Novel Equivalence Principle Tests

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. Non-metric 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 parity-divergent -quartz single crystal test masses. An Equivalence Principle violation >520 times that allowed for opposed composition test masses is predicted.

EXECUTIVE SUMMARY

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 alpha-Quartz
     Figure 4.  log(1-CHI) vs. log(radius): Tellurium,      505,000 Å Radius
     Figure 5.  log(1-CHI) vs. log(radius): alpha-Quartz, 1,010,000 Å Radius
     Figure 6.  log(1-CHI) vs. log(radius): Berlinite,      100,000 Å Radius
     Figure 7.  log(1-CHI) 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

Introduction

mass = (px2 + py2 + pz2 - pt2);	metric signature (+1,+1,+1,-1), Relativity
mass = (pt2 - px2 - py2 - pz2);	metric signature (+1,-1,-1,-1), field theory

1. Local bodies fall identically, because
2. Gravitational mass (F=m_gGM/r²) is indistinguishable from inertial mass (F=m_ia),
3. Regardless of composition,
4. Regardless of geometry (internal structure),
5. Regardless of mass;

The Weak Equivalence Principle extends to the Strong (Einsteinian) Equivalence Principle :

6. Non-rotating 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.
7. Local Lorentz invariance[8] (absolute velocity does not exist) and position invariance. All local free fall frames are equivalent.
8. 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 stress-energy tensor if free fall exists. If no reference frame makes gravitation locally vanish, spacetime curvature is counter-demonstrated 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 (two-slit diffraction, the Einstein-Podolsky-Rosen 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 non-point 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).

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."

*Manifolds with multiple differential structures (e.g., 7-D sphere) have multiple, mutually-exclusive 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 one-to-one correspondence with) the circle, the sphere (a surface) and the "three sphere" (not a ball) respectively. Fields with non-abelian symmetries divide into "electric" (curl-free; e.g., gravitational) and "magnetic" (divergence-free; e.g., inertial) fields as do abelian electromagnetism and the linearized form of Einstein's field equations for weak gravity and slow matter.

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 gauge-field 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, mass-energy, 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).

Newtonian gravitation requires parity invariant conservative forces. General Relativity models gravitation as a parity invariant rank-2 tensor. Nobody has sought empirical exceptions despite acceptable theoretical loopholes[12] including teleparallel gravitation (below).

External symmetry-derived properties (e.g., parity) act on spacetime. A Poincaré group gauge theory can be equivalent to the Einstein-Cartan theory of gravitation[13]. Einstein-Cartan theory operates in Riemann-Cartan spacetime U⁴ - a paracompact, Hausdorff, connected, C, and oriented four-dimensional manifold on which are defined a local Lorentz metric g and a linear affine connection . Curvature and torsion tensors can be obtained from on U⁴:

1. If the torsion tensor vanishes, Riemann-Cartan spacetime becomes pseudo-Riemannian spacetime, V⁴ (General Relativity's description of gravitation);
2. If the curvature tensor vanishes, it becomes Weitzenböck spacetime, A⁴ (in which the teleparallel gravitational energy-momentum pseudotensor is anti-symmetric to parity transformation);
3. If both tensors vanish, it becomes Minkowski spacetime, M⁴.

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 non-hermitian 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 stress-energy (energy-momentum) pseudotensor constructs volume integrals for total gravitational four-momentum and total angular momentum. Teleparallel gravitation can allow a gravitational stress-energy 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 gauge-invariant; not covariant under general coordinate transformations)[17]. This defines an integral energy-momentum 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 Levi-Civita connections is "contorsion" quantifying coframe field twist as measured by the Levi-Civita connection. Contorsion can be expressed as the torsion of the Weitzenböck connection. The Levi-Civita 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.)

Testing the Equivalence Principle

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 gauge-field 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.

Contrasting compositions classically define[36] "different" test masses : baryon number (N+Z), isopin (N-Z)/2, Yukawa forces[37]; gravitational self-energy[38]; electron spin (Dy₆Fe₂₃ and HoFe₃[39]; 94% of Alnico 5 magnetic field) versus electron orbital angular momentum (37% of Sm₂Co₁₇ magnetic field) - a mole of electrons masses an insignificant 5.4858·10^-4 grams; neutrino-antineutrino exchange in different nuclei[40]; and inverse square deviations at small radii (string then M-theory[41]). 400+ years of exquisite tests, notably those derived from Vásárosnaményi Báró Eötvös Loránd[24], null to differential detection limits.

Ferrimagnet Dy₆Fe₂₃ 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 Essex-class 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. Mirror-image 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 mirror-image 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 fiber-suspended 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/r² 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],

and surface inertial centripetal acceleration (sidereal day, WGS 84; sea level unless noted) of Earth's rotation.

r = (6378136.46)[1-([sin²(lat)]/298.257223563)] meters
a = (3.380199)(cosine[lat])/[1-(0.006694380)cosine²(lat)] cm/sec²
   horizontal component of a = a[sin(lat)]

g = (978.032677)[1+(0.00193185139)sin²(lat)]/[1-(0.00669437999)sin²(lat)] cm/sec²
dg/dh = 0.000308766[1-(0.0014665)sin²(lat)] cm/sec²-meter
dg/ds = -0.0008109[cos[(2)(lat)]+(0.0022)cos[(4)(lat)]-(0.0033)cos[2(lat)]sin²(lat)] cm/sec²-km

r = 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]. Drag-shielded free-fall apparatus with ~ 10^-15 is proposed[49].

Symmetry, Chirality, Parity

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:

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 left-handed: cobalt-60 beta-decay parity violation[53]; K-meson[54] and B-meson[55] decay charge-parity 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 reverse-parity 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 self-similar helical geometry in greater active masses, subtended areas, and volumes than spin-polarized test masses[39,62] whose Eötvös experiments null within experimental error. Non-metric gravitation theories also postulate the Equivalence Principle[63].

Metric theories of gravitation postulate[64] spacetime is a Lorentzian manifold, test particles pursue space-time 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 non-commutation 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 divergence-free current (conserved property) arises if the Lagrangian or the action is invariant under continuous transformation.

1. To each continuous symmetry of an action there corresponds a conserved quantity because of the Euler-Lagrange equations of the Lagrangian, and the reverse.
2. To each gauge symmetry of an action there corresponds an identity among Euler-Lagrange equations of the Lagrangian, and the reverse.

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.

Gravitation is modeled by loop quantum theory[74], brane/string/M-theory[75], Lorentzian lattice quantum gravity[76]..., though few predictions exist to be tested. M-theory 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 Michelson-Morley 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 (Colella-Overhauser-Werner[81] and Bonse-Wroblewski[82] neutron interferometers; Kasevich-Chu[83] atom interferometers).

Optical and Geometric Chiralities

Gyrotropy is assigned spiraling toward the observer, geometry spirals away from the observer. Dextrorotatory gyrotropy is levorotatory structure (optically right-handed quartz is space group P3₂21). Distant from optical transitions in ionic crystals[86]:

1. Smaller pitch (repeat distance) of atomic helices gives larger gyrotropy.
   
2. 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)
3. Larger electronic (Pauling) polarizability gives larger gyrotropy.
4. 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):

1. Pseudoscalar, from molecular chirality and persists in disordered solution.
2. Vector. A property of pyroelectric lattices.
3. Pseudodeviator, from lattice symmetry. Silver thiogallate[88], AgGaS₂ with non-polar achiral tetragonal space group I-42d (#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 right-circularly polarized light in a medium. Gyrotropy summed over the electromagnetic spectrum vanishes:

1. Oscillator strength or f-sum rule[89]. The f-values sum to unity for all transitions from a given state;
2. The Thomas-Reiche-Kuhn sum rule[90]. The photoabsorption cross section integral equals the total number of electrons involved in the collective excitation.
3. Kramers-Kronig 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.
4. The sum over (n-1) across the electromagnetic spectrum is zero. The difference between two (n) for orthogonal polarizations sums to zero overall[92].

Geometrically Chiral Test Masses

Chiral crystals are complex objects[86]: Quartz (SiO₂; no detectable gyrotropy 56.16° from crystallographic [0001]), paratellurite (TeO₂; P4₁2₁2, P4₃2₁2), (HgS; P3₁21, P3₂1), langasite[100] (La₃Ga₅SiO₁₄; P321), berlinite (AlPO₄; P3₁21, P3₂21), Bi₁₂(Si, Ge, or Ti)O₂₀ (I23), sodium bromate and chlorate (P2₁3) all possess at least one traceable counter-helix to the dominant atomic helices. Space group P2₁3 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 Bridgman-Stockbarger directional solidification[104]. Good crystals require solution growth as hot solid plastically deforms into lattice dislocations. Molecular solids have undesirable large unit cells.

