Superconductivity the Structure Scale of the Universe (Elastic Resonant Symmetric Medium by Self-Energy) A

Superconductivity The Structure Scale of the Universe (Elastic Resonant Symmetric Medium by Self-Energy) a

Tenth Edition January 15, 2007

Richard D. Saam

525 Louisiana Ave Corpus Christi, Texas 78404 USA e-mail: [email protected] ii

ABSTRACT

A dimensional analysis correlation framework supported by reported experimental evidence (Homes, Harshman along with Voyager and EGRET space platforms and others) is presented which indicates that superconductivity is a self-energy phenomenon and congruent with the concept of the Charge Conjugation, Parity Change and Time Reversal (CPT) theorem (discrete Noether’s theorem). A resonant structure is proposed as an extension of Bardeen Cooper and Schrieffer (BCS) theory, which suspends Lorentz transforms at superluminal velocities in the context of the de Broglie hypothesis. A momentum and energy conserving (elastic) CPT resonant structural lattice scalable over 15 orders of magnitude from nuclear to universe dimensions and associated superconducting theory is postulated whereby nuclear (quark) weak and strong forces, electromagnetic and gravitational forces are mediated by a particle of resonant transformed mass (mt) (110.12275343 x electron mass or 56.2726/c2 Mev) and unit charge congruent with Heisenberg Uncertainty, such that the electron and proton mass and charge are maintained, nuclear density is maintained at 2.34E14 g/cm3, proton charge radius is maintained at 8.75E-14 cm, the universe radius is 2.25E28 cm, the universe mass is 3.02E56 gram, the universe density is 6.38E-30 g/cm3 or 2/3 the critical density and the universe escape velocity is c. Standard model up, down, strange, charm, bottom and top particles are correlated to the developed model at the nuclear scale. The universe time or age is 1.37E10 years and the universe Hubble constant is 2.31E-18/sec (71.2 km/sec-million parsec), which is gravitationally related to the proton charge radius by Planck’s constant. The calculated universe mass and density are based on an isotropic homogeneous media filling the vacuum of space analogous to the 'aether' referred to in the 19th century (but still in conformance with Einstein's relativity theory and extended to nuclear dimensions) and could be considered a candidate for the 'dark matter/energy' in present universe theories. Each 3 particle of mass mt in the proposed dark matter is contained in a volume of 15733 cm . In this context the universe cosmic microwave background radiation (CMBR) black body temperature is linked to universe dark matter/energy superconducting temperature. The model predicts an deceleration value with observed universe expansion and consistent with observed Pioneer 10 &11 deep space translational and rotational deceleration and consistent with the notion that: An object moving through momentum space will slow down. This is evidenced by 56 Mev space sources as detected by Voyager and The Compton Gamma Ray Observatory Mission (EGRET) spacecraft. Also, a reasonable value for the cosmological constant is derived having dimensions of the known universe. Also, a 93 K superconductor should loose .04% of its weight while 100% in superconducting mode, which is close to, reported results by Podkletnov and Nieminen (.05%). Also this trisine model predicts 1, 2 and 3 dimensional superconductors, which has been verified by observed magnesium diboride (MgB2) superconductor critical data. Also dimensional guidelines are provided for design of room temperature superconductors and beyond and which are related to practical goals such as fabricating a superconductor with the energy content equivalent to the combustion energy of gasoline. These dimensional guidelines have been experimentally verified by Homes’ Law and generally fit the parameters of a superconductor operating in the “dirty” limit. Confirmation of the trisine approach is represented by correlation to Koide Lepton relation at nuclear dimensions. Also, the concept of critical volume is introduced to study characteristics associated with CPT symmetry Critical Optical Volume Energy (COVE). A $200,000,000 - 10 year experimental plan is proposed to study this concept. iii

PREVIOUS PUBLICATION

First Edition October 15, 1996 http://xxx.lanl.gov/abs/physics/9705007 Title: Superconductivity, the Structure Scale of the Universe Author: Richard D. Saam Comments: 61 pages, 27 references, 11 figures (.pdf available on request) from [email protected] Subj-class: General Physics

Second Edition May 1, 1999 http://xxx.lanl.gov/abs/physics/9905007, version 6 Title: Superconductivity, the Structure Scale of the Universe Author: Richard D. Saam Comments: 61 pages, 27 references, 16 tables, 16 figures, Subj-class: General Physics

Third Edition February 23, 2002 http://xxx.lanl.gov/abs/physics/9905007 , version 7 Title: Superconductivity, the Structure Scale of the Universe Author: Richard D. Saam Comments: 66 pages, 50 references, 27 tables, 17 figures Subj-class: General Physics

Fourth Edition February 23, 2005 http://xxx.lanl.gov/abs/physics/9905007 , version 9 Title: Superconductivity, the Structure Scale of the Universe Author: Richard D. Saam Comments: 46 columnated pages, 21 figures, 17 tables, 192 equations, 58 references Subj-class: General Physics

Fifth Edition April 15, 2005 http://xxx.lanl.gov/abs/physics/9905007 , version 11 Title: Superconductivity, the Structure Scale of the Universe Author: Richard D. Saam Comments: 46 columnated pages, 21 figures, 17 tables, 192 equations, 58 references Subj-class: General Physics

Sixth Edition June 01, 2005 http://xxx.lanl.gov/abs/physics/9905007 , version 13 Title: Superconductivity, the Structure Scale of the Universe Author: Richard D. Saam Comments: 47 columnated pages, 21 figures, 17 tables, 192 equations, 63 references Subj-class: General Physics

Seventh Edition August 01, 2005 http://xxx.lanl.gov/abs/physics/9905007 , version 14 Title: Superconductivity, the Structure Scale of the Universe Author: Richard D. Saam Comments: 49 columnated pages, 21 figures, 18 tables, 193 equations, 63 references Subj-class: General Physics

Eighth Edition March 15, 2006 http://xxx.lanl.gov/abs/physics/9905007 , version 15 Title: Superconductivity, the Structure Scale of the Universe Author: Richard D. Saam Comments: 52 columnated pages, 21 figures, 18 tables, 201 equations, 73 references Subj-class: General Physics

