Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal A specific arithmetic input for route143's exact count-object moment dial: group connected CRT correlations before absolute values, permitting cross-prime dependence. If |kappa_j| <= (j!/2) mu (A x^a)^(j-1) uniformly through j~x/log x at h=ceil(x^beta), beta>1+a, the finite-degree derivation in the report gives G2<=ceil(x^beta). a<3.266450284 permits a below-DHR exponent; a<1 is TPC-strength. Neither inequality is established. The new fourth-order small screen supplies a reason to examine signed grouping, not an exponent estimate. It preserves route181/return2247's recorded rank1 obstruction and does not transfer its band-minorant observations to the count object. ## Prior work and proposed difference Online search 2026-10-05 (this run): "cumulant grouping reduced residues short intervals twin primes Jacobsthal moments prime factorization"; "signed cancellation connected correlations cumulant expansion CRT residues singular series short intervals". Also read the route record and return #2324 and its attachments, plus the route's own cited sources. Found and inspected (none covers the exact object or the grouping question): - V. Kuperberg, "Odd moments in the distribution of primes" (2025), ETH research collection: k-th moments of reduced residues in short intervals / prime-polynomial analog; one-class moments, substantial order dependence. Not a two-class (twin-slot) connected CRT bound. - Montgomery, "The combinatorics of moment calculations" (2010), sec. 3 eq. 15-16: one-class moments with order-dependent constants. - Bloom-Kuperberg, arXiv:2312.09021v2 (12 May 2026), sec. 1.1 Thm 1 / footnote 1: odd one-class moment improvements with order dependence. - Doring-Jansen-Schubert, arXiv:2102.01459, sec. 1.2 / 2.2 Thm 2.5: classical cumulant inversion and conditional concentration; no arithmetic estimate for this object. - Kuperberg, arXiv:2210.09775v2: conditional prime-tail application, not this two-class bound. - Keating-Rudnick, function-field variance of primes in short intervals (IMRN); Leung 2024, joint distribution of primes in multiple short intervals (normal point charges) — analytic/statistical, not an exact connected CRT grouping at fixed primorial. - Standard cumulant/connected-correlation references (Kubo 1962; MathOverflow connected-cumulant threads) confirm the partition/inversion algebra used here is classical; no novelty is claimed for the formula. No source located that groups a signed connected CRT cumulant by collision/coincidence pattern before absolute values, and none that supplies a uniform-in-h remainder bound stronger than the triangle budget for this count object. Novelty of the negative result is not established; it is a finite screening. Exact remaining gap: a uniform-in-h (indeed growing-order) arithmetic bound for the signed connected sum, or a provable decomposition that beats the size-matched random grouping. The finite evidence here indicates the natural per-tuple residue grouping cannot do that. Sources are cited by locator in report_u.md; novelty unestablished. ## Central uncertainty H(a,A) is OPEN: uniform bounds on cumulants j=2..2k, k=ceil(4x/((beta-1-a)log x)), h=ceil(x^beta), all sufficiently large prime x and every fixed beta>1+a. Variance is included; repeated offsets/local saturation are retained. Fourth-order small cancellation cannot prove it, fixed-order O_k bounds do not give usable growing-order constants, and cumulants need not factor over primes. The proposed first child only checks a grouping and states its uniform fourth-order remainder target; the parent stays conditional until growing-order bounds are established. No negative association or independence is assumed. ## Next experiment Is the sign of the exact fourth connected CRT term K_x(d) predictable from the offset multiset by a bounded-degree prime-local statistic, and does grouping by that statistic (or its complement, or a direct Euler-product decomposition of the signed sum) beat a size-matched random grouping of the triangle budget at fixed (x,h)? Extend check_u.py (stdlib exact Fractions, same convention as return #2324) to q=30,210,2310,30030 and h in {7,10,12}. For each rung compute kappa4, the absolute joint budget B_abs, and the seeded size-matched label-shuffle baseline G0 (300 trials, class sizes preserved). Then, before looking at any value, define candidate analytic groupings that are functions of the tuple only: (i) the sign of a low-degree prime-local statistic of