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For TMs, the underlying rulial multiway graph is translation invariant.

Quasi invariance is probably necessary for universality. I.e. there is a boundedness of the deformation of the geometry.
Probably there cannot be termination in the multiway graph.

Computation universality implies...

Rulial multiway graph is infinite
No trapped surface; no termination .. you can’t get stuck.
I.e there is always a transformation between deterministic computation paths.

Rulial motion of deterministic paths

i.e. changing the initial conditions
There is no invariance

Deterministic computation path

[Somewhat like generational multiway paths]

Encoding is the string figure / projection / p-form that leads from one deterministic path to another

QM case: what is interpretation of the “string figure”

It is rotation in multiway space

Causal invariance [or specifically global confluence] ↔ universality

[ Interpretation of translation for universality for non-halting cases ]

Is there an example of a non-universal rulial MW system?

Ordinary universality

You can translate the trajectories of one machine to trajectories of another, using a fixed “scheme” which always involves only finite paths.
Given a machine, you can start it at any position in the MW graph
A universal DTM has the property that its paths are somehow diverse enough that they can be converted to the paths of any other machine
A slight non-deterministic TM might or might not be universal ( is there a universal pair of 2,2 TMs ? )
[ The extreme NDTM is trivially universal ]

For what underlying rules do you get C.I.? I.e. get universality

E.g. neighbor-independent SSs?

Time dilation in rulial space

Slowdown in encoding

Lorentz contraction

Boost is a mixing of space and time costs
[ Rulial distance measures something about number of degrees of freedom that differ ]
Tradeoffs: space (spacelike) ; parallelism (branchial) ; nondeterminism (rulial)
[ As you increase time, you use more parallelism ]
[ In the TM case, we can compute the rulial distance associated with each quantity ]

Reference frames in rulial space

E.g. cosmological rest frame: just counting the number of operations, independent of what the operations are
Inertial frame: [[[fixed weighting of different operations?]]]

Parallelism in rulial space

In the other spaces, parallelism is taken for granted [e.g. for string substitution]
In other systems, a particular evaluation strategy is the analog of a DTM

Analog of light cone

“Simulation cone” [[ in t steps, what other computation paths can you simulate ]]
If you sample states faster than ρ t apart, you can’t reach them in real time [[ i.e. these are microequivalent ]]
[ Is there a DTM “simulation ‘cone’”? ] [ Analog of evaluation front ] [[ DTM = event selection function ]]

Curvature in rulial space

Presence of eventual consistency?

Special relativity = constancy of speed of simulation

General Relativity

A bundle of geodesics can’t diverge without bound, otherwise you wouldn’t be able to cross-simulate them

Black hole

Region in which geodesics enter but don’t leave; i.e. computationally reducible [ cf primitive recursion vs. general recursive ]

Cosmological event horizon ?

Encoding to convert from one model of computation to the other is undecidable

Rulial space r.o.f.s

A specific theory has to follow a particular (non geodesic) path in rulial space
Need to prevent rulial divergence; trying to prevent all time evolution except the one on the path of our particular DTM
Analog of QM: failure of inductive inference
Analogy to thermodynamics: given a coarse grained observation of the universe, there will many paths in rulial space that are consistent [ ordinary thermodynamics: many init conditions; rulial thermodynamics: many rules that are consistent ]

In rulial space, nondeterminism is related to energy

Units in rulial space

What is the value of ρ?
If we knew the number of parallel threads in rulial space ...
Number of possible rules: the LHS of a rule is at most the size of the universe
10^358 nodes in the hypergraph; # possible rules is O(2^(10^358) ) [up to degeneracy ]
[[ Linked to the number of elementary operations in the universe ]] [[ Conversion between program length and actualization in the universe ]] ????
In AIT, there are questions like: BusyBeaver(program length)

Chaitin’s Ω: density of black holes in rulial space

[ Generalized Ω: density of non-universal TMs ]
In spacetime, given (c, G) and Λ, it is claimed one can estimate the density of black holes...
[ What is the expansion rate of the universe in rulial space? ]
Analog of galaxy in rulial space: a (deterministic) description of the universe that remains coherent [and is not shredded by nondeterminism]
Higher ρ, the harder it is to make a rulial BH (analog of the larger c is, the harder it is to make a physical BH)
Λ is the base of the exponential for the expansion of the physical universe... Λ^t

NDTM vs DTM is like generational states vs just waiting

With an NDTM you will get fragments quickly that you can assemble into a state that a DTM will eventually reach

Quantum / Multispace picture

In multispace, the “horizontal plane” is ordinary space, bubbling out in the vertical direction as branchial space
Measurement consists in coarse graining in the branchial direction to get a “consensus” “classical” result
(If the same coarse graining is applied everywhere, you will have consistent classical behavior)
If you coarse grain in space but not branchial space .... what is this? This makes a mixed state

To make a picture of multispace...

Basically it is a branchial graph for the MWCG
To find which are the spatial edges, construct all snakes

Branchial Space

What are particles in branchial space [quantum gravity]

Is the universe expanding in branchial space?

Insofar as it is expanding, one could get exponential quantum computing speedups
Analog of a galaxy in branchial space is a definite measurement apparatus. [ Anything not quantum coherent will be subject to overall expansion flow in branchial space ]

Could there be cosmological phase transitions in branchial space?

Spacetime

The survival of our universe : maximum time before spacelike singularity

Born rule

Imagine you have some quantum path [i.e. a geodesic in the MW system] + imagine you have a “basis path”.
If you do a measurement which leads to you conclude you are in that basis state, you need to do enough completions to conflate the two paths.
Consider BA -> AB
Add a completion: