Hyperfactorization and ABCD chart

Address geometry is made global by hyperfactorization. Every X≥ 2 receives a unique decomposition X=((A↑↑ C)^B)· D as a tower atom T(A,B,C)=(A↑↑ C)^B multiplied by a residual D. At this point divisibility and internal primality have already been earned. In the τ-statement, A is the largest internal prime atom (under idx, the largest prime divisor of X), C is the maximal tetration height of the A-tower factor, B is the residual tower exponent, and D has only internal prime factors strictly less than A. The resulting ABCD chart, written in later scalar notation as

:ℕ_≥ 2 to P×ℕ_≥ 1^3,

is injective and turns arithmetic objects into typed global addresses. This prevents number from collapsing into a flat additive/multiplicative background and preserves the tower structure needed for boundary polarity downstream.