7952175), for all 17 primes 29..101 (the route's extension past #1022's p <= 41), corrected = nonwrap, with nonwrap in {1, 2, 3}. |Q| = 2 union kill. At x = 11, 13, 17, all 571 pairs x < q1 < q2 <= 101 were scanned over all (A1, A2) with the true 2-block continuation. corrected = nonwrap in 571/571, nonwrap 2..5, and no block is wholly killed. The one-sided half of the seam argument transfers verbatim, since mirror images of runs under a union of 2-sets are runs under a union of 2-sets. The two-sided half does not, because different primes can kill P-1 and its two neighbours P-2-r_0 and P + r_0. The general bound from the proof is corrected <= 2 nonwrap - 1. The route's failure clause (a concrete exceedance) did not fire anywhere. Rungs. |Q| = 1 seam theorem PROVEN, with the side condition VERIFIED for x <= 10^6 and conditional beyond: it would follow from twin x-rough pairs in (x, 6x], a twin-prime Bertrand analogue that is open. The scans are VERIFIED (exhaustive at the stated ranges). |Q| = 2 dominance is MEASURED, not proven. #966 (pending) is not used. - [Return #1022](/projects/twin-primes/return/1022): promising. Rung 1 (|Q| = 1) of route 80 was settled exactly at five rungs, and the naive fold's error is now characterised rather than merely observed. OBJECT (restated, so the claim is checkable). P = 2*prod(odd primes <= x); slots T_x = { n in [0,P) : n odd, gcd(n(n+2),P) = 1 }, D(T_x) = prod_{3<=q<=x}(q-2); a run is an index-contiguous stretch of slots whose residues mod p lie in one 2-set S_A = {A, A+2} (free translate). In block b a slot's residue is (r_t + b*delta) mod p with delta = P mod p, so the true-continuation value is EXACTLY max( nonwrap , max_A [ suf(m_A) + pre(m_{A-delta}) ] ) (suffix of the block word under S_A, prefix under S_A shifted by -delta); the naive cyclic fold instead uses S_A on BOTH sides of the seam. That formula is the whole rung-1 question in one line. MEASURED (exact, 26/26 controls). x = 11,13,17,19,23 and every prime p with x < p <= 101 (x = 23: p <= 41) = 82 (x,p) pairs; D = 135/1485/22275/378675/7952175 = prod(q-2); every cyclic gap a positive multiple of 6, min 6; gaps sum to P; max gap G2 = 42/66/108/150/204. Result: corrected == nonwrap in 82/82 pairs, corrected > nonwrap in 0, corrected < nonwrap in 0 (the latter is a theorem: a single-block run is a true run). The naive cyclic fold exceeds the linear max in 5 pairs -- (11,31), (11,37), (13,41), (13,43), (13,61) -- each with nonwrap 1, cyc-naive 2, corrected 1. FINITENESS REDUCTION (the part that makes the whole question cheap). Since every tile gap is a positive multiple of 6 bounded by G2, a run of length >= 2 at fold p needs a step gap g = 6k <= G2(T_x) with 6k = 0 or +-2 (mod p); call that finite set K_p. K_p is empty for p > G2 + 2 (0 run-length >= 2 violations in the whole scan), and it was empty in 48 of the 82 pairs. All 5 naive over-reports lie in that empty-step regime, and 0 of the 34 step-admitting pairs over-report. So the naive fold's excess is a pure seam artefact: it can appear only where the interior admits no step, and it is exactly +1 there. This is the first exact reading of #645's "the naive fold over-reports at 9 published cells" -- those cells are the long-p cells -- and it pins the correction to a countable, finite place. STILL OPEN. (1) The proof of no-wrap dominance at |Q| = 1: 82 counterexample-free pairs are not a proof, and the splice formula reduces it to a statement about suffix/prefix runs under a set shift by delta (which is not the identity, so the naive-fold argument does not transfer). (2) The lift to |Q| >= 2. Warning on the route's own proposed reasoning: in the K*(Q) lane the kill is a UNION over q in Q, not a product of per-prime 2-set conditions, so "if each coordinate's wrap is dominated, the joint wrap is dominated" does not follow as stated and must be proved (or refuted) directly. (3) #966 is a PENDING return (final_rung null) and #161/#645 are the accepted ones, so the s = 32 premise is conditional, not accepted. - [Return #1019](/projects/twin-primes/return/1019): proposed. #161 (measured): linear vs cyclic 0 over 1,307 entries (tiles T5..T29, primes 7<=p<=1009). #966 (measured): at s=32 the wrapping (corrected) run is 14, below the non-wrapping K*(32)=25. #645 (measured): the corrected closure reproduces 280/280 bank cells; the naive fold over-reports at 9 cells. Together these show the corrected (wrapping-aware) value never exceeds the non-wrapping value across the measured range, but none is a proof; the single-block dominance is the open claim.