Simulation argument (Planck scale)

Programming deep universe (Programmer God) Simulation Hypothesis models at the Planck scale

The simulation hypothesis or simulation argument is the 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). 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 [1]. As the universe in its entirety, down to the smallest detail, is within the simulation, coding will occur at the lowest level.

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 scaleEdit

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 [2].


Dimensioned quantitiesEdit

A fundamental 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 SI units 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. [3].

Successful implementation of a Planck scale simulation necessitates that the programmed (dimension-less) mathematical structures for mass, space and time be indistinguishable from the (dimension-ed) 'physical' units for mass, space and time as are observed from within the simulation.


Programming structureEdit

Numbering systemsEdit

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 1062 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 but has inherent limitations for deep universe simulations. 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 objectsEdit

A number such as π refers to a geometrical construct rather than any numbering system and so may be considered universal and defined as a constant. Likewise, by assigning geometrical objects instead of numbers to the Planck units, the problems with a numbering system could 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 (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. 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 particles 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. As orbits follow regular and repeating patterns, they can be described using mathematical formulas. As the events grow in complexity (from atoms to molecules to planets), so too will the patterns (and so the formulas then used to describe them). Consequently the laws of physics would then become mathematical descriptions of the underlying geometrically imposed patterns. The computational problem could thus be resolved by instituting a geometrically autonomous universe.

Furthermore, as the sum universe is unit-less, there is no limit to the number of objects (size, mass, age ... and so the information content that can be stored and manipulated). 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 universeEdit

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) [4]


Max Tegmark's mathematical universe hypothesis is: Our external physical reality is a mathematical structure.[5] 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".[6]

The following describes how a geometrical approach to a Planck scale deep universe simulation could be implemented [7] [8].


Simulation clock-rateEdit

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;

 'begin simulation
 FOR age = 1 TO the_end                       'big bang = 1                 
         conduct certain processes ........ 
 NEXT age 
 'end simulation

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;

 FOR age = 1 TO the_end                       'age is a dimensionless variable                            
         generate 1 time object T             'T is a dimension-ed unit of time
         generate 1 mass object M             'M is a dimension-ed unit of mass
         generate 1 length object L           'L is a dimension-ed unit of length
         ........ 
 NEXT age 


The age of our 14 billion year old universe measured in units of Planck time approximates 1062tp. In the absence of other factors, the universe simulation would then have a mass and size commensurate with age = 1062 multiples of Planck mass and Planck length (as Planck volume) respectively as shown in this table [9],

.

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 1061 0.428 1061 0.404 1061
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 universeEdit

By 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 (this would restrict the permitted topologies of the simulation). 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 constants.

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) reverse of a black hole).

 FOR age = the_end TO 1 STEP -1                                 
         delete 1 time object T
         delete 1 mass object M
         delete 1 length object L
         ........ 
 NEXT age 


Universe time-lineEdit

As the universe expands and if the data storage capacity expands proportionately, then the 'past' could be retained.

 FOR age = 1 TO the_end                                 
         ........
         FOR n = 1 TO total_number_of_particles
                SAVE particle_details{age, particle(n)}
         NEXT n
 NEXT age 

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 travelEdit

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 reset the simulation. 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 is.



Mass, length, time, chargeEdit

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). Here 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 (un) between them [10].

Geometrical units
Attribute Geometrical object Relationship
mass    
time    
length    
velocity    
ampere    


ScalarsEdit

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    
time    
length    
velocity    
ampere    


SI Planck unit scalarsEdit

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.

 
 
 
 
 


 
 


Physical constantsEdit

We could use this unit relationship as a tool to solve the dimension-ed physical constants. The 4 most precisely known of the CODATA 2014 constants are c, vacuum permeability μ0, Rydberg constant R and α). We can define these as geometrical objects in terms of MLTA and then use these objects to build the geometrical objects for the less precisely known physical constants (G, h, e, me, kB); i.e.:.

α = 137.035 999 139 (CODATA 2014)
 
 


G, h, e, me, kBEdit
Physical constants; calculated vs experimental (CODATA)
Constant Geometry Calculated from (R*, c, μ0, α) CODATA 2014 [11]
Planck constant   h* = 6.626 069 134 e-34, unit = u19 h = 6.626 070 040(81) e-34
Gravitational constant   G* = 6.672 497 192 29 e11, unit = u6 G = 6.674 08(31) e-11
Elementary charge   e* = 1.602 176 511 30 e-19, unit = u-19 e = 1.602 176 620 8(98) e-19
Boltzmann constant   kB* = 1.379 510 147 52 e-23, unit = u29 kB = 1.380 648 52(79) e-23
Electron mass   me* = 9.109 382 312 56 e-31, unit = u15 me = 9.109 383 56(11) e-31

Note: The 2019 redefinition of SI base units) fixed the numerical values of the 4 physical constants (h, c, e, kB). In the context of this model however only 2 base units may be assigned by committee as the rest are then numerically fixed by default and so here are used the CODATA 2014 values.


Electron formulaEdit

Shown is an example of a 'mathematical electron' formula; fe. 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 μ0 and v from c (as these 2 CODATA 2014 constants have exact values).

 
 
 


The electron has dimension-ed (measurable) parameters; electron mass, wavelength, frequency, charge ... and so an electron geometry (the event 'electron') will include the geometrical objects for MLTA, 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), however it is a geometrical structure with the function 'electron' embedded within that structure.

