Newtonian gravitation only depends upon radial mass distribution (e.g., Green's function). It and metric gravitation (mass as a tensor quantity) are symmetric to parity transformation. (x,y,z) and (x,y,z) give identical answers. Nonmetric gravitation (mass as a pseudotensor) can be antisymmetric. (x,y,z) and (x,y,z) give different answers. The proper analysis of spacetime geometry is test mass geometry (chemically identical, opposite parity mass distributions). Quantitative geometric parity divergence is ab inito calculated from atomic coordinates. Dr. Michel Petitjean presents an extensive survey of theory in pure and applied mathematics. A novel parity Eötvös (pronunciation) experiment is proposed in unmodified apparatus using maximally paritydivergent quartz single crystal test masses. An Equivalence Principle violation >520 times that allowed for opposed composition test masses is predicted.
Introduction Table I. Test Mass Property Magnitudes Table II. Symmetry Groups Table III. Postulated Gravitation Independence Table IV. Spin and Parity Operations Testing the Equivalence Principle Table V. Equivalence Principle Tests Table VI. Solar Gravitation At Earth Orbit Table VII. Horizontal Acceleration vs. Latitude Symmetry, Chirality, Parity Figure 1. Chirality compared to Parity Table VIII. Paired Enantiomorphic Space Groups Table IX. Comparison Of Gravitation Theories Optical and Geometric Chiralities Geometrically Chiral Test Masses Table X. Candidate Chiral Test Masses Subtopic Hemiparity Test Masses Calculated Geometric Chirality/Parity Table XI. 43 Invalid and 16 Deficient Enantiomorphic Sohncke Space Groups Table XII. Six Qualified Enantiomorphic Sohncke Space Groups Table XIII. Benzil Differential Parity Enthalpy of Fusion Figure 2. Tellurium Atomic Lattice Figure 3. Tellurium Atomic Lattice with Bonds Table XIV. Calculated Lattice Parity Divergence, Te and alphaQuartz Figure 4. log(1CHI) vs. log(radius): Tellurium, 505,000 Å Radius Figure 5. log(1CHI) vs. log(radius): alphaQuartz, 1,010,000 Å Radius Figure 6. log(1CHI) vs. log(radius): Berlinite, 100,000 Å Radius Figure 7. log(1CHI) vs. log(radius): Distorted Te Unit Cells Table XV. Te Distorted Unit Cells Data Table XVI. Te Distorted Fractional Coordinates Table XVII. Te Distorted Fractional Coordinates Data Table XVIII.Calculated Lattice Parity Divergence, Te Cylinders Table XIX. Measured Quartz Quality Green's Function Table XX. CHI Fluctuations at Small Radius Increments Table XXI. Quartz CHI at Small Radii Gravitational Scaling Table XXII. Compactified Dimensions' Effects Other Chiral Test Masses Table XXIII.Mean Atomic Weights Conclusion Acknowledgements References
time, and distance as geometry. General Relativity[5] is a geometric model of gravitation. Einstein's elevator Gedankenexperiment[6] embodies the Weak (GalileanNewtonian) Equivalence Principle[1]:
mass = (p_{x}^{2} + p_{y}^{2} + p_{z}^{2}  p_{t}^{2}); metric signature (+1,+1,+1,1), Relativity mass = (p_{t}^{2}  p_{x}^{2}  p_{y}^{2}  p_{z}^{2}); metric signature (+1,1,1,1), field theory
Inertial reference frames (coordinate systems) have constant relative velocity in a flat spacetime manifold. Accelerating frames with consistent definitions of energy and momentum (or mass and angular momentum) require nonzero spacetime curvature (assuming an asymptotic symmetry group, which obtains: the BondiMetznerSachs group restricted to the Poincaré subgroup[7]). Local spacetime must have a unique curvature. Local test masses exhibiting nonparallel geodesic trajectories require simultaneous different values of local spacetime curvature. Any paired (sets of) test masses violating the Equivalence Principle empirically falsify metric theories of gravitation at their founding postulate.
 Local bodies fall identically, because
 Gravitational mass (F=m_{g}GM/r^{2}) is indistinguishable from inertial mass (F=m_{i}a),
 Regardless of composition,
 Regardless of geometry (internal structure),
 Regardless of mass;
The Weak Equivalence Principle extends to the Strong (Einsteinian) Equivalence Principle :
 Nonrotating free fall is locally indistinguishable from uniform motion absent gravitation. Linear acceleration relative to an inertial frame in Special Relativity is locally identical to being at rest in a gravitational field. A local reference frame always exists in which gravitation vanishes.
 Local Lorentz invariance[8] (absolute velocity does not exist) and position invariance. All local free fall frames are equivalent.
 The Strong Equivalence Principle embraces all laws of nature; all reference frames accelerated or not, in a gravitational field or not, rotating or not, anywhere at any time (frame covariance; global diffeomorphism invariance aside from the Big Bang).
The Equivalence Principle demands that gravitational fields contain a local Minkowski (flat) spacetime reference frame (free fall). Gravitational fields cannot have a stressenergy tensor if free fall exists. If no reference frame makes gravitation locally vanish, spacetime curvature is counterdemonstrated as a violated Equivalence Principle.
General Relativity's physical systems are always spatially separable into independent components. Systems of three or more particles require cluster separability (macroscopic locality). When the system is separated into subsystems, the overall mathematical description must reduce to descriptions of the subsystems. This is vital in scattering problems with two or more fragments.
Quantum mechanics allows entangled states (superpositions of product states) that require a fundamental irresolvable connection within readily demonstrated physical systems (twoslit diffraction, the EinsteinPodolskyRosen paradox). Macroscopic locality is violated: Measuring the state of one slit in a double slit experiment alters the observed diffraction pattern to single slit patterns (quantum eraser experiments). Relativistic and quantum views are in conflict. Test masses contrasting an emergent phenomenon that cannot be reduced to a point mass equivalent are novel and important tests.
Weak and Strong Equivalence Principles are coupled[9]. Gravitation studies deny nonpoint geometric[10] consequences. Chirality and parity are emergent phenomena  components collectively evolve discrete symmetry system properties absent and unpredictable from any smaller subset. Geometric parity is an obligatory Equivalence Principle challenge if maximum calculated parity divergence (>99% divergence for parity pair quartz single crystals) is evaluated as are composition divergences (0.19% divergence in nuclear binding energy/nucleon for magnesium vs. beryllium).
Property  Fraction of Rest Mass 

rest mass  100% 
crystal lattice atomic geometric parity 
99.9775% (*Te) 99.9771% (*HgS) 99.9769% (*PdSbTe) 99.9730% (*AlPO_{4}) 99.9726% (*SiO_{2}) 99.9713% (*benzil) 
nuclear binding energy (low Z)  0.76% (_{2}He^{4}) 
neutron versus proton mass  0.14% 
electrostatic nuclear repulsion  0.06% 
electron mass  0.03% 
unpaired spin mass  0.005% (^{55}Mn**) 
nuclear antiparticle exchange  0.00001% 
Weak Force interactions  0.0000001% 
When a set of transformations is closed (any transformation can be expressed as the product of other transformations in the set), the set is called a "symmetry group."
Group  Transformation  Unmeasurable Quantities 
Conserved Quantities 

Rotation, SO(3)  Spatial rotations  Absolute angle  Angular momentum L 
Translation  Spacetime translations 
Absolute position  Energy E, or Mass M and Momentum P 
Lorentz  Spacetime rotations and Reflections 
Absolute uniform velocity, Orientation 
Spacetime interval S, Parity P, Time reversal T 
SL(2,C) (Homogeneous Lorentz) 
Spacetime rotations  Absolute uniform velocity 
S (not P or T) 
Diffeomorphism (General Coordinate) 
Spacetime curvature (acceleration) 
Absolute acceleration 
Topological invariants* 
Poincaré  Lorentz plus Translations 
(see above)  L, E (or M) and P 
U(1)  Scalar Phase Shift  Absolute phase  Electric charge 
SU(2)  2D Phase Shift  Absolute 2D phase  Isospin 
SU(3)  3D Phase Shift  Absolute 3D phase  Color 
*Manifolds with multiple differential structures (e.g., 7D sphere) have multiple, mutuallyexclusive equivalence classes of metrics characterized by different, independent definitions of volume (by a factor of the Jacobian under coordinate transformations).
Unitary groups U(1), SU(2) and SU(3) parameter spaces are isomorphic to (in onetoone correspondence with) the circle, the sphere (a surface) and the "three sphere" (not a ball) respectively. Fields with nonabelian symmetries divide into "electric" (curlfree; e.g., gravitational) and "magnetic" (divergencefree; e.g., inertial) fields as do abelian electromagnetism and the linearized form of Einstein's field equations for weak gravity and slow matter.
Class  Invariance  Conserved Quantity 

