Interfaces need only common p-belief
Parent: recursive-harness
Sibling subtrees coordinate through interface contracts, and the epistemics of that coordination is a softened-logic story with a hard impossibility on one side and a usable theorem on the other.
The hard impossibility: two generals
Over an unreliable channel, EXACT common knowledge ("I know that you know that I know…", all the way up) is unattainable: no finite protocol of messages achieves it (the coordinated-attack result from distributed systems theory). Two agents reading each other's messages can never bootstrap certainty about shared understanding. The barrier is absolute, not a matter of engineering better channels or more messages.
The softening: common p-belief
Weaken "know" to "believe with probability ≥ p" at every level of the hierarchy: common p-belief IS attainable (Monderer–Samet 1989), and it suffices for coordination with bounded loss (payoffs degrade gracefully in 1−p). The all-or-nothing epistemic predicate is softened into a dial — the relaxation move applied to epistemic modal-logic. What matters is not absolute certainty but a knob that controls the quality of coordination.
The harness translation
Two sibling agents "agreeing on the interface" is exactly this problem: each reads the contract through a fallible channel (an LLM's noisy comprehension), so exact shared understanding never happens. What raises p:
- A written artifact, not messages: a persistent spec both sides can RE-READ arbitrarily often beats any finite message exchange — this is why written contracts work and implicit/verbal agreements drift;
- Mechanical checks at both ends: each side's gate verifies ITS OWN reading against the artifact (types across the boundary, contract tests, schema validation) — pushing
pup by verification rather than by trust.
The lesson inverts the intuition: more negotiation messages between siblings make things worse, not better. What works is a stable target (the written spec) that both can ground against, and autonomous verification at each end.
The failure mode, re-read
Sibling incompatibility surfacing at reassembly = "p was too low." And the two-generals lesson says the fix is NOT more back-and-forth messages between the siblings — it is a better shared artifact plus gates at both ends. (The root-protocol already encodes this: the parent authors one spec per child; children never negotiate specs peer-to-peer.)
One line
Exact shared understanding between agents is provably unreachable — but a re-readable artifact plus gates at both ends buys arbitrarily high p, and p is all coordination ever needed.