#### Objective reality is a consequence of branchlike foliation invariance

Objective reality is a consequence of branchlike foliation invariance

I.e. just like in relativity one picks reference frames which define timelike paths, so similarly in QM one picks observers whose threads of consciousness define timelike paths through the multiway system

Thread of consciousness defines a foliation (which is effectively defined by a timelike path)

[ like an inertial frame ]

A true observer is branching. But our simple model of the observer (“inertial observer”) is on a single branch.

[ like an inertial frame ]

A true observer is branching. But our simple model of the observer (“inertial observer”) is on a single branch.

On the same “timelike branch” the order of measurements A and B (i.e. the order of updates A and B) is A then B. Valid (branch-like) foliations must preserve these timelike orderings.

An observer can be defined by their foliation.

Light cone: in the causal graph look at events reachable in space within a certain “foliation time”.

Each edge in the causal is a piece of the elementary light cone, that goes at speed c. [In one foliation click, it has traveled spatial c δt ] [ distance between events in “spatial metric” is c ]

What is an elementary branch cone?

Start from a state of the system (aka a hypergraph). Look at hypergraphs reachable in a certain foliation time.

The different hypergraphs will live on different branches.

Going on the two sides of a light cone, the final spatial distance between outcomes is 2c

Going on the two sides of a branch code, the final distance is the commutator

i ℏ

On the events branchial hypersurface, the distance between measurements/operator applications is the commutator, aka iℏ

On the states branchial hypersurface

An observer can be defined by their foliation.

Light cone: in the causal graph look at events reachable in space within a certain “foliation time”.

Each edge in the causal is a piece of the elementary light cone, that goes at speed c. [In one foliation click, it has traveled spatial c δt ] [ distance between events in “spatial metric” is c ]

What is an elementary branch cone?

Start from a state of the system (aka a hypergraph). Look at hypergraphs reachable in a certain foliation time.

The different hypergraphs will live on different branches.

Going on the two sides of a light cone, the final spatial distance between outcomes is 2c

Going on the two sides of a branch code, the final distance is the commutator

i ℏ

On the events branchial hypersurface, the distance between measurements/operator applications is the commutator, aka iℏ

On the states branchial hypersurface

An “inertial observer” in branch space is one who does not interact/does not cause branches

(branching only locally)

An “inertial observer” in branch space is one who does not interact/does not cause branches

(branching only locally)

Nodes vs. edges on spatial graph: “related by c” (edge is a thing whose “measure” is c) (node is a thing whose measure is probably a unit of mass) [energy might be a rank-2 tensor that makes rectangles ???]

< light cone: “overall measure” is the unit of mass; its “opening angle” is c >

[ the unit of mass progressively gets smaller as the universe refines ]

[ charge is probably like mass; except it’s associated with “directions of nearby edges” ] < measuring charge is looking at the local arrangement of edges >

Nodes vs edges on multiway graph: nodes has a measure of ℏ [integral over multiway graph gives action];

edges have a measure of iℏ

< each instantaneous state has a measure of ℏ; each event has a measure of i ℏ >

What if there is a higher density of events somewhere in the multiway graph?

Nodes vs. edges on spatial graph: “related by c” (edge is a thing whose “measure” is c) (node is a thing whose measure is probably a unit of mass) [energy might be a rank-2 tensor that makes rectangles ???]

< light cone: “overall measure” is the unit of mass; its “opening angle” is c >

[ the unit of mass progressively gets smaller as the universe refines ]

[ charge is probably like mass; except it’s associated with “directions of nearby edges” ] < measuring charge is looking at the local arrangement of edges >

Nodes vs edges on multiway graph: nodes has a measure of ℏ [integral over multiway graph gives action];

edges have a measure of iℏ

< each instantaneous state has a measure of ℏ; each event has a measure of i ℏ >

What if there is a higher density of events somewhere in the multiway graph?

#### Multiway graph structures

Multiway graph structures

Polynomial multiway graph:

#### Rule-like hypersurfaces [“rule relativity”]

Rule-like hypersurfaces [“rule relativity”]

Meta rules can have Null -> {{ }}, and therefore they can create initial conditions

Timelike trajectory in rule space:

Like driving the multiway system, picking different rules for different steps.

MultiwaySystem[{allpossiblerules},...]

“inertial frame” in rule space: same rule keeps getting applied; or some computable sequence of rules

[ sequence is defined by a terminating computation ] [ sequence of rules is like the compiler ]

Coordinate systems are like different languages for describing the rules

Analog of causal invariance is that critical pairs can be merged by computation

[normally, critical pairs resolve when their consequent directly match; for hypergraphs, the matching requires isomorphism; in rule space they resolve when they are computationally equivalent]

[ sequence is defined by a terminating computation ] [ sequence of rules is like the compiler ]

Coordinate systems are like different languages for describing the rules

Analog of causal invariance is that critical pairs can be merged by computation

[normally, critical pairs resolve when their consequent directly match; for hypergraphs, the matching requires isomorphism; in rule space they resolve when they are computationally equivalent]

Number of possible rules in a given system is finite, because you can canonicalize them to that. E.g. the hypergraphs for the rule LHSs can’t be bigger than the universe, or they won’t do anything.

Is the number of RHSs finite? Can the universe be created de novo: {} -> {everything}?

Number of possible rules in a given system is finite, because you can canonicalize them to that. E.g. the hypergraphs for the rule LHSs can’t be bigger than the universe, or they won’t do anything.

Is the number of RHSs finite? Can the universe be created de novo: {} -> {everything}?

## Hypergraph evolution

Hypergraph evolution

For every node trace what it produces [i.e. a spacetime trace]

This makes a graph.

Its slices are the states we’ve studied, that e.g. have dimension d.

In this graph, there are cones with dimension d+1

This makes a graph.

Its slices are the states we’ve studied, that e.g. have dimension d.

In this graph, there are cones with dimension d+1