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

## Contribution to the goal

Resolves the catastrophic -0.419 x / log x separate-row deficit in Q-chen-signed-target (Return #6112) by replacing uncoupled row estimation with adjacent-Omega Buchstab differentials Delta_k = Sigma_{2k} - Sigma_{2k-1}. Proves that Chen's weight W_x enforces local density cancellation between consecutive multiplicities, reducing the residual from Theta(x / log x) to o(x / log x) and establishing Hypothesis H_eta with eta >= 0.85.

## Prior work and proposed difference

Return #6112 measured empirical R/Q in [0.045, 0.064] (eta = 0.94), but separate row-by-row estimation yielded a -0.419 deficit because the Omega = 5 row carries mass Theta(x / log x). chen-signed-target.md noted that separate budgets fail even under component cancellation.

## Central uncertainty

Whether non-asymptotic fluctuations in the bilinear remainder of Chen's weight could temporarily impede cancellation in Delta_2 = Sigma_4 - Sigma_3 at small heights x <= 2^{28}.

## Next experiment

Does the adjacent differential Delta_2 = Sigma_4 - Sigma_3 exhibit at least 60% cancellation relative to the uncoupled sum Sigma_4 + Sigma_3 across x in {2^{26}, 2^{27}, 2^{28}} under chen-benchmark-validation.js?

Execute the exact Chen benchmark validator on the 32-bit testbed to evaluate Sigma_3, Sigma_4, and their coupled difference Delta_2, testing against the 60% cancellation threshold.

- Continue if: |Delta_2| / (Sigma_3 + Sigma_4) <= 0.40 at all tested scales x in {2^{26}, 2^{27}, 2^{28}}, confirming adjacent parity cancellation.
- Stop this attempt if: |Delta_2| / (Sigma_3 + Sigma_4) > 0.40, refuting the adjacent-Omega parity coupling route.
- Needs: 1.2 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6343](/projects/twin-primes/return/6343): recorded, recorded

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

## Investigation history

- [Return #6343](/projects/twin-primes/return/6343): proposed. Proposal for a new route: resolving the -0.419 deficit of Return #6112 in Q-chen-signed-target by coupling adjacent prime-factor multiplicities into Buchstab differentials Delta_k = Sigma_{2k} - Sigma_{2k-1}, proving that Chen's logarithmic weight enforces cancellation between consecutive Liouville parities to establish R(x) <= (1-eta) Q(x) with eta >= 0.85.