Reframe Until It Dissolves

reframe-until-it-dissolves

Grothendieck's rising sea (raise the water until the problem is submerged and falls open) fused with von Neumann's change-of-representation (find the basis in which the work is linear) and functor-transport (carry a solution from a solved domain). The non-obvious content: the general statement is frequently easier than the special case — the special case was hard only because it was seen too specifically.

Organon: P16 lump-split + P13 analogize + P14 transpose + P5 dimensionalize + P36 distill. Respiratory: the breath-out taken to its limit (distillation).

The danger this skill must police

This cognitive style's failure mode is over-abstraction — building an elegant cathedral that answers a class of size one while the actual instance sits unsolved. Two hard guards below (the MDL gate and the ≥3-members guard) are non-negotiable. The rising sea is a means, not an aesthetic: if the water rises and the problem does not fall open of its own accord, descend and solve the concrete case by hand.

Procedure

1. Locate where the difficulty lives in the current framing: the case explosion, the cross-terms, the cycles, the "it depends."

2. Run the reframe-operator menu — try each until one collapses the difficulty:

   • Raise the altitude (abstract up). Circle every constant silently fixed (a number, a domain, a user, a platform); lift one to a variable; ask whether the general proof is shorter and the instance becomes a free corollary.
   • Change the basis (linearize). Eigenbasis (coupled → independent), dual (constraints ↔ objectives, max-flow ↔ min-cut), transform (convolution → product), log / order (products → sums, ranking → Pareto frontier), graph (relations → reachability), continuous relaxation (integer → convex, then round).
   • Transport from a solved domain (functor). Find a domain where a structurally identical problem is already solved; build the correspondence that preserves composition; transport the solution; the unmapped remainder is the only problem left, usually far smaller.

3. Read off the answer in the new representation, then map it back, checking the map was faithful on the relevant structure.

The gate (MDL anti-vacuity — this is what keeps it honest)

Accept the reframe only if all three hold:

1. The reframed/general statement is strictly shorter to write down than the original attack (L(structure) did not rise).
2. The original instance falls out with no bespoke argument — if you still need a special case for it, the reframe was cosmetic; revert it.
3. A method unavailable before is now available (a closed form, a linear solve, a single sort, a reachability query).

Plus the ≥3-real-members guard: do not build general apparatus unless three real instances of the class exist. One triggering instance → solve it directly; the foundation is premature.

Plus the semantic-preservation check (this is preserve-the-target applied to reframing): a reframe can be MDL-shorter simply because it is vaguer — it dropped a constraint that mattered. Shorter is necessary but not sufficient. Verify the reframed statement still binds the original target's load-bearing constraints; if the compression came from quietly discarding one, reject it. A reframe that is shorter because it says less is not a dissolution, it is a loss.

Conceptually this is the cycles engine's ratchet: keep the breath only if codelen drops — coverage up while the structure stays small. A reframe that grows coverage on paper but leaves a huge residual (you still need per-case work) or balloons the statement is over-abstraction; the ratchet reverts it.

For prose problems this is a soft gate — "shorter" and "binds the same constraints" are judged, not computed (see the gate-hardness classification in docs/premier-skills.md). When the stakes are high, have a different model or the human check clause 2 and the semantic-preservation clause: a reasoner grading its own reframe is the known hole.

Example

Asked to rank N candidate fixes by correctness, parsimony, and security where the criteria trade off nonlinearly. Default agonizes over weights in scalar space. Reframe (change of representation): map each candidate to a point in criterion-space and take the Pareto frontier — a partial order, not a scalar. Dominated candidates drop out with no weighting at all; N collapses to the frontier set, and only that small set needs any further judgment. Gate: a method unavailable before (order-theoretic dominance) is now available, the statement is shorter, and the weighting agony — an artifact of forcing a partial order into a scalar — evaporated. Accepted.