Gravitational lensing from topological void curvature: geodesic tortuosity, cluster shear proles, and the toroidal annular ring prediction

We demonstrate that topological voids in Regge calculus produce quantitative gravita-tional lensing signals with a unique, falsifiable prediction: an annular convergence ring atthe toroidal void major radius (Rmaj ≈ 100 kpc) that no spherically symmetric dark mat-ter model can produce. This completes the observational case for geometric dark matterestablished in companion papers 1, 2.The toroidal void lensing profile - computed from the physical void shape establishedin 1 (Rmaj = 100 kpc, Rmin = 20 kpc) - produces a convergence κ(R) that peaks at R =Rmaj (not at R = 0), creating a bright annular ring with a suppressed center. The tangentialshear γt(R) exhibits a sign change at R ≈ Rmaj: negative inside the ring (underdense center)and positive outside. The profile is orientation-dependent: face-on galaxies show the annularring while edge-on galaxies show a double-lobed bar at y ≈ ±Rmaj. These features arequalitatively distinct from NFW and constitute the most decisive observational test of thevoid framework. Subtracting a best-fit NFW profile from stacked galaxy-galaxy lensing data(DES, HSC, Euclid), binned by galaxy inclination, would reveal annular residuals if toroidalvoids exist.Supporting evidence establishes the lensing mechanism quantitatively. A thin-lens deec-tion measurement on a 4D-validated Regge block (pentachoral Lorentzian signature verified,N = 5,000 per slice) yields a peak deection of 0.138 arcsec at impact parameter b = 133 kpcfor a 1012M⊙ toroidal void. The peak is 10× smaller than a centrally-concentrated NFWhalo of the same mass because the toroidal geometry distributes mass in a ring; the totalintegrated lensing signal is comparable but spread over the annular area. The off-centerdeection peak (at 1.33Rmaj, not at R = 0) is a direct signature of the annular mass dis-tribution. Geodesic tortuosity through void boundaries is enhanced by Δτ = 2.389 ± 0.755(p = 5.5×10−6, 10/10 seeds positive), confirmed as physical by falsification against randomdamage (p = 0.005). At cluster scales, the void profile shape is less discrepant than NFW atfixed mass (χ2/dof: 42.9 vs. 816.3 for CLASH), but freely-fitted NFW achieves near-perfectfits (χ2/dof 2R0.Orientation averaging reduces the face-on ring contrast from 209,000 : 1 to 3.1 : 1, but abroad bump persists at R ≈ 97 kpc. Euclid can detect the orientation-averaged signal atSNR = 3 with 313,000 stacked lenses (∼2027). Current data (Mandelbaum, CLASH, BulletCluster) shows no excess at Rmaj ( 300,000 face-on galaxies, the toroidal void model is falsified.