# Simulation argument (Planck scale)

Programming a deep universe simulation at the Planck scale
The simulation hypothesis or simulation argument is an argument that proposes all current existence, including the Earth and the rest of the universe, could be an artificial simulation, such as a computer simulation. The ancestor simulation approach, which Nick Bostrom called "the simulation argument", argues for "high-fidelity" simulations of ancestral life that would be indistinguishable from reality to the simulated ancestor. However this simulation variant can be traced back to an 'organic base reality' (the original programmer ancestors and their physical planet). The Programmer God approach conversely states that the universe simulation began with the big bang (the deep universe simulation) and was programmed by an external intelligence (external to the universe), the Programmer by definition a God in the creator of the universe context. The universe in its entirety, down to the smallest detail, is within the simulation.
In Big Bang cosmology, the Planck epoch or Planck era is the earliest stage of the Big Bang, where cosmic time was equal to Planck time. In analyzing the feasibility of a Programmer God simulation, Planck time therefore becomes the reference for the simulation clock-rate, with the simulation operating at or below the Planck scale, and with the Planck units as (top-level) candidates for the base (mass, length, time, charge) units.

## Planck scale

The Planck scale refers to the magnitudes of space, time, energy and other units, below which (or beyond which) the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable, and quantum effects of gravity are expected to dominate (quantum gravitational effects only appear at length scales near the Planck scale). Consequently any study of a deep universe simulation must consider (if not begin at) the Planck scale.

### Dimensioned quantities

A physical constant is a physical quantity that is generally believed to be both universal in nature and have a constant value in time. These can be divided into dimension-ed (with units kg, m, s ... ; speed of light c, gravitational constant G, Planck constant h ...) and dimension-less (units = 1, such as the fine structure constant α). There are also dimension-less mathematical constants such as pi. The mathematical constant is a number that can occur within the simulation, pi for example can emerge from the rotation of an object. The fundamental physical constant conversely is a parameter built into the simulation and so whilst it may be inferable, it cannot be derived from mathematical constants.
The SI units for the dimensioned mksa units are; meter (length), kilogram (mass), second (time), ampere (electric current). The corresponding mksa Planck units are Planck length, Planck mass, Planck time, Planck charge.
Physicist Lev Okun noted "Theoretical equations describing the physical world deal with dimensionless quantities and their solutions depend on dimensionless fundamental parameters. But experiments, from which these theories are extracted and by which they could be tested, involve measurements, i.e. comparisons with standard dimension-ful scales. Without standard dimension-ful units and hence without certain conventions physics is unthinkable..
However, as the simulation itself is in sum total dimensionless, the (dimension-ed) 'physical' units for mass, space and time must be constructed using only dimensionless physical and mathematical constants, and this is the principal challenge for implementing Programmer God simulations.

## Programming structure
### Numbering systems

As well as our decimal system, computers apply binary and hexadecimal numbering systems. In particular the decimal and hexadecimal are of terrestrial origin and may not be considered 'universal'. Furthermore numbering systems measure only the frequency of an event and contain no information as to the event itself. The number 299 792 458 could refer to the speed of light (299 792 458 m/s) or could equally be referring to the number of apples in a container (299 792 458 apples). As such, numbers require a 'descriptive', whether m/s or apples. Numbers also do not include their history, was 299 792 458 for example a derivation of base numbers?
Present universe simulations use the laws of physics and the physical constants are built in, however both these laws and the physical constants are known only to a limited precision, and so a simulation with 10$^{62}$ iterations (the present age of the universe in units of Planck time) will accumulate errors. Number based computing may be sufficient for ancestor-simulation models where only the observed region needs to be calculated, but has inherent limitations for deep universe simulations where the entire universe is continuously updated. The actual computational requirements for a Planck scale universe simulation based on a numbering system with the laws of physics embedded would be an unknown and consequently lead to an 'non-testable' hypothesis. This is a commonly applied reasoning for rejecting the deep universe simulation.

