skills/research-attempt/references/structural-moves.md

# Structural moves for difficult research routes

Read only the moves relevant to the current obstruction. Each should produce
an actual map, invariant, lemma or counterexample test, not an analogy.

| Obstruction | Candidate mathematical move | Decisive obligation |
|---|---|---|
| Agreement only in a bounded range | Isolate an induction, generation, connectivity or vanishing mechanism uniform in the parameter | Cover every remaining case and all exceptional regimes |
| Objectwise equivalence without compatible maps | Construct the comparison functor or diagram and its first coherence obstruction | Naturality/higher compatibility at the level used |
| Homology agreement used for a homotopy claim | Search for operations, local systems, extensions or k-invariants invisible to that homology | An invariant that separates the claimed homotopy types |
| Rational argument needed in characteristic p | Expose division, averaging, exactness or semisimplicity steps | Test primes dividing denominators or stabilizer orders |
| A spectral sequence stops giving information | Identify a possible differential, convergence obstruction or extension class | Nonzero source/target and compatibility with the claimed invariant |
| A local model appears invariant | Identify omitted gluing, tangential or monodromy information | A compatible global reconstruction or witness where it fails |
| The same lemma blocks all routes | Strengthen the input construction or weaken the intermediate lemma | Preserve the original target and prove the new intermediate suffices |
| A computed pattern persists | Derive a recurrence, symmetry, operation or presentation from definitions | Prove its hypotheses instead of extrapolating |

Start counterexample searches from the exact hypotheses. Vary one structural
feature at a time: connectivity, coefficient prime, torsion, action, completion,
dimension, boundary or finiteness. Check that the proposed invariant is invariant
under the equivalence notion being tested. Verify the witness belongs to the
allowed class before minimizing it. “Smallest found” needs a stated search order;
global minimality needs an exhaustive argument over all smaller cases.

For dual targets such as prove invariance or find a counterexample, do not let a
successful bounded result erase the opposite search. Ask what data the proof
uses and which data the proposed invariant can still see. If every candidate
invariant factors through already invariant data, change the invariant before
enlarging the brute-force range.