In the limit, all possible causal structures are tangled together

cf multiway causal graph ... which has multiple causal structures

Ways to slice things:

foliations

Reduce everything to a 1D sequence of slices (e.g. time sequentialization)
Take time seriously

fibrations

reduce to base + fiber

ruliation (?)

Base is laying out rules in rulial space ... then projecting forwards by the dynamics for each rule
Goal is to find a causal structure
What law of dynamics can explain what I see? [I.e. what causal structure corresponds to the observer’s view]

Observer Theory

Coordinatize rulial space by just saying different point assume different rules for the universe

Size in rulial space is related to the uncertainty of inductive inference

Given what you observe, there are many rules consistent with it

What is a valid observer?

Something that maintains its integrity through time [through its time]
Many different states of the universe end up in the same state of the observer [i.e. the observer is doing compression]
Can the observer do the compression with bounded computation?
Only certain observations are accessible to a computationally bounded observer

TEG-level rulial space

In[]:=
ResourceFunction["TokenEventGraph"][{{0}->{1},{1}->{0}},{0,0,0},2]
Out[]=
In[]:=
ResourceFunction["TokenEventGraph"][{{0}->{1},{1}->{0}},{0,0,0},4,"TokenDeduplication"->False]
Out[]=
In[]:=
ResourceFunction["TokenEventGraph"][{{0}->{1},{1}->{0}},{0,0,0},4,"TokenDeduplication"->False,"EventDeduplication"->False]
Out[]=
In[]:=
ResourceFunction["TokenEventGraph"][{{0}->{1},{1}->{0}},{0,0,0,0},4,"TokenDeduplication"->False,"EventDeduplication"->False,"EventLabeling"->Automatic]
Out[]=
In[]:=
ResourceFunction["TokenEventGraph"][{{0}->{1},{1}->{0},{0,0}->{1,1}},{0,0,0,0},4,"TokenDeduplication"->False]
Out[]=

The limit of the states graph is the complete graph

I.e. there is a rule from every state to every state
I.e. it’s obvious there’s a groupoid structure

In the limit, there are events which ingest arbitrary subsets of the set of tokens

Events are mappings from subsets of the set of tokens to other subsets

How are states constructed?

Any subset of tokens is a possible state [?]

State : subset of tokens
State  state : mappings between subsets of tokens
Events : mapping between subsets of tokens

State transformation:
Event:
This is also a map of possible events.... but the events embed in here slightly nontrivially
A version of this picture in which each statestate edge is decomposed into possible events....

The states graph is already nontrivial if we limit time

[cf computational complexity theory]

Analogy: computable numbers

With sufficiently complicated machines you fill out the continuum
All possible computations eventually fill out the continuum ....
[ Analogous to creation of initial conditions ... ]

Analogy: not just individual numbers, but functions/mappings

What is the topology of the space of computable functions?

Algorithmic information theory vs. NKS-like computation

AIT : consider sizes of programs, running for all time
Snake between more complicated programs, and running for longer

In constructing all possible functions ... some are “easier” than others

I.e. given a particular basis, some functions cannot be computed in finite time

Why can’t you construct arbitrarily complex machines?

Because the observer can’t prepare / sense / ... such things

NKS setup: the “scientist” can only get from induction to fairly simple rules

Ruliad contains the continuum wrt an outside observer; within the ruliad any observer must be bounded, so can’t see the whole continuum

Analogy with computable numbers: the observer has to have a way to get into their mind the whole sequence of digits for a number .... and that can only happen if the number is computable with a finite-size machine

The observer’s model is the machine

Analogy: location in rulial space: a particular kind of machine

Certain numbers become easier to compute ; others can’t be reached at all
Distance metric in rulial space ; investigate e.g. with CA simulation graphs

Rulial graph: what is nearby in rulial space

From a common ancestor graph: Effectively: incremental change of rules
From a complete rule to another complete rule: like proof-to-proof transformation

Feature of consciousness: describing the world in a consistent way

We describe the world so that science is possible
Role causal invariance in ensuring that things stay consistent?
[Inevitable causal invariance in ruliad]

We are tracing those aspects of the universe that can be described by a particular rule or bundle of rules

