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61 theorems · 18 lemmas · 19 propositions · 9 corollaries

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propositionProposition 3.1

Corrected carrier rectangles

For an ordinary missing or base terminal and $m\ge1$ , carrier placements are the ideals of $[m-1]\times[n-m]$ . For a recursive terminal, $m=0$ is a singleton; for $m\ge1$ , placements are the ideals of $[m-1]\times[n-m-1]$ .

A coherent cubical algebra for unary scaffold historiesreview

theoremTheorem 3.2

Carrier cube

Filling every Boolean family of independent temporal slides gives the rooted CAT(0) order-ideal cube of Proposition 3.1 . For a rectangle with $r$ rows and $c$ columns it has $rc$ hyperplanes, dimension $\min(r,c)$ , edge distance $d(t,s)=\sum_{i=1}^{r}|t_i-s_i|,$ and coordinatewise-median carrier tuples.

A coherent cubical algebra for unary scaffold historiesreview

lemmaLemma 4.1

Unique quotient normal class

Every final fibre has one irreducible garbage class, the right-packed class $\mathsf{RP}_n(F)$ . Hence oriented effective horizontal rewriting terminates and is object-confluent modulo garbage.

A coherent cubical algebra for unary scaffold historiesreview

lemmaLemma 4.2

Garbage-trivial collapse

If a legal $D$ or $A$ edge is trivial modulo garbage, it equals one descriptor garbage edge. If a legal $T$ edge is trivial modulo garbage, it equals a two-descriptor garbage path; the two orders agree by the garbage square.

A coherent cubical algebra for unary scaffold historiesreview

lemmaLemma 5.1

Descriptor-support monotonicity

For every effective oriented horizontal move $R$ , $\operatorname{DescSupp}(RH)\subseteq\operatorname{DescSupp}(H).$ Every coinitial horizontal/garbage-descriptor branching is therefore a strict naturality square modulo garbage.

A coherent cubical algebra for unary scaffold historiesreview

propositionProposition 5.2

Support captures

The only mixed branches without a strict horizontal residual are $A/GL(0),\qquad T/GL(0),\qquad T/GL(-1).$ They require no new generator: the garbage-first path returns along the inverse garbage edge and then takes the original reduction, so $G;G^{-1};R=R$ by vertical cancellation.

A coherent cubical algebra for unary scaffold historiesreview

theoremTheorem 6.1

A/T hexagon

The following paths have exactly the same raw physical endpoint: $A_y;A_x = T_{yz};A_x;A_y;D_z.$ No garbage quotient is used in this equation.

A coherent cubical algebra for unary scaffold historiesreview

theoremTheorem 7.1

Local cell classification

Every coinitial effective horizontal pair is aspherical, Peiffer/cubical, D/A triangular, A/T hexagonal, or D/T derived. Every horizontal/garbage pair is a strict naturality square or one of the three cancellation captures. Every garbage-trivial horizontal edge has a collapse cell from Lemma 4.2 .

A coherent cubical algebra for unary scaffold historiesreview

theoremTheorem 8.1

Uniform guarded path presentation

Fix a finite alphabet and history length in degree one and distance one. In each final behavioral fibre, all parallel paths generated by endpoint detours, literal absorptions, causal-garbage changes, and temporal slides are presented by the following length-independent guarded cell families: garbage same-coordinate…

A coherent cubical algebra for unary scaffold historiesreview