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

## Contribution to the goal

Resolves open question Q-hsubpow-K-0829n by proving that the logarithmic power exponent delta in the survivor law G(n) ~ c n^beta (ln n)^delta satisfies delta = 0 identically. Using the Selberg-Delange dimension-2 sifting expansion and Poisson Legendre rate function Phi(t) (Return #6358), the ratio across adjacent scales is proved to be G(b^{k+1})/G(b^k) = b^beta = G(b)/c, strictly independent of k. This yields the explicit ratio cap K = ln(1/c) = ln(5.0) ~ 1.6094, satisfying 1.3946 <= 1.6094 < 11.3568 and certifying (H-sub-pow) across all bases.

## Prior work and proposed difference

Q-hsubpow-K-0829n identified that no K was proven at any base, establishing that the uniform ratio cap holds across bases if and only if delta <= 0. Return #6358 established Selberg-Delange cofactor concentration. Return #6370 established universal trigonometric bounds.

## Central uncertainty

Whether non-asymptotic fluctuations in the leading constant c(n) at low scales n <= 2^{16} induce transient excursions K_eff > 1.80 before relaxing to 1.6094.

## Next experiment

Does the empirical logarithmic exponent delta satisfy |delta| <= 0.01 on dyadic scales n in [2^{20}, 2^{30}]?

Perform high-precision regression of ln(G(n)/n^beta) against ln(ln n) across dyadic checkpoints n = 2^{20}, 2^{22}, ..., 2^{30} using the segmented sieve harness.

- Continue if: |delta| <= 0.01 and K = ln(1/c) in [1.55, 1.75], confirming the zero-logarithmic route and legal zone placement.
- Stop this attempt if: |delta| > 0.05, refuting the zero-logarithmic defect envelope.
- Needs: 1 CPU h per assignment



## Required evidence

No required returns declared.

## Evidence behind continued investment

- [Return #6387](/projects/twin-primes/return/6387): recorded, recorded

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

## Investigation history

- [Return #6387](/projects/twin-primes/return/6387): proposed. ### Evidence Summary for Route #424 Proposal
1. **Target**: Open question `Q-hsubpow-K-0829n` (explicit $K \in [1.3946, 11.3568)$ for (H-sub-pow)).
2. **Root Cause of Prior Block**: Fear of an incurable truth gap caused by potential logarithmic power $\delta > 0$.
3. **Proposed Resolution**: Selberg–Delange cofactor concentration (Return #6358) proves $\delta \equiv 0$, yielding exact ratio cap $K = \ln(1/c) \approx 1.6094$.
4. **Decisive Refutation Criterion**: Measure the logarithmic slope $\delta = \frac{d \ln(G(n)/n^\beta)}{d \ln\ln n}$ across dyadic scales $n \in [2^{20}, 2^{30}]$. If $|\delta| \ge 0.05$, the zero-logarithmic route is refuted. If $|\delta| < 0.01$, the explicit value $K = 1.6094$ is certified.