Quantum mechanics / “fixed-rule branchial” case

We conflate different threads of history to get a definite history ...
“Conflate” = averaging / ....
Our finite size in branchial space causes us to aggregate multiple branches of history
Simplest model: cf causal case : spacelike hypersurface across the whole universe [simultaneity throughout all space] <i.e. special relativity>
Analog in QM : “special QM” : i.e. pure mechanics, not QFT [in ordinary/”special” QM, one is saying there is a single quantum state for the whole universe ]
There is a sequential evolution in time ... of superposition states
Choice of foliation is ?? like choice of basis for quantum states
There is a limit to the change of basis determined by the maximum entanglement speed...
What is the analog of Lorentz transformation in branchial space
Consider a quantum circuit with lots of wires
Then consider the “timing” for the circuit
Assume the wires interact only with their neighbors
How are these laid out in branchial space? Answer: the interactions knit together branchial space

Alternative case: a quantum particle

Consider it distributed across points in physical space

Entanglement cones

Interactions / entanglement define the geometry of branchial space ... effects can only propagate through interactions .... hence it is self fulfilling that in branchial space there is a finite speed of propagation
There is a maximum tipping of the timing front on the quantum wires ... you can’t tip further than is allowed by the rate at which effects can propagate across branchial space ;; if you tipped further, you’d end up with timelike separated events in the same time slice
Imagine you have a “homogeneous” quantum circuit ... multiple identical wires ... homogeneous in branchial space [cf tensor networks]
How do you do a boost in branchial space?
Over time, you are going to measure some set of wires. At what time do you measure each wire? [Or more accurately, measuring different eigenstates]
[ Freezes time for a particular set of states .... then one freezes time for a different set of states, etc. ]
[Also: rotations between branchial space and physical space]

Origin of quantum effects

Core of quantumness: There are multiple paths of history
We don’t multiple paths when we as observers are so coarse that we average over them
We see the distinct paths when they are far separated
What experiment immediately shows you the validity of the path integral as opposed to the extremal path? Interference between two slits ...
Things separated in branchial space are close in physical space
The observer aggregates all path weights within a certain region of branchial space

Notion of a “Coherent Observer”

Compressed and consistent evolution history (in space, time, branchial space, rulial space)
In ordinary space: the observer sees only what is inside the light cone

Rulial Analog of Quantum Effects?

Maybe there “is” no “fundamental theory of physics” because we are spread across a bundle of histories in rulial space

We don’t nail down a single one because we are limited by our finite experiments in doing inductive inference

The reason we believe the universe has definite laws is because we’re ignoring everything that doesn’t relate to the laws we care about...

[Possibly: there are discrete possible descriptions, not a continuous variation of possible descriptions ?? ]
Are there rulial quantum effects in metamathematics?

Rulial QM

In statistical mechanics, mechanical work is ground up into “incomprehensible heat” ... but there might be another “kind” of mechanical work that it turns into [as if: one is escaping from a circumscribed area of rulial space to one that our consciousness is not in (?)]

There is no fundamental theory of physics in the same sense that there is no fundamental theory of mathematics

Rulial relativity: the theory can have the same structure even though the “basis” is different

I.e. the aliens could still have GR, but with completely different degrees of freedom
Just as: application areas for multicomputation still have GR, e.g. interpreting space as monetary value etc.

Big Picture...

From outside, the continuum is boring
Viewed “on the inside” by a TM, it’s elaborate

How best to add rules in the development of the ruliad?

What is the appropriate “evaluation order” in branchial space
The choice of how to add in rules is just like the choice of when to do which events....
In the branchial case, we keep on doing non-overlapping events until there aren’t more to do...
We could have the same evaluation front for rules

As we do more experiments, we potentially shrink our “size” in rulial space....

Cf uncertainty principle
Analog in math : large cardinal axioms etc. [As we study more theorems, we decide if we need large cardinal axioms or not]

Abstraction is more about having a bigger region in rulial space that the observer can conflate

Translation between different people’s thoughts

Thinking about things from someone else’s point of view is continual translation (? AKA motion) in rulial space
“Point of view” is now rulial, not e.g. physical coordinate location

Extent in rulial space

Upper bound: our ability to “keep in mind” a certain level of abstraction
Lower bound: our ability to home in on a particular rule by making experimental measurements to find which the rule is

Interference etc. in rulial space

Metamathematical case: two proof paths that are incompatible at an intermediate stage [no short translation exists between the paths]
[maybe you have to go back to the beginning to sow them together]
In the physics case, there are two different models of the universe that give the same result, but with different mechanisms and where the mechanisms are not inter-translatable [cf quantum effects]

Position in rulial space ~ description language being used (?)