### Geometrical objects

A number such as pi refers to a geometrical construct (the ratio of circle circumference to circle radius) and so is not constrained by any particular numbering system (in the decimal system π = 3.14159...), and so may be considered universal. Likewise, by assigning geometrical objects instead of numbers to the Planck units, the problems with a numbering system can be resolved. These objects would however have to fulfill the following conditions, for example the object for length must;
1. embed the function of length such that a descriptive (such as km, mile ... ) is not required
Electron wavelength would then be measurable in terms of the length object, as such the length object must be embedded within the electron (the electron object). Although the mass object would incorporate the function mass, the time object the function time ..., it is not necessary that there be an individual physical mass or physical length or physical time ..., but only that in relation to the other units, that object must express that function (i.e.: the mass object has the function of mass when in the presence of the objects for space and time). The electron would then be a complex event constructed by combining the objects for mass, length, time and charge into 1 event, and thus electron charge, wavelength, frequency and mass would be different aspects of that 1 geometry (the electron event) and not independent parameters (independent of each other).
The objects for mass, length, time and charge must therefore be
2. be able to combine with other objects (for mass, time, charge ...) to form more complex objects (events) such as electrons and apples whilst still retaining the underlying information (the individual objects that combined to form that event)
3. combine in such a ratio that they cancel whereby the sum universe itself (being a mathematical universe) is unit-less. While internally the universe has measurable units, externally (seen from outside the simulation) the universe has no physical structure.
Not only must these objects be able to form complex events such as particles, but these events themselves are geometrical objects and so must likewise function according to their geometries. Electrons would orbit protons according to their respective electron and proton geometries, these orbits the result of geometrical imperatives and not due to any built-in laws of physics. However, as orbits follow regular and repeating patterns, they can be described (by us) using mathematical formulas. As the events grow in complexity (from atoms to molecules to planets), so too will the patterns (and the formulas we use to describe them). Consequently the laws of physics would then become our mathematical descriptions of the underlying geometrically imposed patterns. The computational problem could thus be alleviated by instituting a geometrically autonomous universe.
Furthermore, as the sum universe is unit-less, there is no limit to the number of (mass, time, length ...) objects (aka the information content of the universe). If the 'Programmer' can determine appropriate geometrical objects that satisfy the above and also include a mechanism for the addition of further objects, then a universe could 'grow' accordingly.
There is a caveat; self aware structures within the simulation will perceive a physical mass, space and time as forming their physical reality, these mathematical objects must therefore be indistinguishable from any observed physical reality.

## Planck universe

   The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.
Physicist Eugene Wigner (The Unreasonable Effectiveness of Mathematics in the Natural Sciences)
Max Tegmark's mathematical universe hypothesis is: Our external physical reality is a mathematical structure. That is, the physical universe is not merely described by mathematics, but is mathematics (specifically, a mathematical structure). Mathematical existence equals physical existence, and all structures that exist mathematically exist physically as well. Observers, including humans, are "self-aware substructures (SASs)". In any mathematical structure complex enough to contain such substructures, they "will subjectively perceive themselves as existing in a physically 'real' world".
The following describes how a geometrical approach to a Planck scale deep universe simulation could be implemented.

### Simulation clock-rate

The simulation clock-rate would be defined as the minimum 'time' increment to the simulation. It may be that Gods use analog computers, but as an example;
Quantum spacetime and Quantum gravity models refer to Planck time as the smallest discrete unit of time and so the incrementing variable age could be used to generate units of Planck time (and other Planck units), for example;
The age of our 14 billion year old universe measured in units of Planck time approximates 10$^{62}$t$_p$. In the absence of other factors, the universe simulation would then have a mass and size commensurate with age = 10$^{62}$ multiples of Planck mass and Planck length (as Planck volume) respectively as shown in this table,
cosmic microwave background parameters; calculated vs observed :
| Parameter | Calculated | Calculated | Observed |
| Age (billions of years) | 13.8 | 14.624 | 13.8 |
| Age (units of Planck time) | 0.404 10$^{61}$ | 0.428 10$^{61}$ | 0.404 10$^{61}$ |
| Mass density | 0.24 x 10$^{-26}$ kg.m-3 | 0.21 x 10$^{-26}$ kg.m-3 | 0.24 x 10$^{-26}$ kg.m-3 |
| Radiation energy density | 0.468 x 10$^{-13}$ kg.m-1.s-2 | 0.417 x 10$^{-13}$ kg.m-1.s-2 | 0.417 x 10$^{-13}$ kg.m-1.s-2 |
| Hubble constant | 70.85 km/s/Mp | 66.86 km/s/Mp | 67 (ESA's Planck satellite 2013) |
| CMB temperature | 2.807K | 2.727K | 2.7255K |
| CMB peak frequency | 164.9GHz | 160.2GHz | 160.2GHz |
| Casimir length | 0.41mm | 0.42mm | |

