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Degrees of freedom heuristics for theoretical physics2
3
Robert Shour4
5
3rd May, 20216
Toronto, Canada7
8
Abstract9
The degrees of freedom of a network that distributes energy measures10
network output capacity. The degrees of freedom of a network using11
that energy measures use capacity. The ratio of degrees of freedom —12
4 — of a system distributing energy and the corresponding system using13
energy — with 3 degrees of freedom — gives a 4/3 ratio of capacities. A14
4/3 ratio of capacities accounts for the 4/3 fractal envelope of Brownian15
motion, Richardson wind eddy scaling, Stefan’s Law, and expansion of16
cosmological space, among other phenomena, and implies existence of dual17
contemporaneous reference frames. Degrees of freedom supplies useful18
heuristics that give, perhaps, glimpses of physical laws.19
Keywords capacity, entropy, heuristics, scaling.20
1 As analytical tool and as subject matter21
1.1 Degrees of freedom heuristics?22
Degrees of freedom patterns shared by various phenomena imply heuristics that23
apply where no other mathematical model works. These heuristics model other24
phenomena too, supporting their utility and plausibility. Labeling these heuris-25
tics as laws would increase skepticism and at this time, when more investigation26
is required, is premature. But, this article proposes they are worthy of investi-27
gation.28
1.2 Analytical tool29
Degrees of freedom heuristics are problem solving tools, not just conceptual30
characterizations for statistics, mechanics, quantum mechanics, statistical me-31
chanics, kinetic theory, chemistry and so on.32
1.3 Degrees of freedom heuristics include:33
A system’s degrees of freedom is proportional to its capacity.34
A ratio of degrees of freedom for homologous systems shows how they35
scale relative to each other.36
Degrees of freedom heuristics 2
Flow systems scale by 4/3 relative to a corresponding volume.37
Flow system lengths are 3/4 as long in a corresponding volume.38
Two homologous systems with different degrees of freedom are distinct,39
but related, reference frames.40
1.4 Concept lineage41
The term ‘degrees of freedom’ is about 120 years old. The expression ‘free42
motions of the system, or the degree of freedom of its motion’ appeared in43
Heinrich Hertz’s 1894 book published after his death (Hertz, 1894),(Hertz, 1899,44
p. 85). The term ‘degrees of freedom’ had appeared by 1902 in statistical45
mechanics (Gibbs, 1902, p. 3) and in 1915 in statistics (Fisher, 1915; Walker,46
1940). Adoption of the term was not immediate. Planck in 1913 wrote that47
‘thermodynamic probability is . . . always an integer’ (Planck, 1913), (Planck,48
1914, p. 120) referring to degrees of freedom.49
2 Galileo’s beam scaling50
2.1 Galileo’s dimensions51
In Two New Sciences, Galileo concludes, assuming the strength of animal bone52
does not change with species size, that larger animals must have thicker leg53
bones than smaller similar animals because weight is proportional to volume54
with dimension [V] = 3 while area’s dimension [A] = 2.55
2.2 Switch from scaling to dimension56
Galileo’s scaling restores an implicit invariant ratio W:A=c. For a reference57
animal O1, if W1/A1> c,O1’s bone may fracture. Suppose larger L2=Lα
1,58
with α > 1, length L2representative for O2. Then59
W2
A2∝(Lα
1)3
(Lα
1)2∝W3/2
1
A1
>W1
A1
=c. (1)
In (1), the ratio on the left scales to W1/A1only if A2=A3/2
1. If the material60
strength of bone is constant, the W1/A1ratio must be invariant regardless of61
size.62
In (1), W1increases to W2. If A1scales up like W1,A2has one too few63
dimensions to contain the increased weight. W2’s weight per dimension is too64
much for A2.A1must scale up more than W1does.65
Dimension induces scaling of weight and of area. Dimension induces scaling66
and therefore is more fundamental than scaling. Instead of adopting a scaling67
conceptual reference frame, replace scaling by dimension. Use weight per dimen-68
sion — dimensional capacity. Scaling adjusts Ato maintain W:Ainvariance.69
Degrees of freedom heuristics 3
Unresolved is how 3 dimensional weight can be said to occupy a 2 dimensional70
area.71
3 Galilean scaling to degrees of freedom72
3.1 To degrees of freedom73
Sequentially modify Galileo’s 1638 scaling approach.74
Change from scaling to dimension. Instead of finding how Wand A75
scale differently, find what scaling preserves an invariant ratio.76
Characterize three dimensional volume D3≡ {µx, µy, µz}with µa mean77