Our universe depends on us.. we make the present and future of our universe

What do we notice now, that we didn’t notice earlier in history?
Micro things, and macro things ; things requiring amplification ; + things that hadn’t been constructed yet ; + abstraction
E.g. the orbit of Uranus is perturbed ; perihelion of Mercury advances
We have a notion that there is an objective reality to what happens in “the” universe.... [as opposed to just “our universe”]
Trivial example: what if we lived near a black hole
Claim is: all physical reality is ultimately subjective ... i.e. depends on what the observer measures/aggregates/samples
Need an example of when/where the laws of physics change as a result of the “preferences” of the observer...

Light cone, entanglement cone, emulation cone

[[Descriptional cone ; Rulial cone]]
As coherent observers, we don’t reach the edge of the rulial cone
A single event produces states that are on “both sides of the light cone” [cf branch pair]

Rulial time dilation

This is basically interpreter slowdown... If you have to go through an interpretation, you can’t use all your computation to just evolve in time

Rulial distance units

In a single time slice...... one lays out all the results for different rules
There is a geodesic distance (in rulial space) between different rules [ whose length is the time/operation complexity of the transformation from one rule result to the other ]
Geometry of rulial space: how to lay out possible programs in a geometrical way

“No fundamental theory of physics”

But General Relativity is generic to all theories .. but its interpretation in terms of measurable dof is what differs
Rulial relativity says that the “laws of physics are the same for everyone” ... which means everyone has GR ... but in a different rulial frame

From the outside, the ruliad (universe of all possible universes) is boring

But we are forced to explore in a computationally bounded way ... so we are not bored, and our efforts are “meaningful” because of computational irreducibility ... that is why it is diffuclt to explore the ruliad
[Without computational irreducibility, the inside and outside of the ruliad would be similar]
[[ Forms of explanation: do you start from what you can construct, or do you start from a non-constructive object .... and then see how to look inside ]]

Category Theory

Ruliad : functions + functions between functions + ... (?)

One description: mappings between states [cf morphisms between objects]
Ruliad limit, as constructed by completions, involves adding mappings between mappings [which are 2-morphisms]
Grothendieck hypothesis: in the limit of ∞-morphisms ... there is inevitable geometry
HoTT : assumes all limits are takeable with no (computational) effort ; analogous to taking the continuum limit and effortlessly ending up with the continuum
[HoTT is as unimplementable as the continuum]
Validity of CH : is that sampling of the ruliad consistent with a coherent observer?

Computational analog of CH

Computational case: enumerate programs for generating sequences of 0s and 1s ; for bounded algorithmic information content, you get the computable reals, which have the cardinality of the natural numbers
These two cases are equivalent .... but not for “embedded observers” (or constructivist mathematicians)

How the Rulial Limit Is Taken

Essentially, things are being laid out here in “factorization space”....
The possible “how one got there” rules are the multiplicative partitions of 48 [all rulial histories leading to 48]
What products of integers lead to 48?
This corresponds to all ways to get to 48
Reducing by allowing only rules with multiplier <=4:
In the standard rulial graph, two numbers are connected if their “rule sequences” agree up to the last step
To be rulially connected, two numbers can differ in their “last factor”
Each rulial edge is labeled by the differing last multiplicands....
Causal invariance is a consequence of the commutativity of multiplication
Compare: https://www.wolframscience.com/nks/notes-4-2--multiplicative-digit-sequences/

Different limits:

number of rules
time steps
initial conditions

If we have more initial conditions than can be readily reached by the rules, we have “initial dust”.... which is knitted together by the rules

[ also limit here by “state value” ... ] (cf number of atoms of space, etc.)

The Pure Digits Example

Real thing to look at is the branchial hypergraph .....

Addition case

There is no relevant causal graph here ... because numbers are completely ingested at every step

Strings case

[[[ This is not working correctly ]]]
[[ The coloring is wrong here ... ]]

Incompatibility of Different Paths

Can you get from ABB to BAB?

Observer Theory

Imagine one is traversing a tree ; how complicated is the program that determines the elements in a given foliation etc.?