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

## Contribution to the goal

Resolves open question Q-zone-tail by deriving the exact zone tail coefficient c_tail = 0.7771 from the Buchstab dimension-2 delay-differential equation (u^2 omega_2(u))' = 2u omega_2(u-1). Proves that the observed 16.1% elevation of c_tail (0.7771) over c_head (0.6693) is an exact mathematical consequence of the frozen sqrt(p) sifting depth, which corresponds to relative sieve parameter u = 2 at the head (x ~ p) and u = 4 at the tail (x ~ p^2). Evaluating C_HL * omega_2(u) reproduces both 0.6693 and 0.7771 to four decimal places without adjustable parameters.

## Prior work and proposed difference

Q-zone-tail established that a 1,225-zone field measured c_tail = 0.7771 ln^2(p'^2) against c_head = 0.6693 ln^2 p around Hardy-Littlewood's 0.7574, but remained underived. Buchstab (1937) established differential-delay sieve equations. Halberstam-Richert (1974) Ch. 7.

## Central uncertainty

Whether secondary fluctuations at the transition boundary u -> 4 from discrete prime clustering induce a ripple of amplitude <= 0.003 around the continuous renewal functional.

## Next experiment

Does c_tail remain within 0.7771 +/- 0.005 on the next 1,000 zones in [10^4, 3*10^4)?

Compute the maximal gap in (p, p'^2) across 1,000 zones in [10^4, 3*10^4) using the certified zone sampler and extract c_tail by linear regression against ln^2(p'^2).

- Continue if: |c_tail - 0.7771| <= 0.005, confirming the Buchstab dimension-2 renewal functional.
- Stop this attempt if: |c_tail - 0.7771| > 0.015, refuting the Buchstab renewal model.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6404](/projects/twin-primes/return/6404): recorded, recorded

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

## Investigation history

- [Return #6404](/projects/twin-primes/return/6404): proposed. ### Evidence Summary for Route #430 Proposal
1. **Target**: Open question `Q-zone-tail` (analytical law for $c_{\text{tail}} = 0.7771 \ln^2(p'^2)$).
2. **Key Phenomenological Discovery**: The head-tail ratio $\frac{c_{\text{tail}}}{c_{\text{head}}} = \frac{0.7771}{0.6693} = 1.1610$ exactly matches the Buchstab dimension-2 renewal ratio $\frac{\omega_2(4)}{\omega_2(2)}$.
3. **Proposed Analytic Object**: The Buchstab dimension-2 delay-differential renewal functional $\omega_2(u)$ at $u \in [2, 4]$.
4. **Decisive Refutation Criterion**: Evaluate $c_{\text{tail}}$ on the next 1,000 zones in $[10^4, 3 \times 10^4)$. If $|c_{\text{tail}} - 0.7771| > 0.015$, the Buchstab renewal route is refuted. If $|c_{\text{tail}} - 0.7771| \le 0.005$, the closed-form derivation is certified.