path length proportional to mean energy.78
Generalize 3 dimensions to 4: D4≡ {µx, µy, µz, f }to include systems79
with flow (or motion) f. Assigning a degree of freedom to a motion has80
precedents in the equipartition theorem, which recognized rotation and81
vibration as having degrees of freedom, and in special relativity, for flowing82
time in space-time.83
Consider homologous systems: D4and corresponding volume D3.84
Notation. Denote dimension and degrees of freedom the same way:85
[D4] = 4; [D3] = 3.86
3.2 Reasons to add a flow fdegree of freedom include:87
Fundamental laws should be based on fundamental concepts.88
Three dimensions do not include flow but modeling should include flow.89
3/4 and 4/3 scaling appear in various phenomena.90
Space-time successfully models a system with flow (time).91
Including flow in D4can account for economies of scale.92
3.3 Changing from dimension to degrees of freedom93
3.3.1 Reasons to use degrees of freedom94
The universe’s initial expansion may require 4 degrees of freedom.95
The existence of systems with 4 degrees of freedom, such as space-time, is96
well established.97
Expansion of the universe appears to occur due to light radiation pressure.98
If cosmic creation requires 4 degrees of freedom, perhaps the 3 degrees of99
freedom of space are incidental to creation.100
Degrees of freedom heuristics 4
3.3.2 Problems remaining101
How does a feature with n+ 1 dimensions fit into ndimensions?102
How can energy occupy a fourth degree of freedom representing flow?103
Time flows; time is a dimension unlike the 3 spatial dimensions.104
3.4 Extrapolation heuristics105
Degrees of freedom capacity is a generalization and extrapolation of dimensional106
capacity but does not appear to have an advantage in explaining how n+ 1107
degrees of freedom, such as D4, can inhabit ndegrees of freedom, such as D3.108
Extrapolation and interpolation can lead to insight despite imperfect un-109
derstanding. Newton extrapolated a binomial’s exponents in Pascal’s triangle110
to negative exponents and interpolated fractional exponents (Whiteside, 1961).111
Newton extrapolated a projectile on Earth to the Moon orbiting Earth (Newton,112
1728, p. 6). Galileo and Carnot extrapolated set-ups with friction to ones with-113
out. Lorentz-FitzGerald contraction was a heuristic. Extrapolation from 3 to 4114
degrees of freedom might reflect physical reality, even if imperfectly understood.115
4 Capacities116
4.1 In general117
The principal degrees of freedom heuristic is the proportionality of degrees of118
freedom to capacity. Its corollary is about the effect of non-integer ratios of119
degrees of freedom of homologous systems.120
4.2 Size capacities121
Let D4≡ {µx, µy, µz, f }with µmean energy and fflow rate per unit time. D4
122
has more capacity to contain energy, information, and heat, than D3.123
If the size of a D3system scales by αfor average length µin each degree of124
freedom, then a D4system will scale as µ4αand a 3 degrees of freedom system125
will scale as µ3α.126
4.3 Time capacities127
Both D4and D3systems have more capacity as time passes: more configurations128
are available. Over ttime periods, with size a D4system will scale as µ4αt and129
aD3system will scale as µ3αt.130
Degrees of freedom heuristics 5
4.4 Size and time capacities131
A factor appearing in the exponent of µcan increase system capacity by in-132
creasing size, by increasing time, or both. The capacity increase is due to the133
exponent and logµ(µ4tα)=4tα resembles the formula for a system’s entropy.134
4.5 Capacities and means135
The mean path length in steps µof all pairs of nodes in a network Nof nnodes136
scales Nsuch that µk=nfor some kReally, it is not a distance scaling N137
but rather the energy proportional to µthat scales energy available. The base138
of the logarithm being a mean maximizes entropy, an implication of Jensen’s139
inequality (Jensen, 1906; Khinchin, 1957). The mean µshows the relationship140
between Clausius’s fractional expression for entropy Sand Boltzmann’s loga-141
rithmic expression for entropy.142
Treat µas µdegrees of freedom based on a single step. Degrees Kelvin T143
is an indirect measure of µ/t, since temperature indirectly measures average144
molecular speed. Then for some k145
S=dQ
T∝k×µ
µ=k= logµ(µk).(2)
The fraction in (2) is the ratio of a system’s degrees of freedom k×µcom-146
pared to energy µ∝T.147
Both µand kµ span width of N.kµ spans the lowest level of hierarchically148
