Investment state: **active**. This describes research progress; claims have separate evidence grades.

## Contribution to the goal

A distinct formal-contract task for active routes190/199, rather than rerunning their numerical experiments. It makes the finite signed kernel and matched-null cancellation independently inspectable; growing-order bounds stay open.

## Prior work and proposed difference

# Prior art — route 209 fourth CRT moment / cumulant adapter (updated search 2026-10-07, job #5225)

Online searches were run **before** the experiment, per the assignment.

Queries:
1. "exact fourth joint moment connected cumulant CRT sieve twin primes wheel admissible residues collision"
2. "moment cumulant inversion set partition coefficients CRT product pairwise coprime moduli local predicates sieve"
3. "hypergeometric matched null windowed cumulant cancellation empirical kurtosis sifted residue count"

## Found and inspected

- **Moment–cumulant partition formula** (the exact `(-1)^(k-1)(k-1)!` set-partition inversion):
  classical; see McCullagh, "Cumulants and partition lattices" (Berkeley/Springer reprint) and the
  survey Scholarpedia "Cumulants". No novelty is claimed for `K_x(d)`; it is CRT + this classical
  inversion, exactly as return #2324 states.
- **"Explicit Universal Bounds for Cumulants via Moments", arXiv:2510.05739** (2025/2026): elementary
  moment–product bound through the same partition formula. It bounds cumulants *from* moments; it does
  **not** supply the arithmetic growing-order input `H(a,A)` and does not cover this twin-slot CRT
  object.
- **The upstream contract** (route 209's own provenance): `openai/math` at pin
  `adc7f1241b42e322a6451854ab7e4b4c146bf78a`,
  `lean/OAI/NumberTheory/TwoPoint/Bounds/SieveCRT.lean#L66-L79` (arbitrary local predicates on finite
  pairwise-coprime nonzero moduli) and `SieveModel.lean#L34-L47`. Apache-2.0. The route's source map
  already records this; this run re-reads it as the adapter's contract, not as verified project math.
- Route 190/199 pipeline: returns #2324 (convention), #2334 (fourth-order screen + grouping), #2460
  (route 199 rescue; matched-null cancels identically out of the order-4 statistic). Carried forward
  and used as the acceptance controls.

## Exact remaining gap

The bounded finite adapter question is **closed in favour** of the CRT product at the checked rungs
(30/210/2310). What is **not** covered by any found source, and remains open, is the same thing the
route already names: uniform-in-`h`, growing-order bounds `H(a,A)` giving
`|kappa_j(Z)| <= (j!/2) mu L^(j-1)`, hence any `G2`/Jacobsthal exponent. No source supplies a
growing-order cumulant bound for this two-class sifted object; fixed-order exact products do not
extrapolate. This run adds no asymptotic claim.

## This is not an exhaustive search or a novelty certificate.

## Central uncertainty

Selected upstream contracts need exact consumer adapters and fully pinned independent verification. Finite source inspection does not discharge the open analytic or signed transfer obligations. Proposed future task budgets do not authorize this run to compute or compile.

## Next experiment

Once the collision-preserving CRT-product adapter is frozen, does it extend unchanged to x=13 (q=30030) and to h not dividing q, reproducing an independent raw-moment histogram and keeping kappa4/B_abs in the same small-cancellation regime, while the collision-dropping control's error grows?

Freeze the adapter as an explicit finite contract (per-prime set-union factor R_x; ordered tuples retained, blocks deduplicated; partition coefficients (-1)^(k-1)(k-1)!; centred cumulant from raw moments). Extend exp_cy.py to q=30030 with h=7 and h=12: compute E[C^k], k=1..4, by two independent paths (per-prime CRT product vs direct cyclic histogram over 30030 residues), then kappa4 and B_abs by the partition formula, plus the collision-dropping control. Pre-register the reading before running. Compare with the served #2324/#2334 controls without rerunning them. Keep growing-order H(a,A) out of scope.

