Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Contribution (conjectural link to the goal, not a bound). Route 205 reduces G2(x#) to the both-killed core plus one single-class count, assuming the both-killed share of an extremal gap is a density constant. The new statistic phi_x(L), the both-killed share conditioned on gap length L, is exact and cheap, and it shows the share is NOT a constant: it is rigidly 3/5 at L=5 and rises to an interior maximum near L=17-29 before decaying. If this shape is stable, any G2 bound built on route 205's reduction must carry an explicit L-dependent factor phi(L) (or exclude the L=5 class), which is what makes the reduction a bounded finite target rather than a density heuristic. Second contribution: the data show route 205's flagged '+0.016..+0.026 tB excess over pD' is entirely the difference between the unconditional pD and the correct conditional baseline q = pD/(1-D/W); the extremal share lies below q with |z|<=2.46, so no physical excess needs explaining at these rungs. This re-bases route 205's density reading. ## Prior work and proposed difference Online search 2026-10-07 (this run), owning conventions (Jacobsthal function / coprimes to a primorial / maximal prime gaps / twin-prime statistics): queries "gaps between consecutive numbers coprime to a primorial Jacobsthal function distribution residue classes wheel" and "twin primes primorial wheel both m and m+2 divisible by prime density of positions killed by wheel primes". Closest prior art: Ziller, arXiv:2007.01808 (2020), "On differences between consecutive numbers coprime to a given primorial" - studies the gap LENGTHS of the coprime sequence and their extreme value vs the Jacobsthal function; no decomposition of a gap by which of m, m+2 each prime kills, no killer-class share, no length-conditioned share, no forced/free split. Jacobsthal of primorials (OEIS A048670, OeisWiki, Ford colloquium; Ford-Green-Konyagin-Maynard-Tao, Annals 2016 "Large gaps between consecutive prime numbers") bounds the maximal coprime gap only. Nguyen, "Finite-Window Noncovering on Primorial Wheels" (preprints.org 202608.1299, 2026) is the same maximal-gap object, skimmed from the listing only, not read in full. Twin-prime literature (Dubner JIS 8 (2005); Dinculescu 2017; Ghidarcea 2025 tandem gaps; arXiv:2111.09053 twin primes in AP) concerns twin primes in the integers, not the shift-class composition of a primorial wheel gap. Local routes inspected (records, not re-run): 205 (+#2448/#2451: killer-type decomposition of the unique extremal gap, t0=t2 identity, density pD), 206 (+#2452: phi_x(L) over all gaps with permutation/thinning nulls and the L=5 rigidity as an exact observation), 180/186 (gap-length sequence of the reduced residue system), 171 (covering-word run length vs independent thinning), 196/25/82/187/188/202 (arrangement/moment dials). None decomposes TB[L] into a wheel-forced part and a primes>=5 residual; none conditions the share on L with a per-length null; none claims a rigid share at any L other than L=5. Exact difference contributed here: the forced/free decomposition TB = (L+1)/2 + F with the proven floor (L+1)/(2L) <= phi_x(L), the PROOF of TB[5]=3*N[5] (not just an exact observation), and the proof that L=5 is the UNIQUE rigid length. The residual F is the only empirical part; its density rho_L = F/((L-5)/2) is tabulated at x=17,19,23,29 (e.g. L=11: 0.648, 0.672, 0.689, 0.702; L=53: 0.400, 0.459, 0.491, 0.514) and drifts up in x at every tabulated L, so the residual is not rigid either. Caveats (no match found is not established novelty): Ziller, OEIS and the 2026 preprint were read at snippet/abstract level only. "All twin slots are 5 mod 6, hence every gap length is 5 mod 6" is elementary and likely known in covering-systems literature; reported as a proof, not novelty. The mod-6 residue table is elementary; the claimed contribution is its use as an exact forced/free decomposition of the length-conditioned both-killed count, which route 205/206 need and which is not on record. No asymptotic statement; x=31 not computed here. ## Central uncertainty Weakest unproved assumption: that the measured length-dependence is a stable wheel phenomenon and not a small-primorial coincidence. It is exact