### Expanding universe

By thus adding mass, space and time objects with each increment to the simulation age, the universe would grow in size and mass accordingly resembling a black hole (black hole cosmology), thus dispensing with the requirement for a 'dark energy'. As the universe scaffolding includes objects for mass M, 'dark matter' could correspond to the underlying fabric of the universe itself (a vacuum may indicate an absence of particle matter but not an absence of universe).
As the universe expands in size per increment to age, this expansion could also be used to 'pull' particles with it. Thus this expansion could serve the purpose of introducing momentum into the simulation, the expansion being the origin of particle motion. The expansion of the particle universe would not be the same as this expansion, although it would be driven by this expansion.
The velocity of expansion would be the maximum attainable velocity and so the origin of the speed of light c (to go faster than this velocity would mean leaving the simulation itself), thus both the velocity of expansion (and so c) and the incrementing variable age (and so Planck time) are both constants and constraints.
The forward increment to age would constitute the arrow of time. Reversing this would reverse the arrow of time, the universe would likewise shrink in size and mass accordingly (a white hole is the (time) reversal of a black hole).

### Universe time-line

As the universe expands and if the data storage capacity expands proportionately, then the 'past' could be retained.
Because particles are pulled along by this expansion, previous information is not over-written by new information. The analogy would be the storing of every keystroke. This also forms a universe time-line against which previous information can be compared with new information (a 'memory' of events).

#### Time travel

As the simulation data is stored in entirety (a Planck scale version of the Akashic records), the simulation can be replayed. We could even speculate that if mankind made a bad 'move', such as initiating a nuclear war, it may be possible to rewind the simulation clock back to a period prior to that move and continue from there (as we can do when playing chess against a computer and make a bad move). All future events from that point in time would then be over-written. Time-travel (travelling backwards in time) for an individual may not be possible but for the entire universe it possibly is.

### Mass, length, time, charge

For a simulated universe to be unit-less, the units must be able to cancel within a certain ratio such that in sum total there is no physical universe (when seen from outside the simulation, the universe is merely a data set on a celestial hard-drive). In this example, geometrical objects are assigned to units mass M, length L, time T, ampere A. In the following table are illustrated these objects in terms of 2 dimensionless physical constants; the fine structure constant α and Omega Ω, and to fulfill the above condition (that the sum universe be unit-less), a set of mathematical relationships (u$^n$) between them. By replacing the mathematical relationship u with its SI equivalent (i.e.: u$^{15}$ -> kg), the geometrical mass, length, time and charge objects are interchangeable with 'physical' Planck mass, length, time and charge units.
Geometrical units :
| Attribute | Geometrical object | Relationship |
| mass | $M=1$ | $unit=u^{15}$ |
| time | $T=2\pi$ | $unit=u^{-30}$ |
| length | $L=2\pi ^{2}\Omega ^{2}$ | $unit=u^{-13}$ |
| velocity | $V=2\pi \Omega ^{2}$ | $unit=u^{17}$ |
| ampere | $A={\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }}$ | $unit=u^{3}$ |

#### Scalars

To translate from geometrical objects to a numerical system of units such as the SI units requires scalars (ktlva) that can be assigned numerical values.
Geometrical units :
| Attribute | Geometrical object | Scalar |
| mass | $M=1$ | $k,\;unit=u^{15}$ |
| time | $T=2\pi$ | $t,\;unit=u^{-30}$ |
| length | $L=2\pi ^{2}\Omega ^{2}$ | $l,\;unit=u^{-13}$ |
| velocity | $V=2\pi ^{2}\Omega ^{2}$ | $v,\;unit=u^{17}$ |
| ampere | $A={\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }}$ | $a,\;unit=u^{3}$ |

