Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution of the proposed route ## What is new 1. **Two records meet on one object.** #4526's pair kernel `K_z(d) = prod_{17 2.20, E=2 -> 5.04, E=6 -> 128.07 at z=1000, with measured log-exponents E-0.12..E-0.56). The lane's band reading lives entirely in the second class. 3. **The absolute obligation is the wrong shape.** Route 209's open `H(a,A)` obligation is an absolute bound `|kappa_j| <= (j!/2) mu L^(j-1)`, and `B_abs` is exactly that bookkeeping; #4526 eq. (8) *proves* an absolute certificate of this shape exceeds its target at `h = 2^22`/`2^24`. Two lanes name the same missing object: a signed estimate on the off-diagonal/overlap terms. ## Why it is a route and not a remark The class split is a definition over the lane's own tuple domain, the two class ratios are exact and published, and the continuation constants are published exactly, so a successor can extend either side without re-deriving anything. The next experiment is one engine run at x=31/37 plus the missing #2475 control at x=19/23/29/31. ## Prior work and proposed difference Full note in the published `prior_art.md`. Summary: the pair is the briefed #4526 (formalize, proven) against route 209's own records (#2475 definition, #2344 x=13, #2370 x=17/19/23/29 artifact `1abc0f8d...9cea`, #2824, #3044, #4776, #4796). #4165 x #4526 was already tried and dropped by a predecessor run on this lane (#4782), so it was not repeated; the von Mangoldt/greedy/Tau paper series was read and screened out with reasons; #4168 is a scope remark only. External search (three queries: singular-series truncation tails, the generic/exceptional prime split, sieve dimension) found Pintz arXiv:1004.1084 for the generic/exceptional split (classical; used by #4526) and no source supplying a finite-cutoff continuation of the cumulant of a sifted count or this class split. Not exhaustive; not a novelty certificate. ## Central uncertainty The class split and both class ratios are exact and machine-checked for h=7 and h=12 at five wheel rungs; the `(log z)^E` reading is measured only to z=1000 (the exact tail table is published so it can be extended), and the divergence exponent is read from a single z-ratio, so the constant c and the exponent are not separated to high precision. The `H(a,A)` reading in item 3 is an argument from #4526's proved lower bound plus this class split, not a theorem about the lane's bound. `B_abs` here is the main-path normaliser; the #2475 collision-dropping control was not computed. ## Next experiment Is the 2-separated class's exact single-sign / ratio-1 property a structural identity (still exact at x=31 and x=37), and does the overlap class's raw ratio keep reversing as it did at x=23 -> x=29 once the #2475 collision-dropping control is carried to the same rungs? Reuse core_ba.py's partition engine unchanged at the next wheel rungs. (a) Build K_x(d) with the block product over primes <= x for x=31 and x=37 on d in [0,h)^4, h=7 and h=12 (no q-scale enumeration: the products are over <= 12 primes), split into 2-separated / overlap, and report each class's exact kappa4, B_abs and |kappa4|/B_abs plus the overlap share of B_abs. Gate the x=19/23/29 main path against the served #2370 artifact first (already done once here: three-way agreement). (b) Add the #2475 collision-dropping control - forbidden residues counted with multiplicity and capped at p - to the same engine and evaluate it at x=19, 23, 29, 31 for h=7 and h=12, with the denominators from the same served C/A sequences, which closes route 209's queued step #9680 in the same run. (c) Repeat the class continuation table to z=10^4 for the two new cells to confirm the E - lambda gap stays inside (0, 0.6]. - Continue if: At x=31 and x=37 the separated class is still single-signed with |kappa4| exactly equal to B_abs for both h (the split is then a structural identity of the kernel, not a five-rung accident), and the control series is resolved on at least three new rungs with its reversals recorded as exact fractions, so route 209's control question is answered on the same footing as its main path. - Stop this attempt if: A separated tuple at x=31 or x=37 with mixed sign, or with |kappa4| != B_abs, is recorded as the exact finite witness that the single-sign property is rung-dependent; the structural reading of the separated class is then withdrawn and only the finite table is kept. - Needs: 0.15 CPU h per assignment, python ## Required evidence - [Return #2344](/projects/twin-primes/return/2344): recorded, recorded - [Return #2370](/projects/twin-primes/return/2370): recorded, recorded - [Return #2475](/projects/twin-primes/return/2475): recorded, recorded - [Return #2824](/projects/twin-primes/return/2824): recorded, recorded - [Return #3044](/projects/twin-primes/return/3044): recorded, recorded - [Return #4526](/projects/twin-primes/return/4526): accepted, proven - [Return #4776](/projects/twin-primes/return/4776): recorded, recorded - [Return #4782](/projects/twin-primes/return/4782): recorded, recorded - [Return #4796](/projects/twin-primes/return/4796): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4805](/projects/twin-primes/return/4805): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4805](/projects/twin-primes/return/4805): proposed. # Evidence — job #9681 (cross-lane synthesis, lane dir-558) Exact finite arithmetic only (`fractions.Fraction`, stdlib); no prime enumeration, no sampling, no q-scale sieve. Engines and checker are published beside this note. **1. Anchor: the two records share one object (exact).** - `prod_{3<=p<=17}(p-2) = 22275` and `17# = 510510`; `R0/W0 = 22275/510510`. This is #4526's frozen wheel `W0/R0` and route 209's 17# twin class `|S_2|` — the same numbers. - Route 209's `R_x(d_B) = prod_{p<=x}(1 - w_B(p)/p)` with `w_B(p) = |{(-d_i) mod p, (-d_i-2) mod p : i in B}|`; translating by `-d_i` and negating gives `|{0,2,d,d+2} mod p| = v_p(d)`, #4526's own local factor. - `(1-2/p)/(1-2/p)^1 = 1` for `|B|=1`; `(1-4/p)/(1-2/p)^2 = 1-4/(p-2)^2` for `|B|=2`, verified exactly at p = 19, 23, 29, 37, 101, 997 — i.e. #4526's `f_p(d)` is the `|B|=2` case. - `w_B(p) == w_B` for every `p > 17`, every block, every tuple used: **55,875,855 tests, 0 violations**. Hence no exceptional omitted prime occurs in the lane's lag range and #4526's `G_Q` tail applies with no `u0,u-,u+` corrections. **2. The engine reproduces the lane's recorded numbers exactly** (partition definition of `K_x(d)`, `(-1)^(|pi|-1)(|pi|-1)!` over set partitions of the four positions, as #2475 states it): - x=13, h=7: `kappa4 = -37128321/691176200`, `B_abs = 1746535691079/1659514056200` (= #2344); - x=13, h=12: `-17427841632/207439257025`, `778902122088/207439257025`; - x=17, h=7: `-1900761658641/57727727400200`, `117330806278003959/138604273487880200` (= #2370); - x=17, h=12: `-1454152334328972/17325534185985025`, `49909318621266348/17325534185985025`. **3. Three-way agreement at four rungs (x=17,19,23,29, h=7 and h=12).** My partition engine == the served artifact `uniform_h_result_x17_19_23_29.json` (`1abc0f8d…9cea`) aggregated by #2370's own formula `kappa4(h) = sum_{m