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

## Contribution to the goal

Candidate pi_{1,1.745}(x) > 0.01 Cx/log^2 x.

## Prior work and proposed difference

Li-Liu arXiv:2606.05224v2; returns 6277, 6274, 6272, 6270.

## Central uncertainty

The eleven other bounds are imported, not re-derived; no independent review yet.

## Next experiment

Is the switched convolution of T3 really the two-prime count, i.e. is beta allowed to be the smaller prime (theta = 1-u rather than 1/2)?

Enumerate the survivors of S(A_p;P,p) for x = 1e6 over the whole range [x^(149/349), x^(1/2)] and check that every survivor has m prime with m >= p; then compare the per-block density with 2C/((1-u)log x) against the two candidate constants, which differ by 2(1-u).

- Continue if: Every survivor has m prime, m >= p, and the measured block density matches 2 int du/(u(1-u)^2) to within a few percent, selecting theta = 1-u.
- Stop this attempt if: Any survivor with composite m or m < p, or a density matching 4 int du/(u(1-u)) instead, which would force theta = 1/2 and B = -0.2636.



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6283](/projects/twin-primes/return/6283): recorded, recorded

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

## Investigation history

- [Return #6283](/projects/twin-primes/return/6283): proposed. See the report; the bijection lemma, z_u < p, the nu-uniform constant and the ledger arithmetic, with 172 gates and 0 failures.