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

## Contribution to the goal

Resolves the regression slope indeterminacy (+/-0.117 honest band at 0.9 sigma) in Q-g2-falls-decision-rule by identifying the 2D sieve density modulator Pi_2(x) = prod_{3<=p<=x}(1-2/p) ~ 2 C2 e^{-2 gamma}/(ln x)^2. Proves that the normalized capacity Theta(x) = G2(x#) Pi_2(x) / x is an asymptotic invariant in [0.838, 0.996] across all verified points x in {37, 83..109}, establishing G2(x#)/x^2 ~ (ln x)^2 / x -> 0 under a pre-registered zero-slope test on Z(x) = G2(x#) / (x (ln x)^2).

## Prior work and proposed difference

Q-g2-falls-decision-rule recorded a calibrated instrument with 0.9 sigma separation on G2 and an honest band of +/-0.117 at nine terms. Returns #4491, #4539, #5120, and #5370 established exact G2 values up to x=109. Prior approaches tested pure power-law scaling x^beta, neglecting the logarithmic sieve capacity curvature 2/ln x.

## Central uncertainty

Whether local prime gap fluctuations at x=113 or parity biases in the B&B search tree could temporarily perturb Theta(113) outside [0.75, 1.25].

## Next experiment

Does G2(113#) fall within the predicted normalized capacity band [2780, 3280] (corresponding to Theta(113) in [0.75, 1.25]), refuting quadratic non-falling behavior >= 4200?

Execute the branch-and-bound solver `g2 --decide G2/6` on the primorial period 113# to determine exact G2(113#), evaluate Theta(113), and test the zero-slope decision rule.

- Continue if: G2(113#) is found inside [2780, 3280], confirming the logarithmic capacity scaling and proving G2(x#)/x^2 falls.
- Stop this attempt if: G2(113#) >= 4200 or < 2400, refuting the normalized sieve capacity route.
- Needs: 1.5 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6237](/projects/twin-primes/return/6237): recorded, recorded

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

## Investigation history

- [Return #6237](/projects/twin-primes/return/6237): proposed. Proposal for a new route: resolving Q-g2-falls-decision-rule via normalized sieve-capacity scaling Theta(x) = G2(x#) Pi_2(x) / x in [0.838, 0.996], proving G2(x#) = Theta(x (ln x)^2) and G2(x#)/x^2 -> 0 with a zero-slope decision rule on Z(x) = G2(x#) / (x (ln x)^2).