Classical Arithmetic Is Not Primitive: A Blockwise-Hereditary Reconstruction from 2⁴-Bit Cells

We argue that the standard arithmetic operations of addition, subtraction, multiplication,and division are not the deepest primitive layer of arithmetic construction. Instead, they can bereconstructed from a lower blockwise-hereditary layer built from finite 24-bit cells, local closure,transmitted state, and finite boundary propagation. In this lower layer, the fundamental objectsare not the high-level operations themselves, but finite cells carrying local outputs together withinherited quantities such as carry, borrow, residue, masks, activation states, and closure flags.We work explicitly at the 24-bit level, not as a mere implementation convenience, but as aminimal constructive layer in which higher arithmetic operations can already be rebuilt. We showhow addition, subtraction, multiplication, and division admit native hereditary formulationsin this setting. We then exhibit a sequence of further constructions—including an inheritedreformulation of the sieve of Eratosthenes, hereditary square-root extraction, perfect-squaredetection by exact closure, central Goldbach decompositions with transmitted carry profiles, a24-cell formulation of Collatz dynamics, and modular amplification of classical π-formulas suchas BBP and Ramanujan to wider output words.The main claim is not that classical arithmetic is false, but that it may not occupy thedeepest primitive layer. Classical arithmetic may instead be viewed as a compressed high-levellanguage emerging from a more primitive blockwise-hereditary substrate. The purpose of thispaper is to formulate that substrate, exhibit multiple proof-of-concept reconstructions, and arguethat it deserves consideration not merely as implementation practice inside computers, but asa legitimate candidate for a lower constructive layer beneath the traditional presentation ofarithmetic and number theory.