A Resolution of the Collatz Conjecture

This paper develops a system from first principles. The analysis is built on
four complementary components that together provide a complete description of the
odd–to–odd Collatz dynamics:
1. A finite residue–phase automaton governing all minimally admissible odd-to-
odd Inverse transitions, determined by congruence classes modulo 6 and phase
behavior modulo 3.
2. A Normal–State index in which the removed admissible dyadic factors from
odd iterative minimally admissible output yields a canonical affine arithmetic
skeleton of the families of 4-adic rails generated by the map k→k+2, or
m→m+1.
3. A dyadic slice decomposition indexed by the valuation k = ν_2(3n+1), producing
a disjoint arithmetic partition of the odd integers with exact weights 2−k.
4. A Forward–Inverse locking identity linking the reduced Forward map
T(n)=3m+1/2^ν_2(3m+1)
to the admissible inverse map
R(m;k)=2^k m−1/3,
ensuring that each odd integer has a uniquely determined admissible ancestry.
A principal result of this work is that the affine rails arising from the normal–state
construction coincide exactly with the dyadic slices determined by k = ν2(3m + 1).
Thus the odd integers admit two independent but equivalent global parametrizations:
one affine and one dyadic. This coincidence yields a global arithmetic structure in
which all odd integers arise from admissible lifts above the anchors {1,5}.
Because the residue–phase automaton is finite and the admissible inverse relation
induces a unique Forward parent at each step, the resulting ancestry relation is well
founded. No nontrivial odd cycles can exist, and no infinite runaway is possible.
Consequently, every Forward Collatz trajectory converges to the fixed point 1.