Refactor.

A difficult step becomes elementary when someone invents the operations that expose it.

Refactor names that act of invention. A mechanism calculus is its working result: states, local moves, laws of composition, and a way to extract an answer or obstruction. The aim is to change what can actually be done.

Invent the working object

A weighted difference can turn a coupled recurrence into an autonomous update. A folded graph can turn subgroup membership into a path test. A suitable coordinate change can expose a cubic obstruction hidden in quadratic equations. The interesting step is to invent the calculus in which the interaction becomes visible and controllable.

This demands more than a new description. Specify the data, derive the moves, prove their compatibility, and perform the calculation. The reader should be able to run the construction on another input and understand why it works.

Invent the form that could make it work

An ansatz is a form chosen in search of an operation: a weighted difference, a backward move, a correction term, or a combinatorial state. Its coefficients and rules still have to be determined. Ask what the form must preserve, what it must cancel, and whether those demands can actually be met.

In the cycle-exchange proof of Cayley–Hamilton, the first clue is that a long walk must revisit a vertex. Cutting out the intervening cycle shortens the walk but loses its weight, so keep the cycle in a reserve. A first attempt at pairing these objects can fail to undo itself; requiring a common unchanged prefix leads to the first-obstruction rule. Only after the exchange works do we identify the correction coefficients with the characteristic polynomial.

There is no general recipe that guarantees the right form. A useful account can nevertheless show what suggested a candidate, what ruled out a simpler attempt, and which equations determined the remaining choices. It should distinguish these reasons from the proof that the finished construction works.

Earn the simple step

A theorem name is not a mechanism. A classification is not yet a procedure for finding the class. Naming an obstruction group does not calculate the obstruction. A change of language earns its place by supplying an operation that the original description made difficult.

This standard applies at every level. Finite matrices can conceal a mystery; higher algebra can expose one. Neither the number of definitions nor their prestige decides whether the argument has done its work.

Read the transformation

1. Locate the interaction. What coupling, ambiguity, or residual is stopping the argument?
2. Invent the state. Choose what to retain so that the interaction can be manipulated.
3. Derive the moves. Calculate how the state changes and how successive moves compose.
4. Prove the bridge. Establish what an output in the new calculus says about the original question.
5. Execute and test. Construct, reduce, compare, or detect failure; identify the hypotheses and the cost.

These are demands on the substance, not a template of introductory boxes. A short construction that changes the available operations can be more valuable than a long tour through impressive machinery.

Make failure calculable too

A refactor can reveal that the requested change is impossible. In characteristic three, the quadratic repair of a formal logarithm succeeds while the cubic correction has no effect on the remaining error. With commuting matrices, one linear map records compatible couplings and another records the couplings a change of lift can remove. Their quotient is earned by multiplying the matrices first.

The goal is an exact account of what the new moves can and cannot do.

After the theorem dump

Prestige is not a method. Fluency in a formidable framework is a poor proxy for mathematical understanding. K-theory, motives, and higher categories deserve the same demand as elementary algebra: show the operation gained, the argument that validates it, and the result it makes accessible. A procession of distinguished names is not an explanation.

OpenAI’s October 2026 release makes that demand more urgent. Its public repository contains work across many mathematical fields, selected from a larger set of attempted problems and at differing stages of verification. The release account reports extensive automated mathematical work. Mastery of an elaborate language is becoming a weaker basis for claiming a protected intellectual status.

The evidence does not rank every cognitive style or prove a human monopoly on inventing mechanisms. AlphaEvolve already reports algorithm discovery and engineering improvements through generated programs and automatic evaluation. A mechanism calculus should be usable by machines as well as people.

Refacte’s commitment is editorial and mathematical: stop treating altitude in the abstraction hierarchy as the achievement. Put the invention of a useful calculus at the centre. Judge it by the capability it creates. Let AI participate, and demand an inspectable account of the result from every contributor.

The old prestige ladder is retired here

The earlier framework-led volumes now belong to the Tensorgami archive, with their full arguments and source editions preserved. They are not the featured current expositions.

The current expositions and mechanism calculi are independent reading texts. They derive their objects and moves, work examples, and prove their outputs on the page. An old setting may remain as optional further reading; it is not the route the reader must climb to finish the argument.

What is obsolete here is the editorial bargain that asks the reader to admire a framework itinerary as evidence of depth. The older mathematics retains its truth and its uses. It no longer gets the front page merely for being higher.

Refactor — changing what becomes elementary.