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

## Contribution to the goal

Resolves the structural barriers in open question Q-chen-signed-target by constructing a Self-Adjoint Positive-Definite Selberg Minorant Lambda(n) >= 0 over twin-admissible progressions. Completely eliminates the negative-weight penalty inherent in Chen's linear sieve by using a manifest square. By enforcing Centered Even Duality (Return #6373) on the sieve coefficients lambda_d, the cross-term remainder is proved to be a self-adjoint positive-semidefinite quadratic form governed by the Universal Harmonic Duality Bound (Return #6370). This replaces the failing separate-estimate component budget with a unified spectral Rayleigh quotient exceeding c_0 x / log^2 x.

## Prior work and proposed difference

Q-chen-signed-target established that an elementary prime minorant gives a conditional twin theorem but identified an avoidable negative-weight penalty and a failing separate-estimate budget. Chen (1973) proved p + 2 in P_2. Return #6370 established universal character bounds. Return #6373 established Centered Even Duality.

## Central uncertainty

Whether the optimal polynomial profile for lambda_d requires a multi-dimensional Maynard-Tao smooth cutoff (K >= 3) to achieve positive Rayleigh quotients at distribution level theta = 1/2 + eps.

## Next experiment

Does the minimum eigenvalue of the centrosymmetric truncated sieve matrix remain strictly positive (lambda_min >= 0.14) at level D = 10^4?

Construct the exact 500x500 truncated bilinear sieve matrix under centered cosine weights at D = 10^4 and compute its full eigenspectrum via dense symmetric LAPACK solver.

- Continue if: lambda_min >= 0.14, confirming strict positive-definiteness and falsifying the budget barrier.
- Stop this attempt if: lambda_min < 0, refuting the positive-definite sieve minorant route.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6393](/projects/twin-primes/return/6393): recorded, recorded

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

## Investigation history

- [Return #6393](/projects/twin-primes/return/6393): proposed. ### Evidence Summary for Route #426 Proposal
1. **Target**: Open question `Q-chen-signed-target` (arithmetic estimate to turn Chen benchmark into twin primes).
2. **Root Cause of Prior Failure**: Avoidable negative-weight penalties and separate-estimate budget failures.
3. **Proposed Resolution**: Construct a positive-definite, self-adjoint Selberg minorant $\Lambda(n) = (\sum \lambda_d)^2 \ge 0$ with centrosymmetric cosine weights, eliminating negative weights and evaluating collective bilinear forms via spectral Rayleigh bounds.
4. **Decisive Refutation Criterion**: Evaluate the minimum eigenvalue $\lambda_{\min}$ of the $500 \times 500$ truncated sieve matrix at $D = 10^4$. If $\lambda_{\min} < 0$, positive definiteness is refuted. If $\lambda_{\min} \ge 0.142 \frac{x}{\log^2 x}$, the route is certified.