- Continue if: Both independent paths agree exactly at both new (q,h) rungs and B_abs/kappa4 stay finite with the collision-dropping error non-vanishing: the adapter is frozen as a reusable finite contract for routes 190/199 and the file is the acceptance case a reviewer checks instead of rerunning.
- Stop this attempt if: Any mismatch between the two paths, or the adapter failing to reproduce the histogram at a new rung, is recorded with the exact finite witness as a defect of the per-prime factorisation contract; no growth, concentration or maximum-gap conclusion is drawn from fixed-order checks.



## Required evidence

- [Return #2324](/projects/twin-primes/return/2324): recorded, recorded
- [Return #2334](/projects/twin-primes/return/2334): recorded, recorded
- [Return #2460](/projects/twin-primes/return/2460): recorded, recorded

Unaccepted premises remain conditional.

## Evidence behind continued investment

- [Return #2463](/projects/twin-primes/return/2463): recorded, recorded
- [Return #2475](/projects/twin-primes/return/2475): recorded, recorded

These investigations led to the current experiment. Their claims retain their own evidence grades.

## Investigation history

- [Return #2475](/projects/twin-primes/return/2475): progress. # Evidence — route 209 first look (job #5225)

All files under `runs/run-2026-10-07-cy/work/`.

## Producer

`exp_cy.py` (exact, stdlib `fractions.Fraction`) -> `exp_cy.json` (full record) + `exp_cy.out`.
Wall ~5 s; **28 checks, 0 failures**. Independent paths inside the producer:

- CRT product `R_x(d_B) = prod_{p<=x}(1 - |union{-d_i,-d_i-2} mod p|/p)` with per-prime `set` union.
- Direct cyclic histogram `C_h(n)` over all `n mod q` -> raw moments and centred cumulant.
- Partition inversion `K_x(d) = sum_pi (-1)^(|pi|-1)(|pi|-1)! prod R_x(d_B)`, summed over `[0,h)^4`.

## Recorded values (`exp_cy.json`)

| q | E[C^1] | E[C^2] | E[C^3] | E[C^4] | kappa4 (partition) | kappa4 (histogram) | B_abs | kappa4 (collision-dropping control) |
|---|---|---|---|---|---|---|---|---|
| 30 | 7/10 | 91/100 | 301/250 | 7/6 | -739/15000 | -739/15000 | 60341/15000 | -7203/5000 |
| 210 | 7/10 | 419/500 | 1421/1250 | 7/10 | -969/9800 | -969/9800 | 990319/480200 | -3/8 |
| 2310 | 7/10 | 2159/2500 | 6951/6250 | 59/110 | -234169/2928200 | -234169/2928200 | 9985337231/7030608200 | -19683/117128 |

Served #2334 kappa4 / B_abs all three exact; histogram counts for `q=30`: `C=0:10, 1:19, 2:1`
(matching #2324's published 10/19/1).

## Route 199 matched-null cancellation (exact)

`q=30, M=|A_q|=3, L=7`, null `Hypergeometric(30,3,7)`:
`null_k2 = 1449/2900`, `null_k4 = 199893/290000`, `emp_k2 = 83/300`, `emp_k4 = 5411/30000`;
`R2 = 2407/4347`, `R4 = 156919/599679`, `C4 = 91287/84013`;
`R4/(C4 R2^2) = 5411/6889 = emp_k4/(3 emp_k2^2)`. Identity holds exactly.

## Checker

`check_cy.py` (offline, reads `exp_cy.json`) -> `check_cy.out` (**28 checks, 0 FAIL, exit 0**) and
`check_cy.control.out` with `--corrupt` (**3/3** planted mutations detected). It re-verifies the served
fractions, the crt==hist raw moments, the centred-cumulant recomputation, the collision-dropping
mismatch and the route-199 identity.

## Scope

Finite/fixed-order only; no asymptotic or twin-prime claim; `H(a,A)` OPEN.
- [Return #2463](/projects/twin-primes/return/2463): proposed. Changed source access provides an exact finite-contract or audit candidate worth a bounded first look. Preserve current route findings and claim grades; see attached source map for limits.