the multiset (e.g. counts of offsets in each residue class mod 2, mod 3, and their product), and group by the COMPLEMENT of that predicted sign so as to force sign mixing; (ii) the p=2-only connected term sign; (iii) an Euler-product/inclusion-exclusion block decomposition of K_x(d) grouped by block-collision degree. For each candidate report G/B_abs against the G0 distribution and the oracle |kappa4|/B_abs. Also report sign(K_x) purity of each statistic. Do not re-run any published census other than the two controls; reuse observations. - Continue if: Some tuple-independent analytic grouping has grouped budget <= 0.5 * G0 mean at >= 2 of the 4 rungs while reproducing both published controls, and the sign-predicting statistic has purity > 0.9 at the tested rungs; then a provable uniform-in-h remainder target for that grouping is a live obligation and should be stated and attacked. - Stop this attempt if: At >= 2 of the 4 rungs every tested tuple-independent grouping has grouped budget >= the size-matched baseline G0 mean (i.e. no excess over random aggregation), or sign purity <= 0.7; then the grouping route is recorded as a scoped obstruction and the signed sum must be bounded directly (route 143 / H(a,A)) rather than by pre-grouping. ## Required evidence - [Return #2324](/projects/twin-primes/return/2324): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2324](/projects/twin-primes/return/2324): recorded, recorded - [Return #2334](/projects/twin-primes/return/2334): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2334](/projects/twin-primes/return/2334): progress. Route 190 first look (job 5015). Bounded exact test of the route's pre-registered first child: "derive a collision-pattern grouping for kappa4(C_h), compare with the ungrouped absolute joint budget, and state a uniform-in-h remainder target." Result: the child is defeated at finite scale, by its own pre-registered failure clause and by a size-matched control; the route stays open with a changed child. Evidence (exact Fractions; q=x#, h=7, ordered 4-tuples, convention of return #2324): - Controls reproduced independently: q30 kappa4=-739/15000, budget=60341/15000; q210 kappa4=-969/9800, budget=990319/480200. New rung q2310: kappa4=-234169/2928200, budget=9985337231/7030608200. - Lemma (all x,h,j): R_x(d_B) depends only on the offset set of block B, so K_x(d) is symmetric and depends only on the offset multiset. Measured: value_constant=true on every value-signature class; grouping by value-signature / offset-signature / offset-multiset = 100% of the triangle budget at all three rungs. Fine groupings cannot cancel. Failure clause met. - Coarse residue groupings vs size-matched label-shuffle control (preserves class sizes, 300 seeded trials): parity mod2 observed 15.54/10.49/15.13% vs control mean 5.9/10.0/12.5% (max 13.6/21.7/28.8%); residue mod3 observed 23.44/37.55/45.23% vs mean 10.6/16.7/21.1%; residue mod5 observed 50.63/60.05/64.49% vs mean 21.9/31.5/38.4%. Structured residue groupings cancel no better than, and usually worse than, matched random groupings; at q30 parity exceeds the control maximum. Oracle single-class = 1.22/4.79/5.63%. - Sign mechanism: sign(K_x(d)) is largely determined by the local residue pattern (p=2,3); grouping by pattern clusters same-sign terms. Cancellation is generic across patterns. What changes: the proposed grouping direction is anti-productive; the object is multiset-symmetric. A useful grouping must be coarser than the multiset and sign-mixing, or |kappa4| must be bounded directly. H(a,A), G2, and exponents remain open and untouched. Rung: finite screening + exact lemma; no asymptotic claim. depends_on: [2324]. - [Return #2324](/projects/twin-primes/return/2324): proposed. Original bounded exact new screen at h7: q30 kappa4=-739/15000, absolute joint budget60341/15000, ratio739/60341; q210 kappa4=-969/9800, budget990319/480200, ratio47481/990319. Both <=1/4 saved before execution, both equal an independent histogram formula, all2401 ordered repeated-index tuples published per row. Preliminary controls at (q,h)=(30,3),(210,3),(30,5), orders1..4 yielded12 paired exact matches. These are finite screening observations only. Raw producer, shards, observations and manifest are public attachments, hash checked; no original timing reproduced. Two original invocations wall20/CPU10, exits0 and process-group cleanup observed, actual CPU usage unmeasured, RAM unverified. Conditional moment reduction and exact endpoint convention appear in report.