Ninth Edition November 02, 2006 http://xxx.lanl.gov/abs/physics/9905007 , version 15 Title: Superconductivity, the Structure Scale of the Universe Author: Richard D. Saam Comments: 42 columnated pages, 21 figures, 18 tables, 215 equations, 77 references Subj-class: General Physics

Table of Contents

1. Introduction 2. Trisine Model Development 2.1 Defining Model Relationships 2.2 Trisine Geometry 2.3 Trisine Characteristic Wave Vectors 2.4 Trisine Characteristic De Broglie Velocities And Sagnac Relationship 2.5 Superconductor Resonant Dielectric Constant And Magnetic Permeability 2.6 Fluxoid And Critical Fields 2.7 Superconductor Resonant Internal Pressure, Casimir Force And Deep Space Thrust Force (Pioneer 10 & 11) 2.8 Superconducting Resonant Current, Voltage And Conductance (And Homes’ Law) 2.9 Superconductor Resonant Apparent Weight Reduction In A Gravitational Field 2.10 BCS Verifying Constants 2.11 Superconducting Resonant Cosmological Constant 2.12 Superconducting Resonant Variance At T/Tc 2.13 Superconductor Resonant Energy Content 2.14 Superconductor Resonant Gravitational Energy 2.15 The Concept of Critical Optical Volume Energy (COVE). 3. Discussion 4. Conclusion 5. Variable And Constant Definitions (Including Planck dimensions) Appendices A. Trisine Number, Trisine Mass And B/A Ratio Derivation B. Debye Model Normal And Trisine Reciprocal Lattice Wave Vectors C. One Dimension And Trisine Density Of States D. Ginzburg-Landau Equation And Trisine Structure Relationship E. Heisenberg Uncertainty Within The Context of De Broglie Condition F. Equivalence Principle In The Context Of Work Energy Theorem G. Koide Constant for Leptons - tau, muon and electron H. Authorized For Expenditure (Cost Estimate) 6. References 1

1. Introduction inserted, but for the purposes of this report, six critical The trisine structure is a geometrical lattice model for the temperatures ()Tc are selected as follows: superconductivity phenomena based on Gaussian surfaces o o 1.1 Tc 8.95E+13 K Tb 9.07E14 K (within the context of Maxwell Displacement (D) concept) The upper bounds of the superconductive phenomenon as encompassing superconducting Cooper CPT Charge 3 dictated by the proton density 2.34E14 g/cm and radius B conjugated pairs (discrete Noether’s theorem entities). This of 6.65E-14 cm (8.750E-14 cm NIST reported charge Gaussian surface has the same configuration as a particular radius value) which is defined at a critical temperature Tc matrix geometry as defined in reference [1], and essentially of consists of mirror image non-parallel surface pairs. Originally, m cek32 the main purpose of the engineered lattice [1] was to control t ()b the flow of particles suspended in a fluid stream and generally which is a consequence of a superluminal condition: has been fabricated on a macroscopic scale to remove particles vcdx > 3 in industrial fluid streams on the order of m /sec. In the which is justifiable in elastic resonant condition (see particular configuration discussed in this report, a more equation 2.1.6 and explanation) and generalized conceptual lattice is described wherein there is a ce222== m m v2 c = fine structure constant perfect elastic resonant character to fluid and particle flow. In rt dx() other words there is 100 percent conservation of energy and Which is consistent with the proton (mp) state momentum (elastic or resonant condition), which by definition mc2 == Kcπ Bc pB() we will assume to describe the phenomenon of composed of standard model up quarks (2) and down superconductivity. quark (1) with respective masses of The superconducting model presented herein is a logical 23 mm Kc and 13 mm Kc ()()()et B ()(et )() B translation of this geometry [1] in terms of classical and (5.64 Mev and 2.82 Mev). quantum theory to provide a basis for explaining aspects of the Also the nuclear magneton identity exists as superconducting phenomenon and integration of this 2 phenomenon with nuclear, electromagnetic and gravitational ()2 5emc 2ps= 2.cos 785 g()θ emv() 2 tdx () () forces within the context of references[1-80]. with the ratio 2/5 being the sphere moment of inertia This approach is an attempt to articulate a geometrical model factor and the constant 2.785 being equal to the for superconductivity in order to anticipate dimensional experimentally observed as the nucleus of hydrogen atom, regimes in which one could look for higher performance which has a magnetic moment of 2.79 nuclear magnetons. materials. This approach does not address specific particles At this proton dimension B, gravity energy is related to such as polarons, exitons, magnons, plasmons that may be Hubble constant()HU necessary for the expression of superconductivity in a given 2 hHU= ()32 Gm t/ B proton material. This is similar to acknowledging the existence of This proton dimensional condition conforms to Universe fundamental standard model nuclear particles such as quarks conditions at hundredths of second after the Big Bang. and gluons but only being able to observe resulting unstable o o (inherently inelastic) particles such as pions. Indeed, the 1.2 Tc 2,135 K Tb 4.43E+09 K dimensionally correct (mass, length, time) Trisine structure or The critical temperature of an anticipated superconducting lattice as presented in this report may be expressed in terms of material with the assumed density (6.39 g/cm3) [17] of something very profound and elementary and that is the YBa237 Cu O − x which would express gravitational effects to Charge Conjugate Parity Change Time Reversal (CPT) the extent that it would be weightless in earth's Theorem as a basis for creating resonant structures. The gravitational field (See Table 2.9.1). validity of the Trisine model is related to its dimensional o o 1.3 Tc 1190 K Tb 3.31E+09 K (mass, length, time) correctness or numerical consistency with fundamental constants ( ,, Gc & k). Model numerical The critical temperature of an anticipated material, which b coherency is maintained at less than 1 part in 10,000 on a would have an energy content equivalent to the spreadsheet with name plate calculation precision at 15 combustion energy content of gasoline(see Table 2.13.1). significant digits. o o 1.4 Tc 447 K Tb 2.03E+09 K Superconductivity or resonance is keyed to a critical The critical temperature of an anticipated room temperature()Tc , which is representative of the energy at temperature superconductor, which is calculated to be 1.5 which the superconductive resonant property takes place in a x 298 degree Kelvin such that the superconducting material or medium. The material or medium that is material may have substantial superconducting properties characterized as a superconductor has this property at all at an operating temperature of 298 degree Kelvin. temperatures below the critical temperature. T 93 o K T 9.24E+08 o K In terms of the trisine model, any critical temperature can be 1.5 c b The critical temperature of the YBa237 Cu O − x 2

superconductor as discovered by Paul Chu at the 2.1 Defining Model Relationships University of Houston and which was the basis for the The four defining model equations within the superconductor gravitational shielding effect of .05% as observed by conventional tradition are presented as 2.1.1 - 2.1.4 below: Podkletnov and Nieminen(see Table 2.9.1). dArea()= 4π e (2.1.1) o o ∫∫ ()± 1.6 Tc 8.11E-16 K Tb 2.729 K

