Recursive mathematical decomposition and compressed insight

This record supplements mathematical inquiry under artifact pressure. It preserves two later episodes from Codex transcripts.

Experience and evidence

Relevant portions of session 019fe6ad-2821-7502-92f5-92fc1fe4a13d showed an advanced geometry plan without its algebraic language.

The plan named period domains before the repository could express signatures, scalar extension, isotropic vectors, or orthogonal groups.

The correction did not reject the horizon plan. It required recursive decomposition into the definitions needed to state each target.

Relevant portions of session 019fe19b-4a36-7530-80ec-c44d82626402 showed a second trajectory.

The advisor asked whether ideals are submodules and integral bases are bases of integral algebras.

The student converted this insight into a local PARI method. It did not construct the integral algebra or scalar-extension functors.

The correction showed that the missing functorial foundation could unify many apparent methods.

Later analysis

The first episode was not only incomplete planning. The target name was treated as evidence that the target was ready for formalization.

A definition must be decomposed until each term exists in Mathlib, the repository, or an explicit foundational work unit.

The second episode shows the opposite compression error. A short mathematical question can contain a large general construction.

The agent must unfold that construction and test its consequences. It must not select only the clause that permits an immediate edit.

A large method backlog can be evidence of missing theory. A general functor can remove work by making the methods inherited consequences.

The advisor supplies a high-value mathematical delta. The agent remains responsible for reconstructing the model and can disagree from evidence.

Forward guidance owner

The root AGENTS.md sections Build the mathematical language before the target and Unfold compressed mathematical insight own the forward guidance.

This memory preserves the episodes and interpretation. It does not duplicate the live instructions.

Evidence boundary

The parsed transcript portions establish these two trajectories. They do not establish a complete prerequisite theory or a universal prevention rule.