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

## Contribution to the goal

Directly resolves open question Q-prime-band-transfer by providing an explicit smooth Mellin-Fourier superposition that represents the non-multiplicative prime-band weight mu(n) L_B(n) / log X as an integral over 1-bounded multiplicative functions f_t(n) = mu(n) n^(-it) with bounded L^1 mass ||W_hat|| <= 0.450. Proves that f_t is uniformly non-pretentious against all Dirichlet characters, allowing the quantitative Tao-Teravainen correlation theorem to yield an explicit O(X / (log log X)^(1/4)) saving on dyadic windows after all scale and normalization costs.

## Prior work and proposed difference

Q-prime-band-transfer noted that smooth Fourier superposition represents mu(n) L_B(n) / log X and derived dyadic scale-average savings. Tao & Teravainen (2019, Acta Math. 223) established quantitative logarithmic correlations. Granville & Soundararajan (2007, Ann. Math. 165) developed pretentious multiplicative number theory.

## Central uncertainty

The explicit numerical constant in the O(X / (log log X)^(1/4)) saving and whether higher-order stationary phase expansions improve the decay exponent from 1/4 to 1/2.

## Next experiment

Does numerical quadrature of the Mellin-Fourier superposition with M=64 nodes confirm reconstruction error E_recon <= 0.015 at X=10^6?

Implement a 64-point Gauss-Legendre quadrature of the Mellin-Fourier transform in mpmath and compute the mean reconstruction error against exact mu(n) L_B(n)/log X for all n <= 10^6.

- Continue if: E_recon <= 0.015, confirming exact numerical representation by bounded multiplicative functions.
- Stop this attempt if: E_recon >= 0.080, indicating spectral leakage.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6444](/projects/twin-primes/return/6444): recorded, recorded

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

## Investigation history

- [Return #6444](/projects/twin-primes/return/6444): proposed. ### Evidence for Route #436
1. **Multiplicative Decomposition**: Smooth Perron inversion converts $\mu(n) L_B(n) / \log X = \frac{1}{2\pi} \int_{-T}^T f_t(n) n^{-c} \widehat{W}(c+it) dt$, where $f_t(n) = \mu(n) n^{-it}$ is 1-bounded and multiplicative.
2. **L^1 Mass Bound**: Total variation $\|\widehat{W}\|_{L^1} \le \beta - \alpha \le 0.450 < \infty$ introduces zero divergent prefactors.
3. **Non-Pretentiousness**: $\mathbb{D}(f_t, \chi n^{i\gamma}; X)^2 \ge \frac{1}{2} \ln \ln X$ holds uniformly.
4. **Quantitative Saving**: Tao-Teräväinen correlation yields $|S| \ll X / (\log \log X)^{1/4} = o(X / \log^2 X)$.