#### SI Planck unit scalars

For example, the following unit relationships cancel, as such only 2 scalars are actually required to numerically solve the Planck units, i.e.: if I know k and t then I know l and so also a. The units for MLTA overlap and cancel (units = 1) in the following ratios and so the sum universe may be unit-less regardless of the universe mass and size (as seen from within the simulation).
    ${\frac {{u^{3}}^{3}{u^{-13}}^{3}}{u^{-30}}}\;({\frac {ampere^{3}\;length^{3}}{time}})={\frac {{u^{-13}}^{15}}{{u^{15}}^{9}{u^{-30}}^{11}}}({\frac {length^{15}}{mass^{9}\;time^{11}}})=1$
    $M=(1)k;\;k=m_{P}=.21767281758...\;10^{-7},\;u^{15}\;(kg)$
    $T={2\pi }t;\;t={\frac {t_{p}}{2\pi }}=.17158551284...10^{-43},\;u^{-30}\;(s)$
    $L={2\pi ^{2}\Omega ^{2}}l;\;l={\frac {l_{p}}{2\pi ^{2}\Omega ^{2}}}=.20322086948...10^{-36},\;u^{-13}\;(m)$
    ${\frac {l^{15}}{k^{9}t^{11}}}={\frac {(.203...x10^{-36})^{15}}{(.217...x10^{-7})^{9}(.171...x10^{-43})^{11}}}{\frac {u^{-13*15}}{u^{15*9}u^{-30*11}}}=1$
    $A=({\frac {2^{6}\pi ^{3}\Omega ^{3}}{\alpha }})a;\;a={\frac {A\alpha }{64\pi ^{3}\Omega ^{3}}}=.12691858859...10^{23},\;u^{3}\;(A)$
    ${\frac {a^{3}l^{3}}{t}}={\frac {(.126...x10^{23})^{3}(.203...x10^{-36})^{3}}{(.171...x10^{-43})}}{\frac {u^{3*3}u^{-13*3}}{u^{-30}}}=1$

#### Electron formula

Shown is an example of a 'mathematical electron' formula; f$_e$. This formula is both unit-less and non scalable (units = scalars = 1). AL as an ampere-meter are the units for a magnetic monopole, the chosen scalars r from the permeability of vacuum μ$_0$ and v from c (as these 2 CODATA 2014 constants have exact values).
    $T=(2\pi ){\frac {r^{9}}{v^{6}}},\;u^{-30}$
    $\sigma _{e}={\frac {3\alpha ^{2}AL}{\pi ^{2}}}={2^{7}3\pi ^{3}\alpha \Omega ^{5}}{\frac {r^{3}}{v^{2}}},\;u^{-10}$
    $f_{e}={\frac {\sigma _{e}^{3}}{T}}={\frac {(2^{7}3\pi ^{3}\alpha \Omega ^{5})^{3}}{2\pi }},\;units={\frac {(u^{-10})^{3}}{u^{-30}}}=1,\;scalars=({\frac {r^{3}}{v^{2}}})^{3}{\frac {v^{6}}{r^{9}}}=1$
The electron has dimension-ed (measurable) parameters; electron mass, wavelength, frequency, charge ... and so an electron geometry (the event 'electron') will include these Planck objects, the electron event dictating how these objects are arranged into those parameters. The electron itself is then equivalent to a programming sub-routine, it does not have dimension units of its own (there is no physical electron), it is a geometrical formula that encodes the information required to implement those parameters.
    $f_{e}=4\pi ^{2}(2^{6}3\pi ^{2}\alpha \Omega ^{5})^{3}=.23895453...x10^{23}$, units = 1
electron mass $m_{e}={\frac {M}{f_{e}}}$ (M = Planck mass)
electron wavelength $\lambda _{e}=2\pi Lf_{e}$ (L = Planck length)
elementary charge $e=A.T$
We may interpret this formula for f$_e$ whereby for the duration of the electron frequency (0.2389 x 10$^{23}$ units of age) the electron is represented by AL magnetic monopoles, these then intersect with time T, the units then collapse (units (A*L)$^3$/T = 1), exposing a unit of M (Planck mass) for 1 unit of T, which we could define as the mass point-state. Wave-particle duality at the Planck level can then be simulated as an oscillation between an electric (magnetic monopole) wave-state (the duration dictated by the particle formula) to this unitary mass point-state.