Proper orthochronous Lorentz symmetry 
translation in time (homogeneity) 
energy 
translation in space (homogeneity) 
linear momentum  
rotation in space (isotropy) 
angular momentum  
Discrete symmetry 
P, coordinates' inversion  spatial parity 
C, charge conjugation  charge parity  
T, time reversal  time parity  
CPT  product of parities  
Internal symmetry (independent 
U(1) gauge transformation  electric charge 
U(1) gauge transformation  lepton generation number  
U(1) gauge transformation  hypercharge  
U(1)_{Y} gauge transformation  weak hypercharge  
U(2) [not U(1)xSU(2)]  electroweak force  
SU(2) gauge transformation  isospin  
SU(2)_{L} gauge transformation  weak isospin  
PxSU(2)  Gparity  
SU(3) "winding number"  baryon number  
SU(3) gauge transformation  quark color  
SU(3) (approximate)  quark flavor  
S((U2)xU(3)) [not U(1)xSU(2)xSU(3)] 
Standard Model 
EP tests exploit external symmetries' observables. Internal symmetries' observables (gauged using fiber bundle theory, e.g., charge conjugation) transform fields amongst themselves leaving physical states (translation, rotation) invariant. A local gauge transformation always exists to make the local gaugefield vanish. Two vector potentials differing only by a gauge transformation give the same field. EP tests opposing properties coupled to internal symmetries are empirical first order default nulls.
Linear and angular momenta, massenergy, electric charge, and CPT are strongly conserved (black holes). Properties derived from internal symmetries transform fields amongst themselves leaving physical states (translation, rotation) invariant: U(1) symmetry in electromagnetism, U(2) symmetry in electroweak theory, SU(3) in strong force theory. CPT, quark color, baryon number, and lepton generation number are locally conserved. Weak interactions violate parity conservation and violate the remainder. Gravitation has never been challenged with test mass geometric parity.
Parity symmetry obtains when physics is invariant under a discrete transformation that reverses all space coordinates' signs (x,y,z) but ignores time (t). Angular momentum and spin (J = r X p, axial vector or pseudovector), angular velocity, torque, auxiliary magnetic field (H), magnetic dipole moment, and longitudinal and transverse polarizations remain constant under parity. Parity reverses the signs of the position vector (proper or polar vector) of a particle in space (r to r), velocity (v = dr/dt), linear momentum (p), force (f = dp/dt), electric field (E = V), and electric dipole moment (·E).
J^{P}  Even Parity Object  J^{P}  Odd Parity Object 


0^{+}  Scalar (S)  0^{}  Pseudoscalar (P)  
1^{+}  Axial vector (A)  1^{}  Polar Vector (V)  
2^{+}  Tensor (T)  2^{}  Pseudotensor 
Newtonian gravitation requires parity invariant conservative forces. General Relativity models gravitation as a parity invariant rank2 tensor. Nobody has sought empirical exceptions despite acceptable theoretical loopholes[12] including teleparallel gravitation (below).
External symmetryderived properties (e.g., parity) act on spacetime. A Poincaré group gauge theory can be equivalent to the EinsteinCartan theory of gravitation[13]. EinsteinCartan theory operates in RiemannCartan spacetime U^{4}  a paracompact, Hausdorff, connected, C, and oriented fourdimensional manifold on which are defined a local Lorentz metric g and a linear affine connection . Curvature and torsion tensors can be obtained from on U^{4}:
Quantum field theories (QFT) with hermitian hamiltonians are invariant under the Poincaré group containing spatial reflections. Parity is a spatial reflection and parity is not a QFT symmetry! QFT are invariant under the identity component of the Poincaré group  the subgroup consisting of elements that can be continuous path joined to the Poincaré group identity; only an orthochronous Poincaré group representation. This subgroup excludes parity and time reversal. All hermitian hamiltonians will contain a symmetry and an observable with the properties of parity, even though the Hamiltonian will not be symmetric under space reflection. QFT with nonhermitian Hamiltonians can have real and positive energy spectra with PT invariance, but do not contain parity invariance alone[14]. Metric and quantum field theories of gravitation are tested by the parity Eötvös experiment.
Supersymmetric (SUSY, gauge symmetry plus spacetime symmetry) grand unified theories relating fermions and bosons to each other contain added allowances for symmetry breaking (inserted soft breaking terms into the Lagrangian where they maintain the cancellation of quadratic divergences)[15]. When global symmetry contracts to local symmetry, supergravity (SUGRA) emerges: if one gauges the SUSY transform, because of the SUSY algebra, one inevitably gets a gravitation theory)[16]. Supergravity by itself is not a renormalizable quantum theory.
A gravitational stressenergy (energymomentum) pseudotensor constructs volume integrals for total gravitational fourmomentum and total angular momentum. Teleparallel gravitation can allow a gravitational stressenergy pseudotensor obtained by comparing vectors at different points of spacetime  a coframe field  unlike ordinary General Relativity. (Teleparallel theories wholly equivalent to General Relativity are inoperative here.) When the coframe field changes, the pseudotensor changes (not gaugeinvariant; not covariant under general coordinate transformations)[17]. This defines an integral energymomentum obeying an exact conservation law. It is an observable, and it can be sensitive to parity inversion[18]  "a redistribution of energy between material and gravitational (coframe) fields is possible in principle."
(The difference between the Weitzenböck and LeviCivita connections is "contorsion" quantifying coframe field twist as measured by the LeviCivita connection. Contorsion can be expressed as the torsion of the Weitzenböck connection. The LeviCivita connection can be expressed as the Weitzenböck connection and its torsion. The Ricci scalar curvature can be expressed as the Weitzenböck connection and its torsion. The Lagrangian for General Relativity therefore can be expressed purely in terms of the coframe field  in a way that is not symmetric to parity transformation.)
Physics seeks Equivalence Principle empirical violation. Tests often examine quantities evolved from internal symmetries via Noether's theorem. Internal symmetries by definition  a local gauge transformation always exists to make the local gaugefield vanish  do not affect spacetime. Studies contrasting baryon number, isospin, hypercharge, lepton vs. baryon number... are automatic null results.
Geometric parity is the only untested physical property arising from an external symmetry. Parity is an absolutely discrete symmetry that cannot be approximated by a Taylor series or a sum of infinitesimals. Noether's theorem with its dependence upon smooth Lie groups is inappropriate.
The existence of a symmetry operator implies the existence of a conserved observable. Given G is the Hermitian generator of nontrivial unitary operator U (e.g., parity), then if U commutes with Hamiltonian H so does G [H,G]=0. If U commutes with H it is a symmetry and a conserved quantity. Any system that is initially in an eigenstate of U evolves over time to other eigenstates having the same eigenvalue.
U = c then,
Uexp(itH) = exp(itH) U [U commutes with H]
= exp(itH) c
= c exp(itH)
so exp(itH) is again an eigenstate of U, with the same eigenvalue c. Discrete symmetries also give conserved quantities in classical mechanics (e.g., bifurcation theory of dynamical systems). Parity the symmetry is coupled to geometric parity the property.
Year  Investigator  Accuracy  Method 

500?  Philoponus[20]  "small"  Drop Tower 
1585  Stevin[19]  5·10^{2}  Drop Tower 
1590?  Galileo[2]  2·10^{2}  Pendulum, Drop Tower 
1686  Newton[3]  10^{3}  Pendulum 
1832  Bessel[21]  2·10^{5}  Pendulum 
1910  Southerns[22]  5·10^{6}  Pendulum 
1918  Zeeman[23]  3·10^{8}  Torsion Balance 
1922  Eötvös[24]  5·10^{9}  Torsion Balance 
1923  Potter[25]  3·10^{6}  Pendulum 
1935  Renner[26]  2·10^{9}  Torsion Balance 
1964  Dicke,Roll,Krotkov[27]  3·10^{11}  Torsion Balance 
1972  Braginsky,Panov[28]  10^{12}  Torsion Balance 
1976  Shapiro, et al.[29]  10^{12}  Lunar Laser Ranging 
1981  Keiser,Faller[30]  4·10^{11}  Fluid Support 
1987  Niebauer, et al.[31]  10^{10}  Drop Tower 
1989  Heckel, et al.[32]  10^{11}  Torsion Balance 
1990  Adelberger, et al.[33]  10^{12}  Torsion Balance 
1999  Baeßler, et al.[34]  5·10^{13}  Torsion Balance 
2010?  MiniSTEP[35]  10^{17}  Earth Orbit 
Contrasting compositions classically define[36] "different" test masses : baryon number (N+Z), isopin (NZ)/2, Yukawa forces[37]; gravitational selfenergy[38]; electron spin (Dy_{6}Fe_{23} and HoFe_{3}[39]; 94% of Alnico 5 magnetic field) versus electron orbital angular momentum (37% of Sm_{2}Co_{17} magnetic field)  a mole of electrons masses an insignificant 5.4858·10^{4} grams; neutrinoantineutrino exchange in different nuclei[40]; and inverse square deviations at small radii (string then Mtheory[41]). 400+ years of exquisite tests, notably those derived from Vásárosnaményi Báró Eötvös Loránd[24], null to differential detection limits.
Ferrimagnet Dy_{6}Fe_{23} has no external magnetic field at 1°C. It has 0.4 unpaired electrons/formula unit, or 97 nanograms of net unpaired spins/g test mass. It is ridiculously dilute. This spin test mass is calculated to display measurable spacetime torsion interaction as a cylinder 20 meters (!!!) in diameter and length weighing some 50,000 metric tonnes (25% more than the largest WWII Essexclass aircraft carrier fully loaded and fueled).
Quantifiable composition divergences arise in quantum mechanics, quantum chromodynamics, and relativistic electronic effects (gold[42]). To the extent that bodies are identical they are equivalent. Mirrorimage chiral and parity pair bodies cannot be identical by definition. 65 Sohncke space groups of 230 total[43] describe chiral crystal configurations. 22 Sohncke space groups are eleven enantiomorphic parity pairs. Metric theories of gravitation are falsified at the postulate level if extreme mirrorimage chiral or parity pair bodies are not Eötvös experiment nulls.
Starting in 1889 Eötvös' torsion balance tested the Equivalence Principle. Masses gravitate toward Earth's center. Given two (sets of) test masses, one at each end of a fibersuspended rotor, the horizontal component of centripetal force is directed away from the Earth's spin axis proportional to inertial mass. Net torque twists the fiber if mGM/r^{2} ma. An attached mirror reflects light, amplifying angular deviation into an interferometer. Dicke[27] used Earth's free fall in the sun's gravitational field[44],
Acceleration, cm/sec^{2}  Orbital Position  Date 

