RefacteFind the quantity that moves on its own
Choose a weighted difference, derive the weight, and turn a coupled recurrence into a single update.
Notes and background
The library
Worked refactors, longer arguments, and reference chapters. The mechanism calculi give direct routes to the constructions; this library collects the full studies and their sources.
Go to the mechanism calculi →The place of framework studies
Five complete refactors
The approach↗
A subgroup becomes a machine
Turn generators into loops, fold their shared moves, and read membership and intersections from paths.
A determinant that counts trees
Build the incidence matrix and watch its minors select exactly the spanning trees.
Make first order exact
Choose an arithmetic in which first-order change is exact, then restore the next term to expose a lifting obstruction.
How far can a change of coordinates go?
Cancel a quadratic defect, calculate the next one, and see directly why every cubic repair fails.
Other mathematical routes
Follow a formal identity into an obstruction, compare two polynomial proofs, choose an arithmetic construction, or begin with an analysis argument. The guide connects the notes within each route.
Repair a coordinate and test its limit
Try to turn nonlinear addition into ordinary addition. The first correction works; the next is blocked in characteristic three. Begin with the coefficient calculation, then use the coefficient-and-filtration study to organise its obstruction in a filtered complex.
Make a composition law visible
Compare two ways of exposing the same polynomial constraint. A free-module expansion gives a direct remainder calculation; generic fibres and Frobenius give a second route with different prerequisites.
Turn a numerical question into exact arithmetic
Traces and a recurrence turn rounding into congruences. Factorial tests and valuations produce a prime selector. The first is a finite modular calculation; the second is an exact formula whose purpose is separate from efficient prime search.
Make local control survive aggregation
A product difference becomes separate oscillation bounds. Compactness produces a finite partition, and a measurable weight changes the scale point by point. Follow what each construction makes controllable and which hypotheses permit the passage to a global statement.
Long studies and reference chapters
These volumes contain working calculations alongside wider theory. Enter the active arguments through the mechanism calculi, or open a full study for its context and reference chapters.
The current programme: mechanism calculi
Open the four long studies
Fourier, Symmetry, and Reconstruction
Make the right substitution calculable
Choose root coordinates that respond simply to permutation. Their powers and mixed products turn a cubic or quartic into smaller equations with compatible radical choices.
Begin with centering and the cubic
Follow §4 → §3.1 → §§5.1–5.4 for a complete cubic calculation. The covering construction explains the same choices in families.
What a Boundary Map Can Prove
Turn an edge diagram into arithmetic
An incidence matrix makes cycles, spanning trees, and electrical flows accessible to exact calculation. The same edge data support several different questions.
Start with a worked graph calculation
The companion develops a finite example. Main §§1 and 3 prove the cycle, counting, and probability results directly from the matrices; §2 gives a gluing interpretation.
What a Derivative Retains
Make first-order change an exact operation
Discarding products of increments isolates a derivative. Tangent probes and differentials then organise the rule, while constrained examples show what first-order data leave open.
Calculate with a square-zero increment
The companion starts with polynomial arithmetic. Continue through §§1–5 of the study for a complete first-order route; later questions require additional data.
Quotients, Refinement, and Reconstruction
Make a comparison independent of choices
Keep all possible inputs and outputs in one relation. Quotient its two ambiguities to obtain a canonical comparison, then repeat it across two decompositions.
Work through the six-element comparison
Subgroups and cosets suffice for the companion. Main §1 separately proves basis exchange and the group comparison before the general refinement argument.
Full catalogue
Reading pathsSome notes offer a full web edition; others introduce a PDF argument. Each entry states its background and links to the available reading edition and downloads.
Algebra & geometry
Fourier, Symmetry, and Reconstruction
Long study: direct tensor-chain construction, rank-degree reduction, kernel composition, and local ramification; external geometric identifications are separated. Reading routes
What a Boundary Map Can Prove
Long study: core calculations finish in §12; §§13–15 retain the general Hodge, attachment and period theorems as references. The earlier A Tree Is a Discrete Poincaré Lemma PDF remains available as a historical predecessor; this expanded study is the current edition. Reading routes
What a Derivative Retains
Long study: explicit advanced calculi in §§10–13; general spectral and functor-calculus identifications remain reference inputs. Reading routes
When a Formal Logarithm Fails
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
Quotients, Refinement, and Reconstruction
Long study: finite matrix compatibility, gauge certificates, transporter components, and attachment reconstruction are the operative advanced routes. Reading routes
Six Elements, One Comparison
Keep every output of a comparison relation, remove its two ambiguities, and match the factors of two decompositions of C₆. Seven exercises and worked answers test the construction.
Subgroups and cosets. Eight short sections complete the comparison, with exercises and worked answers. Reading routes
Divisibility of Separated Polynomials
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.
A Galois-Theoretic and Frobenius-Descent Viewpoint on a Polynomial Composition Law
Read a polynomial composition condition through generic fibres and descent. The positive-characteristic argument keeps the role of Frobenius explicit.
Background: Galois theory and polynomials in positive characteristic.
Normality, Coherent Extension, Tensor Operations, and Affineness
Choose algebras, modules, and local sections that make the constructions in Hartshorne II.5.14–18 explicit, then verify the gluing back to geometry.
Background: Schemes, coherent sheaves, and the material of Hartshorne II.5.
Number theory
IBM Ponder This July 2010
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.
Extracting the First Prime Greater than m
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.
Analysis & topology
A Weighting Argument for Pointwise Convergence to Convergence in Measure
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.
A Note on Disjoint Decompositions of [0,1) into Closed Sets
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.
Lebesgue's Criterion for Riemann Integrability
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.
Products Preserve Riemann Integrability
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.
Problems
Selected Problems in Discrete Mathematics and Analysis
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.