The background temperature of the Universe represented 2mkTtbc ()KK by the black body cosmic microwave background 2 = trisine o e − 1 radiation (CMBR) of 2.728 ± .004 K as detected in all Euler (2.1.2) e ()KK 2 directions of space in 60 - 630 GHz frequency range by = = K B the FIRAS instrument in the Cobe satellite and 2.730 ± π sinh()trisine .014 o K as measured at 10.7 GHz from a balloon 1 platform [26]. The most recent NASA estimation of trisine = (2.1.3) CMBR temperature is 2.725 ± .002 K. The particular D()∈t ⋅V temperature of 2.729 o K indicated above which is essentially equal to the CMBR temperature was ⎛ m ⎞ e (2.1.4) f t ± = cavity H established by the trisine model wherein the ⎜ ⎟ c ⎝ mee⎠ 2mvε superconducting Cooper CPT Charge conjugated pair density (see table 2.9.1) (6.38E-30 g/cm3) equals the Equation 2.1.1 is a representation of Gauss's law with a Cooper density of the universe (see equation 2.11.5 and 2.11.16). CPT Charge conjugated pair charge contained within a Also, this temperature is congruent with an reported bounding surface area defining a cell volume or cell cavity. interstellar magnetic field of 2E-6 gauss [78] (see table Equation 2.1.2 and 2.1.3 define the superconducting model as 2.6.2]. developed by Bardeen, Cooper and Schrieffer in reference [2] As a general comment, a superconductor resonator should be and Kittel in reference [4]. operated (or destroyed) at about 2/3 of ()Tc for maximum Recognizing this approach and developing it further with a effect of most properties, which are usually expressed as when resonant concept, it is the further objective herein to select a 2 extrapolated to T = 0 (see section 2.12). In this light, critical particular (wave vector) or (KK) based on trisine geometry temperatures ()Tc in items 1.4 & 1.5 could be operated (or (see figures 2.2.1-2.2.5) that fits the model relationship as 2 destroyed) up to 2/3 of the indicated temperature (793 and indicated in equation 2.1.2 with K B being defined as the trisine 1423 degree Kelvin respectively). Also, the present (wave vector)2 or (KK) associated with superconductivity of development is for one-dimensional superconductivity. Cooper CPT Charge conjugated pairs through the lattice. This Consideration should be given to 2 and 3 dimensional procedure establishes the trisine geometrical dimensions, then superconductivity by 2 or 3 multiplier to Tc ‘s contained herein equation 2.1.4 is used to establish an effective mass function interpret experimental results. f ()mmte of the particles in order for particles to flip in spin when going from trisine cavity to cavity while in 2. Trisine Model Development thermodynamic equilibrium with critical field ()H c . The implementation of this method assumes the conservation of In this report, dimensionally (mass, length and time) momentum ( p ) (2.1.5) and energy (Ε ) (2.1.6) such that: mathematical relationships which link trisine geometry and superconducting theory are developed and then numerical ∑ ∆∆pxnn= 0 values are presented in accordance with these relationships. n=1234,,, (2.1.5) Centimeter, gram and second (cgs) as well as Kelvin ±±KK12 ±± KK 34 temperature units are used unless otherwise specified. The actual model is developed in spreadsheet format with all model ∑ ∆∆Εnnt = 0 n=1234,,, equations computed simultaneously, iteratively and (2.1.6) interactively as required. KK11+=+ KK 2 2 KK 33 KK 4 4 + Q The general approach is: given a particular lattice form (in this Where Q = 0 (reversible process) case trisine), determine lattice momentum wave vectors K , K , 1 2 The numerical value of Q is associated with the equation 2.1.6 K & K so that lattice cell energy (~K2) per volume is at a 3 4 is associated with the non-elastic nature of a process. Q is a minimum. The procedure is analogous to the variational measure of the heat exiting the system. By stating that Q=0, principle used in quantum mechanics. More than four wave the elastic nature of the system is defined. An effective mass vectors could be evaluated at once, but four is deemed must be introduced to maintain the system elastic character. sufficient. Essentially, a reversible process or reaction is defined recognizing that momentum is a vector and energy is a scalar. 3

