Setup

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The universe is modelled as a closed informational system with a fixed total bit count n. From Paper III [meskanen2003], random bit-flip mutations drive the system from a zero-entropy initial state toward full equilibrium, and filters applied to the evolving bitstring extract hierarchical micro-structures whose abundances follow lognormal-like distributions.

At any moment, the n bits are partitioned between two roles:

Spacetime fabric L₀: bits not consumed by any composite structure, constituting the free background metric.
Matter hierarchy L₁,L₂,…: bits bound into composite structures at successive levels (e.g. hadrons, atoms, compounds).

Because the bit budget is fixed, matter structures do not represent new information injected from outside. Every bit locked into a composite entity is a bit withdrawn from the free fabric. Information is strictly conserved:

ρfabric(t) = n − m (t),

(1)

where m(t) = ∑ _kw_k ⋅k(t) is the total number of bits consumed by matter structures, k(t) is the count of composite entities at each level, and w_k is the bit-width of a level-k structure.

2.2 The Relational Scale Factor

An internal observer can only measure distances using the structures available within the system. The resolution of observable space is therefore the total count of independently addressable entities — both free fabric tokens and composite matter structures.

Consider n bits, all initially free fabric, giving resolution n. When one composite structure of width w bits emerges, w fabric tokens are consumed and one matter entity is created. The new resolution is:

(n − w) + 1 = n− (w − 1).

Resolution decreases by w − 1 for every composite entity formed. Generalising, if matter structures collectively consume m(t) bits and produce k(t) composite entities, the total resolution at time t is:

Resolution (t) = (n− m (t))+ k(t).

The two extremes of General Relativity follow immediately from this counting argument:

De Sitter limit (m = 0, k = 0): all bits are free fabric. Resolution = n. An external observer sees pure exponential expansion — the vacuum solution of General Relativity with Λ > 0 and ρ_matter = 0.
Schwarzschild limit (m = n, k = 1): all bits are consumed into a single composite entity. Resolution = 1. The observable space has contracted to a single point — the informational counterpart of the black hole singularity established in Paper II [meskanen2002].

These are not boundary conditions imposed by hand. They are the two extreme values of the same counting equation, separated by the continuous family of states in between.

The relational scale factor, normalised to the maximum resolution n, is:

R (t) = (n-−-m-(t))+-k(t) = ρfabric(t)+-k-(t). n n

(2)

This definition requires no external metric and no postulated background geometry. It is a pure counting statement: the scale factor is the fraction of the bit budget that remains independently addressable.

The equation is intrinsically 1D and linear. To map to observable 3D sphere geometry:

aobs ∝ 4π ⋅R (t)2

This converts the linear bit-depletion curve into sperical and quadric cos²-shaped Schwarzschild interior profile.

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2 Setup

2.1 Information Conservation

2.2 The Relational Scale Factor