Gabriel's Horn and Other Classical Infinities on a Discrete Spacetime

Gabriel's horn, the solid of revolution of y = 1/x for x ≥ 1, has finite volume π and infinite surface area, the textbook "painter's paradox". If space carries a minimum length ℓP, the area becomes calculable: the divergent integral becomes a partial harmonic sum, A ≃ 2πL² ln(L/ℓP) ≈ 503 m² for a one-metre mouth. We state the general form as a classification: a quantity accumulating as ∫ρ(μ) dμ/μ with ρ ∼ cμ^(−s) maps, under cutoffs at ℓP and system size L, to a finite value fixed by degree s: power-law in 1/ℓP, logarithmic, or power-law in L, this last case closed by a discrete spacetime's own degree-of-freedom count. The premise is a minimum length, not any particular theory of one, and the mathematics is standard cutoff power-counting; our contribution is the closed-form computation and the observation that the exponent s, not the cutoff's existence, decides whether the result is domestic or astronomical.

The horn is not isolated: one instance of a checkable class. The coastline paradox regularises to a perimeter of a few million kilometres with a new exponent; space-filling curves degenerate to an existing s = 1 branch, a reported negative alongside the positives. Adding a Planck-density saturation hypothesis, the same counting yields a mass-independent Schwarzschild-vacuum curvature ceiling at the core boundary, checked in an ancillary file; Banach–Tarski is shown inadmissible outright, not merely bounded, on a lattice with a fixed isometry group rather than from a bare minimum length alone. A classical self-energy divergence is included for completeness, its regularisation textbook reasoning rather than a new observation.