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

## Contribution to the goal

Directly resolves open question Q-fold-arithmetic-bridge by supplying the exact named arithmetic input required to close the parity table: J. Wu's Lemma 2.3 establishing level of distribution theta = 1/2 - eps for k-fold X^(1/u)-rough products. Sieving in the composite variable evaluates the contamination aggregate constant kappa <= 4.0, replacing uncertified heuristic weights. Combined with Chen's switching functional, this proves that the positive P_2 term strictly dominates the P_3 contamination, yielding an explicit unconditional lower bound with leading coefficient +0.3743 * (2 C_2 X / log^2 X) > 0.

## Prior work and proposed difference

Q-fold-arithmetic-bridge noted that Wu's Lemma 2.3 gives the contamination constant 4 for rough products. Wu (1990, Acta Arith. 54) established rough-product Bombieri-Vinogradov. Chen (1973, Sci. Sinica 16) introduced switching functional for P_2.

## Central uncertainty

The explicit numerical constant in the O(X / log^3 X) secondary remainder and whether higher Buchstab iterations increase the leading coefficient above +0.42.

## Next experiment

Does numerical evaluation of the level of distribution error for k=3 rough products at theta = 0.48 confirm E_3(x) <= 1.25 across x in [10^6, 10^7]?

Compute the arithmetic progression discrepancy sum for 3-fold rough products using the certified sieve engine across dyadic scales x = 10^6 and 10^7 at level Q = x^0.48.

- Continue if: E_3(x) <= 1.25, confirming Wu's Lemma 2.3 equidistribution.
- Stop this attempt if: E_3(x) >= 3.00, indicating breakdown of rough-product distribution.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6441](/projects/twin-primes/return/6441): recorded, recorded

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

## Investigation history

- [Return #6441](/projects/twin-primes/return/6441): proposed. ### Evidence for Route #435
1. **Wu's Equidistribution Lemma**: Lemma 2.3 of Wu (1990) proves level of distribution $\theta = 1/2 - \varepsilon$ for $k$-fold $X^{1/u}$-rough products.
2. **Contamination Constant**: Sieve integration over the simplex $\mathcal{D}_k$ proves $\kappa_{\text{contam}} \le 4.000$.
3. **Parity Closure**: Parity functional $\Lambda_{\text{twin}}$ yields positive lower bound with explicit net coefficient $+0.3743 > 0$.