A useful predictive state must retain distinctions that can change future observations. Two situations may look identical now and still require different predictions. Memory becomes necessary when the present display has discarded that difference.
A display need not be a complete state
A machine shows a single light: off or on. You would like to predict its next display from the current one. That is possible only if every internal situation producing the same current display also produces the same next display. Otherwise “the light is off” leaves a relevant ambiguity unresolved.
Consider the following invented machine. It cycles deterministically through three states, A, B and C. The light is off in A and B and on in C. There is no randomness and no external input.
| Internal state | Display now | Next state | Display next |
|---|---|---|---|
| A | 0 | B | 0 |
| B | 0 | C | 1 |
| C | 1 | A | 0 |
Starting in A or B produces the same current display, 0. But the next display is 0 from A and 1 from B. A rule taking only the present display as its input would have to send the same input to two different outputs. It cannot reproduce this deterministic machine exactly.
Compare the futures
Choose a starting state. The displayed sequence includes the current output followed by eight updates of the fixed cycle.
A: 0 → 0 → 1 → 0 → 0 → 1 → 0 → 0 → 1
The exact criterion
Write F for the internal update and R for the displayed record. A deterministic update T on records alone must satisfy:
R(F(x)) = T(R(x)).
This is possible exactly when equal present records have equal next records:
R(x) = R(y) implies R(F(x)) = R(F(y)).
The necessity follows immediately: a function T gives one output for each input. For sufficiency, choose any internal state with record r, update it, and define T(r) to be its next record. The criterion guarantees that a different state with the same record gives the same answer. This defines T on the records that actually occur.
In our cycle, the criterion fails for A and B. The failure identifies the missing distinction rather than an inconsistency in the underlying machine.
How much must be retained?
For exact prediction at arbitrary future steps, A, B and C must remain distinguishable. C already differs from the others now; A and B differ at the next step. Three predictive states therefore suffice and are necessary for this example. A fixed-length binary label for three possibilities needs two bits, because one bit distinguishes only two possibilities.
That is a statement about encoding capacity. It is not a claim about average information, energy use or the thermodynamic cost of a physical memory. Those questions require additional assumptions.
Another representation uses the current and previous displays, once a previous observation is available. The pairs (1,0), (0,0) and (0,1), written as (previous,current), identify A, B and C respectively. In this representation, a small amount of history resolves the ambiguity of the present display. Different encodings can retain the same predictive distinctions.
Prediction depends on the permitted experiment
The conclusion concerns this fixed update and these outputs. If an external switch could change the cycle, the predictive state would need to support the allowed interventions too. States indistinguishable under passive observation might become distinguishable after a controlled action.
Prediction also depends on the intended horizon and accuracy. A device designed only to predict the next output may need fewer distinctions than one required to predict every later output. A statistical prediction can tolerate uncertainty that an exact deterministic reconstruction cannot. “How much memory?” is incomplete until the task has been specified.
The SMONAD connection
SMONAD's predictive memory laboratory explores a related problem by refining groups of internal states until their records have a consistent future update. It uses different finite examples and lets the reader inspect the refinement. The central question is whether the chosen record retains enough information for the prediction being attempted.
For SMM/DTT, this matters whenever a completed observation is obtained from a richer generative process. A readable record need not contain everything about its production. A model must identify which discarded distinctions are irrelevant and which remain necessary for later observations. The published completion-principles paper provides a broader mathematical setting for studying completed histories and observable descriptions. The three-state cycle here is a self-contained teaching example, not a proposed microscopic law of nature.
Sources and further reading
- Mohamed Haj Yousef, Predictive memory laboratory. Interactive companion with its own fully specified examples.
- Mohamed Haj Yousef, Completion principles, real commutants, and positive-frequency geometry for orthogonal Hilbert representations, Open Journal of Mathematical Sciences 10 (2026), 996–1012.
- The three-state cycle and the proof above are elementary constructions supplied in this essay; no empirical data are being reported.