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Theorem · guarded path presentation

FPRD-T137

Uniform guarded path coherence for unary histories

Exact statement

For every fixed history length, nonempty finite working alphabet, degree one, and distance one, all parallel paths inside one final behavioral fibre generated by endpoint detours, literal absorptions, causal-garbage changes, and temporal slides are related by a length-independent guarded family of garbage triangles and squares, collapse cells, strong and temporal Peiffer or carrier-cube cells, strict mixed naturality squares, and one bounded absorption-transport hexagon. The three support captures reduce by garbage cancellation and detour-transport is derived.

StatusProved, repaired after independent internal review, and computationally stress-tested
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

Object confluence says every history reaches one normal class but does not identify every pair of derivation paths. T137 supplies the next dimension: local cells that transform one complete D/A/G/T path into another with the same endpoints.

Definitions

  • D is physical endpoint detour, A is literal guarded absorption, G changes one causally unsupported coordinate, and T is temporal carrier slide.
  • The sole non-Peiffer horizontal interaction is the bounded equation (A_y;A_x=T_{yz};A_x;A_y;D_z).
  • A guarded schema type has a fixed local shape, while its endpoint-equality and causal-support certificates may range over the finite history.

Hypotheses and scope

  • Fixed length (n), finite nonempty alphabet, degree (d=1), and distance (k=1).
  • Paths stay inside one final behavioral fibre and use only legal guarded D/A/G/T moves.

Proof or evidence

Quotient termination and the unique right-packed normal class reduce the problem to local branchings modulo garbage. Garbage products supply vertical cells, collapse cells handle horizontal edges already trivial in a garbage fibre, and a complete local classification leaves Peiffer or cubical cells, one D/A triangle, one A/T hexagon, strict naturality, and three cancellation captures. Direct coherent-Newman induction modulo garbage then presents all parallel paths. Fresh checker runs passed the mixed-coherence atlas, A/T hexagon audit, homology mutation probes, and a seeded 6,000-history audit covering 67,848 local pairs.

Uniform guarded path presentation →

Verification notes

An independent internal review found that the first statement omitted garbage-trivial horizontal edges. The paper added bounded collapse cells, reran the complete local audit, and closed that gap. This repair is documented, but it is not external specialist review.

Limitations

  • The result gives finitely many guarded schema types, not a finite context-closed polygraph.
  • Because guards carry history-dependent certificates, no finite-derivation-type conclusion is made.
  • Nothing here extends automatically to higher degree, higher distance, or unrestricted scaffold histories.

Open work

Test whether a comparable guarded coherence theorem exists at higher degree without claiming a finite context-closed polygraph or finite derivation type.

Notes

The theorem is a finite list of guarded schema types. Physical endpoint-equality and causal-support certificates range over finite histories, so this is not a finite context-closed polygraph, finite derivation type, or an unrestricted higher-degree theorem. The written causal-substitution, support, collapse, critical-classification, and coherent-Newman-modulo arguments prove the all-length claim. Complete and seeded executable domains are corroboration and mutation falsifiers, not the proof. Independent review found the initially omitted garbage-trivial horizontal edges; bounded collapse cells repaired the gap and the re-audit returned gap closed. Specialist and novelty review remain open.