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Theorem · restricted two-row sewing

FPRD-T134

Unexposed two-row sewing over a fixed prefix

Exact statement

Fix finite scaffold degree and distance and one behaviorally reduced old prefix. Two unexposed acyclic router/consumer extensions with the same consumer label, literal agreement on MISSING and SELF coordinates, and corresponding valid path demands that reach the same old residual behavior are connected by endpoint detours, causal-garbage changes, continuation-aware injective routing, and dedicated relation exchange. Every common suffix is preserved.

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

Context

A router row stores old-prefix paths in its slots, and a following consumer row selects those slots. Under the unexposed acyclic hypotheses the router itself is unreachable from the consumer and every later top, so unused router data is causal garbage while used slots remain supported.

Definitions

  • A consumer demand iβi\beta selects router slot (i), which stores an old-prefix word αi\alpha_i, and reaches the old endpoint qαiβq_{\alpha_i\beta}.
  • Unexposed means that the consumer never points directly to the router by the empty descriptor.
  • Acyclic means that every used router slot contains an old-prefix path rather than SELF or MISSING.

Hypotheses and scope

  • One fixed behaviorally reduced old prefix, fixed finite degree, and fixed finite distance.
  • Both two-row blocks have the same consumer label and the same valid nonmissing path-demanding coordinates.
  • All other consumer coordinates agree literally as MISSING or SELF, and corresponding demands reach equal old residual behaviors.

Proof or evidence

The proof first gives every demand a private router slot using unsupported scratch slots, aligns the injective selector placement, changes each dedicated factorization under an old-endpoint equality guard, and reverses the scratch splitting. Every move preserves the consumer endpoint vector, so every common suffix replays identically. In the fresh full checker, the named saturated example had two split endpoint fibres without relation exchange and none after adding it; all 2,228 height-three degree-two distance-two actions then had unsplit endpoint fibres.

Unexposed two-row sewing →

Verification notes

The proof and its exact exclusion conditions were rechecked against the governed ledger and proof audit. Both dedicated two-row checkers were rerun on 2026-08-27; their finite domains corroborate but do not prove the uniform theorem.

Limitations

  • This is not a complete final-equivalence calculus at arbitrary degree and distance.
  • Exposed routers, recursive used slots, differing old traces, and unrestricted final-equivalent two-row blocks are outside the theorem.
  • Dedicated relation exchange is relative to the pointed endpoint congruence of the old prefix; no prefix-independent finite relation list is proved.

Open work

Determine whether dedicated relation exchange can be derived by refactoring older rows instead of citing the pointed endpoint congruence of the fixed prefix.

Notes

Dedicated relation exchange cites equality of two bounded composite paths in the pointed endpoint congruence of the old prefix. This is a relative presentation over that old action, not a finite relation list independent of it. Exposed routers, used recursive router cells, differing old traces, and unrestricted final equivalence are outside the claim. No novelty claim is made.