R12 Closed Deliberation No-Go

Status: exact theorem; reject internal self-querying as an information or sample-complexity advantage by itself.

Claim under test

A learner observes a training object Z, then spends T internal rounds choosing questions, answering them from its own state, and updating that state before emitting a hypothesis. The hoped-for claim was that this deliberation could discover target information unavailable to a one-shot learner with the same observations.

Closed-deliberation equivalence theorem

Let Theta denote the unknown target system and R the learner's private randomness. Consider any finite computation

S_0     = encode(Z, R)
Q_t     = choose_t(S_t)
Y_t     = answer_t(Z, R, S_t, Q_t)
S_{t+1} = update_t(S_t, Q_t, Y_t)
H       = decode(S_T).

Assume that no Y_t is answered by Theta or by any channel containing target information beyond Z. Then the complete transcript and final hypothesis are deterministic functions of (Z,R), so

Theta -> (Z,R) -> (S_0,Q_0,Y_0,...,S_T,H)

is a Markov chain and

I(Theta; Q_0,Y_0,...,S_T,H | Z,R) = 0.

Composing the finite updates gives a one-shot algorithm

B(Z,R) = decode(update_{T-1}(...update_0(encode(Z,R))...))

with exactly the same output distribution. It therefore has the same risk and information-theoretic sample complexity. A T-round circuit of size s can be unrolled with size O(Ts); a computational advantage requires an explicit resource restriction on the comparator, not the word "deliberation."

Oracle dichotomy

If round t instead receives Y_t = oracle_Theta(Q_t), the incremental target information is

I(Theta; Y_t | Z,R,Q_0,Y_0,...,Q_t).

If every increment is zero, the theorem above applies. If an increment is positive, the environment supplied new target information and the experiment is active learning, membership/equivalence querying, or another named oracle model. Restricting a comparator to precommitted questions creates a round or adaptivity separation; it does not establish internally generated reasoning.

The smallest strict active-query control is threshold search on four ordered targets: two adaptive binary questions identify the target, while an exact nonadaptive strategy needs three. This is a useful sanity check, not an R12 survivor.

Collapse attacks

Self-review and private debate: every generated critique is a function of the same observations and can be composed into B.
Proof-carrying state: an internal prover and verifier compose into one algorithm. A target-specific prover or verifier is an external channel.
Algebraic closure: closure under a shared hypothesis class is deterministic post-processing. Without that class, an off-support patch survives.
Self-experiment: a target-independent simulator adds no information. A target-answering experiment is an oracle query.
External fixed execution: any fixed solver using only Z can be moved outside the learner without changing behavior.
Conjugacy: closed experiments cannot select a canonical hidden coordinate system when all observations are invariant under relabeling.
Delayed sabotage: two targets that agree on Z and differ on one unseen continuation remain indistinguishable after arbitrary closed deliberation.

Exact boundary

This theorem does not say sequential computation is useless. It can reduce time, space, activation, communication, or circuit description relative to a specified comparator. It says that sequentiality alone cannot improve the information available about Theta or establish a sample-complexity advantage against an equally expressive learner receiving the same prior and data.

The only admissible continuation is a computational generalization theorem: identical samples, identical structural prior, no target oracle, a uniform sequential upper bound, and a lower bound against a precisely named comparator class. That would be a circuit, streaming, data-structure, communication, or proof-complexity result. No CPU falsifier, Shohin fit, or H100 job is authorized for closed self-querying alone.

Prior-art boundary

Target-coupled exact queries are explicit in Angluin's automata-learning model; observable sequential systems are represented by predictive-state and observable-operator formalisms; communication-round advantages include classical pointer-chasing separations. R12's contribution here is the explicit information audit that prevents those channels from being relabeled as internally created evidence.

Primary references:

D. Angluin, Learning Regular Sets from Queries and Counterexamples (1987).
N. Nisan and A. Wigderson, Rounds in Communication Complexity Revisited (1991).