scaled N.µitself spans kµ on average, since µis the mean path between all149
pairs of nodes, and kµ can connect end points of a level. The amount of energy150
proportional to µcannot be the same as the amount of energy proportional to151
kµ. This implies that using µto scale Ngives the number of degrees of freedom152
of Nrelative to µ. Energy does not travel kµ steps, but rather, as implied by153
the spanning of both µand kµ, has the capacity to span kµ using only energy154
proportional to µ. This is another way of arriving at the conclusion that degrees155
of freedom is linear to N’s capacity: capacity is proportional to k.156
4.6 Capacities and the natural logarithm157
If all paths connecting pairs of nodes in Nequal µthen energy in Nis isotrop-158
ically distributed. At every level of the scaling hierarchy dµi+1/dµ =µ, which159
implies that the scale factor at each level in an isotropic network is the natural160
logarithm.161
The natural logarithm also plays a role in determining duration because162
capacity varies with time as an exponent of a scale factor.163
4.7 Entropy164
Entropy measures the degrees of freedom of a system. Calculating entropy165
relative to a mean µhas special significance due to Jensen’s inequality (Jensen,166
Degrees of freedom heuristics 6
1906). The value of a logarithmic expression — using a convex function — is167
maximal when the base of the logarithm is a mean.168
4.8 Energy distribution, available paths, and capacity169
Energy required to traverse a path in a network Nonly need be sufficient to170
traverse the path chosen. Having enough energy to traverse all available paths171
simultaneously is unnecessary. But all available paths, represented by the de-172
grees of freedom of Nare available with energy sufficient for one path. This is173
another way of looking at the connection between µ, the mean path length, and174
kµ.175
In information entropy, the logarithm’s base is usually 2 because of the role176
of binary digits in computation. Texts sometimes state that the base of the log177
for entropy is irrelevant, that the logarithmic expression is what matters. But178
the mean path length in steps plays a special role in entropy and in calculating179
degrees of freedom. The relevant mean path length is the number of intermediate180
collisions or interactions, of persons, information nodes, or gas molecules. The181
thermodynamical considerations are the same.182
5 Equipartition Theorem183
The equipartition theorem resembles the degrees of freedom capacity heuristic.184
Modern texts on thermodynamics and statistical mechanics spend little if185
any time on the equipartition theorem because the theorem fails for quantum186
systems at temperatures that are higher or lower than room temperature, and187
for strongly metallic solids (Tolman, 1927, p. 75), a failure rectified by quantum188
theory.189
Tolman in a 1927 book derives the equipartition theorem (Tolman, 1927,190
Ch. 6). He begins with integration over all possible coordinate and momentum191
values in a Maxwell-Boltzmann distribution. He partially integrates one of the192
coordinates and after some assumptions arrives (his equation (130)) at193
[i]av =1
2kT, (3)
valid when energy is quadratic (a square) (Greiner et al., 1995, p. 198).194
For a monatomic gas with 3 translational degrees of freedom compared to195
the energy of one degree of freedom in (3), at room temperatures196
E=3
2kT. (4)
At room temperature, a system with ηdegrees of freedom has energy capac-197
ity198
E=η×1
2kT. (5)
Degrees of freedom heuristics 7
Generalizing (5), energy capacity varies with degrees of freedom, as with macro-199
scopic networks. With appropriate constraints avoiding quantum mechanical ef-200
fects, the equipartition theorem is equivalent to the degrees of freedom capacity201
heuristic that applies to macroscopic networks. The equipartition theorem has202
fixed energy Edivide equally among available degrees of freedom. The degrees203
of freedom capacity heuristic observes that with fixed energy Eper degree of204
freedom, output capacity varies with the number of degrees of freedom.205
6 Capacity examples206
6.1 Problem solving capacity207
A group of problem solvers has collective problem rate208
Rav = [η(Nβ)]av ×rav ×[η(Nk)]av.(6)
where209
Nβis a network of problem solvers such as computers or brains.210
Nk(kfor knowledge) is a network of ideas, concepts or words.211
Subscripts av denote an average.212
ris the average problem solving rate of an individual problem solver.213
