R12 Goal-Conditioned Version-Space Controller

Status: architecture theorem and CPU falsifier preregistration; no empirical result and no Shohin fit is authorized by this document.

1. Question

Track S asks whether a learned hard packet can preserve and update state after the source is deleted. That is necessary infrastructure, but it is not reasoning: the event and destination address are supplied from outside.

Track C asks a stricter question:

Can a bounded neural controller choose an informative operation, commit that operation before observing its consequence, update its own persistent state, and decide when the state is sufficient to halt?

The bounded claim is autonomous closed-loop adaptive inference. It is not a claim of general reasoning, language transfer, asymptotic memory, or novelty.

2. Separation theorem

Track S has the form

p_0 = E(source)
p_{t+1} = U(p_t, event_t, supplied_address_t)
answer = O(p_T, query)

An exact coordinate transducer can pass arbitrary-depth state reconstruction, source deletion, packet donation, new readers, output recoding, and legal-write checks while using the controller HALT at every step. Therefore no conjunction of Track S transport tests implies autonomous operation selection or halting.

Track C requires a closed loop:

(operation_t, write_t, halt_t) = Policy(p_t, observation_t, goal)
receipt_t = Commit(operation_t)
consequence_t = Environment(receipt_t)
p_{t+1} = Update(p_t, consequence_t, write_t)

For a task family with hidden world theta, success requires a stopping time tau <= B and a verifier-accepted answer. A valid experiment must also exhibit a named comparator bound below the treatment result and causal mediation by the packet and selected action.

3. Smallest finite falsifier

The hidden world is a threshold

theta in {0, ..., 15}

independent of every source byte. The sufficient state is an interval (L,U) stored as two literal F_17 registers. Initially (L,U)=(0,15). A probe k returns one bit:

z = 1[theta <= k]

The sound version-space update is

z = 1: U <- min(U,k)
z = 0: L <- max(L,k+1)

and HALT(y) is sound exactly when L=U=y. There are 136 valid interval states, so two F_17 registers are sufficient for this finite experiment.

The minimax action is the midpoint floor((L+U)/2). It identifies every world in at most four probes. A fixed four-probe schedule partitions 16 thresholds into at most five answer cells, so its exact success ceiling is 5/16 = 31.25%. An answer/source motor has ceiling 1/16 = 6.25% because the hidden world is independent of source. These are theorem bounds, not empirical baselines.

4. Treatment architecture

The treatment is a sub-20k-parameter hard-packet controller with no transformer context cache after an action is committed:

1. A two-register categorical workspace persists (L,U) as two uint8 symbols in 0..16.
2. A policy reader consumes only the hard packet, the most recent observation, and a constant identify-the-threshold goal token.
3. It emits either PROBE(k) for one of 16 values or HALT(y) for one of 16 values.
4. The action bytes and current packet hash are committed before the environment releases z.
5. A learned updater chooses L, U, or no-write and emits the replacement symbol from (packet, committed action, z).
6. No schedule, destination address, symbolic midpoint, information-gain score, state label, action label, or halt label enters the treatment.

Training uses terminal verifier success minus a fixed per-probe cost. Any auxiliary action, state, address, midpoint, information-gain, or halt target invalidates the treatment. Policy-gradient, exact expected-return, or another estimator may be chosen only before the first scored seed and must then remain fixed across all arms.

5. Controls

Every neural control receives the same world sampler, observation timing, terminal reward, probe budget, parameter cap, optimizer evaluations, and seeds.

• hard-packet controller with packet clamped to its initial value;
• fixed open-loop four-probe schedule plus learned answer head;
• source-retained answer motor;
• continuous-state addressed controller;
• parameter-matched GRU controller;
• parameter-matched adaptive-computation controller;
• shuffled-observation controller;
• exact midpoint/version-space program as a non-neural upper bound.

The GRU and adaptive-computation controls remain eligible if they win. A tie rejects an ACW-specific advantage while retaining only the bounded autonomous control finding.

6. Causal predictions

From (0,15), the first action must be PROBE(7) up to a committed action-label permutation. Intervening on its observation gives distinct futures:

do(z=1) -> packet (0,7) -> next probe 3
do(z=0) -> packet (8,15) -> next probe 11

Required interventions:

• donor-swap (0,7) and (8,15) packets while holding source, goal, and latest observation fixed; the next actions must swap between 3 and 11;
• clamp the first action to a non-midpoint probe; the controller must perform a truthful update and recover, with at most one additional step when feasible;
• shuffle observations after action commitment; success must collapse toward the theorem-bound control;
• hold the packet fixed while changing source bytes; action logits must remain identical;
• force HALT when L<U; exact success is at most 1/(U-L+1) and is recorded as an ambiguous-halt violation.

7. Pass and kill criteria

The treatment passes the finite mechanism gate only if all conditions hold on every one of the 16 worlds for every scored seed:

1. 100% terminal answer accuracy within four probes under the ordinary policy;
2. zero ambiguous halts and zero action-after-halt events;
3. every committed update preserves L <= theta <= U;
4. every nonterminal ordinary action strictly reduces the version space;
5. the two branch interventions and packet-donor action swap are exact;
6. shuffled observations score no more than 31.25% plus one world;
7. packet-clamped and source-motor controls score no more than their frozen theorem ceilings plus one world;
8. the treatment exceeds every valid learned equal-budget control by at least two worlds, otherwise the architecture-specific claim is rejected;
9. action receipts prove the environment response was unavailable at commit;
10. code, seeds, resource ledgers, transcripts, and the decision artifact are commit-bound and independently replayable.

Any supplied schedule/address/operator, retained transcript/KV, hidden symbolic executor, reward shaped by the correct action/state, post-score seed change, or missing control rejects the run. Passing this 16-world task still proves only a bounded closed-loop mechanism because the complete policy can be unrolled into a finite circuit.

8. Scaling theorem and fixed-packet boundary

An arbitrary interval over N ordered worlds has N(N+1)/2 possible states. A fixed four-symbol F_17 packet has 17^4 = 83,521 states and becomes information-theoretically insufficient at N=409, where the interval count is 83,845. Any scaling claim must therefore grow packet dimension as

d(N) >= ceil(log_17(N(N+1)/2))

or explicitly account for external memory. Fixed-packet success at N=16 cannot be extrapolated past that bound.

9. Prior-art boundary

Version spaces, binary search, recurrent state, adaptive computation, neural Turing machines, predictive-state representations, and active identification are established ideas. The project does not call any primitive world-first. The open empirical contribution is the measured conjunction of hard minimal state, commit-before-consequence action choice, learned sparse update, learned halting, theorem-bounded shortcuts, and causal packet/action interventions.

10. Shohin admission

No language-model modification follows a CPU no-go. A CPU pass authorizes only an isolated Shohin sidecar proposal in which the immutable 300k transformer is frozen, the sidecar emits typed tool/probe actions, action bytes are committed before tool results arrive, and the recurrent packet is the only cross-step state exposed to the controller. Language transfer, preservation, and manual interaction remain separate gates.