Does one source ensure cosmic homogeneity?

A common origin does not prescribe a spatial pattern

The Single Monad Model motivates the possibility that many observable places arise from one generative source. That can remove the need for many independent fundamental origins. It does not by itself determine how the source distributes records among emergent places.

A single controller can emit the repeating sequence A,A,A,A,B,C,D,D. Over each eight-step cycle, its four spatial labels have frequencies 1/2,1/8,1/8 and 1/4. The process has one source and a perfectly repeatable completion rule, but its completed distribution is not uniform. A different source rule can produce equal frequencies.

The theoretical task is therefore to identify which symmetry, mixing mechanism or selection principle fixes the distribution. The homogeneity and isotropy guide makes this a question about the complete source law and its preparation, rather than the number of fundamental generators.

Uniform source addresses need not give uniform places

In the site’s paired binary-readout example, a signed coordinate is the difference of two bits. Uniform choices of the pair give −1,0,+1 with probabilities 1/4,1/2,1/4. The middle value has two source addresses; each endpoint has one. Equal address weights become unequal spatial weights.

Taking three independent pairs in this particular finite model gives 64 addresses and 27 spatial sites. The central site has probability (1/2)³=1/8, while each corner has probability (1/4)³=1/64. Their ratio is eight. This is an exact property of the specified readout, not a universal prediction of six nested levels.

The 64-address/27-site derivation therefore supplies a useful audit: determine the multiplicity of each completed place and the source measure on its preimage. Uniformity must be proved for the actual physical distribution, not inferred from equal counting at a different level of description.

An explicit rule that does approach homogeneity

Consider four sites on a ring. At each step a record stays at its current site with probability 1/2 and moves to either neighbor with probability 1/4. The transition matrix is invariant under ring translations and preserves the uniform distribution. Starting from p₀=(1,0,0,0), the next two distributions are p₁=(1/2,1/4,0,1/4) and p₂=(3/8,1/4,1/8,1/4).

The four matrix eigenvalues are 1,1/2,1/2 and 0. For n≥1, the distribution is

Its total-variation distance from the uniform state is 2⁻ⁿ⁻¹. Thus the source rule gives a quantitative homogenization rate. Translation symmetry selects a candidate invariant distribution, and decay of the other eigenmodes explains approach to it. Both pieces matter.

This is a finite stochastic illustration. A DTT realization would need to derive the transition probabilities or their effective approximation from its generative state, retained memory and completion instrument.

Symmetry of the law and symmetry of the state differ

The identity transition matrix is also translation invariant, but it never changes the initial distribution. A state concentrated at one site remains concentrated there forever. Thus a symmetric update need not homogenize an asymmetric preparation.

For the ring model, the positive probability of staying put removes a simple alternating-period obstruction and the neighboring moves connect the sites. These dynamical properties make relaxation possible. Changing the allowed moves, introducing conserved site sectors or storing a persistent label can prevent it.

A general source theorem should identify the invariant measure, the conserved information and the decay rates of inhomogeneous modes. If it assumes the uniform state initially, it has constructed a homogeneous solution. Proving why that solution is selected or stable is a further result. This distinction is especially important when a cosmological ansatz begins with a homogeneous geometry.

What must be checked beyond the mean density?

Equal mean density at every site does not determine correlations. For four sites, prepare either the pattern (1,0,1,0) or its translate (0,1,0,1), each with probability 1/2. Every site has mean occupation 1/2. Neighboring sites are always opposite, while opposite sites are identical. Independent fair occupations have the same means but different correlations.

A completed observer can distinguish these sources by joint readings. Consequently a homogeneity claim should specify whether it concerns one-point means, the full probability law or coarse-grained spatial averages in a typical realization. Isotropy additionally concerns directional dependence of the relevant correlations.

For independent samples with finite variance σ², the mean of n samples has variance σ²/n. Persistent correlations can change this reduction. A source that produces the same average density may therefore generate very different fluctuations and approach to large-scale uniformity. Those joint statistics are an important next target for the programme.

Does one source imply a privileged spatial centre?

A source defined before spatial readout is not automatically a point at the centre of completed space. Its ontological role and its assigned spatial coordinate are different concepts. Nevertheless, a particular readout can introduce a preferred site, orientation or boundary. Those choices must be identified in the actual model.

A useful condition is equivariance: transforming the source preparation by an admitted symmetry and then completing it should give the transformed spatial record. This condition can connect source symmetry to readout symmetry. It still needs an invariant preparation or an averaging mechanism to establish the symmetry of the observed state.

The six nested inner roles can be retained in such a construction as dependencies in one generation process. Treating them as independent coordinate directions would change the model and would not solve the measure-selection problem.

A mixing rate becomes physical only after clock calibration

The ring example reaches total-variation distance at most ε after n≥log₂(1/(2ε)) updates, rounded up to an integer and with n≥1. If every completed update takes a fixed measured duration τ, this gives a homogenization time nτ. If update durations depend on the retained state, the same step-count bound does not determine one physical duration.

A joint source model should derive the elapsed-clock distribution along with the spatial transition rule. It can then ask whether faster geometric mixing costs more completion time or other resources. This connects cosmological uniformity to the same source-clock problem that appears in the reconstruction of limiting speed.

A result that proves both quantities together would be more informative than assigning a convenient time unit after computing the mixing matrix. It would expose a definite relation among the programme’s independently defined outputs.

What would make this a cosmological result?

A cosmological model needs completed distances, a clock, matter content and their joint evolution. A scale factor describing distances is an effective summary; the source must determine its dynamics and the fluctuations around it. A homogeneous mean alone does not specify expansion, horizons or the spectrum of perturbations.

The next contribution can remain finite: construct a local source rule with a selected invariant distribution, prove a relaxation bound and compute two-point records with the same parameters. Show how the physical completion rate converts the number of updates into a clock duration. This would connect homogeneity to the source’s geometry and time mechanism and give a controlled starting point for the cosmological model-building programme.

Continue reading

• Homogeneity and isotropy from a common source →
• Worked derivation: 64 binary addresses and 27 spatial sites →
• Building a cosmology from one generative dynamics →
• Generated vacuum: from source histories to cosmic geometry →

Read this discussion as part of a path

• Gravity and a generated cosmos →