For each chord-apex+Kempe colouring (n in [12, 18]), record:
(1) (#L, #R) split of c-edge sides along K_b and K_c. #L == #R only
in 35.43% of colourings (the rest have unbalanced sides --
consistent with the empirical Heawood non-constancy).
(2) Ordered sequence of (i_b mod 2, i_c mod 2) parity pairs at
shared K_b cap K_c vertices in K_b walk order, plus a tally of
transitions in the 4-state space.
Two clean structural observations on the transition matrix:
(A) i_b parity strictly alternates between consecutive shared
K_b-vertices. Every transition goes (0, *) -> (1, *) or
(1, *) -> (0, *); transitions within (0, *) or within (1, *) are
never observed. So shared positions on K_b alternate even/odd in
walk order -- the gap on K_b between consecutive shared vertices
is always odd.
(B) From odd-i_b states, i_c parity must flip too: (1, 0) only
transitions to (0, 1) and (1, 1) only to (0, 0). From even-i_b
states, both i_c outcomes occur.
(B) is explained structurally: at an odd-i_b shared vertex K_b leaves
via the a-edge (which is also on K_c), so K_b and K_c traverse the
same edge and K_c advances exactly one step, flipping i_c. At an
even-i_b shared vertex K_b leaves via the b-edge (off K_c), so K_c
advances at its own pace and i_c can be either parity at the next
shared vertex.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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