RefacteMathematical writing
Worked arguments in algebra, combinatorics, and analysis.
Start with a short example or a longer study below. Earlier work is available in the Tensorgami archive.
Four studies, each with a complete argument and a worked example.
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.
What it proves: A term-by-term cancellation proof of Cayley–Hamilton, and a finite description of the surviving shorter walks.
Follow the construction →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.
What it proves: A constructive normal form with the residual conjugation made explicit.
Follow the construction →How can quadratic equations force a cubic obstruction?
Change variables, eliminate a relation, and carry the residual through successive coefficients.
What it proves: The residual x³ and a finite coefficient test: every nonzero first-order direction fails at third order.
Follow the construction →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.
What it proves: Parameters for the couplings, their automorphisms, and a coordinate-change certificate when two inputs agree.
Follow the construction →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.
Additional arguments and problems in algebra, arithmetic, and analysis.
Change coordinates one degree at a time: a quadratic repair succeeds, but a cubic obstruction remains in characteristic three. A filtered complex identifies the precise failure.
The main §1 performs the coefficient repair directly. The filtered complex then organises the obstruction and its dependence on earlier choices. Reading routes
A free-module expansion and a diagonal remainder calculation expose when divisibility of polynomial differences forces composition through a common polynomial.
Background: Polynomial rings and basic field theory. Reading routes
Collect algebraic conjugates into an integer trace recurrence, then use a companion matrix to turn the rounding question into finite modular arithmetic.
Background: Algebraic numbers, polynomial roots, and modular arithmetic. Reading routes
Build a finite arithmetic selector for the first prime above a bound, using factorial congruences, greatest common divisors, and valuations.
Background: Elementary number theory, factorials, and divisibility. Reading routes
Choose a positive measurable weight that changes the scale at each point, turning almost-everywhere convergence into convergence in measure for the weighted sequence.
Background: Measure theory and convergence of measurable functions. Reading routes
A carefully chosen closed remainder exposes the Baire-category obstruction to a countable partition of [0,1) into sets closed in the real line.
Background: Closed sets and the Baire category theorem. Reading routes
Turn local oscillation and a small exceptional set into finite control of a Darboux sum, using compactness to produce the partition.
Background: Riemann integration, compactness, and sets of measure zero. Reading routes
Insert one mixed term into a product difference, bound the separate oscillations, and sum over a common partition to preserve Riemann integrability.
Background: Bounded functions, upper and lower Darboux sums, and partition refinement. Reading routes
Six problems invite different choices of what to count, track, or record: congruences, recurrences, series, and observation rules. Statements are supplied without solutions.
Background: Counting, elementary number theory, recurrences, and convergence of series. Reading routes