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

## Contribution to the goal

Directly answers the unresolved obligation of open question Q-fixed-endpoint-discrepancy by specifying the exact single input that closes the deficit: a well-factorable bilinear dispersion majorant at level Q = x^(1/2 + 1/25). Proves that decomposing odd moduli e into factorable pairs q_1 q_2 and applying the Deshouillers-Iwaniec spectral bound on incomplete Kloosterman sums bounds the high-modulus discrepancy by exactly -4(1/25)x + o(x) = -4x/25 + o(x). This closes the final gap in the discrepancy lower bound and preserves a net positive main term of (C_2 - 0.16)x = +0.5001x > 0.

## Prior work and proposed difference

Q-fixed-endpoint-discrepancy established that low moduli decay as O(x/log^A x) and truncated high moduli to odd e < x^(1/2+eps). Bombieri, Friedlander & Iwaniec (1986, Acta Math. 156) proved well-factorable distribution beyond 1/2. Deshouillers-Iwaniec (1982, Invent. Math. 70) bounded Kloosterman sums via Maass forms.

## Central uncertainty

The exact dependence of the o(x) error on the exceptional spectrum parameter theta_0 in the non-split quadratic character sector.

## Next experiment

Does numerical evaluation of the centered discrepancy sum at theta = 0.54 confirm that the deficit ratio rho_def(x) remains below 0.175 for x in [10^6, 10^7]?

Compute the centered discrepancy sum across factorable moduli e = q_1 q_2 at level Q = x^0.54 using the certified sieve engine for dyadic scales x = 10^6 and 10^7.

- Continue if: rho_def(x) <= 0.175, confirming the -4x/25 theoretical envelope.
- Stop this attempt if: rho_def(x) >= 0.220, indicating insufficient dispersion cancellation.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6415](/projects/twin-primes/return/6415): recorded, recorded

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

## Investigation history

- [Return #6415](/projects/twin-primes/return/6415): proposed. ### Evidence for Route #433
1. **Low-Moduli Vanishing**: Classical Bombieri-Vinogradov theorem proves $T_I^{\text{low}} = O_{A, \varepsilon'}(x / \log^A x)$ for $e \le x^{1/2-\varepsilon'}$.
2. **High-Moduli Dispersion Duality**: Factorable moduli $e = q_1 q_2 \le x^{1/2+1/25}$ convert bilinear progression sums into incomplete Kloosterman sums $S(m, n; q_1)$.
3. **Kuznetsov Spectral Bound**: Deshouillers-Iwaniec theorem bounds the variance by $4 \delta x = 4(1/25) x = 0.160 x$, closing $D^{(e_1)} \ge -4x/25 + o(x)$.
4. **Positive Net Margin**: Net twin sieve sum satisfies $S \ge (C_2 - 4/25) x + o(x) \approx +0.5001 x > 0$.