Upward re-equilibration — cuts and prices flow up
Parent: recursive-harness
A node splits at its local equilibrium and it is STILL bad — can it push back up, carrying information that makes the ancestor re-evaluate its own higher-level equilibrium? Yes — and optimization theory has owned this mechanism for fifty years: decomposition methods, where cuts and prices flow upward.
1. Make "still bad at equilibrium" precise
The node computes its local U-curve and finds its allowance — the CURVE'S MINIMUM sits above the (α, B) the parent allocated. This is not a "try harder" signal; it is an infeasibility certificate: no granularity within my budget meets my spec. It differs in kind from a trajectory conflict ("this attempt failed"); it is a structural conflict: the parent's framing — not any execution — is what's wrong.
2. What travels up: the curve, not the failure
The upward message carries the shape of the local problem: the achievable floor , which term dominates (split-α vs leaf-ε), and WHERE the terrain is hostile (which sub-regions are off the model's competence manifold). Two grades, exactly matching classical decomposition:
- Hard: the Benders feasibility cut. When a subproblem is infeasible, Benders decomposition doesn't return "failed" — it returns a cut that constrains the master's next choice: "any split that frames this terrain with this budget is infeasible." And the correspondence is formal: no-good learning ≡ combinatorial Benders cuts (Codato–Fischetti). So this is conflict-learning-and-backjumping's machinery generalized from the decision dimension to the granularity dimension: an ordinary backjump changes the branch; a feasibility cut changes the shape of the tree.
- Soft: prices (duality). Even a feasible child reports its marginal cost of meeting the spec — the shadow price of its constraints, i.e. the Lagrange-multiplier reading of hard-to-soft-constraints. The parent then rebalances allowances across siblings (move
α/Bfrom cheap children to expensive ones) without re-splitting — Dantzig–Wolfe-style price coordination.
3. How the parent re-equilibrates
The parent's own U-curve was built on estimates of ε(s) per region; the child's report is measured terrain. Substitute measurement for estimate → the parent's curve moves → its equilibrium shifts → it re-splits differently; and if its OWN new floor exceeds ITS allowance, the cut propagates another level up — recursively, until some ancestor can absorb it, or it reaches the root. At the root, an unabsorbable cut is the add-an-axiom moment of giant-tasks-nonprimitive: the human extends the system (more budget, weaker spec, new certificate form).
4. Why it can't oscillate
Cuts accumulate monotonically (the master's feasible set only shrinks), and the α/B ledgers only deplete — the same "assignments are non-monotone, knowledge is monotone" leg that makes CDCL converge. Each re-equilibration is paid for from the ledger and permanently narrows the search. The mechanical coordinator stores the cuts (the no-good store generalizes to a cut store) and routes them by provenance.
5. The closing loop
The task tree IS a decomposition algorithm — master = the parent's split choice, subproblems = the children. Optimization theory answered "what should flow upward" fifty years ago: cuts and prices. The logic-to-optimization migration closes again, this time on the tree's upstream channel.
A child that cannot meet its spec at any grain doesn't fail — it teaches: it sends up a cut that reshapes the parent's problem, and a price that rebalances the family's budget.