Digit-sum and digit-product loops of the primes and the composites

Send a prime p to the prime whose position in the sequence of primes equals the digit sum of p, and send a composite number to the composite at that position in the same way. We prove that every orbit ends in one of four loops: P1 = (3, 5, 11) and P2 = (7, 17, 19, 29, 31) for the primes, and C1 = (4, 9, 16, 14, 10) and C2 = (8, 15, 12) for the composites. We give an exact census of all numbers up to 10^15 and determine which positions the maps can reach. Since a number and its digit sum are congruent modulo 3, no prime other than 3 has a digit sum divisible by 3. Hence asymptotically one third of all primes are never reached, and the prime 5 is the first loop element of no start other than itself. Each basin and first-entry percentage is a weighted average of a fixed position kernel built from the loops. The composite percentages provably do not converge. The prime percentages do not converge under an explicit hypothesis on the digits of primes, which we test on all primes below 10^10. For the digit-product analogue, almost every start ends in Dead, at a zero digit. The other destinations are the prime loops (2, 3, 5, 11), (17), (73), (2475989), with no further loop below 10^15, and six composite loops, which are complete for all sizes