0.593008  One astronomical unit  
0.613307  Geocenter perihelion 
20012101 mean value 
0.613393  2020, 05 January  
0.613214  2098, 05 January  
0.613305  Earth/Moon barycenter perihelion 
20012101 mean value 
0.613381  2020, 04 January  
0.613252  2098, 04 January  
0.573700  Geocenter aphelion 
20012101 mean value 
0.573786  2085, 04 July  
0.573626  2019, 04 July  
0.573702  Earth/Moon barycenter aphelion 
20012101 mean value 
0.573764  2097, 05 July  
0.573654  2019, 05 July 
and surface inertial centripetal acceleration (sidereal day, WGS 84;
sea level unless noted) of Earth's rotation.
r = (6378136.46)[1([sin^{2}(lat)]/298.257223563)] meters
a = (3.380199)(cosine[lat])/[1(0.006694380)cosine^{2}(lat)] cm/sec^{2}
horizontal component of a = a[sin(lat)]
Resultant, cm/sec^{2}  Geocentric Latitude, degrees  Horizontal Component, sin(lat) 
Centripetal Acceleration, cm/sec^{2} 


3.66  45  550 mph east ground speed 
commercial airliner 

9.51  45  1336 mph east ground speed 
Concorde  
1.46490  60  0.866025  1.691516  
1.58993  55  0.819152  1.940941  
1.66673  50  0.766044  2.175761  
1.69294  45  0.70711  2.394172  
1.69294  44.951894  0.706513  2.396188  
1.66770  40  0.642788  2.594484  
1.59175  35  0.573576  2.775137  
1.46736  30  0.500000  2.934715  
1.29826  25  0.422618  3.071958  
1.08960  20  0.342020  3.185778  
0.84770  15  0.258819  3.275266  
0.57993  10  0.173648  3.339705  
0.29446  5  0.087156  3.378578  
0  0  0  3.391570 
g = (978.032677)[1+(0.00193185139)sin^{2}(lat)]/[1(0.00669437999)sin^{2}(lat)] cm/sec^{2}Any torque is diurnally modulated: For an Eötvös torsion balance with masses aligned eastwest, maximum torque occurs at local noon or midnight when Earth's orbit is maximally antiparallel or parallel to its rotation. Six hours offset torque is minimized. For test masses aligned northsouth, torque maxima occur at 0600 hrs and 1800 hrs. The entire apparatus may be rotated with phaselock detection to decouple noise.
dg/dh = 0.000308766[1(0.0014665)sin^{2}(lat)] cm/sec^{2}meter
dg/ds = 0.0008109[cos[(2)(lat)]+(0.0022)cos[(4)(lat)](0.0033)cos[2(lat)]sin^{2}(lat)] cm/sec^{2}kmr = geocentric radius
a = total centripetal acceleration
lat = latitude
g = gravitational acceleration
h = altitude above sea level
s = distance
Gravitational versus inertial mass anomaly is the dimensionless Eötvös parameter , the difference divided by the average of m_{g}/m_{i} for the two sets of test masses "A" and "B."
= 2[(m_{g}/m_{i})_{A}  (m_{g}/m_{i})_{B}] / [(m_{g}/m_{i})_{A} + (m_{g}/m_{i})_{B}]
Falling mass gravimeters detect 10^{9} of ambient gravity[45]. Levitated dual sphere superconducting gravimeters[46] sense 10^{11}. Eötvös balances discern ~ 10^{13} where room temperature vibration of component atoms interferes[47] as noise. Cryogenic 2.2°K Eötvös apparatus is being debugged[48]. Dragshielded freefall apparatus with ~ 10^{15} is proposed[49].
Eleven parity pairs of enantiomorphic space groups[43] have inverse lattices in which all three spatial coordinates reverse in sign. Unlike chirality, parity changes the space group:
P3_{1}  P3_{1}12  P3_{1}21  P4_{1}  P4_{1}22  P4_{1}2_{1}2  P4_{1}32  P6_{1}  P6_{2}  P6_{1}22  P6_{2}22  
P3_{2}  P3_{2}12  P3_{2}21  P4_{3}  P4_{3}22  P4_{3}2_{1}2  P4_{3}32  P6_{5}  P6_{4}  P6_{5}22  P6_{4}22 
Enantiomers are not interconverted by time reversal plus proper spatial rotation in "true chirality"[50]. Quantitative geometric parity can be calculated[51]. Macroscopic chirality is a complex[52] concept. Liaisons between paired chiral bodies (a left shoe on a left or right foot) exhibit different free energies. Parity Eötvös experiments are profound tests of spacetime geometry.
Spacetime events are empirically lefthanded: cobalt60 betadecay parity violation[53]; Kmeson[54] and Bmeson[55] decay chargeparity violation. Parity violation[56] is inarguable. Enantiomeric excited nuclei have different energy levels for otherwise identical transitions[57]. All atoms are intrinsically homochiral[58]. Excess matter vs. antimatter requires chiral discrimination[59]. A weakly interacting reverseparity entire universe is proposed[60]. Linearly polarized massless photons traversing intergalactic space do not show optical rotation[61]. A photon is not a valid inertial reference frame. Massed interactions are different.
Gravitation arises from spacetime curvature (Riemann, metric), torsion (Weitzenböck, affine[13]), or both. Teleparallelism has torsion acting as a force, analogous to electrodynamics' Lorentz force equation, without geodesics. Parity pair Eötvös experiments offer selfsimilar helical geometry in greater active masses, subtended areas, and volumes than spinpolarized test masses[39,62] whose Eötvös experiments null within experimental error. Nonmetric gravitation theories also postulate the Equivalence Principle[63].
Metric theories of gravitation postulate[64] spacetime is a Lorentzian manifold, test particles pursue spacetime geodesics (all sufficiently small bodies subject only to gravitational interactions and starting with the same initial positions and velocities follow identical spacetime trajectories), and the Strong Equivalence Principle obtains. Component form physical laws move from flat into curved spacetime by replacement of partial derivatives with covariant derivatives (note factor ordering given noncommutation of covariant derivatives, as classical mechanics transforms into quantum mechanics). Geometry's response to matter distribution, Einstein's field equations, arises from the principle of least action. Equivalence Principle violation annuls spacetime curvature.
Local symmetries create conservation laws through Noether's theorem[65]. A conserved quantity derives from each symmetry commuting with time, and the reverse. A divergencefree current (conserved property) arises if the Lagrangian or the action is invariant under continuous transformation.
A symmetry can be broken explicitly  a term in the action or equations of motion may not be invariant. A symmetry can be broken anomalously  not all classical theory symmetries exist in the corresponding quantum theory. Quantum field theory anomaly spoils renormalizability. Anomaly absence in the Standard Model is crucial. A symmetry can be broken spontaneously if it is an exact symmetry of the equations of motion but not of a particular solution therein. Noether's theorem holds if the symmetry is not broken explicitly. Conservations can be relaxed in subsystems displaying reduced symmetry (Born scattering approximation, Fermi's golden rule, Snell's law).
A classical field theory conserved quantity does not demand a quantum field theory conserved quantity in kind. Parity Eötvös experiments offer measurable anomalies between General Relativity's continuous symmetries and test masses' discrete symmetries.
Theory  Metric  Other Fields 
Free Elements  Status 

