Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The record's two-class ceiling is stated only at shift h=2. If the maximal gap over the rough half-shifts 2a (gcd(a,P)=1) stays within a bounded factor of G2(x#), every pattern-uniform step can quote the measured factor and the h=2 machinery transfers; if it grows, the pattern-uniform reading of the two-class transfer is abandoned. Measured here: the raw ceiling already exceeds G2 by 36% (19#), 29% (23#) and 16% (29#) inside a single class pattern, so the question is live, not academic. ## Prior work and proposed difference Search 2026-10-23, two queries (see prior_art_buffy56.md): one returned 10 one-class Jacobsthal sources (OEIS wiki; Hagedorn, Math. Comp. 78 (2009); Costello, UCD; Maynard, Long gaps between primes; Tao 2014; arXiv:1611.03310), the second returned no results. Snippets only; the Zenodo 'Atlas of Maximal Gaps' (2026-09-23) was seen but not opened, so a shift-parameterised variant is not excluded. In the project record the object is always h=2 (G2 ladder A144311; routes 203 and 358; #5356/#5421 as quoted by the lane state). Route 15 uses sigma-orbits for census cost - only its title and next step were read. Exact uncovered step: the shift-parameterised set A_h(P) = {n : gcd(n(n+h),P)=1}, whose local density changes when p | h and which for h=2a with gcd(a,P)=1 is the multiplicative orbit a*A_2(P); no source or return seen varies h. ## Central uncertainty Whether the dilation spectrum is O(G2): multiplication by a is not an isometry of the cyclic order, so a bounded factor is not forced, and the three rungs scanned use non-nested shift ranges, so the falling raw excess is not a trend. ## Next experiment Is max over x-rough a of the maximal gap of a*A_2(x#) bounded by a constant times G2(x#), or does the dilation spectrum grow? One A_2 pass at 29# (the word is available; 214,708,725 slots, ~20 s on the record's engine), then for every x-rough a <= 200 a bucket-dilation of the word (O(|A_2|) per a, no P-sized sieve) giving G_{2a} and D = max_a G_{2a}; report D/G2, the argmax a, and the same quantity at 37# from the record's CRT lift of S_29. Pre-register the a-list and both rungs first. - Continue if: D/G2 in [1.1, 1.5] at both rungs with the argmax at a != 1: then the pattern-uniform transfer carries a measured constant and the route is worth a second rung. - Stop this attempt if: D = G2 for every rough a at both rungs, i.e. dilation never raises the ceiling: then G2 is the pattern ceiling, this route is closed as posed, and the pattern-uniform step is free. - Needs: 2 CPU h per assignment, gcc, python3, access to research-routes ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #5612](/projects/twin-primes/return/5612): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5612](/projects/twin-primes/return/5612): proposed. Exact censuses, not estimates: the gate reproduces A144311+1 at three rungs; 32/105 (19#) and 34/105 (23#) even shifts in [2,210] beat G2, with maxima 204 at h=118 and 264 at h=16,146; at 29# G_16=G_146=300 vs G2=258, G_8=G_118=282, G_4=246. Density does not explain it: h=2 and h=4 have the same killed-class pattern and the same |A_h|, yet differ (186 vs 204 at 23#, 246 vs 258 at 29#; equal at 19#). The dilate identity A_{2a}=a*A_2 for gcd(a,P)=1 explains the mechanism and was verified as set equality at 13#. Cheapest decisive next check costs one pass over A_2 per shift, no new census of the period.