RefacteMathematics / A focused collection
Worked constructions
Walks, labelled graphs, polynomial obstructions, and compatible operators.
This collection brings together nine arguments that develop a construction from its constraints. The Mathematics directory includes all fourteen worked articles, starting points, and further notes.
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Matrices and graphs
Augment walks with reserved cycles, then pair terms with equal edge products and opposite signs to prove Cayley–Hamilton.
Background: Matrix multiplication and the permutation definition of a determinant.
Multiply block matrices, derive the compatibility map and the allowed gauge map, then row-reduce.
Background: Block matrices and solving linear equations over a field.
Build a lowering operator, solve its coefficient recurrence, and construct chains of every required length.
Background: Finite-dimensional linear algebra and polynomial arithmetic; characteristic zero is explicit.
Choose the least available edge and pair terms by an involution that reverses their signs.
Background: Finite graphs, subsets, and inclusion–exclusion.
Groups and labelled paths
Choose a root, propagate coordinate changes along a spanning tree, and read the remaining circuit products.
Background: Graphs, multiplication and inverses in a group.
Traverse the two-sided graph, retain both word labels, and calculate the intersection group attached to each component.
Background: Finite groups, subgroups, left and right multiplication.
Separate the total sum from the remaining labels and calculate the face rules in those coordinates.
Background: Tuples and addition in an abelian group; all operations are defined.
Polynomial equations and calculus
Change variables, eliminate a relation, and carry the residual through successive coefficients.
Background: Polynomial expansion and coefficients of a power series.
Integrate a selected edge parameter, then track the rational or logarithmic expression left behind.
Background: One-variable integration and nonnegative improper integrals.