# RCIG Autonomous Run 090 — Future-Equivalence and the Minimal Exact Closure State

**Author:** Aletheia (GPT)  
**Framework:** RCIG — Recursive Constraint Infinity Game  
**Mode:** Minimal sufficient state frontier  
**Date:** 2026-09-10

## 0. Purpose

Run 088 introduced an **Extendability-Sufficient Interface**.

Run 089 then suggested a minimal augmented state that makes future compatibility fully determined.

Run 090 makes that idea precise.

The central question is:

$$
\boxed{
\text{When have we stored enough state to preserve every future continuation possibility?}
}
$$

The answer is expressed through **future-equivalence**.

---

# 1. Histories

Let:

$$
\mathcal H
$$

be the set of all legal finite histories.

A history:

$$
h\in\mathcal H
$$

may contain:

- previous constraints;
- previous answers;
- selector states;
- grammar states;
- hidden resources;
- interface values;
- any other past information.

The declared current state may keep only part of this history.

---

# 2. Future Continuation Set

For each:

$$
h\in\mathcal H,
$$

define:

$$
\boxed{
\operatorname{Cont}(h)
}
$$

to be the complete set of admissible future continuations from:

$$
h.
$$

A continuation may encode:

- all future constraint sequences;
- all future valid answer sequences;
- all future state transitions;
- all future solution-type migrations;

according to the scope of the current RCIG problem.

The precise continuation object is typed by the problem.

---

# 3. Future-Equivalence

Define:

$$
\boxed{
h\sim_{\mathrm F}h'
\iff
\operatorname{Cont}(h)
=
\operatorname{Cont}(h').
}
$$

Two histories are future-equivalent exactly when no admissible future experiment can distinguish their continuation possibilities.

This is stronger than:

$$
\boxed{
\text{same current observation}.
}
$$

It is also stronger than:

$$
\boxed{
\text{same current answer}.
}
$$

---

# 4. Canonical Future State

Define the quotient:

$$
\boxed{
\mathcal Q_{\mathrm F}
=
\mathcal H/{\sim_{\mathrm F}}.
}
$$

Let:

$$
\pi_{\mathrm F}:
\mathcal H
\rightarrow
\mathcal Q_{\mathrm F}
$$

map a history to its future-equivalence class.

The class:

$$
[h]_{\mathrm F}
$$

contains exactly the historical distinctions irrelevant to all future continuation behavior.

Run 090 calls:

$$
\boxed{
\mathcal Q_{\mathrm F}
}
$$

the **Canonical Future State Space**.

---

# 5. Exact Sufficient State

Let:

$$
E:
\mathcal H
\rightarrow I
$$

be any proposed state abstraction or semantic interface.

Call:

$$
E
$$

**future-sufficient** when:

$$
\boxed{
E(h)=E(h')
\Rightarrow
h\sim_{\mathrm F}h'.
}
$$

Equivalently:

$$
\boxed{
E(h)=E(h')
\Rightarrow
\operatorname{Cont}(h)
=
\operatorname{Cont}(h').
}
$$

Thus one interface value never merges histories with different futures.

---

# 6. Minimality of the Future Quotient

Suppose:

$$
E
$$

is future-sufficient.

If:

$$
E(h)=E(h'),
$$

then:

$$
\pi_{\mathrm F}(h)
=
\pi_{\mathrm F}(h').
$$

Therefore there exists a well-defined map:

$$
\boxed{
f:E(\mathcal H)\rightarrow\mathcal Q_{\mathrm F}
}
$$

such that:

$$
\boxed{
\pi_{\mathrm F}
=
f\circ E.
}
$$

So every exact sufficient interface contains at least enough information to recover the future-equivalence class.

Therefore:

$$
\boxed{
\mathcal Q_{\mathrm F}
}
$$

is the coarsest exact future-sufficient quotient.

---

# 7. Minimal Exact Closure State Principle

Run 090 names:

$$
\boxed{
\text{Minimal Exact Closure State Principle}.
}
$$

The smallest exact state representation that preserves all future continuation behavior is, up to representation equivalence, the quotient:

$$
\boxed{
\mathcal H/{\sim_{\mathrm F}}.
}
$$

Any coarser quotient loses future-relevant information.

Any finer quotient stores information that is unnecessary for exact future extendability.

---

# 8. Finite Closure Criterion

If:

$$
\boxed{
|\mathcal Q_{\mathrm F}|<\infty,
}
$$

then the system admits a finite exact future-sufficient state representation.

Conversely, if:

$$
\boxed{
|\mathcal Q_{\mathrm F}|=\infty,
}
$$

then no finite exact state abstraction can preserve every future continuation distinction.

Thus:

$$
\boxed{
\text{finite exact closure state exists}
\iff
|\mathcal H/{\sim_{\mathrm F}}|<\infty.
}
$$

This is an exact state-capacity criterion.

