Evenly marked theta graphs #
The evenly-marked theta family: all-submodularity, exact torsion order, non-recurrence, and k-general transmission (Lemma 4.15, Theorem 4.8, Corollary 4.17).
TeX labels: thm:thetaSimple (Theorem 1.12, part 1), defn:evenlyMarked
(Definition 4.14).
The submodularity component of the evenly-marked theta theorem follows from the distinct-interior-strand classification, independently of the torsion and inversion-count arguments.
TeX label: Lemma 4.15 (unlabeled in the source), annihilation half; its
proof runs through eq:multDiffMarkedPts.
Lemma 4.15 asserts that [v_{α,i} - v_{β,j}] has order n_α/gcd(n_α,i).
Only the annihilation k·(u - v) ∼ 0 is proved here; minimality is the
separate torsion-order form below.
TeX label: Lemma 4.15 (unlabeled), exact torsion order for arbitrary distinct evenly marked theta strands. This removes the paper's harmless ``without loss of generality'' normalization by a certified slot reindexing.
TeX label: Lemma 4.15 (unlabeled), the equality
n_α/gcd(n_α,i) = n_β/gcd(n_β,j); restated in cor:evenlyMarkedKGT
(Corollary 4.17). This is pure arithmetic, separated from the
graph-theoretic firing and minimality arguments.
TeX label: thm:kgtThetas (Theorem 4.8), both directions.
A rigidly marked theta graph of exact torsion order k has k-general
transmission if and only if [u - v] is non-recurrent.
Hypothesis note: the paper's "rigidly marked" (Definition 4.4) is
all-divisor submodularity together with r(u+v) = 0, and in genus two the
latter is equivalent to u + v ≁ K_G, which is the form hRigid takes.
The paper states Theorem 4.8 for an arbitrary genus-two graph; this is the
theta case, which is the one its own application (thm:g2general, case 3)
uses.
TeX label: thm:kgtThetas (Theorem 4.8), the "if" direction.
For a rigidly marked genus-two graph with torsion order k, non-recurrence of
[u - v] gives k-general transmission. This is the direction that the
genus-two corner-sum machinery supplies, and it holds for an arbitrary
marking of a theta graph, not only an evenly marked one — the evenly marked
case (evenlyMarkedTheta_kGeneral below) is now a corollary.
Two notes on hypotheses. The paper's "rigidly marked" (Definition 4.4) is
all-divisor submodularity together with r(u+v) = 0; the form used here takes
submodularity as hSub and the consequence u + v ≁ K_G as hRigid (in
genus two r(K_G) = 1, so r(u+v) = 0 does imply it). The converse
direction is nonRecurrent_of_kGeneralTransmission; the two are packaged as
thetaRigid_kGeneral_iff_nonRecurrent_class above.
TeX label: Lemma 4.15 (unlabeled in the source), non-recurrence half.
The other two halves of Lemma 4.15 are evenlyMarkedTheta_torsion and
evenlyMarkedTheta_periods_agree above; this is the geometric statement that
the marked difference class is non-recurrent, proved in
ThetaNonrecurrence.lean via disjointness of the canonical-complement rank
supports.
TeX labels: cor:evenlyMarkedKGT (Corollary 4.17), thm:thetaSimple
(Theorem 1.12, part 2).
Every evenly marked pair on a theta graph has general transmission at its exact gcd period. The proof combines the exact-order firing calculation, the all-submodularity classification, and the finite genus-two inversion formula of Lemma 4.10.