The Multiplicative and Exponential Geometry of Digit Polygons: CRT Decomposition, p-Adic Towers, and the Algebraic Closure of the Shoelace Form

This paper extends the digit polygon framework from addition to multiplication and exponentiation of denominators. For distinct primes p and q coprime to the base, the CRT digit identity decomposes the digit sequence of 1/ (pq) exactly in terms of the individual sequences of 1/p and 1/q via the Chinese Remainder Theorem: each digit of 1/ (pq) equals a weighted combination of the individual digits plus a CRT overflow correction determined by the modular inverse idempotents. This yields the multiplicative area law — a four-term decomposition of A (pq) into individual extended areas, a cross-area term controlled by the spectral orthogonality theorem of the additive theory, and a CRT overflow defect. Unlike the additive carry defect, the CRT overflow is non-propagating: each correction term depends only on the local residue pair with no positional memory. For prime powers, the paper establishes the p-adic area scaling law A (1/pⁿ) = p^n−1A (1/p) + Rₙ (p), proves that the normalized area converges to the universal attractor − (b²−1) /24 along the tower, and characterizes Wieferich primes as polygon anomalies — precisely the primes for which the period fails to lift from 1/p to 1/p². The paper establishes that all three algebraic operations (addition, multiplication, exponentiation) converge to the same universal attractor, each through a structurally distinct mechanism: carry propagation, CRT overflow contraction, and p-adic spectral refinement.