ANALYTIC CLASSIFICATION OF THE GAP8/GAP9 ENDPOINT SCREEN
Added after the complete independent numerical audit. This short argument was
cross-checked by noita_structure. It explains the35-pair shape; it is not used
as an unchecked replacement for the explicit transition enumeration.

Normalize R(j)=j+1 mod83 and write exceptional pair(a,b),1<=a<b<=82,
delta=b-a. Track one card just emitted at position0. Until its next output,
an ordinary action cannot select it. Unless a rotation has returned it to0,
ordinary actions therefore leave its position unchanged. Exceptional actions
can shift it by+delta at a or-delta at b. A top return before the intended
endpoint must be followed by an ordinary ejection if output is to be avoided.

Before the first wrap around83, there can be at most one positive jump:
after a->b, the next rotation moves to b+1, and neither endpoint can again
be reached without a wrap. Negative jumps may occur repeatedly, but delay
the first wrap. Therefore at step d before the action, absent a wrap, the
position is congruent to d modulo delta and is at most d+delta.

delta<=73, delta>1:
No wrap is possible by step9 because9+delta<=82. An endpoint ordinary
selection for gap8 must lie in residue8 modulo delta, while gap9 must lie
in residue9 modulo delta. These are distinct. No exceptional endpoint0 is
available because no wrap has occurred. Hence the endpoint sets are disjoint.

delta=74:
The first possible wrap is at step9. Thus gap9 can additionally have endpoint
exception0, while gap8 cannot. All positive endpoints at either gap still
have the distinct residues above. No common endpoint is possible.

delta=1:
If a<=7, jump a->a+1 at step a, then the tracked card is at position9 before
step8; gap8 can emit by ordinary slot9. A path with no exceptional actions
can emit by the same ordinary slot9 at step9. Thus the sets intersect.
If a>=8, no exceptional action affects the card before the gap8 endpoint,
so gap8 has only endpoint8. At gap9, the only possible earlier positive jump
is a=8 at step8, which yields endpoint10; without it, the endpoint is9.
No negative jump can affect the card before step9 when b>=9. Therefore8 is
unavailable for gap9 and the sets are disjoint.

delta>=75:
Let k=83-delta, so2<=k<=8. Because b<=82, a<=k-1. Apply the exception when
the card first reaches a, at step a. It jumps to b and returns to0 at step k.
For desired next-output gap d in{8,9}:
  - if d=k, emit the exceptional action at k;
  - if d>k, eject at k with ordinary slot p=83-(d-k), keep the card unselected
    until step d, then emit the exceptional action when it returns to0.
The ejection slot is nonzero and not1. Other intervening ordinary actions can
always avoid the tracked card with the unrestricted slot alphabet. Both gap
endpoint sets therefore contain exceptional action0.

Consequently the survivors are exactly:
  (a,a+1),1<=a<=7                         :7 pairs
  b-a>=75,1<=a<b<=82                     :7+6+5+4+3+2+1=28 pairs
Total:35.

This proves only one-card endpoint feasibility. It neither supplies a joint
deck trajectory nor guarantees a common limited alphabet for the two paths.
The actual exclusion uses the separately reviewed whole-message propagation.