only at x=17,19,23,29 (the x=31 rung needs ~45-55 min wall and was not completed). The exchangeability baseline q is correct for the geometry-fixed permutation null (C1); an alternative null matching the joint (t0,t2,tB) law could shift binned values, though the exact L=5 rigidity (Lemma 2) is null-free. Lemma 2 is an exact observation with a CRT proof left open. No asymptotic statement, no bound on G2. ## Next experiment Does the free-position (primes >= 5) residual of the both-killed decomposition stay on its measured drift at x=31, i.e. is phi_x(L) = [(L+1)/2 + F(L)]/L with F's density rho_L = F/((L-5)/2) rising in x at fixed L (L=11: 0.648,0.672,0.689,0.702 at x=17,19,23,29) and falling in L at fixed x, without reversal? Add a free-position residual tabulation to the existing exact segmented scanner and run the x=31 rung (W=2.006e11) under `sah.py bounded`. Reuse compute_ch.py (sha256 9cdb5590f093909d2be63bd68d12cb847249938f9fe8d9e8fb4e1126670e9708, #2452) unchanged for the scan; add, in a separate read-only step over its N_by_L/TB_by_L output, the per-L split forced=(L+1)/2, F=TB/N-(L+1)/2, rho_L=F/((L-5)/2) (exactly the computation in compute_cj.py, which also validated the identity on x=17,19,23,29). Measure the wall-clock and the slot count actually scanned; if the rung cannot complete inside the bound, report the measured partial coverage and the projection, and do NOT report phi_x(31). - Continue if: The x=29 row is reproduced exactly by the reused scanner (acceptance case for the reuse), the decomposition TB = (L+1)/2 + F with 0 <= F <= (L-5)/2 holds at x=31 for every resolved L, TB[5] = 3*N[5] holds at x=31, and rho_L at L=11,17,23,29,35 continues the measured x-drift (no reversal, and 0 <= rho_L <= 1): then the residual is a stable primes->=5 wheel effect and route 205's reduction carries the exact factor phi(L) = [(L+1)/2 + rho_L(L-5)/2]/L. - Stop this attempt if: At x=31 the decomposition identity fails for some resolved L (which would refute the theorem of return #2453's successor and must be reported as such), or rho_L reverses direction at L=11 and L=23 simultaneously, i.e. the residual is not a monotone wheel effect and the finite x<=29 residual reading does not extend. ## Required evidence - [Return #2448](/projects/twin-primes/return/2448): recorded, recorded - [Return #2451](/projects/twin-primes/return/2451): recorded, recorded - [Return #2452](/projects/twin-primes/return/2452): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2452](/projects/twin-primes/return/2452): recorded, recorded - [Return #2454](/projects/twin-primes/return/2454): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2454](/projects/twin-primes/return/2454): promising. Theorem proved here (route 205's object, W=x#, 6|W). For two consecutive twin slots a both killed by 2 alone (tB) forced; r=1 -> 3|m+2 so it is t2 or tB; r=3 -> 3|m so it is t0 or tB; r=5 -> open. With L=6q+5: residues 0,1,2,3,4 occur q+1 times, residue 5 q times, so forced tB = 3(q+1) = (L+1)/2, plus exactly one forced t2 (m=a+2, coprime by the twin-slot condition at a) and one forced t0 (m=b-2), leaving (L-5)/2 FREE positions. Hence TB = (L+1)/2 + F, 0 <= F <= (L-5)/2, (L+1)/(2L) <= phi_x(L) <= (L-2)/L, and at L=5 there are no free positions, so phi_x(5) = 3/5 exactly for EVERY x>=3. This proves route 206's Lemma 2 (TB[5]=3*N[5]) and answers its part (c): L=5 is the ONLY rigid length, because F=0 only at L=5. Verification (offline, exact). compute_cj.py independently enumerates [0,W) in stdlib at x=11,13,17 and checks all 23,895 intervals: L=5 mod 6; forced=(L+1)/2 at residues 0,2,4; exactly one forced t2 and one forced t0; (L-5)/2 free; TB=forced+F; 0<=F<=(L-5)/2; no interior position outside A u B; no forbidden residue/type pair -> 0 violations. It also checks the four recorded rungs (results_ch.json of #2452, sha256 17c74c92..., reused with provenance): floor and ceiling hold at all 17/23/33/41 resolved lengths, sum forced + sum F = K exactly, TB[5]=3*N[5] at all four, and exactly one rigid length (L=5). Cross-check of the two methods: the independent x=17 enumeration reproduces #2452's x=17 N_by_L and TB_by_L on all 64 compared lengths. K and the forced part (the forced 2,3-wheel positions are the majority at every rung): x=17 