FPRD-D40 · execution model

Work is a finite history of adaptive reads

Start with finitely many source cells. Derived cells are created in a well-founded order. Each derived cell is computed by a terminating deterministic procedure that may query source cells and earlier derived cells. Its next query may depend on the values already read. A final deterministic observer reads the completed history.

For one source assignment $x$, let $\tau_x(d)$ be the ordered finite sequence of cells actually queried while computing derived cell$d$, and let$\tau_x(O)$ be the observer's ordered query transcript. Repeated queries are retained. Write$Q_x(d)=\operatorname{supp}(\tau_x(d))$and $Q_x(O)=\operatorname{supp}(\tau_x(O))$for the underlying queried-cell sets. The replay support is the least set $R_x$satisfying

This closure is execution-specific. It records the branch that was taken, not every dependency that another execution might use.

FPRD-T150 · replay theorem

Agreement on the slice replays the same work

Let $x$ and$y$ be two source assignments. If they agree on the source cells in$R_x$, then every supported derived cell has the same value and makes the same adaptive queries. The observer therefore follows the same query trace and returns the same result. Recomputing the backward closure gives

The proof is a well-founded induction over supported derived cells. All source queries see the same values by hypothesis; all earlier derived queries see the same values by induction. Determinism then forces the current procedure to repeat its query sequence and value. The observer is the same final induction step.

Composition law

Later work pulls its memory backward through earlier work

Split a history at a downward-closed cut into an earlier block$H$, containing the source cells and every predecessor of its derived cells, and later work$K$. Use the recorded transcripts from one execution throughout. Let$B$ be the old cells directly queried by the supported part of $K$or by the observer. Then the old portion of the composite support is

The completed history need not be retained wholesale. Later work names its old demands, and those demands pull back through the exact dependencies that produced them. This is the precise sense in which prior work is replayable.

Fibre geometry

Irrelevant source coordinates form a product cylinder

When source coordinates vary independently, fixing the supported source cells gives

Every assignment in this cylinder replays the same supported work. The cylinder is contained in a replay-equivalence class; the whole class can be larger. An observer that queries a source cell and returns a constant has the same result and query transcript even when the queried source value changes. With finite coordinate domains, single-coordinate changes within the cylinder generate a Cartesian product of complete graphs. Same-coordinate triangles and distinct-coordinate commuting squares contract every loop.

Independent coordinate domains are essential for the product conclusion. A global validity constraint can cut a non-product subset out of the replay cylinder.

Scope and relation to FPRD

This is a replay theorem, not yet a memory bound

FPRD-T133 is a scaffold specialization. Descriptor evaluation is the adaptive work, the final rooted concrete scaffold supplies the observation roots, and replay-closed causal support is the backward slice. The abstract theorem explains why that proof pattern applies beyond scaffold ports.

The theorem does not say that the slice is the smallest source set preserving only the final output. Algebraic identities, equal values, or another presentation may permit a smaller semantic slice. It also does not bound support uniformly as histories grow, construct a stationary finite-state compressor, resolve FPRD-C02, or imply that every context-free language belongs to PEG.