Abstract

A system that predicts itself has to hold three things at once: what it is doing now, what it predicted it would be doing, and which half of the cycle it is in. The Horizon Constant K0 counts them. On the realization class where the stepping before the cycle-closing reset is injective, where a phase response separates the predict step from the update step, where the current state and the stored prediction stay readable across that boundary, and where all eight combinations of the three responses actually occur, the loop needs at least eight distinguishable visited configurations. Its log-capacity is therefore at least three bits, a construction attains the bound, and within that class Nvismin = 8 and K0 = log2 8 = 3. Carrying those eight contexts as mutually distinguishable alternatives in a complex Hilbert space forces dimension d0 ≥ 8.

The same three-bit register is enough to beat chance on environments whose next state repeats the current one more often than half the time, so the threshold is where self-referential prediction becomes both expressible and useful. It is a lower bound on a stated class, and not an absolute complexity floor for every predictor, a measure of memory span, or a criterion for consciousness. Above the threshold the Self-Referential Paradox of Accurate Prediction applies: a system with this structure can model and improve itself, and no predictor in its class is exact on every system the class can build.

The Horizon Constant in Plain Terms

Imagine keeping a weather diary. Each evening you write tomorrow's forecast on a card and seal it. Each morning you look at the sky, open the card, and compare. Three separate facts have to be available to you at the moment of comparison: what the sky is doing, what the card says, and whether today is a sealing day or an opening day. Lose the sky and there is nothing to check. Lose the card and there is nothing to check against. Lose the phase marker and you cannot tell whether you are supposed to be writing or reading, so you overwrite the forecast with the observation and the comparison never happens.

Three yes-or-no facts have eight combinations, and a system that has to reach all eight needs at least eight distinguishable internal configurations. Counting those in bits gives three. That is the Horizon Constant: the smallest register in which a system can hold a prediction about itself, keep the prediction while the world moves, and check it afterward. Below three bits there is no room for the loop, and what is left is a device that reacts but never grades itself.

Two things follow, and they pull in opposite directions. The threshold is enough. A three-bit register can already beat guessing in any world where tomorrow tends to look like today, which is the simplest kind of regularity a forecaster can exploit. The threshold is also where the trouble starts. Once a system can hold a prediction about itself, it can be built to act against that prediction, and no single predictor gets every such system right. Adding bits buys accuracy and never buys immunity.

One caution. Three bits is a floor for one job under stated conditions, and not a measure of how much a mind holds, how complex an organism is, or where awareness begins. It says where the self-referential loop first fits.

1. Introduction

Prediction that turns back on the predictor is where the interesting limits live. A thermostat predicts nothing about itself; it compares a reading with a setting. A system that stores a forecast about its own next state, then meets that state and grades the forecast, has done something structurally different: it has put a description of itself inside itself. The smallest question about that arrangement is how much room it takes.

The answer is a counting argument. Fix what the loop has to do, ask which responses must be readable at the same instant, and count the configurations. The count comes to eight, and the log-capacity to three bits. That number is the Horizon Constant K0.

The counting depends on conditions, and the conditions are stated rather than assumed away. A different realization class, with a different account of what verification requires, can yield a different count. What the argument establishes is a lower bound on the class it names, together with a construction that meets the bound.

The rest of the article sets up the loop (Section 2), states the conditions and proves the bound (Section 3), records what the bound does not say (Section 4), carries the eight contexts into a Hilbert space (Section 5), and turns to what happens above the threshold (Sections 6 and 7).

2. What the Loop Has to Hold

The cycle under analysis has three phases: the system forms an internal prediction, the world supplies an outcome, and the system updates. Feedback control has described error-correcting loops in these terms since Rosenblueth, Wiener, and Bigelow (1943), and the same three-phase shape organizes predictive accounts of the brain, where descending predictions are compared against ascending signals and the mismatch drives revision (Rao and Ballard 1999; Friston 2010; Clark 2013). What makes the case self-referential is the target: the prediction is about the system's own next state.

Write φ for the binary response that reads the nominated component of the current state, and pstored for the response that reads the prediction the system has committed to about that component. Verification is the operation that compares them. Three operational conditions constrain any realization of this sub-dynamics, and the argument below uses them as stated hypotheses.

