Classical Arithmetic Is Not Primitive: A Blockwise-Hereditary Reconstruction from 2^4-Bit Cells

This paper advances a simple but strong thesis: the standard operations of arithmetic—addition, subtraction, multiplication, and division—need not be taken as the deepest primitivelayer of arithmetic construction. They can instead be rebuilt from a lower blockwise-hereditarylayer composed of finite 2^4-bit cells, local closure, transmitted state, and finite boundarypropagation.At that lower layer, the fundamental objects are not the classical operations themselves,but finite cells carrying local outputs together with inherited quantities such as carry, borrow,residue, masks, activation states, and exact or non-exact closure. Working explicitly at the 2^4-bitlevel, we show that the four classical operations admit native hereditary formulations. We thenextend the same viewpoint to a sequence of higher constructions: an inherited reformulation ofthe sieve of Eratosthenes, hereditary square-root extraction, perfect-square detection by exactclosure, central Goldbach decompositions with transmitted carry profiles, a 24-cell formulationof Collatz dynamics, hereditary constructions of Mersenne and Fermat families, and the modularamplification of classical π-formulas such as BBP and Ramanujan to wider output words.Two concrete addenda are included to clarify the philosophical core of the paper. In symmetricGoldbach pairs around binary centers 2n, primitive decomposition reveals rigid AND/XORidentities that remain nearly invisible in the usual decimal statement of the same equality. Inthe factorial function, the dyadic content of n! can be extracted through the primitive bitwiseform n! ∧ (−n!), yielding the exact law 2^( n−popcount(n)). These examples do not replace classicalarithmetic; they show that certain arithmetic structures become unexpectedly transparent whenviewed from a lower constructive layer.The claim is not that classical arithmetic is false, but that it may not occupy the deepestconstructive level. Classical arithmetic may instead be understood as a compressed high-levellanguage emerging from a more primitive finite hereditary substrate.