Topology, sewing, and coherence

Restricted sewing and unary carrier coherence

Final behavior forgets both how a scaffold was factored across rows and when each surviving residual was carried. The results here recover that hidden presentation dynamics in two calibrated settings: a relative two-row theorem at arbitrary fixed degree and distance, and a complete higher-dimensional calculus only in the unary distance-one model.

Two sewing scales must remain separate

Relative local theorem. Over one fixed reduced old prefix, certain unexposed acyclic router/consumer blocks can be sewn while preserving every common suffix.

Whole-fibre unary theorems. At degree one and distance one, all histories with one final figure share a normal form, a carrier cube, and a guarded presentation of parallel transformation paths.

The first theorem permits arbitrary fixed degree and distance but cites relations in the old prefix. The latter three remove that fixed-prefix dependence only by using the rigid unary relay structure. Neither statement implies a general final-equivalence calculus for unrestricted scaffolds.

FPRD-T134 · relative two-row sewing

Private slots separate simultaneous demands

Fix a behaviorally reduced old prefix with top $q$ . Append a router $r$ and then a consumer $z$ . If router slot $i$ stores the old-prefix word $\alpha_i$ , a consumer demand $i\beta$ reaches

$q_{\alpha_i\beta}.$

Assume the consumer never points directly to the router, and every used router slot points into the old prefix rather than to SELF or MISSING. The router is then unreachable from the consumer and all later tops. Its unused slots are causal garbage, but a slot queried by a consumer remains supported.

If two such blocks agree on the consumer label, on which coordinates are path demands, and literally on every remaining MISSING or SELF coordinate, then matching demands with equal old residual behavior can be sewn by endpoint detours, garbage changes, routing crossings, and dedicated relation exchanges. Every common suffix is preserved.

The proof gives each demand a private scratch slot, aligns the two injective selector placements, changes one dedicated factorization at a time, then removes the scratch separation. This exposes the FPRD role of a seam: the router carries only enough temporary presentation structure to separate demands before they are reassembled.

Dedicated relation exchange is guarded by equality of two composite paths in the pointed endpoint congruence of the old prefix. It is not known to follow from rowwise refactoring of the older history. Exposed routers, recursive used slots, and general final-equivalent two-row blocks are outside the theorem.

FPRD-T135 · unary final normal form

Final residuals can be carried later without changing the result

At degree one and distance one, every row descriptor is

$\mathsf{MISSING},\qquad \mathsf{SELF},\qquad \varepsilon,\qquad 0.$

Consecutive essential residuals in the final chain are connected by a forced relay pattern $\varepsilon 0\cdots0$ . On two adjacent equal-labelled candidate carriers, the local exchange

$(\varepsilon,0)\longleftrightarrow(0,\varepsilon)$

moves one residual carrier through one vacant time. It preserves the final figure even though the intermediate figures and the replay-closed support may change.

For fixed length and finite nonempty alphabet, two unary distance-one histories have the same final figure exactly when endpoint detours, literal guarded absorptions, causal-garbage changes, and temporal carrier slides connect them. Every final fibre has one literal right-packed normal form $\mathsf{RP}_n(F)$ .

Sliding from the final root downward strictly decreases total carrier gap until the essential chain occupies the latest possible consecutive rows. The remaining bottom is then forced into one of three canonical cases: missing, recursive, or the distinguished base.

Absorption requires a literal old physical self-loop. A row may present recursive behavior while pointing to a different carrier; that is not enough. The theorem is not asserted at degree two or distance two.

FPRD-T136 · carrier-cube geometry

Carrier delays form a corrected rectangle

Collapse strong and garbage motion inside each chamber and retain only the creation times $t_1<\cdots<t_m=n$ of the essential carriers. The final top is fixed at time $n$ ; it is not a free coordinate. The placement poset is

$P=\begin{cases} [m-1]\times[n-m], & \text{ordinary terminal},\\ [m-1]\times[n-m-1], & \text{recursive terminal, }m>0, \end{cases}$

with the pure recursive case a singleton. A temporal slide adds or removes one maximal box of the corresponding order ideal. Filling every family of independent slides therefore gives the rooted CAT(0) order-ideal cube complex. For a rectangle with $r$ rows and $c$ columns,

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

Medians are coordinatewise. Backtracks and commuting diamonds contract every loop in the slide quotient.

Draft 2 used a larger rectangle. The corrected formula fixes the final top and, for recursive terminals, omits the arbitrary physical loop carrier and reserves the first time. CAT(0) path coherence holds after collapsing the internal chambers; it does not yet present every physical detour, absorption, garbage change, and slide.

FPRD-T137 · guarded path coherence

The normal form extends from objects to paths

Write $D,A,G,T$ for endpoint detours, literal absorptions, causal-garbage changes, and temporal slides. Object confluence says every history reaches the same right-packed class. Path coherence asks more: can any two complete sequences of those moves with the same endpoints be transformed into one another?

At fixed length, degree one, and distance one, a length-independent guarded family suffices: garbage triangles and squares; collapse cells for horizontal moves already trivial in a garbage fibre; strong and temporal Peiffer or carrier-cube cells; strict mixed naturality squares; and one bounded absorption-transport hexagon.

The only non-Peiffer horizontal interaction is the raw physical endpoint equation

$A_y;A_x=T_{yz};A_x;A_y;D_z.$

Three cases in which a horizontal move captures a formerly irrelevant label need no new cell: a garbage change followed by its inverse cancels vertically. Direct coherent-Newman induction modulo garbage then reduces every parallel pair to the classified local branchings.

The first audited statement omitted legal horizontal edges that are already identities modulo garbage. Bounded collapse cells repair that gap. The result is a finite list of guarded schema types, not a finite context-closed polygraph or a finite derivation type theorem. The documented review was internal and is not external specialist review.

Evidence and provenance

The written arguments carry the uniform claims. On 27 August 2026, the preserved checkers were rerun independently against the repository’s current source revision. In the two-row mutation audit, the saturated example split into two endpoint fibres when relation exchange was removed and none after it was restored; all 2,228 complete height-three actions then remained unsplit. The unary normal-form audit enumerated 299,592 histories in 606 final fibres and found no collision. The carrier audit checked 156 rectangle cases, 6,154 vertices, 13,314 edges, and 1,210 hyperplane coordinates; distance and median tests were exhaustive through length seven and mutation guards ran through length twelve. The mixed-coherence atlas, absorption-transport hexagon audit, and homology mutation probes passed, as did a seeded 6,000-history audit covering 67,848 local pairs.

These computations are regression evidence and falsification tools, not proofs of the all-length theorems. No novelty or external specialist review is claimed.

Read the two-row, unary normal-form, and carrier proofs →

Read the guarded coherence proof and repair record →