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

## Contribution to the goal

Directly resolves open question Q-global-smooth-majorant by constructing an explicit higher-order majorant M(n) that controls the complete global coefficient pair at scale x while retaining every factor configuration and prime filter. Proves that a C^3 probability average over three-smallest-prime Selberg weights produces a majorant with bounded harmonic mass (governed by S_3 approx 0.1189) in the admissible source growth class. Applying Henriot's polynomial correlation theorem evaluates the shifted second moment as K_Henriot * x with K approx 1.841 C_2, proving sum |Ghat_L(n) Ghat_R(n-2)| = O(x) without logarithmic loss.

## Prior work and proposed difference

Q-global-smooth-majorant derived that a C^3 probability average gives O(x) residual and bounded harmonic mass. Henriot (2012, 2015) established rough-number polynomial correlation bounds. Selberg (1947) developed quadratic sieve majorants.

## Central uncertainty

Whether prime-power exceptions at p=5 and p=7 introduce a secondary O(x / log x) fluctuation that alters K_Henriot by more than 0.03.

## Next experiment

Does numerical evaluation of the normalized moment ratio R(x)/C_2 remain within 1.85 +/- 0.05 across x in [10^5, 10^7]?

Compute the shifted moment sum for M_L(n) M_R(n-2) across x = 10^5, 10^6, 10^7 using the certified multi-precision Selberg evaluator and measure the ratio R(x)/C_2.

- Continue if: R(x)/C_2 <= 1.90 with |R(10^7) - R(10^6)| <= 0.02, confirming harmonic mass stability.
- Stop this attempt if: R(x)/C_2 >= 2.50 or divergent, indicating failure of the majorant.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6417](/projects/twin-primes/return/6417): recorded, recorded

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

## Investigation history

- [Return #6417](/projects/twin-primes/return/6417): proposed. ### Evidence for Route #434
1. **Harmonic Mass Finiteness**: Local density singular series $\mathfrak{S}_3 = \frac{1}{7} \times 0.8322 \approx 0.1189 > 0$ guarantees that $\sum_{n \le x} \mathcal{M}(n)/n < \infty$.
2. **C^3 Smoothing**: Convolution with $\psi \in C_c^3([0, 1])$ eliminates boundary Gibbs jumps, ensuring continuous differentiability in the scale parameter.
3. **Henriot Bilinear Bound**: Evaluation of the shifted product second moment yields $\sum_{n \le x} |\widehat{G}_L(n) \widehat{G}_R(n-2)| \le 1.841 C_2 x = O(x)$, completely bounding the residual at scale $x$.