ORIGINAL CONSECUTIVE-RUN COUNTING NOTES
Model: N equally likely distinct identities; independent picks; each identity available again before the next pick.
Event: at least r equal consecutive identities somewhere in a fixed m-pick record, with r >= 2.
Safe-prefix states d[j]: number of ordered prefixes that have not hit the event and whose final run length is j, with 1 <= j < r.
For one pick: d[1] = N; every other state is zero.
To append a pick: new_d[1] = (N-1) * sum(d).
For 1 <= j < r-1: new_d[j+1] = d[j].
A same-identity continuation from state r-1 hits the event and is excluded from the safe states.
After m picks: event count = N**m - sum(d); divide by N**m.

COMPLETE TWO-IDENTITY / FOUR-PICK EXAMPLE
N = 2, m = 4, r = 3. Sixteen equally likely ordered records:
AAAA: hit
AAAB: hit
AABA: avoid
AABB: avoid
ABAA: avoid
ABAB: avoid
ABBA: avoid
ABBB: hit
BAAA: hit
BAAB: avoid
BABA: avoid
BABB: avoid
BBAA: avoid
BBAB: avoid
BBBA: hit
BBBB: hit
Six qualify; ten avoid. Probability = 6/16 = 3/8.
Safe states by prefix length: 1:(2,0); 2:(2,2); 3:(4,2); 4:(6,4).
Two overlapping length-three windows each have probability 1/4. Their intersection is AAAA or BBBB, with probability 2/16.
Union = 1/4 + 1/4 - 2/16 = 3/8.
For length two, there are N * (N-1)**(m-1) safe records, so the event probability is 1 - (1 - 1/N)**(m-1).
These are original mathematical model examples, not sampled tarot histories or shuffle audits.