These conditions of conservation of momentum and energy This geometry fills space lattice with hexagonal cells, each of provide the necessary condition of perfect elastic character, volume (cavity or 2√3AB2) and having property that lattice is which must exist in the superconducting resonant state. In equal to its reciprocal lattice. Now each cell contains a charge addition, the superconducting resonant state may be viewed as pair ()e+ and ()e− in conjunction with cell dimensions ()A boiling of states ∆∆∆∆Εnn,,tpx n , non top of the zero point state and ()B defining a permittivity ()ε and permeability ()km in a coordinated manner. such that: 22 In conjunction with the de Broglie hypothesis [77] providing vck= /(mε ). momentum K and energy KK change with Lorentz transform Then v is necessarily + or – and ()()++=kkmmεε−− written as Now time ()time is defined as interval to traverse cell: 22 KvcK()11− = β time= 2/ B v KK()11−− v22 c 1= KK ()β − 1 then time is necessarily + or - congruent with: Further: given the Heisenberg Uncertainty ∆∆p x≥≥//22 and ∆ E ∆ time mv222 c c11−− v 22 c 1= mv 222 c c()β − 1 ()()() but defined for this model which is equivalent and justified by 2 2 where the energy defined as ‘KK’ contains the factor v /c , the trisine geometry: superluminal iβ and subluminal β Lorentz transform () () ∆∆p x≥≥ h//22 and ∆ E ∆ time h conditions are allowed because in a resonant elastic condition where: h /2 these Lorentz transforms cancel = ()π The parity geometry satisfies following condition: ±±KKββ ± K ββ ± K 11 2 2 33 4 4 ∆∆p x= ∆ E ∆ time≥ h /2 KKββ−1111+ KK− = KK ββ− + KK − 11() 1 2 2() 2 33() 3 4 4() 4 where: ∆∆x==2, B v v , and ∆ time = time and m = constant and is always + ±±Kiββ Ki ± Ki ββ ± Ki t 11 2 2 33 4 4 ∆∆pmv= KK iββ−111+ KK i− = KK i ββ− + KK i −1 t 11() 1 2 2() 2 33() 3 4 4() 4 2 e2 h for all values of velocity v > 0 under normalization ∆∆Emv====2 / 2 ± kT (always +) () tc32B time where K1 ≠ K2 ≠ K3 ≠ K4 and ∆∆∆time= x/ v where ±±KK ±± KK 12 34 the ()B / A ratio must have a specific ratio (2.379760996) and and: KK+ KK≅ KK+ KK 11 2 2 33 4 4 mass()mt a constant + value of 110.12275343 x electron 2 since ()ββ///==ii cc =∞∞ /= 1. mass()me or 56.2726/c Mev to make this work. (see Equations 2.1.5 and 2.1.6 are proven generally valid. In Appendix E for background derivation based on Heisenberg essence, conservation of momentum and energy (elastic Uncertainty and de Broglie condition). condition) is a fundamental property in a resonant elastic state The trisine symmetry (as visually seen in figures 2.2.1-2.2.5) overriding Lorentz transform considerations. allows movement of Cooper CPT Charge conjugated pairs with Conceptually, this model is defined within the context of the center of mass wave vector equal to zero as required by CPT theorem and generalized 3 dimensional parity of superconductor theory as presented in references [2, 4]. Cartesian coordinates (x,y,z) as follows: Wave vectors()K as applied to trisine are assumed to be free xxx xxx particles (ones moving in the absence of any potential field) 11 12 13 21 22 23 and are solutions to the one dimensional Schrödinger equation yyy11 12 13 = − yy21 22223y (2.1.6a) 2.1.7 as originally presented in reference [3]: zzz zzz 2 11 12 13 21 22 23 d ψ 2m t Energy 0 Given this parity condition defined within trisine geometrical 22+=ψ (2.1.7) ds constraints (with dimensions A and B as defined in Section The well known solution to the one dimensional Schrödinger 2.2) which necessarily fulfills determinate identity in equation equation is in terms of a wave function()ψ is: 2.1.6b. 1 ⎛ 2 ⎞ −B 00 00 B ⎛ 2mt ⎞ ψ = Amplitude sin⎜ ⎜ 2 Energy⎟ s⎟ BB BB ⎜ ⎝ ⎠ ⎟ (2.1.8) 0 − = −−0 ⎝ ⎠ (2.1.6b) 33 33 = Amplitude sin() K⋅ s AAAA Wave vector solutions ()KK are constrained to fixed trisine − 0 − 0 2 2 2 2 cell boundaries such that energy eigen values are established by the condition sS== and ψ 0 such that 4

1 1 h 2m 2 3 ⎛ t ⎞ cavity = (2.1.13f) ⎜ 2 Energy⎟ S== K S nπ (2.1.9) ⎝ ⎠ 2πmkTtbc Energy eigen values are then described in terms of what is Bardeen, Cooper and Schrieffer (BCS) [2] theory is generally called the particle in the box relationship as follows: additionally adhered to by equations 2.1.14 – 2.1.18 as verified 22 2 by equation 2.1.13c, array equation 2 in conjunction with n ⎛ π ⎞ equations A.3, A.4, and A.5 as presented in Appendix A. Energy = ⎜ ⎟ (2.1.10) 2mSt ⎝ ⎠ 22 KKBP+ 12⎛ Euler ⎞ For our model development, we assume the quantum number n trisine = = = −−ln⎜ e 1⎟ (2.1.14) KKPB D()∈ V ⎝ π ⎠ = 2 and this quantum number is incorporated into the wave vectors KK as presented as follows: () 3 cavity mtC K 1 22 2 2 2 2 density of states D()∈T ==32 (2.1.15) 2 ⎛ ππ⎞ ⎛ 2 ⎞ 4π kT Energy = = = ()KK (2.1.11) bc 22mS⎝⎜ ⎠⎟ mS⎝⎜ ⎠⎟ 2m The attractive Cooper CPT Charge conjugated pair energy V tt t () Trisine geometry is described in Figures 2.2.1 – 2.2.5 is in is expressed in equation 2.1.16: general characterized by resonant dimensions A and B with a KK V = PB kT (2.1.16) characteristic ratio B/A of 2.379760996 and corresponding 22bc KKBP+ characteristic angle()θ where: Table 2.1.1 with Density of States −−11⎛ A⎞ ⎛ 12⎞ o (2.1.12) θ = tan= tan= 22 . 80 0 ⎜ ⎟ ⎜ ⎟ TK ⎝ B⎠ ⎝ e 3 ⎠ c () 8.95E+13 2,135 1,190 447 93 8.11E-16 TK0 Note that 22.8 ~ 360/8 which indicates a hint to trisine resonant b () 9.07E+14 4.43E+09 3.31E+09 2.03E+09 9.24E+08 2.729 lattice model similarity with reported octal (8) nature of the V 8.09E+01 1.46E-13 8.13E-14 3.05E-14 6.35E-15 5.54E-32 nuclear particle numbers. D()∈ 6.11E-03 3.39E+12 6.09E+12 1.62E+13 7.79E+13 8.93E+30 The superconducting model is described with equations 2.1.13 The equation 2.1.13c array equation elements 1 and 2 along - 2.1.19, with variable definitions described in the remaining with equations 2.1.14, 2.1.15, 2.1.16, 2.1.17 and 2.1.18 are equations. The model again converges around a particular consistent with the BCS weak link relationship: resonant transformed mass m (110.12275343 x electron ()t −1 mass m or 56.2726/c2 Mev) and dimensional ratio B / A of 2eEuler 2 ()e () kT KKe22DV()∈ 2.379760996. bc=+()CDs (2.1.17) π 2mt 2 mvtdx 1 22−1 kTbc==ω (2.1.13a) 2 2 ⎛ 1 ⎞ ⎛ chain ⎞ ⎛ K A ⎞ DV()∈ kTbc= ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ e (2.1.18) ⎝ gst⎠ ⎝ cavity⎠ 2 m ⎝ ⎠ 22 ⎧ K B ⎫ The relationship in equation 2.1.19 is based on the Bohr ⎪ ⎪ 2m magneton emv 2 e ε (which has dimensions of magnetic field x ⎪ t ⎪ ⎪ 2 ⎪ volume) with a material dielectric()ε modified speed of kTbc= ⎨ ⎬ (2.1.13b) light()vε (justifiable in the context of resonant condition) and ⎪ ⎪ mmet 22mmtp establishes the spin flip/flop as Cooper CPT Charge conjugated ⎪ ()2B + ()A ⎪ pairs resonate from one cavity to the next (see figure 2.3.1). ⎩⎪ mmet+ mmtp+ ⎭⎪ The 1/2 spin factor and electron gyromagnetic factor ()ge and superconducting gyro factor ()gs are multipliers of the Bohr ⎧ 22 2 ⎫ ()KKCDs+ 1 magneton where ⎪ ⎪ 2m etrisine 1 ⎪ t − ⎪ 3 me ⎛ cavity mp ⎞ K Dn kTbc= ⎨ ⎬ gs = − = coos()θ 22 2 Euler (2.1.13c) ⎜ ⎟ (2.1.18a) ⎪ KKCD+ s e ⎪ 4 mt ⎝ ()∆∆∆xyz me ⎠ K B () ⎪ 2mπ sinh trisine ⎪ as related to the Dirac Number by ⎩ t ()⎭ (3/4)(g -1)=2π (g -1) . ⎧ ⎛ 1 ⎞ cavity ⎫ sd DE ⎪ ⎜ 2 ⎟ ⎪ Also because particles must resonate in pairs, a symmetry is ⎪ ⎝ gs ⎠ 8π ⎪ kT = created for superconducting current to advance as bosons. bc ⎨ 2 ⎬ (2.1.13d) ⎪ chain e± ⎪ 1 mt e± ⎪ ⎪ gges = cavity H c ⎩ cavity εB ⎭ 22m mv (2.1.19) eeε The proton to electron mass ratio mm is verified through 4 v 1 ()pe dz the Werner Heisenberg Uncertainty Principle volume kTbc= () Unruh (2.1.13e) time± 2π vdx ()∆∆∆xyz⋅⋅ as follows in equations 2.1.20a,b,c): Also, Bose Einstein Condensate criteria is satisfied as: 5