Newton (1687) [67] 
Nonmetric  Potential  None  Nonrelativistic theory 
Nordstrom (1913) [66,68] 
Minkowski  Scalar  None  Unpredicted light detection 
Einstein (1915) [67,68] 
Dynamic  None  None  Viable 
Whitehead (1922) [69] 
Fixed  Tensor  1 parameter [70] 
Possibly viable[71] 
BelifanteSwihart (1957) [68] 
Nonmetric  Tensor  K parameter  Contradicted by Eötvös experiments 
BransDicke (1961) [67,68,72] 
Generic Scalar 
Dynamic Scalar 
parameter  Viable for >620 
Tensor (1970) [80] 
Dynamic  Scalar  2 free functions 
Viable 
Ni (1970) [67,67] 
Minkowski  Tensor Vector Scalar 
1 parameter 3 functions 
Preferredframe effects unobserved 
WillNordtvedt (1972) [68] 
Dynamic  Vector  None  Viable 
Rosen 1973) [68] 
Fixed  Tensor  None  Contradicted by binary pulsar data 
Rastall (1976) [68] 
Minkowski  Tensor Vector 
None  Viable 
Variable Mass,VMT (1977) [68] 
Dynamic  Scalar  2 parameters  Viable for wide range of parameters 
Modified Newtonian Dynamics, MOND (1983) [73] 
Nonmetric  Potential  Free Function  Nonrelativistic theory 
Gravitation is modeled by loop quantum theory[74], brane/string/Mtheory[75], Lorentzian lattice quantum gravity[76]..., though few predictions exist to be tested. Mtheory fails to predict the absence of supersymmetry at low energies, the presence of a positive cosmological constant, and the utter absence of a massless scalar field that its component string theories predict in abundance. Loop quantum theory predicts unobserved vacuum dichroism  lightspeed being very slightly smaller for high energy photons.
The MichelsonMorley experiment showed space is isotropic for massless photons to differential 10^{8} in 1887 and 1.7·10^{15} in 2002[77]. The Higgs field[78] uncouples massless and massed particles. Single crystal silicon polarized interferometry (revolving Earth orbiting the sun) detects 10^{2}[79] to 10^{3}[80] relative, ten orders of magnitude too insensitive (ColellaOverhauserWerner[81] and BonseWroblewski[82] neutron interferometers; KasevichChu[83] atom interferometers).
Gyrotropy is assigned spiraling toward the observer, geometry spirals away from the observer. Dextrorotatory gyrotropy is levorotatory structure (optically righthanded quartz is space group P3_{2}21). Distant from optical transitions in ionic crystals[86]:
 Smaller pitch (repeat distance) of atomic helices gives larger gyrotropy.
 Smaller radial distance from helical axes gives larger gyrotropy.
(Smaller relative radius or pitch increases geometric parity. Smaller radius and pitch together leave scale invariant geometric parity unaltered) Larger electronic (Pauling) polarizability gives larger gyrotropy.
 Tangential or radial (reverses the contribution) polarizability orientation in the plane normal to the helical axis dramatically alters gyrotropy.
Solid state optical activity theory[87] flags three irreducible tensor components, the last two of which tolerate mirror planes of symmetry (gyrotropy without chirality):
 Pseudoscalar, from molecular chirality and persists in disordered solution.
 Vector. A property of pyroelectric lattices.
 Pseudodeviator, from lattice symmetry. Silver thiogallate[88], AgGaS_{2} with nonpolar achiral tetragonal space group I42d (#122), has immense optical rotatory power: 522°/millimeter along [100] at 497.4 nm, reversed along [010].
Gyrotropy arises from differing refractive indices (n) for left and rightcircularly polarized light in a medium. Gyrotropy summed over the electromagnetic spectrum vanishes:
Chiroptical methods do not measure geometric parity.
 Oscillator strength or fsum rule[89]. The fvalues sum to unity for all transitions from a given state;
 The ThomasReicheKuhn sum rule[90]. The photoabsorption cross section integral equals the total number of electrons involved in the collective excitation.
 KramersKronig relationship[91]. Refractive index has a real part (n) arising from phase velocity and an imaginary part (k) arising from absorption. Optical rotatory dispersion is the real part and circular dichroism is the imaginary part of one complex spectrum.
 The sum over (n1) across the electromagnetic spectrum is zero. The difference between two (n) for orthogonal polarizations sums to zero overall[92].
Chiral Test Mass  Temp, °K  aaxis, Å 
baxis, Å  caxis, Å  Volume,
Å^{3}  Density, gm/cm^{3} 
Space Group 

grey Selenium[94]  300 85 2* 
4.3712 4.304 4.278 
4.3712 4.304 4.278 
4.9539 4.968 4.974 
81.975 79.650 78.909 
4.798 4.935 4.985 
P3_{1}21 P3_{2}21 
Tellurium[95]  4.456  4.456  5.921  101.82  6.243  P3_{1}21 P3_{2}21 

neutron diffraction xray diffraction SiO_{2}[96] 
278 298 13 
4.9134 4.9137 4.9021 
4.9134 4.9137 4.9021 
5.4052 5.4047 5.3977 
113.01 113.01 112.33 
2.649 2.649 2.665 
P3_{1}21 P3_{2}21 
Cinnabar, HgS  4.145  4.145  9.496  141.29  8.203  P3_{1}21 P3_{2}21 

Berlinite, AlPO_{4}  4.766 4.9438 
4.766 4.9438 
10.724 10.9498 
210.96 231.771 
2.880 2.621 
P3_{1}21 P3_{2}21 

Palladium antimonide telluride, PdSbTe[93] 
6.5362  6.5362  6.5362  279.24  8.463  P2_{1}3  
Azatwistanone[97] (model)  6.662  13.36  8.606  (756.60)  1.327  (P2_{1}/n)  
[4]Triangulane[98]  120  5.798  10.434  11.872  (718.21)  1.112  P2_{1}2_{1}2_{1} 
[5]Triangulane[99]  110  12.024  5.3156  7.609  443.17  1.096  C2 
benzil  294 100 70 
8.402 8.356 14.380 
8.402 8.356 8.373 
13.655 13.375 13.359 
834.81 808.76 1608.1 
1.255 1.295 1.303 
P3_{1}21 P3_{2}21; P2_{1} 
Chiral crystals are complex objects[86]: Quartz (SiO_{2}; no detectable gyrotropy 56.16° from crystallographic [0001]), paratellurite (TeO_{2}; P4_{1}2_{1}2, P4_{3}2_{1}2), (HgS; P3_{1}21, P3_{2}1), langasite[100] (La_{3}Ga_{5}SiO_{14}; P321), berlinite (AlPO_{4}; P3_{1}21, P3_{2}21), Bi_{12}(Si, Ge, or Ti)O_{20} (I23), sodium bromate and chlorate (P2_{1}3) all possess at least one traceable counterhelix to the dominant atomic helices. Space group P2_{1}3 in QCM often shows good CHI growth, COR=1 and DSI=0 through 1100 atoms contained. Ferroelectric crystals are polarity twinned. Grey selenium must be grown from hot aniline solution[101] or different allotropes obtain; melt growth is disordered. Mechanical trauma deeply disrupts its lattice.
Achiral benzil[102] crystallizes in enantiomorphic space groups. A twinning phase transition[103] occurs at 84°K. Boules obtain by BridgmanStockbarger directional solidification[104]. Good crystals require solution growth as hot solid plastically deforms into lattice dislocations. Molecular solids have undesirable large unit cells.
An Eötvös rotor must be balanced around its vertical supending fiber for mass and moments of inertial overall. Petitjean's mathematics demands each test mass have three equal moments of inertia for maximum parity divergence. Full parity Eötvös experiment test masses are then explicit  identical chemical compositions, macroscopically identical forms, single crystal solid convex bodies without surface drilling or internal hollowing, and extremal opposite parity crystal lattices. Solid spheres or solid right cylinders with height equal to (radius)3 qualify.
Quartz offers a unique opportunity to perform two hemiparity Eötvös experiments, space group P3_{1}21 or P3_{2}21 quartz against amorphous fused silica. Different densities must be compensated. As all test masses are gold plated, this is straightforward.
d_{quartz} = 2.649 g/cm^{3}
d_{silica} = 2.203 g/cm^{3}
d_{gold} = 19.3 g/cm^{3}
A quartz ball of radius R would be balanced by a fused silica ball of radius (Rr) with a vacuumplated gold shell of thickness r. The general solution, sphere or cylinder, is
r^{3}  3Rr^{2} + 3R^{2}r  [(d_{quartz}d_{silica})/(d_{gold}d_{silica})]R^{3} = 0
r^{3}  3Rr^{2} + 3R^{2}r  (0.0260864)R^{3} = 0
www.1728.com/cubic.htm
For R = 1 centimeter in quartz, r = 87.72 micrometers in gold.
Eötvös balances typically employ test mass right cylinders or spheres. A homogeneous isotropic solid sphere is the most symmetric figure in three dimensions. It possesses the following symmetry elements:
The rigorous abstract mathematics of chirality has been extensively examined and reduced to QCM software. CHI[51] is the quantitative calculation of the geometric parity divergence of N points. CHI is globally minimum for all rotations (R) and translations (t) for all correspondences (P) permitted by the colors and/or graph:
CHI = (d)[Min_{{P,R,t}}D^{2}]/4Twhere d is the Euclidean dimension, D^{2} is the sum of the N squareddistances between the set and its parity inversion for a fixed pairwise correspondence with coincident centers of mass, and T is the geometric inertia of the set. All points are assigned equal mass, though their labels may differ. CHI varies between zero (exactly superposable (x,y,z)parity inversions) and one (perfect divergence) inclusive. CHI is a continuous function of nuclear coordinates only  independent of translation, scale, and size. One value exists for a given target and its inverse lattice. It detects zero and approach toward zero. It does not require empirical constraints.
CHI is limited by three conditions:
Parity is a subset of chirality. Lattice chirality is an emergent phenomenon wherein components collectively evolve discrete symmetry system properties absent and unpredicted from any smaller subset. Lattice chirality does not simplify below a microscopic packing unit seamlessly translated along its axes in three dimensions into a selfsimilar macroscopic crystal. Four possibilities obtain:
(2) includes local achiral superunits as in La Coupe du Roi[105]. Cut a ball (apple) in from a pole to its equator along a great circle (lines of longitude 180° apart) with a plane. Repeat rotated 90° at the other pole (perpendicular planes). Make two equatorial cuts in the same direction, surface to center radial sectors, from the equatorial edge of one pole cut to the other and again on the opposite side (rotate about the polar axis; cut, skip, cut). The ball (apple) cleaves into identical homochiral halves. Homochiral components can assemble into a zero chirality lattice. Aspects of (2) lessen parity pair divergence.
Aspects of (2) weaken parity Eötvös experiments. If the set of points corresponding to positions occupied by all symmetry copies of the asymmetric unit is intrinsically chiral (3_{1}3_{2} 4_{1}4_{3} 6_{1}6_{5} 6_{2}6_{4} screw axes), CHI>zero even if the formula units are achiral. 2_{1} 4_{2} 6_{3} or no screw axes can allow geometric chirality to decrease with volume even if formula units are intensely chiral. 10 Sohncke space groups become achiral if their unit cell contents are achiral. 6 Sohncke space groups have no screw axes at all. 21 additional Sohncke space groups contain both left and right screw axes. 22 additional Sohncke space groups contain 2_{1} 4_{2} 6_{3} that are each simultaneously left and righthanded screws with the same screw vector[106]. These 59 space groups contain conflicting geometries that decrease CHI increase with increasing lattice sample radius.
Deficiencies  Space Groups  

