Same-strand endpoint witnesses for Theorem 3.9 #
This file treats the boundary cases omitted from the interior-coordinate
classification. On one normalized strand, the exceptional endpoint/interior
pairs in the paper are exactly (0,n-1) and (1,n) (up to swapping the
marks). Every other endpoint/interior pair has a two-chip witness with rank
table 0, 0, 0, -1.
For the left endpoint and an interior mark v_j with j < n-1, take
D = v_0 + v_(j+1). Prefix firing gives
D - v_j ~ v_1. The right-endpoint witness is the reflected construction,
D = v_(j-1) + v_n.
A non-reflected pair in normalized strand coordinates has rank zero in
every banana of genus at least two. This is the normalized-coordinate wrapper
around rank_same_strand_pair_zero_of_not_reflection_generic; the wrapper is
needed because subdivision slots may be stored in the reverse orientation.
Corrected Theorem 3.9, left-endpoint same-strand branch.
If j is strictly interior and is not the penultimate position, the marking
at the common left endpoint and v_j has a divisor of negative rank
difference. The explicit witness is leftEndpoint + v_(j+1).
Corrected Theorem 3.9, right-endpoint same-strand branch.
If j is strictly interior and is not the first interior position, the
marking at the common right endpoint and v_j has a divisor of negative rank
difference. The explicit witness is v_(j-1) + rightEndpoint.
Repeating any marked vertex on a positive-genus banana is non-submodular. In particular, this handles the two coincident-endpoint cases in the endpoint extension of Theorem 3.9.
The three same-strand exceptional coordinate pairs in Theorem 3.9, expanded to include both orders.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Complete same-strand coordinate branch of corrected Theorem 3.9, including endpoints.
For arbitrary normalized positions on one strand, either their ordered pair is one of the six orientations of the paper's three exceptional pairs, or an explicit divisor has negative marked rank difference.