The Matchstick Game and its Information A Geometric Reading of Euler's Product Formula and Ramanujan's Highly Composite Numbers

We extend the matchstick game from Krause (2026) by a dynamic base and an information reading. The rules S (closing), E (echo), D (new dimension) are kept literally. Thebase of the game is the sequence of primes: every D-move is a prime, and the stage of the game changes exactly at a prime. Inside a stage, the echo count Ω(n) counts the matchsticks on n, and the distinct-prime count ω(n) counts its axes. Three classical objects of analytic number theory emerge as visible geometry: Euler’s product formula, Mertens’ constant e−γ , and Ramanujan’s highly composite numbers. The Boden ladder (mandatory floor) makes the structure theorem of Ramanujan 1915 visible directly. As a control, we report negative findings briefly. The paper is didactic throughout: every classical result is first shown in game language, then identified with its analytic counterpart. The B17 numerology guard is respected: only relations that survive in the limit or under Bonferroni correction are kept