Two Invariant Pair Balances in Fibonacci Carry Lifts

This paper investigates the arithmetic structure of carry sequences arising from Fibonacci additions in base 32, focusing on period-spaced observations and their organization into nonoverlapping binary pairs. Two signed pair imbalances are studied: Delta (δ), measuring the difference between the frequencies of 11 and 00, and Eta (η), measuring the difference between 01 and 10. Both are shown to remain invariant under increasing lift depth, with uniform bounds |δ| ≤ 2 and |η| ≤ 3. An explicit formula for Eta is derived using local 2-adic step sizes and orientation signs. The theoretical analysis is complemented by an exact finite classification of 32-bit carry words. A universal lift representation restricts the possibilities to 48,336 distinct binary words, whose joint Delta–Eta distribution is determined completely. Under cyclic rotation, these words form 1,511 equivalence classes. A canonical normal form is introduced, and a reduced classification by Delta and the absolute value of Eta reveals additional structural relationships. All finite enumerations are supported by reproducible Python code. Together, the results establish exact invariance laws, strong arithmetic restrictions, and a systematic classification of Fibonacci carry patterns across arbitrary digit depths.