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Theorem · quotient geometry and path coherence

FPRD-T136

Corrected carrier rectangles and CAT(0) slide geometry

Exact statement

After quotienting a fixed-length degree-one distance-one final fibre by its internal strong and causal-garbage chambers, temporal carrier slides form the undirected Hasse graph of an order-ideal rectangle. The rectangle is [m-1] x [n-m] for an ordinary terminal and [m-1] x [n-m-1] for a recursive terminal with m positive; the pure recursive case is a singleton. Filling independent slides gives a rooted CAT(0) cube complex whose hyperplanes are delay boxes, dimension is the rectangle width, edge distance is total carrier displacement, and medians are coordinatewise.

StatusProved, corrected, independently audited, and computationally stress-tested
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Once strong and garbage motion inside each chamber is collapsed, a unary final fibre remembers only the construction times of its essential residual carriers. A slide moves one carrier into an adjacent vacant time.

Definitions

  • For ordinary terminals, (t_m=n) and (lambda_i=t_i-i) identify placements with ideals of [m−1]×[n−m][m-1]\times[n-m].
  • For recursive terminals, omitting the arbitrary physical loop carrier reserves time one and (lambda_i=t_i-i-1) gives [m−1]×[n−m−1][m-1]\times[n-m-1].
  • For a rectangle with (r) rows and (c) columns, d(t,s)=∑i=1r∣ti−si∣d(t,s)=\sum_{i=1}^{r}|t_i-s_i|.

Hypotheses and scope

  • Fixed history length, nonempty finite working alphabet, degree one, and distance one.
  • The temporal graph is taken only after quotienting by strong and causal-garbage chambers.

Proof or evidence

Carrier tuples are exactly rectangle ideals and one slide adds or removes one maximal box. The standard order-ideal construction supplies the rooted CAT(0) cube complex; the scaffold-specific proof identifies its vertices and covers. The geometry checker passed 156 rectangle cases, 6,154 vertices, 13,314 edges, 1,210 hyperplane coordinates, exhaustive distance and median tests through length seven, and formula and mutation guards through length twelve.

Carrier-surgery CAT(0) geometry →

Verification notes

The corrected rectangle bounds were checked against both scaffold papers and their audits. The finite carrier and geometry checkers were rerun successfully on 2026-08-27.

Limitations

  • Draft 2 overcounted the rectangle by treating the final top as free and mishandling the recursive terminal; only the corrected formulas are retained.
  • The CAT(0) theorem uses the standard order-ideal cube construction; no novelty claim is made for that background result.
  • Backtracks and diamonds present paths only in the chamber quotient, not all physical D/A/G/T derivations.

Open work

Extend the cubical base through, rather than only below, the strong and garbage presentation fibres.

Notes

The physical carrier of a terminal SELF residual is omitted because strong moves can shift it without changing the chamber. The final top is fixed at time n; Draft 3 corrects the larger unrestricted rectangle printed in Draft 2. CAT(0) uses the standard Ardila--Owen--Sullivant order-ideal construction; the scaffold-specific content is the exact chamber/slide identification. The theorem is path-level coherence only after collapsing internal chambers and does not present all elementary D/A/G/T derivations. Finite checks corroborate the proof and do not prove its all-parameter or CAT(0) statements. No novelty claim is made.