Foundations of Anti-Mathematics and Family Mathematics: Involutions, Anti-Calculus, Generative Networks, and Relational Invariants

This article develops a more detailed formal foundation for two proposed mathematical research programmes, Anti-Mathematics and Family Mathematics. Anti-Mathematics is formulated as the study of explicitly selected involutions, transported operations, operator conjugation, and the distinctions between inversion and order reversal. A finite-interval anti-calculus is defined through negative accumulation and a derivative-based operator with a prescribed base point. Its algebraic decomposition, kernel, range, stability bounds, and behavior under domain reflection are established directly from standard calculus. Family Mathematics is represented by rule-labelled, levelled directed acyclic graphs of mathematical constructions. Parent, ancestor, sibling, cousin, and maximal common-ancestor relations are defined relative to such graphs. Beyond establishing their elementary properties, we give a linear-time procedure for obtaining all maximal common ancestors in finite networks, and show that graph automorphisms preserve the stipulated family relations. A quantitative edge-defect measure distinguishes exact symmetry from imperfect, proposed transformations. The interaction of the programmes is illustrated by involutively paired networks and by explicit worked examples. Relevant connections with universal algebra, calculus, order theory, category theory, and graph algorithms are stated using an expanded bibliography. The contributions are conceptual organization, precise definitions, transparent proofs, and a reproducible research agenda; standard results are not represented as newly discovered, and the naming of a programme does not imply recognition as an established academic discipline.