The Wojciech (WD) Model for Modal Covariance and Positional Rigidity in Fixed-Count Prime Blocks

This work develops the Wojciech (WD) model, a statistical framework for the internal geometry, positional rigidity, and covariance structure of fixed-count blocks of consecutive prime numbers. Motivated by the short-interval prime-count fluctuation theory of Montgomery and Soundararajan, the framework applies a first-order inversion of the local prime-counting process followed by endpoint normalization. The resulting endpoint-pinned field is projected onto shifted-Legendre modes, providing a unified representation of positional fluctuations within individual blocks and statistical dependence between disjoint blocks. The central mathematical object is a modal covariance operator whose diagonal sector characterizes within-block positional rigidity, while its off-diagonal sector describes cross-block correlations. The framework yields continuum covariance laws, finite-grid Toeplitz operators, explicit modal sign relations, and asymptotic multipole expansions governing the decay of correlations with block separation. The theory is further extended to heterogeneous block configurations with unequal spans, prime counts, and modal orders through exact cell integration of the underlying continuum kernel. Finite-count and finite-density refinements are developed using centered Hardy–Littlewood singular-series correlations, producing parameter-free pair-level arithmetic corrections to both modal variances and cross-block covariance. A formal span-marginalized higher-cluster construction extends the treatment beyond pair order while retaining the random physical span as an intrinsic component of the fixed-count ensemble. Extensive numerical benchmarks evaluate the model across multiple prime scales. The finite-density pair refinement achieves a mean absolute relative error of 0.234% in the single-block modal-variance benchmark. A separate high-scale covariance study spanning (Q=10^{12})–(10^{15}) and 684 modal configurations yields a pooled measured-to-predicted amplitude slope of 1.00154 ± 0.0092, with all 232 statistically resolved signals above the three-standard-error threshold exhibiting the predicted sign. Additional heterogeneous-block benchmarks examine 192 configurations across four prime scales, supplemented by independently implemented computational diagnostics. These experiments support the predicted sign structure in statistically resolved cases while revealing limitations in the amplitude calibration of unequal-block covariance, particularly the distinction between observed-span conditioning and common-density rank-grid evaluation. The resulting limitations are explicitly retained rather than absorbed into fitted correction parameters. The framework provides a coherent, quantitatively testable statistical description of consecutive-prime geometry. Its prime-specific predictions remain conditional on the underlying short-interval covariance assumptions and inverse-counting approximations, while the higher-cluster construction is formal rather than a complete asymptotic proof. The deposit includes the complete research paper, LaTeX source, bibliography, numerical benchmark datasets, computational implementations, reproducibility materials, and associated supporting files.