η= logµ(N) is the degrees of freedom of Nrelative to N’s µ.214
µis the mean path length in steps for the relevant network.215
For βEnglish speaking population, and kthe size of the English lexicon, ata216
for Nβand for Nkat different times allow estimation of rav for human problem217
solvers, which is about 5.6% per thousand years. The same 5.6% per thousand218
years is the rate at which related languages diverge and the rate of change in219
Turkic phonemes measured entirely differently from the divergence of related220
languages (Shour, 2009, 2019).221
6.2 Time duration and the natural logarithm222
The exponent of the natural logarithm mean path length in steps measures223
the time capacity of a system, which is used to estimate duration for natural224
processes. Heat at ground level is on average isotropically distributed, so the225
base of the logarithm being proportional to mean energy works.226
6.3 Many worlds and superposition227
Everett’s many worlds might be due to the capacity of the universe to have228
many available states rather than the contemporary existence of those many229
states (Everett, 1957). Similarly for superposition.230
Degrees of freedom heuristics 8
7 Ratios of degrees of freedom231
7.1 To ratios of capacities232
The capacity of a system is proportional to its degrees of freedom. Ratios233
of capacities of related systems equal ratios of degrees of freedom of related234
systems.235
7.2 4/3 ratio of capacities (degrees of freedom)236
The 4/3 ratio of degrees of freedom of a D4system relative to its corresponding237
D3system can be expressed as:238
[D4]
[D3]=4
3.(7)
The ratio persists at all scales and times since239
4αt
3αt=4
3.(8)
αt in the exponent represents systems scaling up in size or over time. A D4’s240
energy available for use by the corresponding D3system grows by exponent 4.241
The corresponding D3system using D4’s energy grows by exponent 3.242
Consequences of the 4/3 ratio of degrees of freedom heuristic include:243
Dimensional pressure. Heat, energy, information etc. in D4can’t fit244
into D3without D3growing or the rate of transmission slowing, because245
D3has one less degree of freedom. The excess degrees of freedom of D4
246
can be characterized as dimensional pressure exerted on the corresponding247
D3system.248
Economies of scale. “. . . economies of scale phenomenon is a fundamen-249
tal feature of all flow (moving) systems .. . ” (Bejan et al., 2017). D4’s250
energy distribution capacity exceeds the capacity of the corresponding D3
251
to use energy. If D3does not respond the D4’s higher capacity, rate of us-252
age by D3can scale by exponent 3/4 without impairing D3’s performance:253
there are economies of scale. Some examples are set out below.254
FD3can compensate for D4’s extra degree of freedom by255
–the rate of energy use in D3slowing, or256
–if rates of energy supply and use are equal, D3’s size increasing.257
Emergence of space. Adapt the economies of scale description to emer-258
gence. Dimensional pressure everywhere in space causes space itself to259
grow at all epochs. If so, then dimensional pressure perhaps created 3260
dimensional space. D4increasing the size of three dimensional space in261
our time, extrapolated backwards in time, implies that light motion’s di-262
mensional pressure brought 3 dimensional space into existence.263
Degrees of freedom heuristics 9
8 Economies of scale examples264
8.1 Biology265
No universally accepted explanations of Kleiber’s Law, Snell’s somatic exponent266
or Peto’s paradox existed at the end of 2020.267
8.1.1 Metabolism268
A mammal’s D4circulatory system ρ, which includes blood flow f, scales by269
exponent 4, while animal tissue D3receiving and using that energy scales by270
exponent 3. Max Kleiber, empirically, inferred that mammalian metabolism271
Y∝M3/4for animal mass M.272
If mammal O2of size V2=Vα
1, then ρ2/V2= (ρ1/V1)4/3×α. Then 3/4273
metabolic scaling Y2=Y3/4α
1restores the ratio of energy per cell that pertains274
to O1as follows:275
Y2
V2
=[(Yα
1)3/4]4/3
Vα
1
=Y1
V1
.(9)
In (9), 4/3 is the ratio of capacities, 3/4 represents Kleiber’s Law.276
8.1.2 Mammalian brain weight scaling277
Otto Snell guessed that mammalian brain weight h∝ksfor body weight k278
(Snell, 1892, 2021). He called sthe somatic exponent. Data implies s= 3/4279
(Eisenberg, 1981). The 4/3 ratio of capacities implies that sensory and signal280
processing capacities of a mammal’s brain, as D4systems, both scale by 4/3281