1 1 ⎛ B 1 ⎞ π ∆x == = ∆⎜ ⎟ (2.1.20a) vd = (2.1.26) 2 ∆∆pK2 ⎝ 2 π ⎠ mBt xB() 2 ⋅ B 1 13⎛ B 1 ⎞ time = ∆y == = ∆⎜ ⎟ (2.1.20b) (2.1.27) 2 ∆∆pK2 ⎝ 232π ⎠ vd yP() The relationship between ()vdx and ()vvεεxy, & v ε z is per the 1 1 ⎛ A 1 ⎞ group/phase velocity relationship presented in equation 2.1.30. ∆z == = ∆⎜ ⎟ (2.1.20c) 2 ∆∆pK2 ⎝ 2 2π ⎠ 2 zA() (group velocity )⋅ ( phase veloocity) = c 1 ⎧ ⎛ A 1 v2 ⎞ ⎫ ∆∆∆xyz⋅⋅ = ⎧ 2 ⎫ ⎪ εx ⎪ () (2.1.21) ⎛ 1 vεx ⎞ ()vdx ⋅⋅⎜ 1 2 c⎟ ()()()222KKKBPA ⎪()vdx ⋅⋅c ⎪ ⎪ ⎜ B v ⎟ ⎪ ⎜ 2 ⎟ ⎝ 4 dx ⎠ ⎪ ⎝ π vdx ⎠ ⎪ ⎪ 3 ⎪ m ⎪ 2 ⎪ ⎪ 2 ⎪ p 3 1 cavity mt ⎛ v ⎞ ⎛ v ⎞ = − (2.1.22) ⎪ 1 εy ⎪ ⎪ A 1 εy ⎪ 2 mg4 xyz m ⎨()vdx ⋅ 2 ⋅ c ⎬=⎨()vdx ⋅⋅⎜ 1 2 c⎟ ⎬ = c (2.1.30) es()∆∆∆ e ⎜ v ⎟ B v ⎪ ⎝ π dy ⎠ ⎪ ⎪ ⎜ 4 dy ⎟ ⎪ Equations 2.1.20 - 2.1.22 provides us with the confidence ⎝ 3 ⎠ ⎪ 2 ⎪ ⎪ ⎪ ( ∆∆∆xyz⋅⋅ is well contained within the cavity ) that we can ⎛ 1 vεz ⎞ 2 ⎪()v ⋅⋅c ⎪ ⎪ ⎛ A 1 v ⎞ ⎪ proceed in a semi-classical manner outside the envelop of the ⎪ dx ⎜ 2 ⎟ ⎪ ⎪(v )⋅⋅⎜ εz c⎟ ⎪ ⎩ ⎝ π vdz ⎠ ⎭ dx 1 2 ⎪ ⎜ B 4 vdz ⎟ ⎪ uncertainty principle with the resonant transformed mass ()mt ⎩ ⎝ 3 ⎠ ⎭ and de Broglie velocities()vd used in this trisine superconductor model or in other words equation 2.1.23 holds. 2 2.2 Trisine Geometry ⎛ 2 KK ⎞ 1 mv () ⎛ 2 ⎞ ()td The characteristic trisine resonant dimensions A and B are = mvtd = (2.1.23) ⎜ 2m ⎟ ⎝⎜ 2 ⎠⎟ m ⎝ t ⎠ t indicated as a function of temperature in table 2.2.1 as The Meissner condition as defined by total magnetic field computed from the equation 2.2.1 exclusion from trisine chain as indicated by diamagnetic 22 22 2 K B K A ⎛ B⎞ (2.2.1) susceptibility()Χ = −14π , as per the following equation: kTbc== 22m m ⎝⎜ A⎠⎟ kT 1 t t Χ = − bc = − Trisine Geometry is subject to Charge conjugation Parity chain H 2 4π (2.1.24) c change Time reversal (CPT) symmetry which is a fundamental Equation 2.1.25 provides a representation of the resonant symmetry of physical laws under transformations that involve transformed mass ()mt the inversions of charge, parity and time simultaneously. The

