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Normalized Sequential Orderings Have Bounded End Variation

Claim/Theorem

Let \(M\) be a sequential matroid of rank and corank at least 3, neither a wheel nor a whirl, and normalize sequential orderings using the canonical ends from [[sequential-matroid-has-canonical-left-right-ends.md]].

If \(\Sigma\) and \(\Sigma'\) are normalized sequential orderings of \(M\), then the left ends have the same refined type and satisfy:

  1. if \(L(\Sigma)\) is a triangle or triad end, then the first three elements of \(\Sigma'\) lie in \(L(\Sigma)\);
  2. if \(L(\Sigma)\) is a segment or cosegment end, then the first
\[ |L(\Sigma)|-1 \]

elements of \(\Sigma'\) lie in \(L(\Sigma)\); 3. if \(L(\Sigma)\) is a fan end, then either the first

\[ |L(\Sigma)| \]

elements of \(\Sigma'\) lie in \(L(\Sigma)\), or the first

\[ |L(\Sigma)|+1 \]

elements of \(\Sigma'\) lie in a maximal fan whose internal elements are exactly \(L(\Sigma)\).

The analogous statement holds on the right.

Equivalently, once a sequential ordering is normalized, its variability near either end is bounded to a very small local template determined by the end type. Here the triangle and triad cases are the 3-element boundary cases from the finer Theorem 1.4 language, not a separate abstract classification of fixed ends.

Dependencies

  • [[sequential-matroid-has-canonical-left-right-ends.md]]

Conflicts/Gaps

  • This theorem still does not classify the whole interior of a sequential ordering.
  • It gives endpoint rigidity, not a contradiction with expansion or with Quantum Tanner geometry.
  • For Conjecture 3, it converts the sequential cut-rank-2 loophole into a much smaller template-exclusion problem around canonical ends and local fan/segment/cosegment structure.

Sources

  • 10.1016/j.ejc.2005.10.005 (Theorem 1.4, pp. 966-967)