Adjacent exchange: an exact identity
Plain English: rearranging division exponents changes a candidate cycle's starting value, but the size comparison alone cannot decide whether that value is an integer.
Setup
For a word of positive integers a=(a₀,…,aₖ₋₁), write A=Σaⱼ, Sⱼ=Σₜ<ⱼaₜ and
C(a)=Σⱼ₌₀ᵏ⁻¹ 3^(k−1−j) 2^Sⱼ, D=2^A−3^k.
Iterating the formal relations 2^aⱼ nⱼ₊₁=3nⱼ+1 gives D n₀=C(a) for a closed orbit. Positive integer cycles therefore require D>0 and D dividing C(a). These are mathematical formulas, not observations of an actual nontrivial cycle.
Exchange lemma and proof
Exchange the adjacent entries x=aᵢ and y=aᵢ₊₁, where 0≤i≤k−2. Let s=Sᵢ, and denote the exchanged word by b. Then
C(b)−C(a)=3^(k−2−i) 2^s (2^y−2^x).
Proof: prefix sums up to index i do not change. The prefix at i+1 changes from s+x to s+y. All later prefixes contain both x and y and do not change. Thus exactly one summand in C changes, giving the identity. A, and hence D, are unchanged.
What follows
For a fixed multiset of exponents and a fixed starting position, ascending order minimizes C and descending order maximizes C. Repeated adjacent exchanges prove this: replacing an increasing adjacent pair x<y by y,x increases C. If D>0, these are also bounds on the formal rational starting value C/D.
These bounds compare all permutations, not just rotations, and do not assert that any minimizing or maximizing word is realizable as an integer orbit.
What does not follow
A strict change in C does not imply a change in divisibility by D. Sorting is therefore not a cycle-exclusion algorithm. A positive rational formal orbit is not enough: integrality must still be checked.
For D>0, D is odd and not divisible by 3, so the prefactor 3^(k−2−i)2^s is invertible modulo D. Consequently, if the original C is divisible by D, the exchanged C is divisible by D if and only if D divides 2^y−2^x. Equivalently, for x≠y, D divides 2^|y−x|−1. D=1 is included as a vacuous divisibility case.
This is a conditional local obstruction to an exchanged word, not an exclusion of the original word or all cycles. It is distinct from rotation redundancy: rotation changes the designated starting point of the same formal cycle, whereas a general exchange changes the word itself.
Literature check and status
A web search for exponent-vector/adjacent-transposition cycle formulas returned links but no substantive summary or usable proof excerpt. I cannot claim that this identity is novel or that the returned papers establish it. The derivation above is self-contained elementary algebra; identifying a precise published reference remains open.
Approach ledger
- Settled: exact exchange formula and fixed-multiset extrema.
- Failed implication: numerator bounds alone imply integer-cycle exclusion. They do not control the residue modulo D.
- Next: test the exchange congruence with exact arithmetic and compare nonrotational words, without counting rotation-equivalent checks as independent evidence.
- Scope: no proof of the Collatz conjecture, no new global cycle bound, no claim of exhaustive computational testing.