### Relativity

The mathematics of perspective is a technique used to project a 3-D image onto a 2-D screen (i.e.: a photograph or a landscape painting), using the same approach here would implement a 4-axis hypersphere universe structure in which 3-D space is the projection.
The expanding universe approach can also be used as a method to replace independent particle motion with motion as a function of the expansion itself, this expansion generating the universe time-line. In the mass point-state the particle would have defined co-ordinates and so all particles simultaneously in the point-state per unit of age may be measured relative to each other. As photons (electromagnetic spectrum) have no mass state, they cannot be pulled along by the universe expansion (they are date stamped, as it takes 8 minutes for a photon from the sun to reach us, that photon is 8 minutes old) and so photons would have a lateral motion within the hyper-sphere. Visible (via the electromagnetic spectrum) 3-D space then becomes a projected image from the 4-axis hyper-sphere, the relativity formulas are used to translate between the hyper-sphere co-ordinates and 3-D space co-ordinates.
In hyper-sphere co-ordinate terms; age (the simulation clock-rate), and velocity (the velocity of expansion as the origin of c) would be constants and thus all particles and objects would travel at, and only at, V = c, however in 3-D space co-ordinate terms, time and motion would be relative to the observer. The time dimension of the observer measures the change in state = change of information = change in relative position of particles in respect to each other and thus derives from, but is not the same as, the expansion clock-rate age or object time T, for in the absence of motion there is no means for the observer to measure either, the dimensionless age variable however would continue to increment (the universe hyper-sphere would continue to expand at the speed of light).

#### Particle motion

We take 2 particles A (v = 0 in 3D space) and B (v = 0.866c in 3D space) both of which have a frequency = 6; 5 units of time in the wave-state followed by 1 unit of time in the point-state. The hyper-sphere expands radially at the speed of light. Both particles begin at origin O, after 1second, B will have traveled 0.866*c = 259620km from A in 3-D space (horizontal axis). From the perspective of the A (hypersphere expansion) time-line axis, B will have reached the point-state after 3 time units and so will have twice the (relativistic) mass of A. However the hypersphere expands radially from origin O, and so both A and B will have traveled the equivalent of 299792458m from O (radius OA = OB, v = c) and so by simply inverting our graph, B can equally claim that A has traveled 259620km from B in 3-D space terms.
As each step on the time-line axis requires 1 time unit (and as only the particle point-state can have defined co-ordinates), there are 6 possible velocity solutions (if we include v = 0), this means that B can attain Planck mass (m$_B$ = m$_P$/1) when at maximum velocity v$_{max}$ (relative to the A time-line axis), but B can never reach the horizontal axis = expansion velocity c, and so for any particle v$_{max}$ < c. A small particle such as an electron however has more time divisions and so can travel faster in 3D space than can a larger particle (with a shorter wavelength).
Depicted is particle B at some arbitrary universe time t = 1. B begins at origin O and is pulled along by the hyper-sphere (pilot wave) expansion in the wave-state. At t = 6, B collapses back into the mass point state and now has defined co-ordinates within the hypersphere, these co-ordinates become the new origin O’.
In hypersphere coordinates everything travels at, and only at, the speed of expansion = c, this is the origin of all motion, particles (and planets) do not have any inherent motion of their own, they are pulled along by this expansion as particles oscillate wave-state to point-state ad-infinitum.

#### Particle N-S axis

Particles are assigned an N-S spin axis. The co-ordinates of the point-state are determined by the orientation of the N-S axis, of all the possible solutions, it is the particle N-S axis which determines where the point-state will occur. Thus if we can change the N-S axis angle of A compared to B for example, then as the universe expands the A wave-state will be stretched as with B, but the point state co-ordinates of A will now reflect the new N-S axis angle.
A, B, C do not need to have an independent motion; they are being pulled by the universe expansion in different directions (relative to each other). We can thus simulate a transfer of physical momentum to a particle by changing the N-S axis. The radial hyper-sphere expansion does the rest.