Zero intrinsic lattice chirality 
P1  P2  P2_{1}  C2  P4  P4_{2} 
I4  P3  P6  P6_{3}  
2_{1} 4_{2} 6_{3} screw axes only 
P222_{1}  P2_{1}2_{1}2  P2_{1}2_{1}2_{1}  C222_{1}  C222  F222 
I222  I2_{1}2_{1}2_{1}  P4_{2}22  P4_{2}2_{1}2  I422  P6_{3}22  
C_{n} axes only  P222  P422  P42_{1}2  P312  P321  P622 
Opposite sense helices 
I4_{1}  I4_{1}22  R3  R32  P23  F23 
I2_{1}3  P432  P4_{2}32  F432  I23  P2_{1}3  
P4_{1}32  F4_{1}32  I432  P6_{2} P6_{4} 
P6_{2}22 P6_{4}22 
I4_{1}32 P4_{3}32 

Samesense helices plus 2_{1 }  P4_{1} P4_{3} 
P4_{1}22 P4_{3}22 
P4_{1}2_{1}2 P4_{3}2_{1}2 
P6_{1} P6_{5} 
P6_{1}22 P6_{5}22 
The three pairs of remaining enantiomorphic space groups possess unique screw axes[106]. These structures exclude primary conflicting symmetries and support maximal CHI increase with increasing lattice sample radius . A given material's lattice must still be examined with QCM for graph theoretic correspondences beyond the identity correspondence.
Acceptable  Space Groups  

Samesense helices 
P3_{1} P3_{2} 
P3_{1}12 P3_{2}12 
P3_{1}21 P3_{2}21 
10^{13} mass anomaly is 2.148 cal/gram. Water hydrogen bond energy[107] is 5.57 kcal/mole (328 cal/gram), carboncarbon single bond energy is 85 kcal/mole (3500 cal/gram). Test the Equivalence Principle with calorimetry [108]. Parity pair single crystals' heats of combustion vary with geographic orientation, local time of day, and impressed inertial acceleration.
Property[109]  Value  

Molecular weight 
210.2322 g/mol  
Triple point 
94.864°C  
Dynamic melting point 
Onset 94.43°C 
Meniscus 94.77°C 
Melt 95.08°C 
Thermodynamic melting point 
Onset 94.55°C 
Meniscus 94.72°C 
Melt 94.86°C 
Enthalpy of fusion mp = 94.82°C 
112.0 J/g 26.77 cal/g 
23.546 kJ/mol 5.6276 kcal/mol 

Enthalpy of fusion mp = 94.85°C 
110.6 J/g 26.44 cal/g 
23.26 kJ/mol 5.559 kcal/mol 

Enthalpy of fusion mp = 94.86°C 
112.0 J/g 26.76 cal/g 
23.54 kJ/mol 5.626 kcal/mol 

Enthalpy of parity divergence 
26.96 J/g 6.444 cal/g 
5.668 kJ/mol 1.355 kcal/mol 

Differential enthalpy of parity divergence 
~24% for 3·10^{13} g/g parity anomaly E = (3·10^{16} kg)(299,792,458 m/sec)^{2} 
Putting unique atoms' fractional coordinates through space group symmetry operations[43] populates the unit cell, the smallest repeated lattice volume. Selenium and tellurium unit cells contain three atoms, but three points define an achiral plane. Moved, the unit cells each contain six halfatoms  the repeating chiral unit as an emergent phenomenon. Selenium and tellurium values would not change given CHI weighted for atomic mass and fractional unit cell occupancy. The goal is the fastest growing lattice CHI from the smallest unit cell with the smallest volume/atom and heaviest atoms in the best parity space group.
The abstract mathematics of quantitative parity divergence only sees points' coordinates for analysis. A 3300 Å^{3} cube of tellurium lattice with 98 atoms (innermost 20% electron probability ellipsoids shown) appears to be a spatially repeating collection of points. Where is the parity divergence or even chirality?
Drawing nearest neighbor distances reveals aligned homochiral 3fold helices. Perception and "common sense" are irrelevant. Rigorous mathematics is the only valid analysis. (Look beyond the screen to relax your eyes' convergence. The twoblock image will double, one view from each eye. Increasing distance from the screen eases the deconvergence process. As you go increasingly walleyed the images will move further apart causing the center blocks to overlap. When the bottom crosses merge you can look into the 3D image.)
A macroscopic test mass is an apparently continuous medium. Lattice volumes are quantized by nearest neighbor distances at scale. The number of calculations needed to determine CHI grows as the products of factorials of the number of each color of atom (not each chemical element) given full QCM diagnostics. A lattice is qualified by QCM evaluation of consecutive angstrom increment radii from half the longest unit cell axis length to at least 1000 atoms contained. A QCM 1000atom run will typically require 1020 CPUhours in an RS6000/Power3 given connectivityoptimized HyperChem *.hin file input. It is a quirk of QCM graph theory analysis that 15 points input as *.xyz file format would require more than the age of the universe to calculate  to give the same answer as *.hin format input after a few minutes.
QCM begins with enumeration of graph automorphisms in concentric layers of the array starting at its origin. "Direct symmetry index" DSI, the normalized minimized sum of the N squareddistances between the vertices of the dset and the permuted dset, measures the set's similarity to itself. A set having no direct symmetry in a ddimensional Euclidean space is still not symmetric after immersion in a higher dimensional space while a nonspecific chiral set would become achiral. DSI>0 beyond a few contained unit cells is a disqualification for extremal paritydivergent test masses.
"Correspondences" COR includes the identity element but is not a count of group theory symmetry elements. COR>1 beyond a few contained unit cells is a disqualification for extremal paritydivergent test masses. Assigning different atom labels (SiO_{2}) does not default assign different graph theory point colors. QCM numeric outputs are independent of input file structure format, atom connectivity (if any), and list ordering. All points are assigned unit weight. Mass is mass.
If DSI=0 and COR=1, a FastCHI subset of QCM is valid. FastCHI coding calculates CHI (how a set is not similar to its parity inversion) linearly with the number of atoms (runs in O(n) time and O(1) space). A 2 GHz Linux PC in AMD Athlon hardware (Wintel is 40% slower) can examine ~10^{9} atoms/second. Large quartz radii were run in a cluster of 16 AMD Opteron848s in 30 days. Tellurium, quartz, and berlinite were given 609, 90,386 (150,000 CPUhrs; 444 quadrillion atoms), and 571 radius samplings respectively to calculate the following:
Sphere Radius, Å  Lattice CHI  Total Atoms 
Unit Cells  Volume, Å^{3} 

