A standalone proof that, above a semantic-compression threshold, any system that computes evaluative signals but does not allow them to causally steer selection/control cannot remain generally competent under novelty.
We prove a necessity theorem (a no-go result), not an identity claim.
Hot Zombie Failure Theorem (informal).
When a task/environment contains more potentially relevant information than a system can represent at once, the system must continuously select what to represent (attention / retrieval / compute allocation). If the system computes evaluation (error/loss/surprise/value) but evaluation is causally inert for selection/control, then there exists a novelty regime in which the system necessarily fails (persistent error / breakdown of competence).
This result does not assume or conclude that evaluation is consciousness. It only establishes that evaluative leverage is required for stable competence under severe compression.
Let the environment contain an n-dimensional latent state at each discrete time step t:
Interpretation: each coordinate Zt(i) is a potentially task-relevant factor.
The system has bounded representational/control capacity k, with k < n.
This is the super-threshold / compression regime: there
are more degrees of freedom than can be concurrently tracked.
At each time t, the system allocates its limited resources to a subset of indices:
Interpretation: the system can actively track (or operate on) at most k latent factors at a time.
Remark (Generality).
You can read At as attention,
retrieval selection, compute routing, branching, tool-calling focus, or
any other capacity-allocation mechanism. Nothing in the
theorem depends on “attention” specifically; it depends only on
bounded selection.
The system maintains an internal state ht, constructed from its observation stream and selection history:
We assume only the following capacity property:
(Capacity property) If i ∉ At, then the system cannot reliably encode the current value of Zt(i) into its state ht.
This is the minimal meaning of “bounded working context”.
At each time step t the system must predict a target Yt and outputs a prediction Ŷt.
Loss is any nonnegative error measure ℓ(Ŷt, Yt). For concreteness, think of:
The system computes an evaluative signal Et that includes (at least) loss/error/surprise:
No other structure of Et is needed.
A general system updates its selection policy and other control variables via an update function:
Here Ct represents all other internal control variables (e.g., policies, gates, temperatures, routes, memory-write rules).
The system has evaluative leverage if evaluation can affect future control:
In short: evaluation is part of the feedback loop.
A hot zombie computes evaluation but evaluation is causally inert for control:
Equivalently: the system may compute, broadcast, log, or internally
represent an arbitrarily rich evaluation Et,
but Et
cannot influence any future selection/control variable.
So it may “generate heat” (loss exists), but it cannot channel it into control.
We require only a weak competence condition:
Nontrivial novelty (moving relevance).
The system must remain competent in environments where which latent
factor matters can change over time in a way not perfectly predictable
from a fixed internal schedule, so that the only reliable way to track
what matters is through performance feedback.
This distinguishes “general competence” from “works only in one fixed situation forever”.
Remark (worst-case intent).
The theorem is a worst-case novelty theorem: it allows environments to
be interactive (the task may depend on the system’s
current allocation).
This is not a trick; it formalizes “moving relevance” in its strongest
form.
(Optional note.) A closely related non-interactive version can be obtained by replacing the adaptive choice of jt with an unpredictable schedule of relevance that is independent of the agent but not computable from the agent’s fixed internal clock.
Assume n > k. For any hot-zombie system (evaluation-blind control) with selection capacity |At| ≤ k, there exists a novelty environment in which the system incurs persistent nontrivial loss (cannot maintain competent performance).
Equivalently: in the compression regime, any architecture that aims to be generally competent across novelty must allow evaluation to causally steer selection/control (must have evaluative leverage).
Fix an arbitrary hot-zombie system. Because the system is evaluation-blind, its selection/control sequence cannot be driven by error feedback.
We now construct an environment that forces persistent error.
At each time step t:
System chooses selection At ⊆ {1..n} with |At| ≤ k.
Environment samples latent state Zt uniformly at random from {0, 1}n (all bits independent fair coins).
System observes only selected coordinates Zt(i) for i ∈ At and forms its state ht.
Environment chooses the target from an unselected
coordinate.
Because n > k,
there exists at least one index not selected at time t.
Choose any jt ∈ {1..n} \ At
and define:
System predicts Ŷt based on ht.
This environment exhibits “moving relevance”: the relevant coordinate is always outside the currently allocated capacity.
Conditioned on everything the system knows at time t (its state ht), the bit Zt(jt) is still a fair coin.
Reason:
- Zt was
sampled uniformly.
- jt was
chosen to be an unobserved coordinate.
- by the capacity property, ht does not
contain reliable information about Zt(jt).
Therefore:
- P(Yt = 1 ∣ ht) = 1/2.
No prediction strategy can beat chance on this target. Hence the best achievable expected 0-1 loss is:
For log-loss, the best expected loss is at least log (2).
So for every time step t:
for some constant c > 0 independent of t.
Because the above lower bound holds at every time step, loss cannot converge toward 0. The system is persistently wrong on an O(1) fraction of steps.
Thus the system cannot remain competent on this novelty environment.
This proves that for every hot-zombie architecture, there exists a (simple) novelty regime that forces persistent nontrivial error.
Therefore, in the compression regime (n > k), evaluation-blind selection/control cannot support general competence across novelty.
QED.
The theorem’s contrapositive gives the necessity statement:
If a bounded-capacity system (k < n) is required to remain competent across novelty regimes, then its evaluative signal must have causal influence on future selection/control.
Equivalently:
General competence across novelty ⇒ evaluative leverage.
This is the formal sense in which “hot zombies” are unstable: once information pressure exceeds working capacity, there must be a feedback channel from mismatch/error to selection/control.
In the super-threshold regime (n > k), a system that cannot use its own evaluation to steer selection/control cannot be generally competent under novelty; therefore hot zombies necessarily fail.