FPRD-T155: The support-quotient theorem for replayable memory

1. Three kinds of memory

Let

$$X=\prod_{i\in I}X_i$$

be a finite product of nonempty source alphabets, and let $F:X\to Y$ be the complete answer demanded from the source. In a language cut, $F(x)$ can be the vector of answers to a finite continuation family.

Three different objects arise.

1. Operational support. A replay trace records source coordinates actually queried by an execution.
2. Semantic support. A coordinate is essential when changing only that coordinate can change $F$.
3. Quotient memory. Sources are identified when they have the same value of $F$. The number of equivalence classes is $|F(X)|$.

2. The theorem

Define

$$E(F)=\{i\in I:\exists x,x'\text{ differing only at }i \text{ with }F(x)\ne F(x')\}.$$

A set $S\subseteq I$ is sufficient when agreement on $S$ forces equal answers:

$$x|_S=x'|_S\Longrightarrow F(x)=F(x').$$

Then:

1. $S$ is sufficient if and only if $E(F)\subseteq S$. Hence $E(F)$ is the unique least semantic coordinate support.

2. The minimum number of exact deterministic summary states is $|F(X)|$.

3. The quotient is bounded by the semantic carrier:

   $$|F(X)|\le\prod_{i\in E(F)}|X_i|.$$

   Equality holds exactly when the induced map on the essential-coordinate product is injective.

4. If $F=(F_c)_{c\in C}$ is a family of future answers, then

   $$E(F)=\bigcup_{c\in C}E(F_c).$$

5. Suppose a deterministic adaptive execution on source $x$ has a stable replay support $Q(x)$: agreement on $Q(x)$ reproduces the same answer. Then

   $$E(F)\subseteq\bigcup_{x\in X}Q(x).$$

   For a future family, take the union over both sources and futures.

3. Proof

If $S$ is sufficient and $i\notin S$, two sources differing only at $i$ agree on $S$, so they have equal answers. Thus every essential coordinate is in $S$.

Conversely, suppose $E(F)\subseteq S$. If two sources agree on $S$, change their coordinates outside $S$ one at a time. None of those coordinates is essential, so no change alters $F$. The two sources therefore have the same answer. This proves the first claim.

For the second claim, any exact summary must separate sources with different answers. The value $F(x)$ itself is a matching summary. Thus the exact minimum is $|F(X)|$.

The first claim makes $F$ factor through the projection to $E(F)$. Its image therefore has at most as many elements as the essential-coordinate product. Equality is exactly injectivity of the induced factor.

A coordinate changes a vector-valued map exactly when it changes at least one component, proving the union formula.

Finally, let $U=\bigcup_xQ(x)$. If two sources agree on $U$, the second agrees with the first on $Q(x)$, so stability gives equal answers. Thus $U$ is sufficient, and the first claim gives $E(F)\subseteq U$.

4. The gaps can all be strict

On $X=\{0,1\}^n$:

• A program can query every coordinate and return only $x_1$. Its operational envelope has size $n$, while semantic support and quotient memory both have size one bit.
• Parity depends essentially on all $n$ coordinates but has only two answers. Semantic support has size $n$, while quotient memory is one bit.
• The vector of coordinate queries $F(x)=(x_1,\ldots,x_n)$ has all coordinates essential and $2^n$ quotient states. Here the semantic carrier bound is tight.

These examples show that trace size, semantic coordinate support, and compressed answer information are genuinely different resources.

5. Consequence for FPRD

FPRD-T150 supplies stable operational replay supports. This theorem extracts their semantic core. FPRD-T152 identifies the exact quotient memory of a future family. The remaining resource is dynamic:

How much local work is required to update, name, transport, split, and merge the quotient as the input cut advances?

Persistent scaffolds can retain a large quotient with bounded work when its addresses move along a simple cursor. FPRD-C02 asks whether projected pushdown frontiers always admit such a stationary transport mechanism.

The theorem uses standard finite-function support and quotient facts. Its role is to organize the replayable-memory program, not to claim literature priority.