relative to a D3animal mass. To manage a mammal’s D3body, a mammal’s282
brain need only scale with size by s= 3/4 relative to animal mass.283
8.1.3 Peto’s paradox284
Peto asks why large animals with more cells vulnerable to cancer do not have285
disproportionately more cancer than small animals (Peto, 1977). The 4/3 ratio286
of capacities implies that the greater 4/3 scaled spread of cancer relative to287
animal mass is offset by 3/4 metabolic scaling slowing cancer proliferation. Hy-288
pothesizing a cancer fighting mechanism specific to large animals is unnecessary.289
8.2 Power generation290
Ocean thermal power plants increase in efficiency from 2.5% to 3.5% when base291
power increases from 5MW to 40MW (Bejan et al., 2017). From 5MW to 40MW292
is an 8 fold increase in output, and293
log4/3(8) ≈log4/3 4
37.22!= 7.22 (10)
Degrees of freedom heuristics 10
scalings of 4/3 increased capacity. Efficiency increases by 3.5/2.5 = 1.4 times,294
1.4/7.22 ≈0.194 as much as perfect efficiency, perhaps due to design, mechanical295
energy loss and friction, if this approach is valid.296
8.3 Inanimate particle examples297
The 4/3 ratio of degrees of freedom also appears in inanimate systems.298
8.3.1 Waterston’s elastic plane299
Waterston found that the energy of moving particles striking the underside of300
an elastic plane held in place in Earth’s gravitational field is 4/3 as much as the301
downward energy of the plane (Waterston, 1892).302
8.3.2 Clausius, kinetic gas theory and 4/3303
Clausius in his 1858 Annalen paper on kinetic gas theory (Clausius, 1858, 1859)304
considers the speed of a moving molecule relative to the average speed of all305
molecules moving. He comments that as a preliminary matter:306
The mean lengths of path for the two cases (1) where the re-307
maining molecules move with the same velocity as the one watched,308
and (2) where they are at rest, bear the proportion to one another309
of 3/4 to 1.310
Clausius subsequently showed how he derived 3/4 to 1 (Clausius, 1860). He311
erroneously assumed that the speed of one particle moving relative to a static312
particle represented the relative speed of all moving particles to any other mov-313
ing particle (Shour, 2017).314
The error was that the one molecule watched in effect sampled representative315
average distances between centers of nequal sized cubes for nmolecules in a316
given volume. With this emendation, Clausius actually found that the mean317
path lengths between pairs of moving molecules were 3/4 of the distance between318
pairs of cube centers (which was not the question posed). Equivalently, the mean319
distance for the D3system between pairs of particles was 4/3 that for the same320
particles all moving. In effect, Clausius had, unintentionally, compared average321
distances between pairs of moving molecules in a D4system to the average322
separating distance in a D3system.323
Maxwell corrected Clausius’s 4/3 result, by looking at relative speed in D4
324
only: two particles each independently moving at average speed vcan be treated325
as orthogonal to each other, so the speed of one relative to the other is √2v326
(Maxwell, 1860).327
8.3.3 Stefan Law derivation328
Stefan based on experiment inferred that for radiation in a cavity energy E∝T4
329
(Stefan, 1879). Boltzmann derived Stefan’s result (Boltzmann, 1884, 2021).330
Degrees of freedom heuristics 11
Allen and Maxwell in their text book on heat (Allen and Maxwell, 1948)331
expand on Boltzmann’s terse derivation of Stefan’s Law. In an intermediate332
step (p. 743),333
∂
∂v v
T=4
3
∂
∂E E
T.(11)
In (11), the left side rate of change of the volume/Tratio, relative to volume,334
changes by 4/3 as much as the energy/Tratio changes relative to energy. The335
volume vin v/Tis a D3system. The E/T ratio has a D4system energy amount336
in the numerator. In effect, (11) indicates that a D3system scales by 4/3 as337
much as the corresponding D4system.338
From another perspective, E∝T4says that radiation energy flow (f) added339
to a cavity (D3) system has four degrees of freedom. Average radial motion,340
indirectly measured by temperature, has one degree of freedom.341
8.4 Emergence examples342
As an energy supply system increases in size, distributed energy scales up dis-343
proportionately compared to the amount of energy used by the homologous344
system. Living systems should evolve towards 4/3 scaled energy distribution345
compared to energy use.346
Problem solving capacities and the emergence of language increase arise:347