Hemiparity Test Masses

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 P3₁21 or P3₂21 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/cm³
d_silica   =  2.203 g/cm³
d_gold   = 19.3 g/cm³

A quartz ball of radius R would be balanced by a fused silica ball of radius (R-r) with a vacuum-plated gold shell of thickness r. The general solution, sphere or cylinder, is

r³ - 3Rr² + 3R²r - [(d_quartz-d_silica)/(d_gold-d_silica)]R³ = 0
r³ - 3Rr² + 3R²r - (0.0260864)R³ = 0
www.1728.com/cubic.htm

For R = 1 centimeter in quartz, r = 87.72 micrometers in gold.

Calculated Geometric Chirality/Parity

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:

1. the identity element;
2. one point of inversion (its center);
3. an infinity of C_n rotation axes each passing through the inversion point;
4. an infinity of mirror planes each containing the inversion point and a rotation axis;
5. an infinity of mirror planes each containing the inversion point and normal to a rotation axis;
6. an infinity of S_n improper axes each passing through the inversion point.

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}D²]/4T

CHI is limited by three conditions:

1. Intrinsic parity divergence. Perfect parity divergence delivers CHI=1 given two other conditions:
2. Inertial moment disparity. CHI=1 can obtain only if  I_X = I_Y = I_Z exactly, and
3. Inertial moment products. CHI=1 can obtain only if
   _i (I_Xi*I_Yi) = 0
   _i (I_Xi*I_Zi) = 0
   _i (I_Yi*I_Zi) = 0

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 self-similar macroscopic crystal. Four possibilities obtain:

1. Unit cell contents are achiral, crystal is achiral
2. Unit cell contents are   chiral, crystal is achiral
3. Unit cell contents are achiral, crystal is   chiral
4. Unit cell contents are   chiral, crystal is   chiral

(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 (3₁3₂ 4₁4₃ 6₁6₅ 6₂6₄ screw axes), CHI>zero even if the formula units are achiral. 2₁ 4₂ 6₃ 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 2₁ 4₂ 6₃ that are each simultaneously left- and right-handed screws with the same screw vector[106]. These 59 space groups contain conflicting geometries that decrease CHI increase with increasing lattice sample radius.

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.

10^-13 mass anomaly is 2.148 cal/gram. Water hydrogen bond energy[107] is 5.57 kcal/mole (328 cal/gram), carbon-carbon 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.

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 half-atoms - 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 ų 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 3-fold helices. Perception and "common sense" are irrelevant. Rigorous mathematics is the only valid analysis. (Look beyond the screen to relax your eyes' convergence. The two-block 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 3-D 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 1000-atom run will typically require 10-20 CPU-hours in an RS6000/Power3 given connectivity-optimized 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 squared-distances between the vertices of the d-set and the permuted d-set, measures the set's similarity to itself. A set having no direct symmetry in a d-dimensional 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 parity-divergent 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 parity-divergent test masses. Assigning different atom labels (SiO₂) 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 ~10⁹ atoms/second. Large quartz radii were run in a cluster of 16 AMD Opteron-848s in 30 days. Tellurium, quartz, and berlinite were given 609, 90,386 (150,000 CPU-hrs; 444 quadrillion atoms), and 571 radius samplings respectively to calculate the following:

Table XIV. LATTICE PARITY DIVERGENCE, Te, alpha-Quartz, and Berlinite

Sphere Radius, Å	Lattice CHI	Total Atoms	Unit Cells	Volume, Å3
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	4.11·104 3.71·104 1.99·104	4.19·106 4.19·106 4.19·106
500	0.999991634437844450 0.999980456860055180 0.999989604104047446	15,427,878 41,696,416 44,676,376	5.14·106 4.63·106 2.48·106	5.24·108 5.24·108 5.24·108
1,000	0.999993481890267883 0.999997440127579185 0.999998427421108820	123,422,301 333,575,254 357,408,684	4.11·107 3.71·107 1.98·107	4.19·109 4.19·109 4.19·109
2,000	0.999998519283409702 0.999999080483513838 0.999999455994921283	987,378,232 2,668,598,322 2,859,270,260	3.29·108 2.97·108 1.58·108	3.35·1010 3.35·1010 3.35·1010
5,000	0.999999871340933588 0.999999789636193490 0.999999929783070689	15,427,793,636 41,696,845,260 44,676,133,458	5.14·109 4.63·109 2.48·109	5.24·1011 5.24·1011 5.24·1011
10,000	0.999999897747013216 0.999999968983088517 0.999999980194705342	123,422,354,560 333,574,731,752 357,409,097,084	4.11·1010 3.71·1010 1.99·1010	4.19·1012 4.19·1012 4.19·1012
20,000	0.999999983730030373 0.999999992565814146 0.999999990115051648	987,378,792,095 2,668,597,813,994 2,859,272,764,336	3.29·1011 2.97·1011 1.59·1011	3.35·1013 3.35·1013 3.35·1013
100,000	0.999999999546726956 0.999999999648281730 0.999999999778529238	123,422,348,782,767 333,574,726,196,900 357,409,096,311,540	4.11·1013 3.71·1013 1.98·1013	4.19·1015 4.19·1015 4.19·1015
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	3.29·1014 1.11·1015 2.63·1015 5.30·1015	3.35·1016 1.13·1017 2.68·1017 5.39·1017
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	2.97·1014 1.00·1015 2.37·1015 4.63·1015 1.56·1016 3.82·1016	3.35·1016 1.13·1017 2.68·1017 5.24·1017 1.77·1018 4.32·1018

*Te unit cells contain six half-atoms; quartz seven atoms and four half-atoms,
berlinite thirteen atoms and ten half-atoms. All lattice points are calculated whole.
(Berlinite, Z. Kristallogr. 192 119 (1990))

log(1-CHI) vs. log(radius): Tellurium to 505,000 Å Radius

FastCHI Programming, Matthew Francey
80-bit BigCHI and statistics package, Tony Lapen
559 CPU-hrs, 1.8 GHz Opteron, Linux, John Edward Scott

log(1-CHI) vs. log(radius): alpha-Quartz to 1,100,000 Å Radius

FastCHI Programming, Matthew Francey
80-bit BigCHI upgrade and statistics package, Tony Lapen
Parallelized CHIpir, John Edward Scott
115,000 CPU-hrs in a 178-Xeon cluster + 12,000 CPU-hrs in a 16 Opteron-848 cluster + 20,000 PC CPU-hrs

log(1-CHI) vs. log(radius): Berlinite AlPO₄ to 100,000 Å Radius

FastCHI Programming, Matthew Francey
80-bit BigCHI upgrade and statistics package, Tony Lapen
157.2 CPU-hrs, Wintel, John Hooper

The berlinite unit cell is the -quartz structure with a doubled c-axis, Si-O- vs. Al-O-P-0-. If atoms are taken to be anonymous mass the c-axis 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(1-CHI) vs. log(radius).

Te unit cell axes are a,b=4.456 Å, c=5.921 Å. The a,b-axes or c-axis were given large arbitrary variations and CHI was progressively calculated from 5.1-13 Å radius up to a 10,000 Å radius ball with 390-430 radius samplings for the test cases. (The mathematics of CHI is sensitive to shape but not to scale.)

log(1-CHI) vs. log(radius): Distorted Te Unit Cells

FastCHI Programming by Matthew Francey
80-bit BigCHI upgrade and statistics package, Tony Lapen

Table XV. Te DISTORTED UNIT CELLS DATA

Lattice	a,b-axes, Å	c-axis, Å	Unit Cell Volume, Å3	Slope	Intercept	Total Atoms
Theory, Te				-2	arbitrary
Tellurium	4.456	5.921	101.816	-1.99950	0.786973	1.05·1015
Short a,b	0.913	5.921	4.274	-1.97716	0.000013	2.94·1012
Short c	4.456	1.184	20.360	-2.00775	0.302008	6.17·1011
Long a,b	17.287	5.921	1532.374	-1.97145	1.46827	8.20·109
Long c	4.456	30.307	521.151	-1.98165	1.38036	2.41·1010

Te unit cell fractional coordinates are -0.2636, 0.0 ,1/3. The symmetry operations of space group P3₁21 then populate the unit cell with three atoms total (six half-atoms 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 84-94 atoms to a 10,000 Å radius ball containing 123.4 billion atoms in each test case.

