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3 Mapping to General Relativity

The Friedmann equation governs the expansion of a homogeneous, isotropic universe in General Relativity:

( ) a˙ 2 8πG- Λ- a = 3 ρmatter + 3 .

The present framework maps onto this equation under the identifications:

∑ ρmatter ←→ --j kj(t), Λ ←→ ρfabric(t), a ← → R (t). n n
(3)

These identifications are not free parameters. They follow from the counting argument: matter entity density maps to ρmatter because matter entities brake expansion by consuming fabric; free fabric density maps to Λ because it drives expansion by providing addressable resolution. The cosmological constant is not a parameter of the vacuum energy — it is the fraction of the bit budget that remains unbound.

The two GR boundary solutions are recovered exactly:

Limit Information picture GR solution
k = 0, m = 0 All bits free, R = 1 De Sitter (Λ > 0, ρ = 0)
k = 1, m = n One entity, R = 1∕n Schwarzschild singularity

Numerical analysis shows the spatial contraction profile matches also across all states.

PIC

Figure 1: The purple (counting equation) and dashed orange (Friedmann mapping) curves perfectly overlap, demonstrating that general relativity’s geometric contraction under the presence of matter is fundamentally equivalent to an internal observer’s loss of background resolution.

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