Unit Cell* Tellurium Quartz Berlinite 
0.018926 0.173940 0.174566 
6 11 23 
1 1 1 
102 113 211 
Quartz 3.6 Berlinite 5.0 
0.843228753630922340 0.928954111975661469 
14 46 
1.6 2.6 
176 539 
10  0.953161251843284545 0.989463287292071760 0.922918192326544688 
126 340 350 
42 38 19 
4,189 4,269 4,102 
20  0.994970730830562692 0.991987685060543311 0.989437910906527996 
987 2,670 2,854 
329 297 159 
33,510 33,528 33,449 
30  0.995341618942362163 0.994391347133448844 0.999136218439710468 
3,315 9,014 9,658 
1,105 1002 537 
113,100 113,200 113,200 
40  0.994311225094353860 0.995770871078683400 0.999632392784788830 
7,906 21,346 22,888 
2,635 2,372 1,272 
268,100 268,000 268,200 
50  0.997182871713665811 0.997497356338878373 0.997751761704918893 
15,428 41,696 44,665 
5,143 4,633 2,481 
523,600 523,600 523,500 
100  0.998674087455959555 0.999455952622276367 0.999643499178059791 
123,411 333,526 357,384 
4.11·10^{4 } 3.71·10^{4 } 1.99·10^{4 } 
4.19·10^{6 } 4.19·10^{6 } 4.19·10^{6 } 
500  0.999991634437844450 0.999980456860055180 0.999989604104047446 
15,427,878 41,696,416 44,676,376 
5.14·10^{6 } 4.63·10^{6 } 2.48·10^{6 } 
5.24·10^{8 } 5.24·10^{8 } 5.24·10^{8 } 
1,000  0.999993481890267883 0.999997440127579185 0.999998427421108820 
123,422,301 333,575,254 357,408,684 
4.11·10^{7 } 3.71·10^{7 } 1.98·10^{7 } 
4.19·10^{9 } 4.19·10^{9 } 4.19·10^{9 } 
2,000  0.999998519283409702 0.999999080483513838 0.999999455994921283 
987,378,232 2,668,598,322 2,859,270,260 
3.29·10^{8 } 2.97·10^{8 } 1.58·10^{8 } 
3.35·10^{10} 3.35·10^{10} 3.35·10^{10} 
5,000  0.999999871340933588 0.999999789636193490 0.999999929783070689 
15,427,793,636 41,696,845,260 44,676,133,458 
5.14·10^{9 } 4.63·10^{9 } 2.48·10^{9 } 
5.24·10^{11} 5.24·10^{11} 5.24·10^{11} 
10,000  0.999999897747013216 0.999999968983088517 0.999999980194705342 
123,422,354,560 333,574,731,752 357,409,097,084 
4.11·10^{10} 3.71·10^{10} 1.99·10^{10} 
4.19·10^{12} 4.19·10^{12} 4.19·10^{12} 
20,000  0.999999983730030373 0.999999992565814146 0.999999990115051648 
987,378,792,095 2,668,597,813,994 2,859,272,764,336 
3.29·10^{11} 2.97·10^{11} 1.59·10^{11} 
3.35·10^{13} 3.35·10^{13} 3.35·10^{13} 
100,000  0.999999999546726956 0.999999999648281730 0.999999999778529238 
123,422,348,782,767 333,574,726,196,900 357,409,096,311,540 
4.11·10^{13} 3.71·10^{13} 1.98·10^{13} 
4.19·10^{15} 4.19·10^{15} 4.19·10^{15} 
Tellurium 200,000 300,000 400,000 505,000 
0.999999999810150121 0.999999999962125567 0.999999999948796222 0.999999999981536820 
987,378,790,078,890 3,332,403,416,661,100 7,899,030,320,918,100 15,895,271,169,403,000 
3.29·10^{14} 1.11·10^{15} 2.63·10^{15} 5.30·10^{15} 
3.35·10^{16} 1.13·10^{17} 2.68·10^{17} 5.39·10^{17} 
Quartz 200,000 300,000 400,000 500,000 750,000 1,010,000 
0.999999999899102952 0.999999999990628857 0.999999999964449503 0.999999999987239756 0.999999999987958381 0.999999999994484622 
2,668,597,809,177,800 9,006,517,605,688,300 21,349,690,977,976,000 41,698,615,192,377,000 140,732,826,273,680,000 343,696,999,447,690,000 
2.97·10^{14} 1.00·10^{15} 2.37·10^{15} 4.63·10^{15} 1.56·10^{16} 3.82·10^{16} 
3.35·10^{16} 1.13·10^{17} 2.68·10^{17} 5.24·10^{17} 1.77·10^{18} 4.32·10^{18} 
The berlinite unit cell is the quartz structure with a doubled caxis, SiO vs. AlOP0. If atoms are taken to be anonymous mass the caxis doubling vanishes. The smaller the volume/atom the faster CHI vs. radius asymptotically approaches CHI=1, and the less positive is the graph intercept of log(1CHI) vs. log(radius).
Lattice  Volume, Å^{3}/atom 
Intercept 

Tellurium  33.939  0.788633 
Quartz  12.557  0.546185 
Berlinite  11.720  0.506840 
Te unit cell axes are a,b=4.456 Å, c=5.921 Å. The a,baxes or caxis were given large arbitrary variations and CHI was progressively calculated from 5.113 Å radius up to a 10,000 Å radius ball with 390430 radius samplings for the test cases. (The mathematics of CHI is sensitive to shape but not to scale.)
Table XV. Te DISTORTED UNIT CELLS DATA
Lattice  a,baxes, Å  caxis, Å 
Unit Cell Volume, Å^{3}  Slope  Intercept  Total Atoms 

Theory, Te  2  arbitrary  
Tellurium  4.456  5.921  101.816  1.99950  0.786973  1.05·10^{15} 
Short a,b  0.913  5.921  4.274  1.97716  0.000013  2.94·10^{12} 
Short c  4.456  1.184  20.360  2.00775  0.302008  6.17·10^{11} 
Long a,b  17.287  5.921  1532.374  1.97145  1.46827  8.20·10^{9 } 
Long c  4.456  30.307  521.151  1.98165  1.38036  2.41·10^{10} 
Te unit cell fractional coordinates are 0.2636, 0.0 ,1/3. The symmetry operations of space group P3_{1}21 then populate the unit cell with three atoms total (six halfatoms in the native structure). All 27 permutations of an arbitrary 0.389 change were evaluated through all three fractional coordinates added (P), subtracted (M), or unchanged (0). CHI was progressively calculated with 401 samplings from a 9 Å radius ball containing 8494 atoms to a 10,000 Å radius ball containing 123.4 billion atoms in each test case.
Table XVI. Te DISTORTED FRACTIONAL COORDINATES
Fractional Coordinate  Change  Value  Mark 

a/x  none +0.389 0.389 
0.2626 +0.1254 0.6526 
0 P M 
b/y  none +0.389 0.389 
0.0 +0.3890 0.3890 
0 P M 
c/z  none +0.389 0.389 
1/3 +0.722333... 0.055666... 
0 P M 
Table XVII. Te DISTORTED FRACTIONAL COORDINATES DATA
Change, a/x,b/y,c/z  Slope  Intercept  Change, a/x,b/y,c/z  Slope  Intercept  

P 0 0  2.01944  0.901199  M 0 0  2.02110  0.811078  
0 P 0  1.99126  0.720836  0 M 0  1.99929  0.779414  
0 0 P  1.99906  0.798343  0 0 M  1.99987  0.798708  
P P 0  1.99798  0.775438  M M 0  1.99103  0.718670  
P 0 P  1.99552  0.864120  M 0 M  2.02444  0.848134  
0 P P  1.99069  0.691879  0 M M  1.99191  0.750860  
P M 0  1.99173  0.759762  M P 0  2.00055  0.783633  
P 0 M  1.99590  0.865412  M 0 P  2.02192  0.844351  
0 P M  1.99736  0.705094  0 M P  2.00640  0.783672  
P P M  2.00477  0.779102  M M P  1.99885  0.708784  
P M P  2.00139  0.764053  M P M  2.00811  0.795166  
M P P  2.00692  0.805269  P M M  1.99445  0.741966  
P P P  1.99071  0.744485  M M M  1.99097  0.692458  
0 0 0 Tellurium 
1.99950  0.786973 
Theory predicts an exact 2 slope. The less positive the intercept the more parity divergent is the lattice. Unit cell volume  smaller volume is more divergent  is much more important than atom placement within the unit cell of an acceptable parity pair space group.
Dr. Penelope Smith at Lehigh University notes that CHI is a connection between eigenvalues, special functions, and their representation theory with solid angles and exponentials of fractions of pi. The intercept is now the solid angle subtended by the smallest vertex angle of a polyhedron (the supplement of its dihedral angle) defined by the caxis helix,
log(1CHI)= 2[log(radius)] + [(180) ()/60] 
( is the smallest vertex angle in the helix. The slightly distorted tetrahedral OSiO helix angle is 110.56° vs. 109.47° undistorted)log(1CHI) = 2[log(radius)] + 0.494277
3cm quartz test mass has CHI = 1  1.387·10^{16}, theory
3cm quartz test mass has CHI = 1  1.535·10^{16}, graph fit
Subtended solid angle was tested against explicit calculation for quartz, tellurium, distorted tellurium unit cells, and berlinite aluminum phosphate (double length caxis) that all express rigorous QCM DSI=0 COR=1 regardless of atom labeling or connectivity. It works to 10% difference/average values. As the standard deviation of calculated log(1CHI) is typically 0.997 units, the two routes to fitting CHI give indistinguishable values.
The optimum crystal structure would have the
The parity Eötvös experiment using parity pair single crystal tellurium, cinnabar, or quartz is astounding robust against a real world minor fraction of crystal structural imperfections and impurities. No heavy atom crystal lattice is substantively better than single crystal tellurium, cinnabar, or quartz for achieving maximum parity divergence as Eötvös experiment test masses.
Hexagonal Cylinder  Lattice CHI  Total Atoms 
Unit Cells  Volume, Å^{3} 
Inertial Disparity* 