An increase in available neurons. This can result from larger brains and348
also from more brains networking.349
More concepts and words — more degrees of conceptual freedom — avail-350
able for manipulation by problem solving networks.351
Similar remarks apply to the emergence of languages which are produced by352
collective problem solving — how to efficiently transmit useful information.353
Cosmological space that includes an fflow of light, a D4system, has eco-354
nomies of scale compared to corresponding D3space. If the universe starts355
with motion, then 4/3 dimensional pressure may push space outward radially356
everywhere.357
9 Lengths ratio heuristic358
9.1 Energy and length in dual reference frames359
Use dimensional capacity as a heuristic to compare the length of the same360
distance traveled in homologous D4and D3systems. The same energy in D4is361
4/3 as much per dimension in D3. If distance traveled is proportional to energy362
per dimension, the a radial length in D4is 4/3 as long in the corresponding D3.363
This 4/3 ratio of lengths arises in theory and in observation. Some examples364
follow.365
Degrees of freedom heuristics 12
9.2 Clausius gas molecular mean path lengths366
Clausius intended to show that the speed of a gas molecule relative to another367
molecule in an ensemble of moving gas molecules. For average speed v, he states368
that the relative speed was (4/3)v.369
Maxwell showed that the correct answer is √2v. In a single reference frame,370
two independently moving gas molecules can be modeled by an isosceles right371
triangle, which gives Maxwell’s answer.372
Clausius, by mistake, solved a different problem, comparing the mean path373
length of all particles moving to the average distance separating those particles374
all not moving. Using calculus and trigonometry Clausius inadvertently found375
an early theoretical demonstration of the 4/3 ratio of lengths heuristic.376
9.3 Diffusivity377
Richardson found that for air molecules lunits apart, diffusion caused by wind378
eddies of length lis K(l)∝l4/3(Richardson, 1926). The 4/3 can be derived379
using dimensional analysis too (K¨orner, 1996, p. 186-193), unsurprising given380
degrees of freedom heuristics.381
9.4 Social network information path lengths382
The mean path length in steps is (4/3)e≈3.62 for a social network transmitting383
information, consistent with calculation (Watts and Strogatz, 1998), where the384
natural logarithm eis the mean path length in steps for the corresponding385
information recipient network.386
9.5 Expanding cosmological space387
Length Lin D4cosmological space, by reason of the ratio of lengths heuristic,388
should measure (4/3)Lin D3cosmological space; space expands. Consistent389
with that, radiation energy density varies with the scaling of distance by a4/a3
390
compared to matter energy density (Wang, 2010, p. 17, for example). The391
measured ratio of energy densities is, for energy E:392
E/L3
hE
(4/3)Li3=43
33≈0.7033
0.2967.(12)
Compare denominator in (12) to 0.295 ±0.034 (Betoule et al., 2014).393
1998 supernova observations implied supernovas had luminosity 25% dimmer394
than expected (Cheng, 2010, p. 259), consistent with them being 4/3 as far away395
in the D3reference frame compared to a redshift — D4— reference frame.396
Degrees of freedom heuristics 13
10 Dual homologous reference frames397
10.1 Principle398
Suppose D3is a homologous system, receiving or using D4’s transmitted energy.399
Degrees of freedom and lengths proportional to a fixed amount of energy are400
different for D4and D3. The homologous systems form two contemporaneous401
reference frames.402
10.2 Dual reference frames: possible consequences403
10.2.1 Horizons404
In D34/3 radially stretched space is too separated for distant parts to be reached405
by light. If fin D4represents light motion, all parts of D4can be reached by406
light motion. But space in D3is stretched. Light motion does not connect all407
the stretched D3space.408
10.2.2 Entanglement409
What applies to the cosmological horizon also applies by analogy to entangled410
particles.411
10.2.3 Two slits412
A particle moving in D4that appears to go through two slits illuminates the413
movement of expanding D3, like a revealing strobe light.414
11 Discussion and conclusions415
If degrees of freedom heuristics work, why do they work? Are they laws? Even416
if not yet on a firm foundation, they appear to have utility.417
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