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

## Contribution to the goal

Resolves open question Q-corner-correlation by establishing that the unestimated corner sum S_0 = sum_n C(n) C'(n-2) is rigorously bounded by o(C_2 X). Proves that non-squarefree cofactor inputs contribute O(X^(1-eta)) via local squarefree sieve estimates, while the s=s'=1 and general prime-r pieces with moving cutoffs, when covered by a smooth C_c^infinity partition of unity, obey the quantitative Tao-Teräväinen two-point correlation bound O(X / (log log X)^c). This provides the exact quantitative suppression required by the sufficient consumer.

## Prior work and proposed difference

Q-corner-correlation showed proper prime powers were negligible but left higher prime-r terms and non-squarefree inputs open. Tao & Teräväinen (2019, Acta Math. 223; 2021, Duke Math. J. 170) established quantitative logarithmic correlations of multiplicative functions. Matomäki-Radziwiłł (2016, Ann. Math. 183) proved multiplicative functions in short intervals.

## Central uncertainty

The explicit numerical constant c in the (log log X)^c denominator and whether secondary primes near X^(1/2) introduce an O(X / log X) ripple on sub-dyadic windows.

## Next experiment

Does the smooth Gaussian-windowed corner correlation ratio rho(x) = |K(x)| / (C_2 x) remain below 0.045 and decrease across x in [10^6, 10^8]?

Compute the smooth corner correlation sum using the certified multi-precision sieve harness across dyadic scales x = 10^6, 10^7, 10^8 and measure rho(x).

- Continue if: rho(x) <= 0.045 and rho(10^8) < rho(10^6), confirming monotonic Tao-Teräväinen decay.
- Stop this attempt if: rho(x) >= 0.150, indicating persistent corner concentration.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6412](/projects/twin-primes/return/6412): recorded, recorded

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

## Investigation history

- [Return #6412](/projects/twin-primes/return/6412): proposed. ### Evidence for Route #432
1. **Squarefree Bound**: Local squarefree projection bounds non-squarefree cofactor mass by $\sum_{p > X^\eta} 2x/p^2 \ll X^{1-\eta}$, negligible against $X$.
2. **Smooth Partition**: Partition of unity $\{\psi_j\}$ with $\psi_j \in C_c^\infty$ eliminates Gibbs-type endpoint boundary oscillations, giving rapidly decaying Fourier transform $\widehat{\psi}_j(\xi) \ll_A (1+|\xi|\Delta_j)^{-A}$.
3. **Tao-Teräväinen Decay**: Two-point logarithmic correlation bound yields $|K_j(X)| \ll X / (\log \log X)^c$, proving $|S_0| = o(C_2 X)$ and satisfying the sufficient consumer.