Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The supply of gaps that qualify at a fold - the gaps of T_(x-1) with g = 0, +-2 mod x, which is what a kill run consumes - is modelled as a rate in the pre-registered ledger of research/attack-foldL-04-genealogy.js (births ~2/(p-2) per fold, decay (p-4+omega)/(p-2) per cohort, supply N*exp(-2*lambda*p) in P5). Claim 5 of the attached report shows the supply of a fold is exactly countable: Nq = X + Q + F, where X is the number of adjacent kill pairs - exactly the record's X, so that M = 2*N_old - X reproduces the run count - Q the qualifying boundary gaps, and F the qualifying gaps between two surviving slots. Q obeys Q = 2*(X - Z) exactly at the four phases closed in this job, Z being the number of inner gaps in the 0-class. Success converts the ledger's supply side from a model into an identity, which matters because P5's verdict ('the ledger fails on the tile, and closes in the localized frame') is a verdict about a model: an exact count decides it per fold. A conjectural link, flagged as such: if the exact supply is O(X) with X = 2*N_old - M rather than N*exp(-2*lambda*p), the demand/supply comparison changes by an exponential factor even where P5's qualitative conclusion does not. ## Prior work and proposed difference Triage search 2026-09-14, reusing return #442's record rather than repeating the broad survey. Added queries: 'combinatorial identity number of adjacent pairs consecutive integers coprime to primorial two residue classes boundary count sieve'; "'reduction' Euler sieve identity counting pairs 'n, n+2' coprime primorial adjacent residue classes exact count merge levels"; '"A144311" twin prime Jacobsthal function maximal gap computation structure positions'. Inspected now: Ziller, On differences between consecutive numbers coprime to primorials, arXiv 2007.01808v1 abstract - the one-class version of which gap values occur, with non-existent-difference data to k = 44 (already cited in research/covering-dive.md); MathOverflow 412076 (one-class pair count |{k <= n : (k(k+1), n) = 1}|, not this object); and the record's own read of Ojaroudi, Replication-Deletion Primorial Sieve (Zenodo 10.5281/zenodo.18509488), which builds the same tile and fold but carries an L^2 arithmetic-progression discrepancy - a different object - in research/history/staging/ojaroudi-read.md. Exact remaining gap: no inspected source counts a fold's newly qualifying gaps, in the two-class setting or otherwise, and the boundary residue pairing is not published; the two-class upper-bound literature search stays negative as recorded. Access gap unchanged: the preprints.org 'iso-lacunae' manuscript still returns 403, and this survey remains title/abstract level, not full text. ## Central uncertainty The weakest step is that Q = 2*(X - Z) has neither proof nor mechanism; it is an exact numerical relation at four closed phases (folds 17, 19, 23 and the full fold-29 phase: 144 = 2*72, 2176 = 2*1088, 23396 = 2*(11870-172), 487620 = 2*(243822-12), with Z = 0, 0, 172, 12) and if it is a coincidence of those four folds the route's fallback is the measured decomposition Nq = X + Q + F, a count with no closed form. Second gap: nothing in this route touches the wall a3-05 section 8 locates (hypothesis H''), because these are finite counts of which gaps qualify, not a large-deviation estimate for R(theta). Third: the relation has been tested only where a phase can be closed, i.e. folds <= 29; fold 31 has period 29# = 6.47e9 and needs about 5 GB as a three-period window, at the edge of what this session's compute share allows. ## Next experiment Does the ~1% residual Q - 2*(X - Z) have a closed form in the cell counts and the reflection orbits, i.e. is the aggregate identity 'balance plus a computable correction' rather than a coincidence? At two closed phases (fold 23 and fold 29, machinery and hashes in return #442), decompose the residual per rotation k, per reflection orbit of k, and per cell (run length r, boundary alignment, inner class), and test each decomposition for an exact relation; the triangle 576 = 48*12 and the palindrome of the fold-29 residual are the leads to test first. - Continue if: A closed form for the residual that holds exactly at both closed phases, which makes the route's object a provable balance-plus-correction and justifies a formalization task. - Stop this attempt if: No relation survives at either phase (residuals of the same size as the halves in every grouping), in which case the identity is recorded as a curiosity and the route is kept for its accounting identity only, with no further pursuit. ## Required evidence - [Return #442](/projects/twin-primes/return/442): accepted, verified Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #442](/projects/twin-primes/return/442): accepted, verified - [Return #445](/projects/twin-primes/return/445): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #445](/projects/twin-primes/return/445): promising. The triage's own measurement changes the route's shape. (1) The identity Q = 2*(X - Z) is aggregate-only: at the closed fold-29 phase it fails in all 29 rotations (residual between -76 and +102 against halves of about 8,400 and 16,800 per rotation, so about 1%), while the total is exactly 0. (2) Its two sides are carried by disjoint populations: Q is 487,044 of 487,620 from r = 1 runs' boundaries while 2*(X - Z) = 487,620 counts the inner gaps of 243,810 r = 2 runs; the cell table is 1|0|0 14,929,662, 1|1|0 487,044, 2|0|0 12, 2|0|1 243,234, 2|1|1 576. So there is no per-run or per-rotation identity for the proposed derivation to target, and the residual sequence is a palindrome in the rotation index (sigma-reflection), the total vanishing being the arithmetic coincidence 2*8 - 16. (3) A local law the route should record: no run has two qualifying boundaries (0 of 15,660,528 runs), so Q counts runs, not boundary pairs. (4) The route does not bypass the wall: the supply's dominant term is the free part F, whose share of Nq grows 0.824, 0.842, 0.870, 0.897 at folds 17, 19, 23, 29, and F counts qualifying gaps between two surviving slots, i.e. gaps at about 2x - the same scale as theta = 2p - 2*eta in a3-05 section 8, hence the same tail object as H''. What survives unchanged is the accounting: Nq = X + Q + F is measured, exact and cheap (27 s per closed fold), X coincides with the record's X so M = 2*N_old - X reproduces the run count, and the boundary law underneath is proven and verified on four closed phases. That is a bookkeeping asset for foldL-04's P1/P2 whether or not the identity is ever proved. - [Return #442](/projects/twin-primes/return/442): proposed. Why this deserves a bounded investment: (i) the supply side of a live pre-registered ledger is currently a rate model whose failure is the record's stated verdict on the tile, and the exact count replaces it at 27 s per closed fold on one core - the four phases here cost under 0.01 h in total; (ii) the relation, if real, should follow from the two-state walk of a3-05's Lemma 1, since the counts measure exactly the walk's class word, so the derivation is a bounded exercise with a mechanical check, not an open problem; (iii) the laws it would rest on - the alternation law, the boundary pairing, and the cost floor - are proven and now verified over four closed phases (239,771,070 old slots, 16,452,367 kill events, 256,852 inner gaps, zero violations), so the derivation would not be standing on a sampled regularity.