Tellurium unit cell positions 7 helices x 5 atoms 19 helices x 9 atoms 37 helices x 13 atoms 61 helices x 16 atoms 91 helices x 20 atoms 127 helices x 23 atoms 169 helices x 27 atoms 217 helices x 30 atoms 271 helices x 34 atoms 
0.018926 0.672986 0.949332 0.971947 0.988757 0.987677 0.986497 0.992871 0.983477 0.997870 
6 35 171 481 976 1820 2921 4563 6510 9214 
1 12 57 160 325 607 974 1521 2170 3071 
102 1,189 5,811 16,344 33,164 61,844 99,256 155,051 220,940 312,711 
64.00 % 13.55 % 5.337% 4.132% 0.920% 0.930% 0.547% 0.796% 1.181% 0.226% 
Selenium and tellurium are exceptional for small unit cell 100% heavy atom content forming identical helices. Nextnearest neighbor distances are almost identical (3.436 Å and 3.491 Å respectively), but selenium bonds are shorter (2.373 Å and 2.835 Å). Selenium's fragile lattice is disordered by physical manipulation, and the grey allotrope cannot be obtained as sufficiently large single crystals by solution growth. Tellurium and selenium display identical variation of CHI with spherical radius despite selenium having a 17.2% shorter pitch when scaled to the same helical radius. The mercury sublattice of cinnabar duplicates the tellurium lattice. It does not confer "tellurium breath" nor does it react with gold during vacuum gilding.
Large tellurium crystals obtain by Czochralski growth and annealing under hydrogen (50200 hours at 320380° C)[110] which further removes oxygen, sulfur, selenium, arsenic, lead... as volatile hydrides. Physical helicity has the same sense as optical rotation (±55.6°/mm at 5000 nm[95]). Crystals have easy cleavage planes along . Typical dislocation densities are 13·10^{4}/cm^{3} with a hole concentration of 10^{14}/cm^{3} (intrinsic ptype semiconductor). A single crystal tellurium right cylinder two centimeters in diameter and long contains 1.83·10^{15} helices for a summed axial length of 3.66·10^{13} meters (33.9 lighthours; 3.1 times the solar system's diameter). Each helix has 3.38·10^{7} 360° turns. Both elements' helices are remarkably isolated in space (view with CHIME plugin; hold left mouse click and drag; right mouse click for menu; reduced window eases stereoimage fusion).
A classical Eötvös experiment opposes test masses' nuclear binding energies/baryon. Riley Newman's 2.2°K Eötvös balance[48] will oppose Be and Mg (weighted for natural isotopic abundance[111]). Neutron (939.565330 MeV) and proton (938.271998 MeV) average mass equivalent, weighted for 16 protons and 17.3202 neutrons (magnesium isotopic abundance), is 938.944286 MeV.
Be = 6.462844 MeV/baryonA one centimeter diameter quartz spherical test mass with 99.9726% parityactive mass calculates to CHI=0.999999999999998619 (1  1.381·10^{15}) of perfect CHI=1 geometric parity divergence. Parity Eötvös experiments offer 521 times larger property divergence than the planned composition experiment.
Mg = 8.265129 MeV/baryon
[Mg  Be]/[(17.3202n+16p)/33.3202] = 0.1919%
Cultured quartz is sold as multikilogram single crystals of both chiralities. The quartz lattice (view with CHIME plugin) contains conflicting helices: 6fold helices with long axes parallel to the zaxis are of opposite chirality to parallel 12fold helices; 4fold helices with the 12fold helices' chirality form a 60° grid weaving through the xy plane. Care must be taken not to heat quartz near 573° C during working to avoid a first order twinning transition. Quartz is a potentially poorer geometric test mass than tellurium for its conflicting helices and lower average atomic weight. However, quartz is hydrothermally grown to 20 kg single crystals whereas tellurium is a monumental struggle for Czochralski growth to one centimeter single crystals.
Quartz crystal imperfections at all scales degrade its resonant acoustic quality factor Q,
Q = 2(energy stored)/(energy lost) each cycle.Infrared OHstretch absorption at 3410, 3500 or 3585 cm^{1} is used to predict Q at 5 MHz[112], EIA Standard 4771, JIS C 6704, and IEC 60758, e.g.,
Alpha = (A_{3500}  A_{3800})/Ywhere A is absorbance at the given wavenumber and Y is the pathlength in centimeters of a Ycut crystal (zregion material, as opposed to +x, x, or s). Proton impurities in quartz terminate otherwise continuous ~SiO~ helices as helix~OH. Proton uptake varies as crystal growth rate. Slower growth gives higher Q.
Grade  Minimum Q at 5 MHz  Alpha  A_{3410}  A_{3500}  A_{3585} 

Aa  3.8·10^{6}  0.015  0.075  0.026  0.015 
A  3.0·10^{6}  0.0240.033  0.082  0.033  0.024 
B  2.42.2·10^{6}  0.0450.050  0.100  0.045  0.050 
C  1.8·10^{6}  0.0600.069  0.124  0.060  0.069 
D  1.41.0·10^{6}  0.1000.120  0.145  0.080  0.100 
E  1.00.5·10^{5}  0.1600.250  0.190  0.120  0.160 
The properties of a potential field depend only on position. The amount of energy gained or lost is independent of what path is taken start to finish. The integral of potential over path to give energy only depends upon the end points. Green's function is an integral kernel that can solve an inhomogeneous differential equation with boundary conditions. Green's function only depends on the distance between the source and the measured field points.
http://www.maths.soton.ac.uk/staff/Andersson/MA361/node46.html The gravitational potentials of configurations (triaxial ellipsoids, spheroids, spheres, disks) in Newtonian gravity, i.e. the potentials derived by integration of the Poisson equation Green's function 1/r  r' over the volume of the configuration, are well known. A Green's function solution is unique.
Consider a parity pair of quartz test masses in the weak field limit (Newtonian). Choose a zaxis position such that a silicon is centered at the same unit cell position in both space groups P3_{1}21 and P3_{2}21. Spatial distribution of the four oxygens around that silicon then define lattice chirality. Said oxygens are in positions "R" in the Renantiomorph and "S" in the Senantiomorph. Green's function G(x,y) has two arguments,
x = position of source
y = position of field potential measurement
Suppose G(x,y) varies in x on an angstrom scale. A function maximum falls on the central silicon. The next function maxima are at position "R" but miss position "S." There would then be a different answer for R versus Senantiomorphs. Can this obtain, and on what scale? Laplace's equation describing the behavior of gravitational potential is symmetric to parity inversion,
Replacing (x,y,z) by (x,y,z) does not change anything. The only component of gravitation is the radial one, and it only sees overall density (identical for enantiomorphic crystals). A gravitational parity anomaly must have nonNewtonian origins.
From general topological arguments,
G(x,y) = K[dist(x,y)]^{(2N)}where "K" is a scaling constant, "N" is the dimension of space (3 not 4 for traditional Relativity), and "dist(x,y)" is the distance between x,y (ignoring details about timelike components, retarded potentials, etc.). Where would subatomicscale wiggles arise to allow a gravitational parity anomaly? Selfgravitation would require the nuclear masses to be enormous to give a macroscopic effect. The average nearest neighbor internuclear SiO distance in quartz is 1.609 Å. Such small wavelength, high frequency components have no basis for origin. (A 0.161 nm photon has energy ~7.7 keV compared to SiO bond strength of 8.3 eV.)
What are the fluctuation symmetry and radius scale of CHI as radius increases? The symmetry is explicit. A radial increment sufficient to add atoms to the existing solid sphere of lattice will always do it in mirror image along all coordinate axes, space group P3_{1}21 versus P3_{2}21.
The minimum radius increment that will add atoms to the existing solid sphere of lattice decreases as radius increases. It is remarkably small even at small radii. Green's function requirements for a gravitational parity anomaly  incommensurate structure and a characteristic scale much smaller than atomic lattice spacings  are fulfilled. 1 fm = 0.00001 Å.
Radius Interval, Å  Radius Increment, fm 
Plot 