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 c-axis helix,

log(1-CHI)= -2[log(radius)] + [(180-) ()/60] -
( is the smallest vertex angle in the helix. The slightly distorted tetrahedral O-Si-O helix angle is 110.56° vs. 109.47° undistorted)

log(1-CHI) = -2[log(radius)] + 0.494277
3-cm quartz test mass has CHI = 1 - 1.387·10^-16, theory
3-cm 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 c-axis) 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(1-CHI) is typically 0.997 units, the two routes to fitting CHI give indistinguishable values.

The optimum crystal structure would have the

1. parity space groups P3_(1,2) P3_(1,2)12 or P3_(1,2)21
2. smallest unit cell volume,
3. smallest lattice volume/atom,
4. unit cell with most equal inertial moments (all atoms weighted equally),
5. largest helix angle extended along the c-axis,
6. smallest a,b-plane helical radius,
7. heaviest atoms.

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.

Selenium and tellurium are exceptional for small unit cell 100% heavy atom content forming identical helices. Next-nearest 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 (50-200 hours at 320-380° 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 1-3·10⁴/cm³ with a hole concentration of 10¹⁴/cm³ (intrinsic p-type semiconductor). A single crystal tellurium right cylinder two centimeters in diameter and long contains 1.83·10¹⁵ helices for a summed axial length of 3.66·10¹³ meters (33.9 light-hours; 3.1 times the solar system's diameter). Each helix has 3.38·10⁷ 360° turns. Both elements' helices are remarkably isolated in space (view with CHIME plug-in; 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/baryon
Mg = 8.265129 MeV/baryon
[Mg - Be]/[(17.3202n+16p)/33.3202] = 0.1919%

Cultured quartz is sold as multi-kilogram single crystals of both chiralities. The -quartz lattice (view with CHIME plug-in) contains conflicting helices: 6-fold helices with long axes parallel to the z-axis are of opposite chirality to parallel 12-fold helices; 4-fold helices with the 12-fold 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.

Alpha = (A₃₅₀₀ - A₃₈₀₀)/Y

Green's Function

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 z-axis position such that a silicon is centered at the same unit cell position in both space groups P3₁21 and P3₂21. Spatial distribution of the four oxygens around that silicon then define lattice chirality. Said oxygens are in positions "R" in the R-enantiomorph and "S" in the S-enantiomorph. 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 S-enantiomorphs. 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 non-Newtonian origins.

From general topological arguments,

G(x,y) = K[dist(x,y)]^(2-N)

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 P3₁21 versus P3₂21.

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.

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 = 10⁶/(R_Å)²

Chirality is not a point phenomenon. It is demonstrated that enantiomorphic centimeter-diameter 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 body-centered SiO₄ tetrahedron, or a radius of 01.609 Å.

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 parts-per-trillion gravitation parity anomalies.

Gravitational Scaling

R = [c]/(M_sc²)](M_p/M_s)^(2/n)

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 non-classically 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.

Gravitation empirically varies as 1/r² 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 n-space 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 close-packed bowling balls and a deep bed of monodisperse 1-micron 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 close-packed identical balls with radius 1, tetrahedral holes will contain a ball with radius [(6)/2]-1 or 0.2247. Thus 1-micron 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 non-interaction cannot be arithmetically discounted by "common sense."

For three mutually tangent spheres with radii r₁ r₂ r₃ there is the Soddy circle solution:

r₄ = (r₁r₂r₃)/{r₁r₂ + r₁r₃ + r₂r₃ (±)2[r₁r₂r₃(r₁ + r₂ + r₃)]}

If gravitation did have an antisymmetric interaction with maximal parity-pair 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 3-D diffusion.

Other Chiral Test Masses

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:

Chiral polymers[121] have loosely packed crystal lattices. Aggregated helicenes[122] can have []_D=170,000° (1400°/mm gyrotropy neat) as can neat cholesteric (nematic-C) 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 platinum-silver P6₁ helix[128] has a 7898.1 ų unit cell. A claimed intensely chiral octahedral complex[129], tris[1,2-dithiolatophenylenetungsten(IV)][130], has a 5649.0 ų unit cell.

Conclusion

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 non-null parity Eötvös experiment supercedes parity-violating 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/c³)^1/2 or 1.616·10^-26 nm and thus implies a spherical volume approximating 2.21·10^-78 nm³. 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 half-atoms in the walls of a 0.1018 nm³ irreducible configuration. A non-null parity Eötvös experiment breaches "point phenomenon" by a volu