This paper develops the multiplicative and prime-power geometry of digit polygons, building on the companion exact single-polygon theory and the repaired addition theory for carry propagation. For distinct primes p and q coprime to the base b, an exact Chinese Remainder Theorem digit identity is proved on the common period equal to the lcm of the two individual periods, expressing the digits of 1/ (pq) as a specific linear combination of the two lifted digit blocks plus a CRT overflow correction. This yields an exact multiplicative area law for the signed area of the digit polygon of 1/ (pq), decomposing it into scaled individual areas, an interaction term governed by the polarization form, and an exact CRT defect expressible as a Dirichlet energy difference. Exact locality, bounds, and mean identities for the overflow vector are also established. For prime-power towers, the paper first proves an exact order-lifting theorem giving the multiplicative order of b modulo pⁿ in terms of the base period and the p-adic valuation of bᵏ - 1, with Wieferich primes emerging as the degenerate case where period growth is delayed. On the digit side, an exact orthogonal decomposition of the area is proved: the digit block of 1/pⁿ splits into a block-average component inherited from the base period and an orthogonal refinement component, and the area splits correspondingly. The inherited component is shown to become asymptotically flat, while the full normalized area satisfies a prime-power attractor theorem: the area density along the p-adic tower converges to negative one-half the digit variance, equal to - (b² - 1) /24, with an explicit error rate of order p^ (- (n-t) ). A false scaling law from an earlier draft claiming that the area of 1/pⁿ equals p^ (n-1) times the area of 1/p plus a refinement is explicitly corrected with a concrete numerical counterexample. All results are proved in full.
Kevin Fathi (Sun,) studied this question.