221 CPT symmetry provides the basis for creating particular mcrtdd=> mv where v 0 (2.1.25) 2 resonant structures with this trisine geometrical character. Table 2.1.2 with m , m . Frequency 2k T ω & ()t ()r ()b c () Table 2.2.1 Dimensions B and A with B vs. Tc plot Wave Length 2πc frequency or ()c⋅ time table & plot. 0 () TK 0 c () TK 8.95E+13 2,135 1,190 447 93 8.11E-16 c () 8.95E+13 2,135 1,190 447 93 8.11E-16 TK0 0 b () TK 9.07E+14 3.38E+09 3.31E+09 2.03E+09 9.24E+08 2.729 b () 9.07E+14 4.43E+09 3.31E+09 2.03E+09 9.24E+08 2.729 A (cm) 2.80E-14 5.73E-09 7.67E-09 1.25E-08 2.74E-08 9.29 mt 1.00E-25 1.00E-25 1.00E-25 1.00E-25 1.00E-25 1.00E-25 B (cm) 6.65E-14 1.36E-08 1.82E-08 2.98E-08 6.53E-08 22.11 mr 1.37E-23 3.28E-34 1.83E-34 6.87E-35 1.43E-35 1.25E-52 1E-06 ω 2.34E+25 5.59E+14 3.12E+14 1.17E+14 2.44E+13 2.12E-04

Wave Length 8.04E-15 3.37E-04 6.05E-04 1.61E-03 7.74E-03 8.87E+14 1E-07

1E+00 1E-08

1E-01 Length B (cm) 1E-02 1E-09 1 10 100 1000 10000 1E-03 Critical Temperature Tc (K) 1E-04 Wave Length (cm) 1E-05 These structures have an analog in various resonant structures 1 10 100 1000 10000 postulated in the field of chemistry to explain properties of Critical Temperature Tc (K) delocalized π electrons in benzene, ozone and a myriad of other molecules and incorporate the Noether theorem concepts Where ()vd is the de Broglie velocity as in equation 2.1.26 and of energy – time and momentum – length symmetry. also related to resonant CPT time± by equation 2.1.27. 6

The approach is the trisine cell cavity projection on to the y, z Figure 2.2.1 Trisine Steric (Mirror Image) Parity Forms plane as indicated in figure 2.2.3. Steric Form Steric Form One Two 1 3B approach ==23AAB (2.2.6) Rotation 2 3 2A z 0 Degree Figure 2.2.3 Trisine approach from both cavity A2 + B2 = C 2 approaches 120 Degree C A

B 240 Degree A

2B 2B 3 3 The side is the trisine cell cavity projection on to the x, z plane Trisine characteristic volumes with variable names cavity and as indicated in figure 2.2.4. chain as well as characteristic areas with variable names 1 (2.2.7) section, approach and side are defined in equations 2.2.3 - side==22 A B 2 AB 2 2.2.7. See figures 2.2.1 - 2.2.4 for a visual description of these parameters. The mirror image forms in figure 2.2.1 are in Figure 2.2.4 Trisine side view from both cavity sides conformance with parity requirement in Charge conjugation Parity change Time reversal (CPT) theorem as established by determinant identity in equation 2.1.6b and trivially obvious in cross – product expression in equation 2.2.2 in which case an orthogonal coordinate system is right-handed, or left-handed for result 2AB which is later defined as trisine cavity side in equation 2.2.7. Essentially, there are two mirror image sides to each cavity. At the nuclear scale, such chiral geometry is Figure 2.2.5 Trisine Geometry consistent with the standard model (quarks, gluons etc). C A y 22 22 2 2 2 x + = ()22BABBAB×+()=+sin()θ = 2 AB(2.2.2) B A B C 2 cavity= 23 AB (2.2.3) 2B 3 2 chain= cavity (2.2.4) 3 2B 3 Figure 2.2.2 Trisine Cavity and Chain Geometry from Steric (Mirror Image) Forms 2B 3

2B Cell Section 3

2B B B Indicates Trisine Peak Each Peak is raised by height A y 2B x Table 2.2.2 with cavity and section plots Indicates Trisine Peak 0 TKc 8.95E+13 2,135 1,190 447 93 8.11E-16 2 2 2 () 2B A + B = C TK0 3 b () 9.07E+14 3.38E+09 3.31E+09 2.03E+09 9.24E+08 2.729 C A cavity (cm3) 4.29E-40 3.68E-24 8.85E-24 3.84E-23 4.05E-22 15733 B chain (cm3) 2.86E-40 2.45E-24 5.90E-24 2.56E-23 2.70E-22 10489

section (cm2) 1.53E-26 6.43E-16 1.15E-15 3.07E-15 1.48E-14 1693 The section is the trisine cell cavity projection on to the x, y 2 plane as indicated in figure 2.2.2. approach (cm ) 3.22E-27 1.35E-16 2.42E-16 6.45E-16 3.10E-15 356 side (cm2) 3.72E-27 1.56E-16 2.80E-16 7.45E-16 3.58E-15 411 section= 23 B2 (2.2.5) 7

2 1E-18 ⎛ ⎞ K B ⎛ B ⎞ 1E-19 ⎜ + ⎜ ⎟ ⎟ 2B ⎝ 3 ⎠ 1E-20 ⎜ ⎟ 2 1E-21 ⎜ ⎟ cavity KP ⎛ P⎞ kTbc= ⎜ + ⎟ (2.3.8) 1E-22 chain time 22P ⎝⎜ ⎠⎟ 1E-23 ± ⎜ ⎟