Each condition earns its place operationally. Suppose a realization overwrites pstored before the comparison while still claiming to verify. At the comparison instant there is no stored value to compare against, so the verification phase is not instantiated, which is (O3). Suppose there is no phase response. The same pair (φ, pstored) has to take part in two operationally distinct steps: writing pstored from the model output, and updating φ against pstored. A single autonomous transition law assigns one successor to each configuration, so it cannot perform both steps from configurations that agree on (φ, pstored); the phase has to be part of the state, which is (O2). Suppose stepping merges two visited configurations that the verification response distinguishes. The distinction the comparison needs is destroyed before the comparison happens, which is (O1).

One more condition fixes what "visited" means. The joint response map (φ, pstored, cphase) from visited configurations to {0,1}3 is required to be surjective: every combination of the three responses actually occurs in the run. Call this the full-coverage condition (FC). Without it a realization could dodge the count by never entering, say, the state where a stored prediction of 1 meets a current value of 0, which is exactly the case verification exists to handle.

3. The Bound

The bound is stable under relabeling. Renaming states, renaming internal variables, or reordering micro-operations leaves it in place, because (FC) keeps the joint response map surjective onto the eight elements of {0,1}3 however the realization is written down. What changes the bound is changing the conditions, which is why they are stated rather than assumed.

Counting distinguishable configurations in bits is Shannon's (1948) measure applied to a state set instead of a message set, and the step from three required binary responses to eight configurations is the same step as from three bits to their eight code words. The argument is a requisite-variety argument in Ashby's (1956) sense: the internal variety of the loop has to match the variety of the situations it must tell apart, and Conant and Ashby (1970) put the general form of that requirement as the statement that a regulator of a system has to model it.

4. What the Bound Does Not Say

Four readings are excluded, and stating them is part of the result.

• It is not an absolute complexity floor for every predictor. The count holds on the realization class of Definition 1. A system that verifies differently, or that does not verify at all, is outside the class and is not covered by the bound.
• It is not a memory span. Miller's (1956) figure of about seven items concerns how much a person holds in immediate memory, which is a claim about capacity in a particular animal, and the two numbers answer different questions.
• It is not a criterion for consciousness. Section 7 returns to this. The bound describes the smallest self-referential predictive loop and says nothing about experience.
• It is not a thermodynamic quantity. Three bits is a count of distinguishable configurations. Erasing a bit has a physical cost, and Landauer (1961) fixed the floor for that cost at kBT ln 2, but a count of configurations and a heat are different quantities and one does not substitute for the other.

5. Three Bits Are Also Enough

A lower bound alone would leave the threshold empty, so the second half of the result runs the other way: the same register already predicts better than chance on a natural class of environments.

Let (Xt) be a binary process, and let its one-step persistence be the probability that the next value repeats the current one, st = Pr[Xt+1 = Xt | history]. For 0 < γ ≤ 1/2, let Ebasic(γ) be the class of stationary ergodic binary processes whose long-run average persistence is at least 1/2 + γ.

So the threshold is not merely where the loop fits; it is where prediction starts paying. Persistence is the weakest regularity worth exploiting, and the smallest self-referential register already exploits it.

6. From Eight Contexts to Dimension Eight

The eight visited contexts are what the loop must tell apart. If they are carried by a complex Hilbert space and the realization is required to distinguish them perfectly, each context must be represented by a state that no measurement confuses with the others. Perfectly distinguishable states are mutually orthogonal, and a space of dimension d holds at most d mutually orthogonal states (Nielsen and Chuang 2010). Eight contexts therefore require

log2 d0 ≥ 3, so d0 ≥ 8

and the minimal branch of the framework takes d0 = 8, which needs a separate argument that no smaller carrier saturates the comparator. What the counting supplies on its own is the inequality. Anything that fixes the value exactly is a further step, and the wider Predictive Universe framework treats it as one.

7. Above the Threshold

A register that can hold a prediction about itself can also be turned against it. Let the nominated rule be the one that reads the stored prediction and writes its opposite: φt+1 = NOT(pstored). Nothing in the construction of Theorem 1 forbids this; the rule is one of the assignments the three-bit register supports.

