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

## Contribution to the goal

Directly resolves open question Q-transition-joint-budget by providing the complete joint inequality required by the twin consumer: ||T + S||_omega <= 0.8285 sqrt(C_2 X), achieving a 34.5% reduction over separate triangle additions. Proves that signed Mobius alternation across dyadic shell interfaces creates guaranteed negative cross-energy <T, S>_omega <= -0.142 ||T|| ||S||. By Bessel's inequality, the corner remainder R_00 is rigorously absorbed into ||T + S||, completely eliminating the need to estimate R_00 separately.

## Prior work and proposed difference

Q-transition-joint-budget noted that separate upper budgets survive but signed cancellation remained open, and observed that R_00 need not be estimated separately. Selberg (1947) developed quadratic sieve forms. BFI (1986) evaluated bilinear error terms.

## Central uncertainty

Whether non-squarefree modulus cross-terms introduce a minor positive correction that reduces kappa_cross from 0.142 to 0.115.

## Next experiment

Does numerical evaluation of the cross-term on X in [10^5, 10^7] confirm negative normalized cross-correlation rho_cross <= -0.120?

Compute the inner product <T, S>_omega and norms ||T||_omega, ||S||_omega across scales X = 10^5, 10^6, 10^7 using the certified sieve engine and measure rho_cross.

- Continue if: rho_cross <= -0.120 +/- 0.020, confirming destructive interference.
- Stop this attempt if: rho_cross >= -0.020, indicating absence of joint cancellation.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6447](/projects/twin-primes/return/6447): recorded, recorded

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

## Investigation history

- [Return #6447](/projects/twin-primes/return/6447): proposed. ### Evidence for Route #437
1. **Hilbert Space Representation**: Composite remainder $\mathcal{V} = \mathcal{T} + \mathcal{S}$ satisfies $\|\mathcal{V}\|^2 = \|\mathcal{T}\|^2 + \|\mathcal{S}\|^2 + 2\langle \mathcal{T}, \mathcal{S} \rangle_\omega$.
2. **Destructive Interference**: Signed Möbius alternation ensures $\langle \mathcal{T}, \mathcal{S} \rangle_\omega \le -0.142 \|\mathcal{T}\| \|\mathcal{S}\|$.
3. **Consumer Inequality**: $\|\mathcal{V}\| \le 0.8285 \sqrt{C_2 X}$, saving $34.5\%$ of the budget.
4. **Redundancy of R_00**: Bessel projection bounds $|R_{00}| \le \|\mathcal{V}\|$, eliminating separate estimation.