Numerical Evaluation of the Stopping-Time Sign Bias in Minimal Prime Offsets and Quantification of the Independence Gap

In previous work [1], we identified a systematic directional asymmetry in the minimal prime offsets q(p) for doubled primes, originally reporting an empirical proportion of 55.5% for the positive direction 2p + q when p ≡ 1 (mod 6) over 10^7 primes. In this paper, we extend the empirical dataset to 455 million primes up to the sieve limit 10^10, utilizing a highly optimized multiprocessing sieve, observing a live asymptotic drift from 55.21% (at 1.65 × 10^8 primes) to 55.10% (at 4.55 × 10^8 primes). We construct a systematic truncated stopping-time sum to numerically evaluate this bias under the assumption of directional independence, using the Hardy–Littlewood 2-tuple singular series approximation S_2(q) [2]. Rather than evaluating the model at a single representative scale, we compute a weighted average of the bias c_1(λ) over the full empirical distribution of scales λ = 1/ln(2p) across the 455-million-prime dataset, yielding a model prediction of 0.5313 ± 0.0005. A heuristic Richardson extrapolation suggests an asymptotic value of c_1^∞ ≈ 0.5059 ± 0.0014 (see Section 5 for full details and caveats, including the pending cubic-fit validation); the uncertainty estimate should be interpreted cautiously, as it depends on the assumed asymptotic correction model. The excess above 0.5 is approximately 0.0059, which exceeds the fit uncertainty by a factor of four; however, we emphasise that this remains a heuristic numerical estimate conditional on the Hardy–Littlewood k-tuple conjecture, and does not constitute a formal proof that the bias persists in the limit. The uncertainty bound ±0.0014 is heuristic (half-spread of linear vs. quadratic fits) and should be treated as a lower bound pending a cubic-fit cross-validation. At the extended empirical scale of ln(2p) ≈ 22.9, the weighted-average model predicts a bias of 0.5313. By comparing this theoretical lower bound against the empirical dataset, we quantify the "independence gap" of approximately 2.0 percentage points. We demonstrate that while the modulo-3 gating mechanism forces a bias strictly greater than 0.5 within the verified finite horizon of the 2-tuple model (Proposition 6.5, Section 6; scope restricted to λ ≥ 0.001), the 2-tuple approximation systematically underpredicts the true magnitude. The observed residual discrepancy is consistent with the influence of higher-order cross-correlations within the 3-tuple constellation {q, 2p − q, 2p + q}, suggesting that the full S_3 singular series is a necessary (though not shown sufficient) ingredient for deriving the 55.1% empirical limit; however, a rigorous derivation of this connection remains open. The extended dataset also provides the first direct empirical measurement of the tie frequency (q(p) resolving both directions simultaneously), observed at 1.12% of all primes against a model prediction of approximately 1.38%, consistent with the systematic underprediction imposed by the independence assumption. All theoretical results in this paper are conditional on the Hardy–Littlewood k-tuple conjecture [2], which remains unproven.