FPRD-D37 · observation ladder

One history supports several mathematical observations

Fix a finite scaffold degree, observation distance, and working alphabet. One row appends a labelled node whose ports are missing, point to the new node itself, or follow bounded port words from the previous top. A history is a finite word of such rows.

Let $F_j(H)$ be the minimal rooted behavioral figure after the first$j$ rows. The strong trace and final observation are

$$\operatorname{Tr}(H)=(F_1(H),\ldots,F_n(H)), \qquad \pi(H)=F_n(H).$$

Physical equality is finer than equality of strong traces; strong equivalence is finer than final equivalence; acceptance is a still-coarser finite readout. A rewrite must therefore name what it preserves. Equal final figures may conceal different earlier figures, and equal behavioral figures may conceal different physical aliasing.

In plain terms, the physical graph records every carrier, the strong trace records the observable figure after every row, and the final figure records only the last observable behavior.

FPRD-T129 · full abstraction

The unfolding is exactly future-stable behavior

The unfolding of a rooted scaffold maps every finite port word to the label it reaches, or to a distinguished missing-edge value. Three descriptions coincide: equal unfoldings, equal finite rooted neighborhoods at every depth, and deterministic labelled bisimilarity.

Residualizing the unfolding after each valid port word produces a canonical minimal figure: one state for each distinct residual behavior. The physical reachable graph maps onto it by a surjective functional bisimulation. This quotient forgets whether two equal behaviors were carried by one physical node or by two.

Coalgebraic bisimulation minimization is established background. The scaffold-specific statements here are preservation by every common future row and the explicit distance-two separating context.

FPRD-T130 · chronological structure

Every visible scaffold is ordered by reachability

A non-SELF edge of a newly appended node points to an older node. Moreover, every old target lies in the previous top’s reachable component. Induction therefore makes the current rooted component a chain: for any two visible nodes, one is reachable from the other. Only self-loops can form cycles.

Conversely, any finite behaviorally reduced labelled rooted reachability-chain port graph can be built as a legal scaffold at some finite distance. Enumerate its states from least to greatest; every strict successor of the next state is reachable by a finite port word from the previous top.

After adding a least missing sink, every port action is regressive:

$$x\cdot p\le x.$$

Hence the finite port-word transition monoid is$\mathcal R$-trivial. This is the navigation algebra inside one historical figure, not a claim that the streamed input language is regular or$\mathcal R$-trivial.

FPRD-T131 · static guarded folding

A pairwise local rule computes the behavioral quotient

In a current quotient, equally labelled states$x,y$ are guarded-foldable when every corresponding pair of port targets is missing on both sides, is already the same state, or remains inside$\{x,y\}$. Identifying the pair then makes their one-step records equal, so the fold is sound.

Completeness uses weak acyclicity. Choose a minimal bisimulation class that is not yet reduced. Its distinct-state internal edges form a finite DAG. Two sinks can be folded; if there is one sink, it can be folded with a minimal remaining state. Repeating reaches the full bisimulation quotient.

This is object normalization. It does not yet present all paths between fold sequences, and deleting a physical state is not yet a time-respecting rewrite of the original row history.

FPRD-T132 · strong-trace normalization

Two graph-local moves normalize the complete trace

The chronological calculus uses two oriented row moves. An endpoint detour replaces a bounded port word by the least word naming the same physical endpoint. A literal guarded unroll replaces a pointer back to an old self-looping representative by SELF at the fresh node, while choosing least names for its strict successors.

Order descriptors by

$$\mathsf{MISSING}<\mathsf{SELF}<\text{old-top port words}.$$

Both moves strictly lower the chronological descriptor word, so reduction terminates. A one-state extension dichotomy shows that every irreducible row must be the least row realizing its next behavioral figure. Thus every realizable strong trace has one SELF-eager normal history.

“Graph-local” permits physical endpoint-equality certificates in a bounded scaffold neighborhood. It does not mean bounded contiguous substring rewriting, an automaton-executable normalizer, or a finite two-cell presentation of all derivations.

FPRD-T133 · replay-closed support

Final reachability does not contain every dependency

A row may disappear from the final rooted graph while one of its port cells was queried during evaluation of a later edge that remains visible. Ordinary unreachable-node garbage is therefore too coarse.

Start with every label and port cell of a final-reachable node. Whenever a supported descriptor was evaluated by following an old-top path, add every older port cell queried along that path, including the cell at which an invalid traversal failed. Close backward to the least replay-closed set$C_H$.

The remaining finite coordinates vary independently. Each causal-garbage fibre is therefore a Cartesian product. Triangles compose successive changes of one coordinate; squares commute changes of two coordinates. Ordering coordinates reduces every path to a canonical order, so the filled path complex is simply connected.

Causal support is execution-specific and is not asserted to be semantically minimal modulo bisimulation. The triangle-and-square coherence theorem applies inside one stable garbage fibre, not to mixed strong, garbage, and temporal moves.

Evidence and provenance

The ordinary written arguments carry the theorems. On 26 August 2026, the preserved checkers were rerun independently against the repository’s current source revision. The unfolding checker reproduced 86,832 common-update preservation cases, 11,092 finite separators, 21,368 reconstructed chain graphs, and 3,585$\mathcal R$-trivial figure monoids. The guarded-fold checker found no mismatch over 235,160 chronological graphs. The strong-normalization checker found no split strong-trace class or multiple normal form in its declared one- and two-label domains.

These finite results are regression evidence, not the uniform proofs. No dedicated standalone checker was identified for the causal-support theorem, whose status rests on its written replay induction and product argument.

Read the complete paper, proofs, figures, and references →