An endpoint is not a history

Rewriting semantics often identifies computations that reach the same result. That is appropriate when only the endpoint matters, but it erases which local events occurred, which events were independent, and how one history can be transformed into another.

$$\text{same endpoint} \;\not\Rightarrow\; \text{same history or coherent transformation}.$$

FPRD keeps several observation levels available: the semantic endpoint, the sequence of visible states, occurrence-labelled events, and the higher cells relating paths of transformations. The appropriate quotient depends on the question being asked.

Worked case · one rewrite rule, two counts

Independent scheduling hides factorially many histories

Consider the terminating confluent rule$aa\to a$, while retaining the occurrence contracted at each step. A complete reduction from$a^n$ to$a$ deletes the$n-1$ original gaps in some order. Hence there are exactly

$$(n-1)!\quad\text{occurrence-labelled complete histories}.$$

Swapping consecutive contractions with disjoint residual supports changes only their schedule. Quotienting by those swaps leaves the planar full binary merge tree, so the number of schedule classes is

$$C_{n-1}=\frac1n\binom{2n-2}{n-1}.$$

If only the successive words were recorded, every one of these histories would collapse to the same chain$a^n\to a^{n-1}\to\cdots\to a$. That representation is adequate for reachability but too coarse for independence or coherence.

Worked case · the first coherence loop

Connected transformations can still contain a hole

At source $a^4$, the six concrete histories and their elementary disjoint or overlap moves form a hexagon$C_6$. Contracting the one edge that merely exchanges independent scheduling gives the pentagon$C_5$ of the five binary trees on four leaves.

$$C_6\longrightarrow C_5, \qquad \pi_1(C_6)\cong\pi_1(C_5)\cong\mathbb Z.$$

Both graphs are connected, but the surviving loop means that two paths of transformations need not yet be identified. In this fixed component, one higher cell along the raw hexagon—or equivalently the quotient pentagon—is necessary and sufficient to fill the loop.

This is the distinction between confluence, connectedness, and coherence in its smallest useful form. Read the fixed-arity cell argument →

Worked case · native higher geometry

Pentagons and squares assemble into a sphere

At five leaves, independent rotations appear alongside associativity pentagons. The schedule-quotient rotation complex$K_5$ has

$$14\ \text{vertices},\quad 21\ \text{edges},\quad 6\ \text{pentagons},\quad 3\ \text{squares}.$$

The nine faces are indexed by the proper contiguous clusters of the five leaves. Every edge lies in exactly two faces. Removing one pentagon and collapsing the other eight faces leaves a tree, which gives the constructive conclusion

The homology calculation corroborates the result but does not by itself prove simple connectedness; that earlier inference was explicitly withdrawn. The theorem concerns this five-leaf complex, not an all-arity coherence classification.

Worked case · choosing the observation

A final graph can forget how its behavior was built

A scaffold history has several legitimate semantics. Its physical graph records every node and endpoint; its strong trace records the minimal behavioral figure after every prefix; its final figure keeps only the last one. Writing $F_j(H)$ for the figure after prefix $j$,

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

Equal strong traces remain equal under every common continuation, and two graph-local moves normalize exactly that equivalence. Final equivalence is harder: it can erase earlier differences and even hide nodes whose port cells were queried while constructing a surviving edge. The relevant dependency slice is therefore replay-closed causal support, not ordinary final reachability.

Foundational theorem · FPRD-T150

Replayable work is a backward slice, not the whole history

A finite work history may choose later reads from values returned by earlier reads. For one completed execution, begin with the records queried by the final observer and close backward under the query traces that produced them. Call the resulting set $R_x$.

$$y|_{R_x\cap I}=x|_{R_x\cap I} \quad\Longrightarrow\quad \operatorname{output}(y)=\operatorname{output}(x), \quad R_y=R_x.$$

A well-founded replay induction proves more than output equality: every supported derived record repeats the same adaptive queries and value. Under sequential composition, later demands pull back exactly through the earlier dependency graph. With independent source domains, changes outside the slice form a product cylinder.

This is a per-execution trace theorem. It does not bound slice size over growing histories, construct a stationary compressor, or resolve FPRD-C02. No literature novelty claim is made.

Boundary theorem · FPRD-T151

A tiny completed slice can hide a large future family

Replay support is execution-specific. Before the future is known, a prefix may need to remain ready for many different slices. In the middle-bit language, every completed decision queries one old symbol, yet $r$ continuations select$r$ different symbols and recover all $2^r$ response profiles.

This is not a lower bound against scaffolding automata or PEGs. Persistent roots can sometimes transport many possible addresses by a bounded local rule.

Worked case · semantics of persistent histories

Four local moves recover final behavioral equivalence

A scaffold history can contain endpoint detours, absorbed literal work, causally irrelevant garbage, and changes in when a surviving carrier is transported. These are operationally different histories that may leave the same final behavior.

The theorem turns a semantic equality into a bounded local calculus. After strong and garbage motion inside each chamber is collapsed, the possible carrier times form the ideals of a corrected rectangle. If that rectangle has $r$ rows and$c$ columns, its temporal-slide complex satisfies

$$\dim X=\min(r,c), \qquad d(t,s)=\sum_{i=1}^{r}|t_i-s_i|.$$

FPRD-T137 then goes one dimension higher: parallel paths made from all four moves are related by a length-independent guarded family of triangles, squares, carrier cubes, and one bounded absorption-transport hexagon.

See the sewing theorem, corrected carrier geometry, and coherence proof →

These whole-fibre statements are restricted to degree one and distance one. They do not establish a finite presentation for unrestricted scaffold histories, and no external specialist review is recorded.

Worked case · rewriting as ideal membership

Reversibility turns reachability into algebra

A reversible data VASS has orbit-finitely many transition rules, each usable in both directions. If representative rules are$(u_i,v_i)$, form the equivariant binomial ideal

$$I_T=\left\langle \iota(v_i-u_i): i=1,\ldots,m,\ \iota\in\operatorname{Emb}(\mathcal A) \right\rangle.$$

Here an unbounded family of rewrite paths receives a compact algebraic interface. For homogeneous data structures, finite embedding renamings can equivalently be realized by automorphisms. But finite equivariant generation is not automatically an algorithm for finding a basis. A stronger “nicely orderable” hypothesis supports a Gröbner-basis decision procedure; the weaker labelled-WQO formulation recorded on the site remains open. The Rado graph is not a positive example for that weaker hypothesis: its ordered age already has a chordless-cycle antichain.

Where the area is moving

Current work asks which local moves generate semantic equality, which higher cells make those moves coherent, and when an operational system admits a finite normal form or algebraic certificate. Failed quotients are retained because they reveal exactly which history information a proposed semantics erased too early.

Working answer to the driving question. Local transformations generate the same global behavior when they preserve the chosen observation and a complete relation calculus connects every history in each semantic fibre. Connectedness handles objects; coherence additionally controls the paths between them.