Mathematics reference
A self-contained tour of the algebra, order theory, category theory and process-theoretic machinery used to formalize industrial-materials R&D — built up from sets to effectful categories, each idea tied back to a concrete problem.
Modules
Ten self-contained modules — read in order, or jump to what you need.
Orientation & notation
How to use this reference, the notation conventions, the P-/RD-code legend, and the concept→problem index that ties every idea back to the materials-R&D problems.
Sets, orders & lattices
Sets and functions and their canonical factorization; equivalence and quotients; partial orders, lattices, Pareto antichains and Galois connections — the order-theoretic spine.
Categories & universal properties
Objects and morphisms; iso/mono/epi; functors and natural transformations; products, limits, adjunctions and Yoneda — structure described by universal properties.
Monoids, groups & actions
Binary operations and the invertibility axis; why processing is a monoid not a group; presentations and the word problem; group actions and symmetry.
Rings, modules & multilinear algebra
Rings and modules; direct sum vs tensor product (non-interacting vs fully coupled); symmetric/exterior algebra; canonical forms and invariants; the Grothendieck group.
Homological algebra
Chain complexes, exact sequences and homology as the measure of obstruction; Tor and Ext as derived functors — conservation laws and what blocks them.
Fields & Galois theory
Field extensions, the Galois correspondence, the algebra↔geometry dictionary (Nullstellensatz), and impossibility-by-invariant — when something simply cannot be done.
Monoidal & process categories
Monoidal categories and string diagrams; premonoidal/effectful categories for path-dependence; resource theories; a blueprint process category of paint-making. (Beyond the book.)
Computation, locality & knowledge
Rewriting and confluence; trace monoids for concurrency; sheaves and cohomological gluing; Formal Concept Analysis and ologs — locality, knowledge and data fusion. (Beyond the book.)
The materials bridge
Where the abstract machinery touches materials science: PSP linkages, identifiability, sloppy models, mixture designs, scaling laws, equivariance — and the honest limits.
Learning paths
Curated routes through the modules for different goals.
The core throughline
Orders → categories → monoids → processes: the spine of the formalization.
Hidden state & obstruction
Why performance factors through something you can’t see — and what blocks invertibility.
Knowledge & locality
Transfer, scope, and fusing partial knowledge across a catalogue.
The codes. The P# chips are the thirteen problem characteristics and the RD# chips the eight research directions from the synthesis. The full legend and the concept→problem index live on the orientation page.
Part of a four-document set: the ARiSE draft (problem + AI solution), this modular Mathematics reference, the companion materials reference, and the synthesis. Generated from modular Markdown with a custom static-site builder.
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