BFUT Simulation Deposit Documentation (detailed code-facing methodology)

Standardized geometry used in all three simulations
- Finite spherical universe radius R = 13.8 billion light years (Gly)
- Perfect observational reach D = 13.8 Gly in all directions
- Observer positions, where randomized, are sampled uniformly in volume, not uniformly in radius

Reason for standardization
The simulations are intended to test the finite-age / finite-visible-origin interpretation, i.e. the claim that at approximately 13.8 Gly lookback one is effectively observing the beginning. For this purpose, the relevant finite sphere is taken to be the 13.8 Gly sphere itself, rather than the later 46 Gly comoving reinterpretation. The same geometry is used in all codes to avoid any ambiguity or parameter switching.

Simulation A: Off-Centre Boundary-Depth / Isotropy Constraint
Purpose
To test how restrictive a finite spherical universe becomes if one requires even loose near-isotropy in line-of-sight depth to the boundary.

Method
For an observer at offset r from the centre of a finite sphere of radius R, the nearest boundary depth is R - r and the farthest boundary depth is R + r. The anisotropy ratio is therefore:
A = (R + r) / (R - r)

For a tolerance delta, acceptable isotropy is defined as:
A <= 1 + delta

This yields the exact allowable offset:
r_max = R * delta / (2 + delta)

The allowed volume fraction is:
f_V = (r_max / R)^3

A Monte Carlo realization with 1,000,000 random observers is included to numerically confirm the analytic result.

Simulation B: Off-Centre Observation-Overlap Paradox
Purpose
To test the geometric contradiction that, in a finite spherical universe, a generic off-centre observer with perfect visibility to the full 13.8 Gly radius simultaneously sees beyond the model while failing to see most of the model.

Method
Two equal-radius spheres are considered:
1. The finite-universe sphere of radius R = 13.8 Gly
2. The observer-centred observational sphere of radius D = 13.8 Gly

For an observer at offset r from the universe centre, the exact sphere-sphere overlap volume is computed using the standard closed-form intersection formula.

The following quantities are computed:
- Fraction of observed volume lying inside the finite sphere
- Fraction of observed volume lying outside the finite sphere
- Fraction of the finite sphere that remains unobserved

Because R = D, the “outside observed” and “unseen finite sphere” fractions are equal by symmetry.

The paradox zone condition is:
R - r < D

Since R = D, this reduces to r > 0, meaning every non-central observer lies in the paradox regime.

Simulation C: Structured Matter-Map Reprojection Instability
Purpose
To test whether a fixed finite structured 3D universe that appears broadly coherent from one vantage point remains observationally stable when the observer is moved to generic other locations.

Synthetic matter-field construction
This simulation does NOT attempt to reconstruct the actual observed sky or to claim an exact galaxy catalog. Instead, it constructs a deliberately structured, topology-agnostic mock large-scale universe containing the minimum qualitative ingredients that any realistic structured finite cosmos would have.

Three components are used:
1. Cluster nodes
   - Dense concentration centres randomly seeded inside the finite sphere
   - These act as analogues of cluster / supercluster hubs

2. Filamentary bridges
   - Selected pairs of cluster nodes are linked
   - Points are populated along the bridge paths with local scatter
   - This creates cosmic-web-like anisotropic structure

3. Diffuse background population
   - A lower-density random population fills the sphere
   - This prevents the model from being only clumps and lines

This design intentionally produces clustered regions, filaments, void-like underdense zones, and directional anisotropy.

Comparison protocol
- The reference observer is placed at the centre
- Additional observer positions are randomly sampled uniformly in volume
- For each observer, the observable matter within D = 13.8 Gly is converted into:
  a) an angular sky map
  b) a radial shell distribution
  c) a dipole / directional asymmetry vector

A combined similarity score is then computed relative to the reference observer. The key question is not whether the sky is exactly identical, but whether it remains broadly stable under generic observer relocation. It does not.

Important disclosure
The synthetic map is conservative. It is not fitted to the real sky. Its role is to test the geometry of observer relocation in any finite structured universe. If anything, a more realistic and more strongly structured cosmic web would generally increase reprojection instability rather than reduce it.
