Introducing the Janus Coprime Algebra: A Case Study on the LEO Kessler Critical Surface

We introduce the Janus Coprime Algebra (JCA): a structured operator class on the coprime residue lattice (Z/m) * for squarefree primorial m, defined by two independent axioms — bi-involutive equivariance and exact rank-4 invariant subspace V₄. The CREM commutator C (m) = Dₛym, Pₜau is the first known member; this paper documents three more: the sigma-tau tensor decomposition operator pair, the cyclic sigma-flux drive Hamiltonian family, and the tau-equivariant embedding family. The two axioms are independent: a kernel substitution test against five non-tent kernels shows bi-involutive equivariance does not force rank-4 collapse. V₄ is exactly invariant — relative leakage stays at the IEEE 754 floor (10^-15) at every primorial through the 8th. The 4D braid evaluates in approximately 280 microseconds at every primorial tested, including phi (m) = 1, 658, 880, with 9 percent wall-time variation across 3, 456-fold substrate growth. The Kessler critical-surface case study deploys JCA on the live ESA DISCOSweb LEO catalog (49, 998 objects). The conjunction graph is 116 times super-critical; halving the giant component requires deorbiting 2, 424 objects (2. 9x leverage over random). Fiedler eigenvalue 1. 01 times 10^-5 identifies Anderson-localization-consistent quantum-walk dynamics. Substrate-native Grover at K = ceiling (pi/4 times square-root of 92, 160) = 238 exact iterations identifies the maximum-exposure target with textbook query-count quadratic speedup. The bundle includes 24 reproducibility scripts — including substratedaily. py for daily-refresh Fiedler telemetry against the live catalog — and the full pipeline runs in 7. 3 minutes on a consumer laptop. Patent reservation: U. S. Provisional Patent Applications 64/031, 440, 64/033, 689, 64/048, 617, and 64/054, 093.