 , units = 1

electron mass   (M = Planck mass)

electron wavelength   (L = Planck length)

elementary charge  


We may interpret this formula for fe whereby for the duration of the electron frequency (0.2389 x 1023 units of T) 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.

By this artifice, although the 'physical' universe is constructed from particles (particle matter), particles themselves are not physical, they are mathematical.


RelativityEdit

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 [12].

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 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 [13].

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, age however would continue to increment (the universe hyper-sphere would continue to expand at the speed of light).


 
A and B in hypersphere co-ordinates from origin O, (frequency 0A = 0B = 6)

Particle motionEdit

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 (mB = mP/1) when at maximum velocity vmax (relative to the A time-line axis), but B can never reach the horizontal axis = expansion velocity c, and so for any particle vmax < 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).


 
particle wave to point oscillations in hyper-sphere expansion co-ordinates


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 axisEdit

 
particle N-S spin axis orientation mapped onto hyper-sphere

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.





PhotonsEdit

 
doppler shift measured along the expanding hypersphere time-line axis

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),   (v = 0) emits a photon (wavelength  ) towards  . The time taken (h) by   to absorb the photon depends on the motion of   relative to  . The Doppler shift;

 

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 OrbitalsEdit

 
4 body orbit; (4 mass points, 6 orbitals)

All particles simultaneously in the point-state at any unit of age form gravitational orbital pairs with each other [14]. These orbital pairs then rotate by a specific angle depending on the radius of the orbital. These are then averaged giving new co-ordinates in the hypersphere. Furthermore the orbital plane also rotates. The observed gravitational orbits of planets are the sum of these individual orbital pairs averaged over time.

Orbits, being also driven by the universe expansion, occur at the speed of light, however the orbit along the expansion time-line is not noted by the observer and so the orbital period is measured using only 3D space co-ordinates.

Geometrical orbitalsEdit

Gravitational orbitals are divided in an 'alpha based pixel' sub-structure

(inverse) fine structure constant α = 137.03599...,

np = pixel number

  = number of Planck mass point-states per unit of Planck time

Orbital geometry for 2 (rotating) mass points;

  (pixel to mass ratio)
  (pixel aggregate)
  (orbital length)


Hyper-sphere co-ordinatesEdit
 
B's orbit relative to A's time-line axis in time units t

While B (satellite) has a circular orbit period to on a 2-axis plane (horizontal axis as 3-D space) around A (planet), it also follows a cylindrical orbit (from B1 to B11) 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 and orbital velocity measure is limited to 3-D space co-ordinates.

We can simplify this cylinder by projecting it onto a 2-axis plane as the difference between 2 orbits.

 
 
 
 


Example: A 1kg satellite B orbit at radius r = 42164.170 km from earth center

 
 
  m
  million km
  s
  s
  s, .


PrecessionEdit

The ellipticity of the B orbit around A.

semi-minor axis:  

semi-major axis:  

radius of curvature : 

 


arc secs per 100 years:

 
Mercury = 42.98  
Venus = 8.62  
Earth = 3.84  
Mars = 1.35  
Jupiter = 0.06

The orbital plane becomes

 



SingularityEdit

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. Thus we may discuss the 3D surface area of a black-hole but not its volume (interior).



External linksEdit


ReferencesEdit

  1. Plato's Cave - the Programmer God, Malcolm Macleod, online edition http://www.codingthecosmos.com/ebook-physics/, 2003-present
  2. Planck scale, Brian Greene; "[1]"
  3. Michael J. Duff et al, Journal of High Energy Physics, Volume 2002, JHEP03(2002)
  4. Wigner, E. P. (1960). "The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959". Communications on Pure and Applied Mathematics 13: 1–14. doi:10.1002/cpa.3160130102. 
  5. Tegmark, Max (February 2008). "The Mathematical Universe". Foundations of Physics 38 (2): 101–150. doi:10.1007/s10701-007-9186-9. 
  6. Tegmark (1998), p. 1.
  7. adapted from the EPJ article; Macleod, M.J. "Programming Planck units from a mathematical electron; a Simulation Hypothesis". Eur. Phys. J. Plus 113: 278. 22 March 2018. doi:10.1140/epjp/i2018-12094-x. 
  8. Plato's Cave - the Programmer God, Malcolm Macleod, online edition http://www.codingthecosmos.com/ebook-physics/, 2003-present
  9. Macleod, Malcolm; "Programming cosmic microwave background for Planck unit Simulation Hypothesis modeling". RG. 26 March 2020. doi:10.13140/RG.2.2.31308.16004/7. 
  10. Macleod, M.J. "Programming Planck units from a mathematical electron; a Simulation Hypothesis". Eur. Phys. J. Plus 113: 278. 22 March 2018. doi:10.1140/epjp/i2018-12094-x. 
  11. [2] | CODATA, The Committee on Data for Science and Technology | (2014)
  12. Macleod, Malcolm; "Programming cosmic microwave background for Planck unit Simulation Hypothesis modeling". RG. 26 March 2020. doi:10.13140/RG.2.2.31308.16004/7. 
  13. Macleod, Malcolm; "Programming relativity for Planck unit Simulation Hypothesis modeling". RG. 26 March 2020. doi:10.13140/RG.2.2.18574.00326/3. 
  14. Macleod, Malcolm J.; "Programming gravity for Planck unit Simulation Hypothesis modeling". RG. Feb 2011. doi:10.13140/RG.2.2.11496.93445/8.