Boundary submodularity on theta graphs #
The interior part of paper Theorem 3.4 was already formalized in
ThetaExceptionalArithmetic. This file treats the genuinely informative
boundary cases. We first work in the subdivision spec's stored path
coordinates; normalized-coordinate wrappers are supplied below.
Projection-clean wrapper around the genus-two one-chip reduction. Keeping
this lemma abstract prevents the elaborator from unfolding oneChip on a
concrete subdivision vertex type merely to compare mark G u v with G.
Abstract mark wrapper for the pointwise form of all-divisor
submodularity.
All-divisor submodularity is symmetric in the two marks. This local generic wrapper avoids unfolding concrete banana divisors during transport.
Distinct strictly interior theta strands give an all-submodular marking. This is the first case of Corollary 3.6, kept here independently of the statement ledger so the complete iff below has no circular import.
Paper sources: thm-NonSubmodGenus2 (Theorem 3.4) and
cor-allSubmodSameStrand (Corollary 3.6), boundary pair (0,n-1).
On a theta strand of length at least two, marking the stored-coordinate
initial endpoint and the penultimate vertex makes every divisor submodular.
The proof follows the paper's genus-two reduction. A hypothetical negative
rank difference gives a one-chip auxiliary vertex w; the same-strand
interval theorem forces w to be either the penultimate vertex or the other
endpoint, while the reduction excludes both.
Paper sources: thm-NonSubmodGenus2 (Theorem 3.4) and
cor-allSubmodSameStrand (Corollary 3.6), boundary pair (1,n).
On a theta strand of length at least two, marking the first interior vertex
and the stored-coordinate terminal endpoint makes every divisor submodular.
For an auxiliary vertex on another strand, cor:suppUV rules out the needed
rank-zero deletion at the endpoint; the on-strand case is again excluded by
the interval calculation.
Paper sources: thm-NonSubmodGenus2 (Theorem 3.4) and
cor-allSubmodSameStrand (Corollary 3.6), endpoint pair (0,n).
The two endpoints of a theta graph make every divisor submodular. When the
auxiliary vertex lies on a strand stored in the same direction as the marked
strand, the exceptional-position interval is empty. In the opposite stored
direction, the elementary endpoint-path rank calculation gives rank -1
directly.
Normalized-coordinate boundary theorems #
Corollary 3.6 boundary (0,n-1), in the paper's normalized strand
coordinates, independent of the subdivision slot's stored orientation.
Corollary 3.6 boundary (1,n), in normalized strand coordinates.
The endpoint pair, in normalized coordinates.
Corollary 3.6, endpoint/penultimate boundary in intrinsic endpoint notation.
Corollary 3.6, first-interior/endpoint boundary in intrinsic endpoint notation.
Corollary 3.6, the two multivalent endpoints.
The complete same-strand classification #
Corollary 3.6, same-strand case, with the paper's i < j
normalization. These are exactly the three all-submodular boundary pairs.
Theorem 3.4, existence equivalence for arbitrary ordered positions on one theta strand, including all three boundary cases. This removes the interiority restriction of the earlier ledger theorem.
Full endpoint-safe form of Corollary 3.6 #
The coordinate alternatives in Corollary 3.6, stated so that endpoint vertices may be presented on any strand. The first clause is the genuinely different-strand case (both coordinates must be interior). The second says that the two physical marked vertices are, in either order, one of the three normalized same-strand boundary pairs.
Equations
- One or more equations did not get rendered due to their size.
Instances For
An all-submodular marking on a positive-genus banana has distinct marks.
Corollary 3.6, complete and endpoint-safe.
For arbitrary normalized coordinate presentations of the two marks, every divisor is submodular exactly when the marks are distinct interior points on different strands, or their physical vertices form one of the three same-strand boundary pairs.