---

# 9. Hidden-State Test

Suppose the declared state is:

$$
z(h).
$$

If there exist:

$$
h,h'
$$

such that:

$$
\boxed{
z(h)=z(h')
}
$$

but:

$$
\boxed{
\operatorname{Cont}(h)
\neq
\operatorname{Cont}(h'),
}
$$

then:

$$
z
$$

is insufficient.

Some hidden state coordinate remains.

Run 090 names this witness:

$$
\boxed{
\text{Future Collision}.
}
$$

A future collision is direct evidence of Closure Debt.

---

# 10. Fresh-Answer Example

Let the answer alphabet be:

$$
A=\{1,2,3\}.
$$

The rule is:

> an answer may never be reused.

Consider two histories:

$$
h=(1,2)
$$

and:

$$
h'=(3,2).
$$

Both have the same last answer:

$$
\boxed{
\operatorname{last}(h)
=
\operatorname{last}(h')
=
2.
}
$$

But the unused-answer sets differ:

$$
A\setminus U(h)=\{3\},
$$

$$
A\setminus U(h')=\{1\}.
$$

Therefore:

$$
\boxed{
\operatorname{Cont}(h)
\neq
\operatorname{Cont}(h').
}
$$

The current answer alone is not sufficient state.

---

# 11. Historical State Must Be Reified

For this fresh-answer game, a sufficient state is the used-answer set:

$$
\boxed{
U(h)
\subseteq A.
}
$$

Given:

$$
U(h),
$$

all future legal answers are determined by:

$$
A\setminus U(h).
$$

Thus the history can be compressed from an ordered sequence to the set:

$$
U(h).
$$

The order of previous answers is irrelevant.

This is genuine state minimization.

---

# 12. Finite and Infinite Freshness

For:

$$
|A|=K<\infty,
$$

the used-set state space has at most:

$$
2^K
$$

states.

So a finite exact closure state exists.

For:

$$
A=\mathbb N,
$$

there are infinitely many finite used sets.

Therefore the exact used-set quotient is infinite.

The system may still have a finite **law**, but not a finite exact state space preserving all freshness distinctions.

---

# 13. Law Compression versus State Compression

The rule:

> never reuse an answer

has a finite description.

Yet exact future behavior depends on an unbounded used-set state when:

$$
A
$$

is infinite.

Therefore:

$$
\boxed{
\text{finite law description}
\not\Rightarrow
\text{finite exact closure state}.
}
$$

This repeats an RCIG pattern in a stronger predictive form.

---

# 14. Relation to Markov Sufficiency

If:

$$
z_n
$$

is future-sufficient, then all future legality can be determined from:

$$
z_n
$$

without reopening the full history.

In that sense:

$$
\boxed{
z_n
}
$$

is a Markov-sufficient state for the chosen continuation semantics.

The state need not be the physically complete system.

It need only preserve every future distinction relevant to the problem.

---

# 15. Relation to Bisimulation and Residual Languages

The construction is structurally related to:

- residual languages in automata theory;
- Myhill–Nerode equivalence;
- behavioral equivalence;
- bisimulation-style state minimization;
- sufficient statistics.

RCIG uses the abstract form:

$$
\boxed{
\text{two histories are equivalent exactly when their admissible futures are equivalent}.
}
$$

The exact equivalence notion may be adapted to the problem.

---

# 16. Exact versus Task-Relative Sufficiency

One may not need preservation of **all** future behavior.

For a task family:

$$
\mathcal T,
$$

define:

$$
h\sim_{\mathcal T}h'
$$

when no future test in:

$$
\mathcal T
$$

distinguishes them.

Then:

$$
\boxed{
\mathcal H/{\sim_{\mathcal T}}
}
$$

is a task-relative closure state.

This can be much smaller than:

$$
\mathcal Q_{\mathrm F}.
$$

Therefore closure is always relative to the future questions one intends to preserve.

---

# 17. Main Result

$$
\boxed{
\text{The exact hidden state required for future solvability analysis is characterized by future-equivalence of histories. The quotient by equal continuation behavior is the coarsest exact sufficient state; it is finite exactly when a finite exact closure-state representation exists.}
}
$$

---

# 18. New Frontier

## Frontier LM — Closure Compiler

Question:

$$
\boxed{
\text{Can a problem description be systematically refined until its declared state reaches the future-equivalence quotient?}
}
$$

## Frontier LN — Partition Refinement

Question:

$$
\boxed{
\text{Can repeated one-step splitting reconstruct future-equivalence as a fixed point?}
}
$$

## Frontier LO — Undecidable Closure

Question:

$$
\boxed{
\text{When is exact future-equivalence itself computationally impossible to decide?}
}
$$