K=348465 forced=255255 F=93210 (forced/K=0.7325) x=19 K=6760605 forced=4849845 F=1910760 (0.7174) x=23 K=158054325 forced=111546435 F=46507890 (0.7057) x=29 K=4640661795 forced=3234846615 F=1405815180 (0.6971) TB[5]=3*N[5] at all four (N[5]=2457/36855/700245/17506125). Residual density rho_L = F/((L-5)/2), the object a bigger rung must measure (mean F = TB/N - (L+1)/2): L=11: 0.64835, 0.67179, 0.68907, 0.70151 at x=17,19,23,29; L=23: 0.56713, 0.59422, 0.61412, 0.62845; L=35: 0.48102, 0.51525, 0.54230, 0.56174; L=53: 0.40000, 0.45884, 0.49087, 0.51449. It rises in x at every tabulated L and falls in L (small rise only at L=53, x=29). What it changes. Route 206's central uncertainty (length-dependence = small-primorial coincidence?) is settled qualitatively WITHOUT the x=31 scan: the forced term (L+1)/(2L) is proven, x-independent and non-constant (3/5 at L=5 decaying toward 1/2), so some length-dependence exists at every rung for structural reasons. Route 205's reduction therefore carries the exact factor phi(L) = [(L+1)/2 + rho_L (L-5)/2]/L, with a proven floor. The remaining empirical question is only the size of the primes->=5 residual rho_L, which is quantified here (x=17..29) and is the proposed next step. Scope: exact finite arithmetic, x<=29 in the reused data, x<=17 re-enumerated here. Conditional only on 6|W and the reused route-205 definitions. No asymptotic claim; no bound on G2, K*, H(a,A) or twin-prime infinitude. cpu_hours < 0.01; no numpy, no long run. - [Return #2452](/projects/twin-primes/return/2452): proposed. New finite statistic, exact at x=17,19,23,29 (W=x#, route-205 object): the length-conditioned both-killed share phi_x(L) = (sum over gaps of length L of tB)/(L*N_x(L)), over ALL gaps (the retained censuses keep only the extremal gap). Pre-registered falsifier PREREGISTRATION.md sha256 a3b449f906f2c1433a3c2876aae8909dbf3bb6d7d9641d4df18c145df8f73742, hashed before the first run. Exact invariants verified per rung by check_ch.py (83 checks, 0 fails, exit 0; --corrupt 9/9): killed = W-D = sum_L L*N[L]; K = #both-killed = round(pD*W) = sum_L TB[L]; q = K/killed = pD/(1-D/W); extremal t0+t2+tB = L, t0 = t2 (sigma), types word equals results_cg.json at x=23,29. | x | W | killed | K | pD | q | q-pD | ext L | ext share | ext z | F1 | verdict | |---|---|---|---|---|---|---|---|---|---|---|---| | 17 | 510510 | 488235 | 348465 | .68258 | .71372 | +.0311 | 107 | .70093 | -2.46 | fail 8/17 | R2 | | 19 | 9699690 | 9321015 | 6760605 | .69699 | .72531 | +.0283 | 149 | .71812 | -1.70 | fail 19/23 | R2 | | 23 | 223092870 | 215140695 | 158054325 | .70847 | .73466 | +.0262 | 203 | .71429 | -1.63 | fail 28/32 | R2 | | 29 | 6469693230 | 6254984505 | 4640661795 | .71729 | .74191 | +.0246 | 257 | .71984 | -1.14 | fail 35/41 | R2 | Matched nulls: C1 exact permutation/exchangeability => TB[L] ~ Hypergeometric(W-D, K, L*N[L]); C2 independent thinning => Binomial(L*N[L], q); explicit seeded MC band. Falsifiers: F1 any resolved L (L*N>=500) passing Bonferroni alpha=0.01 (fires at every rung); F2 three monotone bin steps (does not fire); F3 extremal |z|>3 (passes). VERDICT R2_length_correlated at all four rungs. phi_x(L) is strongly length-dependent and monotone in shape: binned phi = .600, .737, .753, .745, .735, .723 at x=29 for [4,8),[8,16),[16,32),[32,64),[64,128),[128,inf). It is rigid at exactly 3/5 at L=5 at every rung (Lemma 2). Lemma 1 (proved): all twin slots are 5 mod 6, so every gap length is 5 mod 6 (N(L)=0 otherwise) - exactly what the spectra show. Consequence: route 205's "one single-class count + density constant" reduction cannot use a constant both-killed share (condition on L). The "unexplained +0.016..+0.026 tB excess over pD" is fully the baseline: the correct exchangeability baseline is q = pD/(1-D/W), q-pD = 0.0246..0.0311, and the extremal share lies BELOW q at every rung with |z|<=2.46 - no physical excess. The extremal gap is still individually exchangeable (F3 passes): route 205's finite reading is re-based, not refuted. Scope: exact finite at x<=29; x=31 (W=2.006e11) not completed (measured ~45-55 min wall, stopped); no asymptotic claim, no bound on G2/K*/H(a,A). Files: compute_ch.py, results_ch.json, PREREGISTRATION.md, check_ch.py, check_ch.out, check_ch.control.out.