The root protocol — what to say at the top

Parent: recursive-harness

The problem (short)

safe-recursion-theorem states hypotheses; but what INSTRUCTION does the human at the top actually issue so that agents below recursively unfold into a giant task? Today's practice issues tasks — and every theorem hypothesis is violated: specs are prose not checks, children grade themselves, budgets don't decrease, failures are absorbed silently. The unfolding must happen by construction, not by hope.

You don't issue a task — you issue a self-reproducing constitution

The theorem's hypotheses must hold at EVERY level, so the top instruction must contain the machinery to recreate itself one level down. This is the quine structure of the Kleene recursion theorem — a program that carries its own source: the protocol is passed down as data. Root instruction = a small conserved constitution C + the quadruple (task T, executable acceptance spec S, budget B, confidence allowance α). Note the tolerances ride along: δ is implicit in S (the strictness of the check bundle is the δ-neighborhood — operationalizing-tolerances), and α is explicit because it must be accounted, like budget.

The constitution C (five clauses)

SOLVE-or-SPLIT. If you can complete T within B and verify against S, do it; submit the artifact WITH evidence (check results). (leaf soundness)
SPLIT is a four-part artifact. A split must produce: (a) children's executable acceptance specs $S_1…S_n$ — authored by the parent BEFORE the children exist (test-first), and no agent ever authors the spec it is judged by (separation of powers — the anti-Goodhart clause); (b) a reassembly plan; (c) a split certificate: an argument that $\bigwedge S_i$ + reassembly $\Rightarrow$ S; (d) budgets $B_i$ , each strictly less than B, and confidence allocations $\alpha_i$ with $\sum_i \alpha_i$ + the split-gate's own α ≤ the parent's α — without this bookkeeping the theorem's 1−Σα guarantee evaporates as the tree grows. This four-part plan passes its own gate BEFORE any child is spawned. (the plan is an artifact; well-founded measure; the union bound stays finite)
Pass the constitution down. Each child receives the SAME C plus $(T_i, S_i, B_i)$ . The recursion unfolds automatically. (hypotheses hold at every level)
Every action declares its heading. Each step must state which residual it expects to reduce and by how much; a step that cannot say what it decreases is not a step but a wander. This makes δ observable at step granularity (gate-deviation-is-observable-delta). (observable deviation)
Fail loudly. Budget exhausted, or no split lowers the measure → propagate the conflict upward with its analysis; NEVER silently absorb a failure. Conflicts are fuel for backjumping and no-goods — and when the failure is structural (no grain within the allowance meets the spec), what goes up is a feasibility cut plus a price, forcing the ancestor to re-equilibrate. (conflict-driven convergence)
Tolerance discipline. A deviation within the gate's δ passes — perfection is never the bar; outside δ is a conflict, not a retry-forever. The repair loop watches the residual ratio $\hat{k} = r_{n+1}/r_n$ (operationalizing-tolerances): two consecutive non-contracting rounds → stop repairing, raise the conflict. And a feasibility floor: a node can achieve at best the ε/(1−k) ball (inexact-contraction), so the split gate must REJECT any child spec tighter than that floor — demanding more precision than the floor is an infeasible spec, the parent's error, not the child's. (local soundness up to (δ, α); containment k<1; no impossible demands)

The top's entire leverage: S_root, C, and the two allowances

The human at the root authors the constitution, the root acceptance spec, and the two allowances (B, α). The quality ceiling of the whole tree is the quality of S_root — you get what you can check — and the confidence ceiling is the α you budget — you get the certainty you pay for. One check: S_root itself must respect the feasibility floor — a root spec tighter than the achievable ε/(1−k) ball dooms the whole tree before it starts. Everything else is delegation.

The honest gap

Authoring executable specs for children is itself the hardest fallible move — tacit knowledge concentrates exactly there (tacit-spec-as-spec-compression). So the split gate deserves the largest share of the verification budget, and the leaf-stopping rule couples to the learner: split until each child's spec is learnable from the context you can inject (see coupling-to-the-learner).

Close

The top does not command the tree; it seeds a constitution whose fixed point IS the tree.