Same-strand interior witnesses in arbitrary genus #
This is the same-strand branch of the corrected Theorem 3.9. The paper refers back to the theta calculation, but the witness and its four rank values are in fact genus-independent once there are at least three strands.
For normalized interior coordinates i < j, put k = j - i and
D = v_i + v_k. The pair v_i + v_k is non-reflected and has rank zero;
after deleting v_j it slides to one of the two endpoint chips. The two
other deletions are respectively one chip and a nonprincipal degree-zero
divisor. Thus rankDelta D = -1.
Distinct vertices on a positive-genus banana determine a nonprincipal
degree-zero divisor, hence a divisor of rank -1.
Corrected Theorem 3.9, same-strand interior branch.
On a banana of genus at least two, any two distinct normalized interior marks
on one strand admit the explicit negative-rank-difference witness
D = v_i + v_(j-i). The theorem is stated for genus at least two because
that is the natural range of the rank calculation; Theorem 3.9 uses it with
3 ≤ g.
Equal interior marks also admit a negative rank-difference witness in positive genus. Here the witness is the marked chip plus the normalized left endpoint; deleting the (repeated) mark leaves one endpoint chip, while deleting it twice leaves a nonprincipal degree-zero divisor.
Full same-strand interior branch of corrected Theorem 3.9, including both coordinate orders and coincident marks.