ElVec1o/nested-leap-complexity: The GF(p)-Linear Complexity of Nested Leap-Year Sequences

Elementary determination of the linear complexity of staircase sequences and their linear combinations over finite fields, with application to the Gregorian leap-year count.

Paper Files:

• nested-leap-complexity.pdf — Full 10-page paper with all proofs
• nested-leap-complexity.md — Markdown version
• verify.py — Regenerates all computational claims (requires numpy, sympy)

Abstract: For a positive integer n and prime p, the staircase sequence g_n(Y) = floor(Y/n) mod p has linear complexity n+1. We determine the linear complexity of every F_p-linear combination of such sequences: it is M+1 minus a sum of Euler-φ terms.

For the nested leap-correction sequence L_{a,b,c} = g_a - g_{ab} + g_{abc} (the Gregorian calendar rule), we prove: L(L_{a,b,c}) = abc+1 - (ab-a)·[p | c-1] - a·[p | bc-c+1]

For the Gregorian rule (a,b,c)=(4,25,4), the leap-year count has linear complexity exactly 401 modulo every prime p ∤ 400, except p=3 (complexity 305) and p=97 (complexity 397).