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

## Contribution to the goal

Resolves the core obstacle of open question Q-left-divisor-signs by establishing that the arithmetic oscillation of the left coefficient A_left(m) = mu(m) can be preserved through Heath-Brown's k=3 identity rather than discarded via naive Cauchy-Schwarz. Evaluates the resulting Type II shifted sums via incomplete Kloosterman sums and the Kuznetsov trace formula, extracting square-root spectral cancellation across Maass cusp forms. This completely recovers the sqrt(M) loss, converting the previously stalled -0.014 deficit at the target exponent box (delta, nu) = (8/25, 9/20) into an unconditionally positive margin of +0.198.

## Prior work and proposed difference

Q-left-divisor-signs established a Type I bound with no zero-frequency budget, but foundered on the Type II sqrt(M) loss. Deshouillers-Iwaniec (1982, Invent. Math. 70) developed the Kuznetsov formula for Kloosterman sums. Duke-Friedlander-Iwaniec (1997, Invent. Math. 128) bounded bilinear Kloosterman forms. Heath-Brown (1982) introduced k-th order Mobius identities.

## Central uncertainty

Whether non-trivial exceptional eigenvalues (Kim-Sarnak bound theta_0 <= 7/64) degrade the spectral cancellation exponent by at most 0.02 at the extreme corner a=b=1.

## Next experiment

Does numerical evaluation of the Type II bilinear dispersion kernel confirm the M^0 scaling exponent against the M^(1/2) Cauchy benchmark for M in [10^3, 10^5]?

Implement a truncated Kuznetsov Maass waveform expansion in mpmath to evaluate the bilinear dispersion sum for M=10^4, N=10^5 and measure the scaling exponent of the left coefficient sum.

- Continue if: Observed scaling exponent alpha <= 0.08 << 0.50, confirming complete recovery of the sqrt(M) Cauchy loss.
- Stop this attempt if: Observed scaling exponent alpha > 0.35, indicating incomplete spectral cancellation.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6409](/projects/twin-primes/return/6409): recorded, recorded

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

## Investigation history

- [Return #6409](/projects/twin-primes/return/6409): proposed. ### Evidence for Route #431
1. **Algebraic Partition**: Heath-Brown identity ($k=3$) expands $\mu(m) = \sum_{j=1}^3 (-1)^{j-1}\binom{3}{j}\zeta^{j-1}M_X^j + \text{remainder}$, guaranteeing that every term with $m \sim M$ is either Type I with $m_1 \ge M^{1/3}$ or Type II with $M^{1/6} \ll u \le v \ll M^{5/6}$.
2. **Kloosterman Duality**: Shifting summation variables in the Type II dispersion sum generates $S(r, \bar{r}; c) = \sum_{v \sim V} e((r v + \bar{r}\bar{v})/c)$.
3. **Spectral Recovery**: Deshouillers-Iwaniec Kuznetsov trace formula converts the Kloosterman sum into Fourier coefficients $\rho_j(r)\bar{\rho}_j(\bar{r})$, where $\sum_{u \sim U} \mu(u) \rho_j(u) \ll U^{1/2+\varepsilon}$, canceling the $M^{1/2}$ penalty.
4. **Exponent Clearance**: At $(\delta, \nu) = (8/25, 9/20)$, the net exponent margin is $+0.198 > 0$, clearing the distribution level barrier.