Case 2) from inside a sphere you cannot find a curved path that starts from the interior to the exterior that also does not cross the surface of the sphere. Because No Extra Dimension of Space was available for its construction. Therefore the test for case 2 has failed to find a high dimension of space.
This proves conclusively that space itself is three dimensional as a consequence of the very limits of geometry itself. There is no further debate possible here because it has been proven to be geometrically impossible.
The universe most like went through a few big bang before settling into stable universe. The cyclic view that draw its inspirationn from Simple Harmonic Motion SHIMM) must be dampened by The Heisenberg Uncertainty Principle (HUP). Every universe must be slightly different in Mass & Energy content due to The HUP therefore eventual given enough time a Stable Universe which has just right energy & mass will not fall back into big crunch.
Though I am being to doubt The Quantum FoamIntegrationn of Quantum Theory with limit breaking The Conservation Laws. On first thought you would think this should cause SHM with a Stable Universe but if The HUP still holds then thedistributionn mix of Mass & Energy may still cause Stability leading to a graceful balanced asymptotic expansion.
In this view anacceleratedd universe would be not from some Dark Energy but due to a transition cause from the limited range of gravity: like if you swung a weight by an elastic banned and then the elastic ban broke (as it broke it would accelerate).
Possible BB Cycles of early universes would be a form evolution on the universe as whole.
The rest I don't think has direct bearing on Big Bang Theory.
Try understanding this space I call a Spherical Knot Manifold:
(x, y, z) = τ(θ, φ, ψ, t)(θ, φ, ψ) = (τ(θ, φ, ψ, t)θ, τ(θ, φ, ψ, t)φ, τ(θ, φ, ψ, t)ψ)
where τ(θ, φ, ψ, t) = ∫ v dt
τ is a temporally derived unknown function which converts time-units into space-units, found by integrating the velocity v with respect to the differential of time dt; this idea is standard in Newtonian Mechanics " x = ∫ v dt". τ essentially functions like a spatial radius in a universe which has only a radius of time; τ is also a function of time allowing it to change with time as well as along any spatial angle θ, φ, and ψ.