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3 Resolution of spacetime

Because only n bits of information exist, any emergent geometry can possess only finitely many distinguishable spatial cells and only finitely many distinguishable temporal ticks. This is the principle of resolution of spacetime:

The spatial resolution of any region and the temporal resolution of any clock are bounded by the finite informational budget of the underlying bit-string.

No external rule dictates how that budget is to be partitioned into space and time. The only admissible answer is typicality: the partition that dominates the measure under the Ehrenfest dynamics.

3.1 Maximal theoretical resolution

In the absence of any emergent structure the entire entropy budget is available for geometry. The maximal theoretical resolution is therefore the pure entropy curve itself:

S(τ)- ( ) Rmax (τ) = n = H2 p(τ ).
(8)

This is the resolution an internal observer would measure if the bit-string remained completely unstructured.

3.2 Spatial resolution – the chain of knots

Bits may be pictured as structureless spacetime fabric—the links of a chain. Microstructures form when local windows of the chain adopt particular compositions. Each microstructure is a knot: it is made from the same links that previously contributed to the length of the chain, and once the knot is tied those links are no longer available to resolve distance. If a knot consumes w bits, the observable spatial resolution drops by w for every such knot. The residual budget

∑ RQ (τ) = Rmax (τ)− 1- (bits locked in knots) n
(9)

is the only quantity still available to an internal observer as spatial resolution.

3.3 Temporal resolution – differential aging

A structure of width w can advance its own proper time only after a full set of w elementary flips has occurred. Its retarded clock is therefore the sample-and-hold quantisation

⌊ ⌋ τlocal(w) = w ⋅ τ∕w ,
(10)

with associated lapse 1∕w . Heavier knots age more slowly with respect to the universal raw coordinate τ . The informational backlog that the raw clock has already registered but the retarded clock has not yet seen is the pending entropy of that scale. Pending bits remain part of the finite budget; they simply have not yet been experienced by the local observer.

The same chain-of-knots analogy applies temporally: each knot introduces a coarser ticking rate, stretching the perceived duration of events that occur at that scale.

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