Conditional Expansions for Fixed Consecutive Prime Gap Words

Let an ordered gap word consist of positive even integers. We study two related but distinct counting problems: the general pattern count, which requires primality at the prescribed cumulative offsets, and the consecutive count, which additionally requires the resulting primes to form a block of consecutive primes. For a fixed offset set we first establish an exact finite inclusion–exclusion identity and derive computable Bonferroni bounds. Under an explicit uniform hypothesis on a finite extension family that is stronger than the usual leading-term Hardy–Littlewood prediction, the ratio of consecutive to general counts is expressed as a finite sum involving normalized extension-layer singular series ratios E_r(H) and ratios of weighted logarithmic integrals. Repeated integration by parts together with formal power-series division yields a unified recurrence for coefficients of arbitrary fixed order. Here “arbitrary fixed order” means that the truncation order is fixed before taking X → ∞; no convergence of the resulting formal series is asserted. Since (s)ⱼ₊₁/(s)_j = s + j → ∞, the corresponding formal series has radius of convergence exactly zero; increasing the order at a fixed finite L need not improve the approximation. The complete second-order coefficient is E_2 − E_1, the term −E_1 arising from the continuing contribution of one-point extensions at order L^−2; the third-order coefficient is −E_3 + 2E_2 − (k + 2)E_1. We also prove that offset reflection preserves every E_r and every conditional coefficient. Hence the local coefficients in the expansion cannot produce a persistent directional effect under reversal of the gap word. The numerical part covers the 1554 patterns formed from even gaps 2, 4, …, 12 with k = 2, …, 5 prime points, and compares theoretical models with empirical baselines at the four scales 10^4, 10^5, 10^6, 10^7. On the 376 holdout observations with general count at least 20, the RMSE values of the second-order benchmark retaining only E_2/L^2, the complete second-order model, and the third-order model are 0.1657, 0.1017 and 0.0506 respectively. Before any count at scale 10^8 was generated, 41 representative patterns together with their selection rules were frozen by a hash manifest; at X = 10^8 the corresponding RMSE values are 0.0759, 0.0463 and 0.0119. An independent reference implementation, which does not use the main computational core or NumPy, recomputed all patterns at X = 10^5 and agreed exactly with the main implementation on every integer count. For 17 non-palindromic reversal pairs the mean absolute difference of consecutive ratios falls from 0.01973 at 10^7 to 0.00824 at 10^8; the data examined here provide no evidence of a stable directional reversal effect. The paper develops an extension-layer coefficient system for fixed consecutive multi-gap words and derives a conditional recurrence to arbitrary fixed order. This framework separates the contributions of the different extension layers at second and higher orders, yielding in particular the complete second-order coefficient E_2 − E_1. We also prove reflection invariance of all conditional coefficients and provide a reproducible finite computational study based on a hash-frozen validation set, independent recomputation, and sensitivity analysis.