Projection-Induced Coherence in Prime Gap Statistics: Baseline Geometry, Operator Alignment, and Residual Effects

This work motivates a broader, domain-agnostic view in which observed regularities may reflect projection through aligned measurement frameworks rather than intrinsic structure alone. Although established here in the arithmetic setting, the same mechanism could plausibly operate in other relational systems, including physical theories where observables depend on shared geometric or operator structure. This paper develops a diagnostic framework to analyze apparent structure in prime gap statistics by explicitly separating intrinsic properties from projection-induced effects. Using the Primorial Coverage Coherence (PCC) statistic, we show that the majority (~94%) of observed cross-depth coherence in prime gaps arises from baseline integer-lattice geometry—specifically, the generic divisibility structure of the natural numbers—independent of primality. A smaller residual (~6%) appears only when the operator used to select primes (divisibility avoidance) aligns with the operator used to measure gap structure (divisibility coverage). Through controlled numerical experiments up to N = 10^6, we demonstrate that this residual is not a conditioning cost or intrinsic multiplicative signature but a projection artifact caused by operator alignment. We falsify the conditioning-cost hypothesis by constructing synthetic survivor processes under non-arithmetic partitions, which produce negative residuals. We further show that non-multiplicative relations, such as digit-sum partitions, can reproduce or exceed prime PCC values, confirming that PCC primarily measures partition geometry rather than arithmetic essence. To reduce projection bias, we introduce a smallest-prime-factor (SPF) conditioned analysis that isolates attribution structure rather than cumulative coverage. While SPF-based statistics yield cleaner separation between primes and random processes, they reflect tautological properties of prime gaps rather than novel arithmetic structure. Overall, we establish projection-induced coherence as a measurable, falsifiable phenomenon and provide explicit criteria for distinguishing genuine structure from artifacts introduced by aligned selection and measurement operators. The framework applies broadly to layered relational systems beyond number theory, offering a general methodology for diagnosing spurious coherence induced by projection.