This tree has been building a theory of when computation must refuse, using MFG as the test bed, and describing itself as an MFG certification lab. Six kinds of refusal are already sitting in it, unnamed as a class.
Nothing on this page is claimed, certified, enclosed or proved. No kernel, no certificate, no falsifier, no ledger record, and no literature gate has run on it. It names a direction and what would have to be true. A prospectus that reads like a result is the defect, so this one says so at the top and is not published.
The geometric-vs-representation split found on 2026-07-30 was recorded as “a distinction”. It is the first two entries of a classification. What makes it research rather than a survey is that each kind has a DIFFERENT remedy — so a decidable discriminator between them is an instrument, not a description. Everything else in this set of five becomes easier to state once this vocabulary exists.
| Kind | What it means | Instance |
|---|---|---|
| Geometric | No isolated point exists; the solution set has positive dimension. The refusal is a theorem. | S1 — ker(J) exactly the 6-dimensional equilibrium face, nullity 6, no gauge |
| Representation | The solution is isolated and real, but the basis cannot reach it within budget. The refusal says nothing geometric. | mfg-congest — σmin = 0.645, the ℓ¹ν tail running out; flat across five norms |
| Conditioning | Isolated and representable, but the enclosure widens faster than the residual shrinks. | the spectral-conditioning question; the seaport det ≠ 0 reading |
| Arithmetic | The object is fine; the number system cannot decide it. | Apéry — integrality invisible to any continuum limit; interval.js:94 refusing non-integer exponents |
| Structural | The theorem's hypothesis fails, so nothing follows even from a clean computation. | the radiiPolynomial soundness gap — a self-map was checked, a contraction was not |
| Approximation-induced | The simplification made to gain tractability is itself what manufactures the obstruction. | mean-field variational inference — Wainwright–Jordan: the optimisation becomes MORE non-convex as more dependencies are broken. Added 2026-07-31 from the MFT survey; see edge-of-chaos.html |
The sixth arrived from outside MFG entirely, which is the first evidence that the classification is about computation rather than about this tree's subject matter.
Given a failed enclosure, the kind of refusal is decidable — there exists a test, computable from artefacts the solver already produces, that separates geometric from representation refusal without human judgement. Candidate: nullity of the linearisation under exact rational arithmetic (geometric) versus norm-invariance of the failure locus under a change of weight (representation, already measured flat across five norms).
Two instances the test cannot separate, or one it separates wrongly — a case with positive nullity whose failure nonetheless retreats when the basis is enriched.
N: LOW. Classification is not deep mathematics. A: VERY HIGH and unverified. The asset is that this tree has collected refusals for months, which nobody does, because everybody ships successes.
A taxonomy with no discriminator is a survey with a table. The whole thing stands or falls on at least one decidable test. If no kind can be told from another except by reading the transcript, this is prose and should be labelled prose.