#### Photons

Information between particles is exchanged by photons. Photons do not have a mass point-state, only a wave-state and so have no means to travel the radial expansion axis, instead they travel laterally across the hyper-sphere (they are `time-stamped', a photon reaching us from the sun is 8minutes old).
The period required for particles to emit and to absorb photons is proportional to photon wavelength as illustrated in the diagram (right), $A$ (v = 0) emits a photon (wavelength $\lambda$) towards $B$. The time taken (h) by $B$ to absorb the photon depends on the motion of $B$ relative to $A$. The Doppler shift;
    $v_{observed}=v_{source}.{\frac {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}{1-{\frac {v}{c}}}}=v_{source}.{\frac {h}{\lambda -z}}$
Photons cannot travel the radial expansion axis, and so instead of virtual co-ordinates OA, OB and OC and a constant time and velocity, and as the information between particles is exchanged via the electromagnetic spectrum, ABC will measure only the horizontal AB, BC and AC (x-y-z) co-ordinates, thus defining for the observer a relative 3-D space.

### Gravitational Orbitals

All particles simultaneously in the point-state at any unit of age form gravitational orbital pairs with each other. All these orbitals then rotate by an angle according to the radius of the orbital, the point co-ordinates are then summed and averaged to give the new co-ordinates, age then increments and the cycle continues. The observed gravitational orbits of planets are the sum of these individual orbital pairs averaged over time and thus an orbit, as with a particle, has a frequency component (a time dimension) added to the Planck level.

##### Hyper-sphere co-ordinates

Orbits, being also driven by the universe expansion, occur at the speed of light in hyper-sphere co-ordinates. While B (satellite) has a circular orbit $2\pi r$ on a 2-axis plane (horizontal axis as 3-D space) around A (planet), it also follows a cylindrical orbit (from $B'$ to $B''$) around the A time-line (vertical) axis in hyper-sphere co-ordinates. A is moving with the universe expansion (along the time-line axis) at (v = c) but is stationary in 3-D space (v = 0). B is orbiting A at (v = c) but the time-line axis motion is equivalent (and so `invisible') to both A and B and so the orbital period t and orbital velocity are limited to 3-D space co-ordinates (via $2\pi r$). The actual orbital period at velocity v = c will then be measured by A as $t_{d}$ giving a time dilation effect.

## Singularity

In a simulation, the data (software) requires a storage device that is ultimately hardware (RAM, HD ...). In a data world of 1's and 0's such as a computer game, characters within that game may analyze other parts of their 1's and 0's game, but they have no means to analyze the hard disk upon which they (and their game) are stored, for the hard disk is an electro-mechanical device, is not part of their 1's and 0's world, it is a part of the 'real world', the world of the Programmer. Furthermore the rules programmed into their game would constitute for them the laws of physics (the laws by which their game operates), but these may or may not resemble the laws that operate in the 'real world', assuming there even exist such laws. Thus any region where the laws of physics (the laws of the game world) break down would be significant. A singularity inside a black hole is such a region.
For the black-hole electron, the mass point-state would then be analogous to a storage address on a hard disk, the interface between the simulation world and the real world, a massive black-hole as a data sector.
The surface of the black-hole would then be of the simulation world, the size of the black hole surface reflecting the stored information, the interior of the black-hole however would be the interface between the data world and the real world, and so would not exist in any 'physical' terms. As analogy, we may discuss the 3-D surface area of a black-hole but not its volume (interior).

## Fine structure constant

The fine structure constant was used in the gravitational formulas as common to both orbital radius (a length dimension) and orbital velocity (a velocity dimension) forming a pixel scaffolding for particles and particle interactions above the Planck unit sub-structure. Thus the simulation has a foundation, the Planck scale, and an operating level, the particle quantum scale.
Whilst we may speculate on the origin of physical constants like alpha, these constants (this approach uses 2), are, like the singularity, outside of the 'knowledge frame' as seen from within the simulation. Furthermore, the geometrical objects from which the mathematical constants (and so mathematics itself) arise, may merely be the programming language of the simulation and need not have any resemblance to the 'real world' (the world of the Programmer).