REPRODUCIBLE MODEL · FINITE DYNAMICS
When does a record predict its own future?
Two source states can look identical now and produce different observations next. This laboratory finds the smallest predictive distinction for a supplied finite update.
Updated · Exact finite example
What is established here: a mathematical test of record sufficiency. The update and observation rule are inputs. This example does not establish that nature follows this update or that a second observable time is required.
Inspect the hidden distinction
Begin with classes that retain only the present binary output. Each refinement separates states whose successors fall into different current classes. Stop when every class has one well-defined successor class.
The default example below is complete. Interactive controls become available when the script loads.
Current partition
{A, B} | {C, D}
Round 0: two present-record classes. A and B both display 0, but their next outputs are 1 and 0. The stable partition is {A} | {B} | {C, D}: three predictive states, requiring two fixed-length bits in total.
| State | Output R(x) | Successor F(x) | Next output | Class now | Stable class |
|---|---|---|---|---|---|
| A | 0 | C | 1 | 1 | 1 |
| B | 0 | A | 0 | 1 | 2 |
| C | 1 | C | 1 | 2 | 3 |
| D | 1 | D | 1 | 2 | 3 |
| Initial state | t=0 | t=1 | t=2 | t=3 | t=4 | t=5 | t=6 | t=7 |
|---|---|---|---|---|---|---|---|---|
| A | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| B | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |
| C | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| D | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Complete refinement history
| Round | Classes | Number of classes |
|---|---|---|
| 0 | {A, B} | {C, D} | 2 |
| 1 | {A} | {B} | {C, D} | 3 |
Exports are calculated examples, not experimental data. No information is uploaded.
The exact closure criterion
Let X be a finite state set, F:X→X a deterministic update, and R:X→{0,1} the displayed output. A reduced update T on present records exists exactly when equal present records always have equal next records:
Equivalently, R∘F=T∘R on the allowed source states.
Necessity follows by applying T to the shared record. For sufficiency, define T(r)=R(F(x)) for any state with R(x)=r. The criterion makes the choice of x irrelevant. In the default example A and B violate the criterion, so the binary display alone cannot evolve autonomously.
Adding a predictive distinction between A and B repairs the quotient. States C and D remain merged because both display an unending sequence of ones. The coarsest predictive state need not identify the full microscopic state.
Why refinement finds the minimal predictive partition
Begin with P₀(x)=R(x). Refine by the ordered pair consisting of the current class and the successor class:
By induction, two states are in the same class after round k exactly when their displayed sequences agree from time 0 through k. Every changing round splits at least one finite class. Consequently the procedure stops after finitely many splits. At stability, equal classes have equal successor classes, so the induced update predicts every later output.
Conversely, states with identical complete future output sequences can never be separated by this procedure. The stable partition is therefore exactly the coarsest equivalence preserving the present output and all its future continuations under this one update.
The eight-output table is an illustration. The stability test, rather than the finite table alone, establishes agreement for every future step. If external interventions are admitted, the partition must preserve successor classes under every allowed operation; the current widget does not perform that larger test.
A difference that takes three updates to appear
In the second example, A→B→C→D→E, while E→E and F→F. States A–D display 0; E and F display 1. Its partitions are:
→ {A,B,C} | {D} | {E,F}
→ {A,B} | {C} | {D} | {E,F}
→ {A} | {B} | {C} | {D} | {E,F}.
A and B agree at times 0, 1 and 2, then differ at time 3. Checking only the next output of this pair would miss their distinction. The stable model has five predictive classes. Its minimum fixed-length class label requires ⌈log₂5⌉=3 bits. This is an encoding-capacity bound, not an average entropy, energy cost or thermodynamic claim.
What this contributes to the source programme
A spatial readout or completion record can discard distinctions needed for later predictions. The constructive task is to specify what remains in the source state and show why the retained state is sufficient. A visible display does not by itself provide that proof.
The six nested inner roles of SMM/DTT can organize a proposed generation process, but their nesting does not automatically supply this closure property. Test the chosen completion map against the actual autonomous source update. The model specification guide turns those choices into a reviewable model; the reproducibility guide explains how to preserve the calculation.
Continue with one source and predictive records, memory and repeatability tests, and the research claim map. For the published completion framework, see DOI 10.30538/oms2026.0328; the finite systems on this page are self-contained teaching examples.