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Advance Actual Native Boundaries with Retained Route Identities

Abstract

Advance Actual Native Boundaries with Retained Route Identities.

Theorem 1.1 (Earn the Updated Boundary from Actual Slicing).

Lean statement: D5/S3/Combinatorics/Graph/PrefixReversal/NativeActualBoundaryAdvance.native_actual_even_odd_boundary_advance

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Graph/PrefixReversal/NativeActualBoundaryAdvance.native_actual_even_odd_boundary_advance (✓ std3). ∎

Source. Repository-derived.

Commentary.

At order at least three, an actual old boundary path, disjoint restored pieces and a concrete updated joint partition on the same Hamilton cycle determine the next native even or odd boundary. Literal native ports transfer the endpoint template through the actual two-path restoration; the next endpoint law is earned from the supplied old data.

Theorem 1.2 (Retain the Distinguished First-Cut Factorization).

Lean statement: D5/S3/Combinatorics/Graph/PrefixReversal/NativeActualBoundaryAdvance.native_actual_first_cut_even_odd_boundary_advance

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Graph/PrefixReversal/NativeActualBoundaryAdvance.native_actual_first_cut_even_odd_boundary_advance (✓ std3). ∎

Source. Repository-derived.

Commentary.

The actual first-cut update preserves the concrete route prefix and the same old boundary inventory while advancing the native endpoint template. Every supplied cycle, path and cut identity remains explicit; no arbitrary recursive existence premise closes an all-order theorem.

References