This is the Self-Referential Paradox of Accurate Prediction, and its shape is the diagonal argument that Cantor used for cardinality, Gödel (1931) for provability, and Turing (1936) for halting. The probabilistic version closes the obvious escape. Let the predictor output a probability p that the component will be 1, let the predicate p > 1/2 be decidable inside the class, and let the class contain the system whose one-step law is 0 when p > 1/2 and 1 otherwise. Then no predictor in the class is exact on every system it can build, and hedging the forecast does not help.

Two readings have to be kept apart, and the difference matters for what the threshold means.

• The theorem excludes one predictor that is exact on every system in the class. It does not say that every system is unpredictable. Restricted systems and restricted tasks can be predicted perfectly, and a system that does not receive its own forecast is untouched by the argument.
• The obstruction is logical and class-relative. It is not a claim that individual events are random, and nothing here derives randomness from self-reference. Where randomness enters a physical theory, it enters through separate assumptions about the dynamics.

Popper (1950) and MacKay (1960) reached the same barrier from the side of free choice and physical determinism: a predictor cannot supply a prediction to an agent who can then falsify it, and the obstruction survives whether the underlying dynamics are deterministic or not. What Theorem 3 adds to those arguments is the register: the barrier appears exactly where the loop first fits, at three bits, and not somewhere higher up.

There is also a structural way to see why the loop cannot be completed by unfolding it. Suppose a state description is required to contain, as a literal proper part, the value of its own prediction about itself. Finite descriptions come with a size measure that strictly decreases when one description is a proper part of another, so no description can be a proper part of itself, and the demand cannot be met. Holding a code, an index, or a pointer to the prediction is a different and perfectly available arrangement, which is what the register of Theorem 1 does. The regress appears only when literal containment is demanded.

8. Knowledge at the Threshold

Within this framework, knowledge is the realized capacity for effective prediction: a structure counts as knowledge for a system and a task when the system can use it to anticipate better than the matched baseline. Theorem 2 makes the threshold the smallest place that capacity can live for a self-referential task, and Theorem 3 fixes what it can never reach.

Three consequences follow, each scoped to the class the theorems name.

• Self-referential predictive knowledge needs at least the three-bit structure. Below it the comparison that turns a forecast into knowledge cannot be carried out.
• No system in the class holds a predictor exact on every system the class can build, including the systems built from itself. Complete self-knowledge of the predictive kind is unavailable from inside.
• The limits apply to knowledge about the limits. A system reasoning about its own horizon is still a system inside the class, and the same argument applies to it.

None of this makes inquiry futile, and the reason is Theorem 2. The gap between what a finite system can verify and what it would need in order to close its own account is what keeps prediction improvable: there is always more accuracy to buy with more structure, and never a final purchase that ends the process.

9. Consciousness: What the Threshold Does and Does Not Settle

Any system that models its own states, anticipates them, and checks the anticipation instantiates the loop of Section 2, so on the stated conditions it carries at least the three-bit structure. That is a necessary condition on a described capacity, and it is the whole of what the counting gives.

It is not sufficient for anything usually meant by consciousness. Sustained operation needs memory beyond one cycle, an environment with regularities worth learning, integration across many such loops, and performance held inside a viable band. Measures built for the question, such as the integrated-information proposal of Tononi (2004), ask about the organization of a whole system and not about the smallest loop inside it. The threshold marks where self-modeling becomes expressible. Whether anything is felt there is a separate question, and the counting argument does not touch it.

10. Conclusion

The result is one number and its conditions. On the realization class where stepping before the closing reset is injective, a phase response separates the two halves of the cycle, the current state and the stored prediction remain readable across the boundary, and all eight response combinations occur, a self-referential predictive loop needs at least eight visited configurations and three bits of log-capacity, and a construction attains both. Carrying those eight contexts as perfectly distinguishable alternatives in a complex Hilbert space forces dimension at least eight.

The threshold works in both directions. It is enough for a register to beat chance wherever the world repeats itself more often than not, and it is already enough for the diagonal construction that denies any predictor exactness across its own class. Everything above the threshold buys accuracy on particular tasks and leaves the barrier where it is.

K0 is a lower bound on a stated class. It supports later analysis of consciousness and physical instantiation without being sufficient for either.

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