Inheritance as a Linear Operator, and What Replaces It: From Punnett squares to a complete probabilistic kernel

Abstract

This is a preprint. It has not been peer reviewed. A unified, audited and extended treatment of Mendelian inheritance modelling, in seven parts. Parts I and II preserve the mathematics of an earlier 54-page study of sickle-cell, ABO and ABO x Rh inheritance as linear algebra, and then audit it against its own MATLAB source. The audit changes the premise: the original programs already looped over all 36 and all 324 ordered parental pairs, so the reported 83.62% and 67.83% coverage figures describe the prose and displayed tables rather than the executed model, and no coverage-restoration claim is available. The displayed sickle-cell trajectory is reproducible but sums to 93.75%, because it zeroes the affected class without renormalising. Part III proves G(n) = 3^n by induction, generalises to arbitrary allele counts, derives U = G(G+1)/2 unordered parental pairs and exactly (15^n + 5^n)/2 supported transitions, then replaces the square matrix with a complete rectangular kernel implemented as dense, CSR, hash-adjacency and streamed representations. All four agree to 6.94e-17. At five loci CSR retains 4,575,976 payload bytes against dense's 57,631,824. Part IV tests the population assumptions against public rs334 genotype calls for 2,504 individuals across 26 populations, with exact Hardy-Weinberg tests and a Holm adjustment, and compares the new representation against the square-matrix baseline under an explicit taxonomy separating structural coverage, probability mass and predictive accuracy. Part V adds eye colour as a two-locus epistatic trait driven by real rs12913832 genotype calls, audits it, and compares it against two published phenotyped cohorts; then treats height; then states P(n), the general n-locus problem, with output-size lower bounds showing that materialising the complete kernel is Omega(15^n) for any implementation, that returning all children of a heterozygous pair is Omega(3^n), and that the implemented factored single-child query is Theta(n) and therefore asymptotically optimal. Part VI derives, without implementing, an extension in which parental age enters the transmission kernel through de novo mutation and an individual's age enters the genotype-to-phenotype map through methylation and histone acetylation, including an expression gate with its cross-entropy objective, gradient, convexity proof and step-size bound. Part VII evaluates the whole. Scope. This work establishes representational completeness, numerical agreement between implementations, measured performance on one machine, and asymptotic bounds. It establishes no predictive accuracy for any trait, because no held-out phenotype data was available for any model built here. Part VI is derivation only. Nothing in it is medical, diagnostic or reproductive advice. Reproducibility. Every number in the paper is bound at build time to a retained machine-readable result. Source code, tests, frozen datasets with SHA-256 digests, raw benchmark output, figures and the source ledger are included. The software is MIT licensed; the written work and figures are CC BY 4.0. The included aggregate genotype datasets remain subject to the terms of their originating projects and are not relicensed. AI assistance. Computation, figure generation, manuscript preparation and adversarial review were assisted by AI agents. Scientific authorship, interpretation and responsibility for every claim are the author's.