Cavity (cm^3) ⎜ 2 ⎟ 1E-24 K A ⎜ A ⎛ ⎞ ⎟ 1E-25 + ⎜ ⎟ ⎝⎜ 22A ⎝ ⎠ ⎠⎟ 1 10 100 1000 10000 Critical Temperature Tc (K) The wave vector K B , being the lowest energy, is the carrier of the superconducting energy and in accordance with derivation in appendix A. All of the other wave vectors are contained 2.3 Trisine Characteristic Wave Vectors within the cell cavity with many relationships with one Superconducting resonant model trisine characteristic wave indicated in 2.3.9 with the additional wave vector ()KC defined in 2.3.10 vectors KKDn, Ds and K C are defined in equations 2.3.1 - 2.3.3 below. mm 111mm 1 et tp gm2 (2.3.9) 1 22+ = st 2 2 (2.3.1) mmKetB+ mmKtpA+ 6 KC ⎛ 6π ⎞ 3 K = Dn ⎝⎜ cavity⎠⎟ 12π K = C 22 (2.3.10) The relationship in equation 2.3.1 represents the normal Debye gs AB+ wave vector K assuming the cavity volume conforming to ()Dn The wave vector K C is considered the hypotenuse vector a sphere in K space and also defined in terms of because of its relationship to KA and KB. KK, and K as defined in equations 2.3.4, 2.3.5 and 2.3.6. AP B The conservation of momentum and energy relationships in See Appendix B for the derivation of equation 2.3.1. 2.3.11, 2.3.12 and 2.1.22 result in a convergence to a trisine 1 lattice B/A ratio of 2.379760996 and trisine mass ()mt of ⎛ 8π 3 ⎞ 3 1 3 110.12275343 x electron mass me . Essentially a reversible K Ds= = ()KKK A B P (2.3.2) () ⎝⎜ cavity⎠⎟ process or reaction is defined recognizing that momentum is a The relationship in equation 2.3.2 represents the normal Debye vector and energy is a scalar. wave vector()K Ds assuming the cavity volume conforming to a Conservation ∆∆px= 0 characteristic trisine cell in K space. ∑ nn Of nBCDsDn= ,, , (2.3.11) See Appendix B for the derivation of equation 2.3.2. gKK±± ±± K K Momentum sBC() DsDn 4π (2.3.3) KC = Conservation 33A ∑ ∆∆Etnn= 0 Of nBCDsDn= ,, , The relationship in equation 2.3.3 represents the result of (2.3.12) Momentum 22 2 2 equating one-dimensional and trisine density of states. See KKgKKBCsDsDn+=() + Appendix C for the derivation of equation 2.3.3. The momentum and energy are in equilibrium and define an Equations 2.3.4, 2.3.5, and 2.3.6 translate the trisine wave elastic state in 1 part in ~100,000 after trisine model iteration of several thousand steps which results in a convergence of vectors KC , K Ds , and K Dn into x, y, z Cartesian coordinates as trisine mass (m )and (B/A) values and concluding in the represented by K B , KP , and K A respectively. The t superconducting current is in the x direction. following: 2π The momentum ratio calculates to: K B = (2.3.4) 2B ±±KKgKKDs Dn() s() ±± B C = 0.999865427 The energy ratio calculates to: 23π 22 22 KP = (2.3.5) gKsDsDnB()+ K() K+ K C= 1.000160309 3B for an average = 1.000012868 2π Another perspective of this convergence is contained in K A = (2.3.6) A Appendix A. Note that the sorted energy relationships of wave vectors are as follows: Numerical values of wave vectors are listed in Table 2.3.1. 222 22 KKKACDsPB>> >> KK (2.3.7) Equation 2.3.8 relates addition of wave function amplitudes (B/√3), (P/2), (B/2) in terms of superconducting resonate energy()kTbc . 8

Table 2.3.1 with Trisine wave vectors Figure 2.3.2 Trisine Steric CPT Geometry in the TK0 c () 8.95E+13 2,135 1,190 447 93 8.11E-16 Superconducting Mode showing the relationship between 0 layered superconducting planes and wave vectors . TK K B b () 9.07E+14 3.38E+09 3.31E+09 2.03E+09 9.24E+08 2.729 Layers are constructed in x, y and z directions making up 4.72E+13 2.31E+08 1.72E+08 1.06E+08 4.81E+07 1.42E-01 K B (/cm) an scaled lattice or Gaussian surface volume xyz. 5.17E+13 2.52E+08 1.88E+08 1.16E+08 5.27E+07 1.56E-01 K Dn (/cm) 8.33E+13 4.07E+08 3.04E+08 1.86E+08 8.49E+07 2.51E-01 K Ds (/cm) 8.65E+13 4.22E+08 3.15E+08 1.93E+08 8.82E+07 2.60E-01 KC (/cm) 5.45E+13 2.66E+08 1.99E+08 1.22E+08 5.56E+07 1.64E-01 KP (/cm) 2.25E+14 1.10E+09 8.19E+08 5.02E+08 2.29E+08 6.76E-01 K A (/cm) Figure 2.3.1 in conjunction with mirror image parity images in Figure 2.2.1 clearly depict the trisine symmetry congruent with the Charge conjugation Parity change Time reversal (CPT) theorem. Negative and positive charge reversal as well as time± reversal takes place in mirror image parity pairs as is more visually seen in perspective presented in Figure 2.3.2.

