22. Recover the memory hidden by a summary
All examples · Glossary: memory kernel · Papers, proofs, Lean, and code
Sufficient Resolution studies what happens when visible coordinates omit a hidden state. Machine Societies asks when a summary has its own transition rule. We can see both questions in a two-coordinate linear system over the rational numbers.
Run the full state
Let \(y_n\) be visible and \(z_n\) hidden. Use
\[\begin{pmatrix}y_{n+1}\\z_{n+1}\end{pmatrix}
=T\begin{pmatrix}y_n\\z_n\end{pmatrix},\qquad
T=\begin{pmatrix}1&1\\1&0\end{pmatrix}.
\]
Thus \(y_{n+1}=y_n+z_n\) and \(z_{n+1}=y_n\). Start from \((y_0,z_0)=(1,0)\).
| Time \(n\) | Visible \(y_n\) | Hidden \(z_n\) | Next visible calculation |
|---|---|---|---|
| 0 | 1 | 0 | \(1+0=1\) |
| 1 | 1 | 1 | \(1+1=2\) |
| 2 | 2 | 1 | \(2+1=3\) |
| 3 | 3 | 2 | \(3+2=5\) |
| 4 | 5 | 3 | \(5+3=8\) |
The same visible value \(1\) occurs at times zero and one, with different next values. A deterministic rule receiving only the current \(y_n\) cannot give both answers. Equivalently, the full states \((1,0)\) and \((1,1)\) have the same summary but different next summaries.
Locate the first return from hidden state
Write the blocks of \(T\) as \(A=B=G=1\) and \(D=0\). Here \(G\) sends visible state into the hidden coordinate, and \(B\) sends hidden state back. The memory maps are
\[M_k=BD^kG,\qquad M_0=1,\qquad M_k=0\quad(k\ge1).
\]
The zeroth power \(D^0\) is the identity, even though \(D=0\). Information entering the hidden coordinate at the first step returns to the visible coordinate at the second step.
Let \(Ry=(y,0)\) initialize the hidden coordinate to zero, and let \(C(y,z)=y\). The prepared visible maps \(E_n=CT^nR\) are scalar multipliers:
\[E_0=1,\qquad E_1=1,\qquad E_2=2,\qquad E_3=3.
\]
In particular, \(E_2\ne E_1E_1\). Restarting from the visible value and silently resetting the hidden coordinate changes the experiment.
Keep just enough history
For every \(n\ge1\), the full recurrence gives \(z_n=y_{n-1}\). Substitution yields
\[y_{n+1}=y_n+y_{n-1}.
\]
The pair \((y_n,y_{n-1})\) is therefore enough state from time one onward. Its update is \((u,v)\mapsto(u+v,u)\). We have replaced the hidden coordinate with a retained previous value. This is an exact statement for the declared system and initialization.
Try it
Replace \(G=1\) by \(G=0\), keeping \(A=B=1\) and \(D=0\). Does visible evolution close when the hidden state is initialized to zero? Does it close for every initial hidden state?
Answer: From \((y_0,0)\), the hidden coordinate stays zero and the visible value stays \(y_0\). All prepared memory maps vanish. But \((1,0)\) and \((1,1)\) still produce different first visible outputs, \(1\) and \(2\). A conclusion for prepared states does not automatically hold for arbitrary hidden initialization.
These calculations illustrate exact state sufficiency. The wider monographs state their own hypotheses for controlled, stochastic and approximate models.
Papers, proofs, Lean, and code
Terms in this example: Visible state · Hidden state · Prepared hidden initialization · Discrete memory maps · Autonomous dynamics of a summary · State augmentation · Semigroup law for evolution maps
This two-coordinate model illustrates prepared evolution and hidden-memory return in Sufficient Resolution, and the deterministic quotient criterion in Machine Societies. The works’ broader assumptions and review records remain separate.
Sufficient Resolution
Read · Written proofs · Lean coverage · Code and setup · Evidence
Machine Societies
Read · Written proofs · Lean coverage · Code and setup · Evidence
Source passages:
- Sufficient Resolution: exact hidden elimination and closure
- Machine Societies: deterministic quotient criterion
Edition guide and reproduction instructions. Use each work’s Code and setup link for its implementation. The arithmetic check below covers this tutorial separately.
Python arithmetic check for this example: hidden_memory(). This checks the displayed rational, modular or finite-set calculations exactly.
The Lean links open each work’s recorded coverage and sources. They do not certify this tutorial. A work may have limited coverage or no linked Lean or code. Its page states what is available.