The Universe Has No Boundary - Three Geometric Tests

A finite spherical universe of radius R = 13.8 Gly is assumed, with perfect observational reach D = 13.8 Gly in every direction. Each test below places observers at random positions inside that sphere (uniform in volume) and asks whether the universe would look the same - isotropic, self-consistent - from those positions as it does from the centre. It would not, except from a vanishingly small privileged region.

A - Isotropy Constraint
B - Overlap Paradox
C - Sky-Map Instability
Finite universe boundary (R = 13.8 Gly) Observer position (off-centre) Privileged near-isotropic zone

Off-Centre Boundary-Depth / Isotropy Constraint

For an observer at distance r from the centre, the ratio of the deepest to the shallowest line of sight to the boundary is A = (R+r)/(R−r). Near-isotropy (A close to 1) requires r to be extremely small relative to R.

Analytic result: the allowed radius is r_max = R·t/(2+t). The allowed volume fraction scales as (r_max/R)³ - a cube - so even a loose 10% tolerance permits only about 0.011% of the universe's volume as an acceptable observer location.
Finite universe (R = 13.8 Gly) Observable sphere (D = 13.8 Gly, centred on observer) Observer

Off-Centre Observation-Overlap Paradox

Every observer's reach D equals the universe radius R. For an off-centre observer, the observable sphere and the finite universe sphere are not the same sphere. Part of what the observer "sees" lies outside the finite universe; part of the finite universe lies beyond what the observer can see.

A generic (randomly positioned) observer is not near the centre. The mean offset is about 75% of the universe radius, and for such an observer, roughly half of everything observed lies outside the assumed finite sphere, while roughly half of the finite sphere goes completely unobserved.
Cluster nodes Filament bridges Diffuse background Test observer

Structured Matter-Map Reprojection Instability

A synthetic structured universe is built from cluster nodes, filamentary bridges between them, and a diffuse background - a topology-agnostic stand-in for any structured cosmic web, not a reconstruction of the real sky. The angular, radial, and dipole "sky signature" seen from the centre is compared against the signature seen from many relocated observers.

Disclosure: this is intentionally a generic structured mock universe, not an attempt to reproduce the actual observed sky. Its purpose is to test whether any finite structured universe that looks broadly coherent from one point remains observationally stable when the observer is moved. It does not: only a small privileged central region preserves a similar-looking sky.