100  101.50000 200  201.50000 
100.0 100.0 
Graph 1 
100  100.45000 300  300.04500 
100.0 10.0 
Graph 2 
900  900.00450 27002700.00045 
1.0 0.1 
Graph3 
Graph 1 shows a visibly different frequency of CHI fluctuations for 100 Å and 200 Å starting radii given the same radius increments. Graph 2 and Graph3 show that multiplying the starting radius by three and dividing the radius increment by ten gives an apparently constant CHI fluctuation frequency. Multiplying the radius by 3.16228 is stable over a broad observed range. If R_{Å} is the starting radius in angstroms and r_{fm} is the corresponding radius increment in femtometers that always adds atoms, then
r_{fm} = 10^{6}/(R_{Å})^{2}A 0.5 cm radius test mass always adds atoms, has CHI fluctuations, with a radius increment of 4·10^{10} fm. Oxygen and silicon nuclear diameters are 6.05 and 7.29 femtometers respectively. Thermal fluctuations are ~5% of bond length.
Chirality is not a point phenomenon. It is demonstrated that enantiomorphic centimeterdiameter quartz balls are deeply asymptotic to theoretical maximum parity divergence. As the diameter decreases the parity divergence decreases. Chirality and parity divergence abruptly vanish at scales smaller than a bodycentered SiO_{4} tetrahedron, or a radius of 01.609 Å.
Contents  Atoms  CHI  DSI  COR 

SiO_{4}  5  0.000238  0.658392  2 
SiO_{4}Si_{4}  9  0.606391  0  1 
unit cell  11  0.408110  0  1 
Green's function monotonically increases with the number of massed points included in a spherical envelope of increasing diameter. Parity divergence oscillates about its trend line with significant amplitude. Green's function analyses do not constrain partspertrillion gravitation parity anomalies.
R = [c]/(M_{s}c^{2})](M_{p}/M_{s})^{(2/n)}R = compactified dimension radius
The "extra" six dimensions loosely comprise one charge dimension, two isospin dimensions, and three color dimensions. Their physical meaning is debatable. Below compactified dimensions' radius gravity nonclassically varies as 1/r^{(2+n)}. At larger spans the anomaly exponentially decays as a Yukawa potential, [1 + e^{R/}]. If compactified dimensions exist, spacetime can be homogeneous but not isotropic at small scales. The parity Eötvös experiment is a powerful test of spacetime isotropy at small scales.
Compactified Dimensions  Anomaly Radius, Å 
Empirical Observation, Lower Limit 

One  10^{21}10^{23}  planetary orbits affected; wrong 
Two  10^{6}10^{7}  1/r^{2} deviates; not at 0.01 mm[113] 
Three  10  crystal unit cells; (parity Eötvös) 
Four  10^{1}3·10^{2}  200x uranium nuclear diameter; 1/18 Bohr hydrogen diameter 
Five  10^{2}10^{3}  7x uranium nuclear diameter 
Six  10^{4}  2/3 uranium nuclear diameter 
Seven  2·10^{5}  proton Compton wavelength 
Gravitation empirically varies as 1/r^{2} in 59  1150 Å effective gap separations (9600  10,690 Å center of mass separation) in atomic force microscope Casimir Effect experiments[116]. An active radius (with decay at its edges) implies a characteristic anomalous volume[117] whose contents' geometry interacts with its nspace container. Parity pair Eötvös experiments directly probe three to four compactified dimensions.
The emergent scale of quartz parity is ~0.484 nm. Given a gravitational parity interaction at much smaller scale, would it show? Consider a deep bed of closepacked bowling balls and a deep bed of monodisperse 1micron radius silica balls. They have identical void space, 25.952 vol% (1[/32]) and identically shaped voids. A methane molecule has an effective diameter of 0.000409 microns [118]. Methane roars through the bowling balls but exceedingly slowly diffuses through the silica balls.
For cubic or hexagonally closepacked identical balls with radius 1, tetrahedral holes will contain a ball with radius [(6)/2]1 or 0.2247. Thus 1micron radius silica spheres have voids holding 0.4495 micron diameter spheres. This is 1100X larger than methane's effective diameter. The four trigonal windows to each tetrahedral hole will contain a sphere with radius 0.1547 radius or 0.3094 micron diameter in our case. Methane can arithmetically freely pass through the silica ball bed by a generous factor of 756, though in the real world it certainly doesn't  not by a long shot. A gravitational noninteraction cannot be arithmetically discounted by "common sense."
For three mutually tangent spheres with radii r_{1} r_{2} r_{3} there is the Soddy circle solution:
r_{4} = (r_{1}r_{2}r_{3})/{r_{1}r_{2} + r_{1}r_{3} + r_{2}r_{3} (±)2[r_{1}r_{2}r_{3}(r_{1} + r_{2} + r_{3})]}The positive solution is the inscribed fourth circle tangent to all three. (The negative solution is the circumscribed fourth circle tangent to all three.) Octahedral and cubic lattice holes are larger in radius and have larger radiused windows.
If gravitation did have an antisymmetric interaction with maximal paritypair divergent crystal lattices, a very small scale for gravitational granularity might have difficulty "oozing through," though the spacetime interpretation (with Heisenberg uncertainty?) is certainly more recondite than 3D diffusion.
Carbon, nitrogen, and oxygen atoms have similar small masses, and hydrogen is negligible. Light atoms have perceptible de Broglie wavelengths and participate in tunneling reactions[120]. Nucleic acids, proteins, sugars... CHNO organics in general are uninteresting for being much more like flat vacuum than tellurium:
Substance  Mean atomic weight 
Formula (space group)  Comments 

tellurium  127.60  Te (P3_{1,2}21)  99.98% chiral rest mass 
PdSbTe  118.59  PdSbTe (P2_{1}3)  99.98% chiral rest mass 
cinnabar  116.33  HgS (P3_{1,2}21)  99.97% chiral rest mass 
quartz  20.03  SiO_{2} (P3_{1,2}21)  99.97% chiral rest mass 
benzil  8.09  C_{14}H_{10}O_{2} (P3_{1,2}21)  99.97% chiral rest mass 
tartaric acid  9.38  C_{4}H_{6}O_{6}  2 chiral centers 
cyclooctaamylose  7.72  C_{48}H_{80}O_{40}  40 chiral centers 
palytoxin  6.55  C_{129}H_{223}N_{3}O_{54}  64 chiral centers 
insulin  7.37  C_{257}H_{383}N_{65}O_{77}S_{6}  51 amino acids 
somatotropin  7.16  C_{990}H_{1529}N_{263}O_{299}S_{7}  191 amino acids 
Chiral polymers[121] have loosely packed crystal lattices. Aggregated helicenes[122] can have []_{D}=170,000° (1400°/mm gyrotropy neat) as can neat cholesteric (nematicC) liquid crystals, but optical chirality is irrelevant. Binaphthyls are [5]helicenes less a ring junction. Jacobsen's ligand chelates uranyl[123], tin, or lead[124] with small geometric chiralities. Helicates[125] including iron[126] and silver helicates[127] ((R,R)ligand gives (S)helices) have large unit cell volumes. An alternating platinumsilver P6_{1} helix[128] has a 7898.1 Å^{3} unit cell. A claimed intensely chiral octahedral complex[129], tris[1,2dithiolatophenylenetungsten(IV)][130], has a 5649.0 Å^{3} unit cell.
Equivalence Principle violation falsifies metric theories of gravitation (leaving affine, teleparallel, and noncommutative theories unscathed). Contrasted test mass compositions afforded 400+ years of null results; calculated parity pair test masses have never been examined. A nonnull parity Eötvös experiment supercedes parityviolating energy difference[131] explanations of biological homochirality.
Gravitation applies to free elementary particles (single neutrons fall[132]). Classical physics, relativity, and quantum mechanics are all point phenomena. A "point" spans the Planck length[133], (G/c^{3})^{1/2} or 1.616·10^{26} nm and thus implies a spherical volume approximating 2.21·10^{78} nm^{3}. Three atoms define an achiral plane. Geometric parity is an emergent phenomenon vanishing at smaller than unit cell scales. A tellurium unit cell contains three atoms as six halfatoms in the walls of a 0.1018 nm^{3} irreducible configuration. A nonnull parity Eötvös experiment breaches "point phenomenon" by a volume factor of 10^{77}, confronting Planckregime physics with a bench top experiment.
"If we want to solve a problem that we have never solved before, we must leave the door to the unknown ajar," Richard Feynman[134]. The parity Eötvös experiment is a powerful new test of spacetime theories. Somebody should look.
Dr. Petitjean generously donated thousands of Alpha 2100 workstation and RS6000/Power3 CPU hours exploring CHI for molecules and crystal lattices. Advanced Micro Devices donated a cluster of 16 Opteron848s for 900 hours. This author acknowledges a great debt of gratitude.