RefacteFind the quantity that moves on its own
Choose a weighted difference, derive the weight, and turn a coupled recurrence into a single update.
Mathematical writing by Henry Shin
Refacte
Changing what
becomes elementary.
Invent a small calculus: states, moves, and laws that make the difficult interaction calculable.
A working example
Choose what moves.
Start with sn+1 = sn + sn−1. Try un = sn − c sn−1.
un+1 = (1 − c)un + (1 + c − c²)sn−1
At c = 1, the extra input remains. The ordinary difference has not separated the recurrence.
Derive the weight, step by stepMechanism calculi
The elementary is something mathematics can produce. Refactor the problem until its difficult interaction has a working calculus—and prove that the new moves answer the original question.
Earn the simple step →Complete first encounters
See the change happen.
Five worked entries.
Begin with familiar operations.
↗
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.
The current programme
Build the mechanism.
Construct every chain. Expose a hidden cubic. Classify compatible couplings. Remove a graph's coordinate choices. The advanced arguments themselves now perform these operations.
Construct every tensor chain
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 →Make a hidden cubic appear
How can quadratic equations force a cubic obstruction?
Change variables, eliminate a relation, and carry the residual through successive coefficients.
Output: The higher operation is calculated from the equations; its name comes after the calculation.
Work through the calculus →Classify compatible couplings
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 →Remove the tree, retain the circuits
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 →After the theorem dump
Prestige is not a method.
Command of a formidable framework is a poor substitute for inventing a useful move. The older framework-led expositions are now background. The centre is the construction that makes a problem work.
The argument, and what the AI evidence actually supports →The wider library
Notes and background.
Polynomial comparisons, exact arithmetic, local estimates, and problems to work on. Every text retains its sources and mathematical scope.
The background library →Change the comparisonDivisibility of Separated Polynomials
Make an arithmetic testIBM Ponder This July 2010
Change the measure of errorA Weighting Argument for Pointwise Convergence to Convergence in Measure
Choose by the questionReading paths
Essays
What understanding makes possible.
What a New Proof Makes Possible
How new proofs of familiar theorems expose mechanisms, improve estimates, and connect mathematical subjects. Comparing the arguments reveals why their constructions work, where they generalize, and what further questions they open.
When a Description Becomes a Verdict
How to test whether a description retains what a conclusion needs, through eigenvalues, naturality, and the scope of judgment.
Mathematical Life in the Age of AI
How people acquire the means to understand and pursue mathematics, and how AI changes the responsibilities of institutions that cultivate and judge those capacities.
Research
Research papers.
Current and earlier work in group theory, algebra, additive combinatorics, and geometric analysis.
Explore the research →Finite generation of permutation twisted cohomology
Representation spheres give multigraded section rings for permutation modules. This paper proves a finite-module theorem over arbitrary commutative Noetherian bases and uses the doubled regular representation to construct an explicit Noetherian model for the tensor-triangular geometry of permutation complexes.
A useful test: does the new description retain the operation?
Compare the identity with the shear J(x, y) = (x + y, y). Both have eigenvalues 1, 1. Iterating the shear gives Jⁿ(0, 1) = (n, 1), whereas the identity leaves (0, 1) fixed.
The eigenvalue list cannot calculate this difference. Retaining the off-diagonal action can. A refactor earns its simpler operations by keeping the information those operations need.
Work through the operator comparison (§12.3), or recover nilpotent chain lengths (§10).
About Henry Shin
Mechanism calculi, mathematical work, and correspondence.
Other interests
Patterns on large twisty cubes and notes on playing Brood War and Overwatch.
Editions and sources
Fixed editions, original downloads, and the publication record.