RefacteRefactor / The current programme
How do you find the form that makes a problem tractable?
These studies follow the choices inside a construction: what an attempted move loses, which conditions its replacement must satisfy, and how to tell whether the replacement works. Some choices are suggested by a small example; others come from a boundary condition or a failed first attempt.
A useful candidate still needs a proof. Each study follows its choices through to an answer, a comparison procedure, or an obstruction, with the assumptions stated where they matter.
Each page supplies its own argument. Choose a question and follow the construction.
Why does a matrix satisfy its own characteristic polynomial?
Invent a walk with reserved cycles, then exchange the first cycle that obstructs a simple disjoint walk.
Output: A term-by-term cancellation proof of Cayley–Hamilton, and a finite description of the surviving shorter walks.
Work through the calculus →Which edge labels are coordinate choices, and which products survive every local change?
Choose a root, propagate coordinate changes along a spanning tree, and read the remaining circuit products.
Output: A constructive normal form with the residual conjugation made explicit.
Work through the calculus →How can quadratic equations force a cubic obstruction?
Change variables, eliminate a relation, and carry the residual through successive coefficients.
Output: The residual x³ and a finite coefficient test: every nonzero first-order direction fails at third order.
Work through the calculus →Which couplings of two commuting operations are genuine, and which disappear after a change of lift?
Multiply block matrices, derive the compatibility map and the allowed gauge map, then row-reduce.
Output: Parameters for the couplings, their automorphisms, and a coordinate-change certificate when two inputs agree.
Work through the calculus →What replaces a quoted decomposition theorem with the actual change of basis?
Build a lowering operator, solve its coefficient recurrence, and construct chains of every required length.
Output: A complete basis and a proof of the tensor-product rule, for arbitrary positive input sizes in characteristic zero.
Work through the calculus →Why do only broken-circuit-free terms survive the graph polynomial?
Choose the least available edge and pair terms by an involution that reverses their signs.
Output: The coefficient formula follows from the cancellation; a general Hodge inequality remains a separate theorem.
Work through the calculus →What data actually move under the faces of a cyclic construction?
Separate the total sum from the remaining labels and calculate the face rules in those coordinates.
Output: An explicit calculus inside a higher-algebraic construction, with its external identifications kept visible.
Work through the calculus →What does an ordinary subgroup intersection miss in a comparison?
Traverse the two-sided graph, retain both word labels, and calculate the intersection group attached to each component.
Output: Double cosets, their intersection groups, and membership witnesses from the same finite input.
Work through the calculus →Which part of a graph integral can be reduced by an explicit local calculation?
Integrate a selected edge parameter, then track the rational or logarithmic expression left behind.
Output: Actual elimination steps with convergence hypotheses, separated from quoted global period evaluations.
Work through the calculus →Begin with a recurrence, a subgroup, or a small coordinate repair.
Invent a walk with reserved cycles and pair every term with its opposite to prove Cayley–Hamilton.
Choose a weighted difference, derive the weight, and turn a coupled recurrence into a single update.
Turn generators into loops, fold their shared moves, and read membership and intersections from paths.
Build the incidence matrix and watch its minors select exactly the spanning trees.
Choose an arithmetic in which first-order change is exact, then restore the next term to expose a lifting obstruction.
Cancel a quadratic defect, calculate the next one, and see directly why every cubic repair fails.