Topological voids in Regge calculus: boundary curvature, action comparison, and the GaussBonnet cosmological constraint

We show that excising tetrahedra from a Regge triangulation concentrates all curvature
at the void boundary - none remains in the interior (f∂ = 1.000 to ∼10−12) - and
that the Gauss-Bonnet theorem uniquely selects toroidal topology (χ = 0) as the only
cosmologically viable void shape, by twelve orders of magnitude (α ∼ 109). This geometric
mechanism requires no dark matter particles and no modified gravity: the curvature budget
of a toroidal void in discrete spacetime, governed by the same GaussBonnet theorem that
constrains smooth manifolds, provides a complete gravitational potential equivalent to a
dark matter halo.
The curvature balance C+ = C− is conrmed by Richardson extrapolation on sharp
toroidal voids (N = 5,000-320,000, 10 seeds per resolution): continuum limit |O∞| = 1.026±
0.113 (95% CI [0.913, 1.139], R2 = 0.998). The Regge action difference ΔSR < 0 at all void
radii with 100% sign consistency across 80 realizations, establishing that voids are lower-
action states. SPARC calibration yields the physical void scale (ℓ = 5.0 kpc, Rmaj =
100 kpc, Rmin = 20 kpc), and a BIC comparison on 171 SPARC galaxies finds the void
model preferred over NFW (90 vs. 69 galaxies at equal parameter count k = 2). A 58-test
verification suite (50 pass, 6 caveats, 2 fail) provides independent numerical confirmation.
The Hopf link-breaking experiment confirms ΔC = 4πΔχ in the discrete setting to |ε| <
10−6.