A Global Rigid Halting Clock from an Adelic Scalar

Abstract | The same single spacetime-dependent scalar π(x^μ) ∈ ℝ × ℤ₂ × ℤ₃ × ℤ₅ × ℤ₇ × ℤ₁₁ that defines the global proper-time clock in Paper I, the global proper-scale clock in Paper II, and the global proper-color clock in Paper III now defines a fourth and final rigid clock: the global rigid halting clock. The infinite arithmetic tape T formed by concatenating the digit sequences of the three preceding clocks has halting probability Ω_T = 1 as the self-verifying bootstrap outcome for a halting witness in the fractal layer. The halting (``witness'') flip 0 → 1 is therefore not an axiom but the bootstrap necessity on the unique tape T derived from the scalar and the observed number of generations. The four papers together constitute a complete, parameter-free description of spacetime, gravity, gauge forces, flavor, dark-matter abundance, and the existence of a halting clock. No new axioms or free parameters are introduced beyond the observed generations; all physics and the halting witness existence follow mathematically.