BOOK I — Of Witnessed Facts

Proposition 1 (The Reduction Theorem). Every admissible proof-of-behaviour terminates in a signature check.

Proof. By Law I the Verifier has no other faculty. Whatever intermediate apparatus a scheme employs — enclaves, zero-knowledge circuits, juries — its final gift to the Verifier is a number the Verifier checks against a trusted key. Hence the design of any scheme reduces to a single question: which key, already trusted, signs the last step, and what did signing it require? ∎

Proposition 2 (The Impossibility of the Naïve Hash). An Agent cannot prove faithful exercise of B by exhibiting hash(agent) = hash(B).

Proof, in three cuts.

1. The recipe is not the meal. For the Verifier to check hash(B), the canonical bytes of B must be public; hence hash(B) is public, and any party may present it having executed nothing. A hash of code proves knowledge of the source, never faithful execution. (Contrast a signature, which proves possession of a secret the world does not hold.)
2. Opacity (Law II). Even given the Agent’s true bytes, deciding whether they compute B is undecidable. The test is therefore at once too strict — it rejects an Agent that computes B correctly but was compiled differently — and too weak — it accepts an Agent that merely contains B’s bytes yet never calls them, or calls them and discards the result.
3. The absent world. B_dns depends on live DNS (Def. III–IV). No static artifact — no hash, however canonical — contains the state of the world’s DNS at the instant of asking. The very datum in dispute is not in the code. ∎

Scholium (the rescue). Proposition 2 does not bury the conjecture; it locates its error. The private key was never implicit in the Agent’s binary — it is implicit in the terminal fact. Domain ownership already has a secret key somewhere: the DNSSEC zone-signing key, the TLS certificate key, or operational control over the resolver’s answer. The task is not to invent a code→bytes→hash language, but to notice that the deed ends in a fact that already possesses a key, and to carry that key’s signature to the chain. Book II enumerates the ways. Two of its members (Propositions 6 and 8) vindicate the conjecture’s spirit exactly — one by binding the hash to live hardware, the other by making a secret that can only be derived by actually performing the deed.

Proposition 3 (The Witness Dichotomy). Behaviours partition into the provable and the unprovable by a single test: does the terminal fact carry a witness?

Discussion. This is Law III restated as a working classifier, and it is the most useful single tool in the treatise. Applied to our system:

• B_dns — provable. DNS ownership terminates in a fact with (at least) three candidate witnesses: a DNSSEC signature chain, a TLS server certificate, or a quorum’s signed observation.
• “This mail server honestly authenticated the sender before relaying” — not provable by witness; its terminal fact (an operator’s diligence) carries no key. This is precisely why the honest remedy there is bonds and reputation, not a proof. The dichotomy predicts the shape of the honest answer before we write a line of it.

Scholium — the ladder of witnesses. Not all witnesses are equally cold. Ascending in trust-coldness: (a) a single operator’s signature (today’s postmaster — one warm key); (b) an attested enclave’s key (audited code + one vendor); (c) a threshold of independent operators (a plural warm set, no one of which suffices); (d) the DNS root’s own signature chain (a key the fact is already defined by — the coldest, because trusting it adds nothing not already assumed by the word “domain”); (e) a pure mathematical proof (trusting only an assumption about number theory). Book II is, in effect, a climb up this ladder.