The Quaternionic Telegraph: Quantum Structure, Spacetime Signature, and Induced Gravity from a Counted Substrate

We ask how much physics is forced by one primitive: lightlike walkers on a line, reversing direction at random ticks, each carrying an ordered product of quaternion units as its only record. From this substrate we derive, with graded receipts — kernel-checked where formal, lattice- or data-checked where numerical: quantum amplitudes (the Born weight as the unique stable bookkeeping within the homogeneous family); three spatial dimensions, Lorentz signature and spin-1 2 from the mark algebra, conditional on one displayed factorization principle and the lift postulate; the charge structure of the three gauge sectors in one algebraic home — an exact U(1) phase, a temporal SU(2), and su(3) as the stabilizer of the same unit, with the Casimir ratio 4/9 verified at zero axioms and confinement appearing as Casimir-scaling census rent (several supporting components synthesize known results and are graded as such); and a gravitational field equation — in every region, geometry must expect exactly the modular energy that matter carries — whose static solution is Newtonian, q2ϕ∗ = 24πM/c (up to a chirality normalization U ≈2; O11), with a coupling test-body universal across four Gaussian systems; a binding interacting fifth test is reported, including its registered failure and its zero-new-parameter reconciliation. The modular ingredients of that equation are checked against published trapped-ion data; one sharply-committed variant of the framework is excluded at 28–52σ by cosmological data; and a neutrino-sector kill condition is pre-registered. Scope is stated throughout: the gravity law is static and scalar; gauging the charge slots and the tensor-sector road to Einstein are deferred to future work.Machine-verified artifacts (Lean 4), numerical pipelines, and the registration record are archived at doi:10.5281/zenodo.21987955 (v2).