Figure 2.3.1 Trisine Steric Charge Conjugate Pair 2.4 Trisine Characteristic De Broglie Velocities Change Time Reversal CPT Geometry. The Cartesian de Broglie velocities()vdx , ()vdy , and ()vdz as well as ()vdC are computed with the trisine Cooper CPT Charge conjugated pair residence resonant CPT time± and characteristic frequency()ω in phase with trisine superconducting dimension()2B in equations 2.4.1-2.4.5. K 2B ω eH v ==B =2B =c gB6 dx m time 2π mv s (2.4.1) t ± e ε In equation 2.4.1, note that mvecε eH is the electron spin axis precession rate in the critical magnetic field()H c . This electron spin rate is in tune with the electron moving with de

Broglie velocity()vdx with a CPT residence time± in each cavity. In other words the Cooper CPT Charge conjugated pair flip spin twice per cavity and because each Cooper CPT Charge conjugated pair flips simultaneously, the quantum is an 212() or integer which corresponds to a boson. K v = P dy m (2.4.2) t K v = A dz m (2.4.3) t K v = C dC m (2.4.4) t The vector sum of the x, y, z and C de Broglie velocity components are used to compute a three dimensional helical or

tangential de Broglie velocity()vdT as follows: vvvv=++222 It is important to reiterate that these triangular lattice dT dx dy dz representations are virtual and equivalent to a virtual reciprocal =++KKK222 lattice but as geometric entities, do describe and correlate the ABP mt (2.4.5) physical data quite well. 3 = cos()θ vdC gs It is important to note that the one degree of freedom time ± (2.96E+04 sec or (164 x 3) minutes) in table 2.4.1 corresponds 9 to a ubiquitous 160 minute resonant signal in the 1 3 2 2 ⎛ 8 mtεθcos()⎞ ⎛ 3BB2 ⎞ Universe.[68,69] This would be an indication that the time 4 (2.4.8) ± = ⎜ 2 ⎟ ⎜ − ⎟ momentum and energy conserving (elastic) space lattice ⎝ 81 Μae± ⎠ ⎝ 3 3 ⎠ existing in universe space is dynamically in tune with stellar, where the Madelung constant()M y in the y direction is galactic objects. This appears logical, based on the fact the calculated as: major part of the universe mass consists of this space lattice n=∞ which can be defined in other terms as dark energy matter. ⎛ 1 1 ⎞ M y = 3∑ ⎜ − ⎟ n=0 ⎝ 31()+ 2nn− 1 31()+ 2+ 1⎠ (2.4.9) Table 2.4.1 Listing of de Broglie velocities, trisine cell pair = 0.523598679 residence resonant CPT time± and characteristic frequency()ω And the Thomas scattering formula[43] holds in terms of the as a function of selected critical temperature (Tc) and Debye black body temperature (Tb) along with a time± plot. following equation: 2 TK0 8.95E+13 2,135 1,190 447 93 8.11E-16 2 c () ⎛ ⎞ ⎛ K B ⎞ ⎛ 8π ⎞ e± 0 side = TK 9.07E+14 3.38E+09 3.31E+09 2.03E+09 9.24E+08 2.729 ⎜ ⎟ ⎜ 2 ⎟ (2.4.10) b () ⎜ ⎟ ⎝ K Ds⎠ ⎝ 3 ⎠ ⎝ Mmv yε t dy ⎠

vdx (cm/sec) 4.96E+11 2.42E+06 1.81E+06 1.11E+06 5.06E+05 1.49E-03 A sense of the rotational character of the de Broglie ()vdT vdy (cm/sec) 5.73E+11 2.80E+06 2.09E+06 1.28E+06 5.84E+05 1.72E-03 velocity can be attained by the following energy equation:

v (cm/sec) 2.36E+12 1.15E+07 8.61E+06 5.28E+06 2.41E+06 7.11E-03 1 11chain 2 dz mv2 cos mB2 2 tdT= ()θω()t ( ) vdC (cm/sec) 9.09E+11 4.44E+06 3.31E+06 2.03E+06 9.27E+05 2.74E-03 2 gs cavity 2 (2.4.11) v (cm/sec) 2.48E+12 1.21E+07 9.05E+06 5.54E+06 2.53E+06 7.47E-03 11chain 2 dT + cos mA 2 ()θ ()t ( ) ω time± (sec) 2.68E-25 1.12E-14 2.02E-14 5.37E-14 2.58E-13 2.96E+04 gs cavity 2 ω (/sec) 2.34E+25 5.59E+14 3.12E+14 1.17E+14 2.44E+13 2.12E-04 Equation 2.4.12 provides an expression for the Sagnac time. 22 1E-10 AB+ 2π section 1 time± = 2 (2.4.12) 1E-11 B time± 6 vdx 1E-12 1E-13 2.5 Superconductor Resonant Dielectric Constant And Time (sec) 1E-14 Magnetic Permeability 1E-15 The material dielectric is computed by determining a 1 10 100 1000 10000 displacement()D (equation 2.5.1) and electric field()E Critical Temperature Tc (K) (equation 2.5.2) and which establishes a Equation 2.4.6 represents a check on the trisine cell residence dielectric()ε or DE/ (equation 2.5.3) and modified speed of light v (equations 2.5.4, 2.5.5). Then the superconducting resonant CPT time as computed from the precession of each ()ε ± fluxoid (equation 2.5.6) can be calculated. electron in a Cooper CPT Charge conjugated pair under the ()Φε The electric field(E) is calculated by taking the translational influence of the perpendicular critical field()H c . Note that the energy (m v v /2) (which is equivalent to kT ) over a precession is based on the electron mass()me and not trisine t dx dx bc distance, which is the trisine cell volume to surface ratio. This resonant transformed mass()mt . 11mv concept of the surface to volume ratio is used extensively in time = e ε fluid mechanics for computing energies of fluids passing ± g6 eH B (2.4.6) s ± c through various geometric forms and is called the hydraulic Another check on the trisine cell residence resonant CPT time± ratio. is computed on the position of the Cooper CPT Charge ⎛ 1 ⎞ ⎛ m v2 ⎞ ⎛ 2 section⎞ ⎛ 1 ⎞ conjugated pair() e particles under the charge influence of E tdx = ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ each other within the dielectric()ε as they travel in the y ⎝ gs ⎠ ⎝ 2 ⎠ ⎝ cos()θ ⎠ ⎝ e± cavity⎠ direction as expressed in equation 2.4.7. 1 2 2 2 2 ⎛ 1 ⎞ ⎛ ⎛ B⎞ ⎞ ⎛ mvtdx⎞ ⎛ 1 ⎞ dy Μae± 1 21 (2.5.1) m = 2 () ⎜ + ⎜ ⎟ ⎟ t 2 = 2 (2.4.7) ⎜ g ⎟ ⎝ A⎠ ⎜ time⎟ ⎜ e ⎟ dt εθcos()y ⎝ s ⎠ ⎝ ⎠ ⎝ ±±⎠ ⎝ ⎠ From figure 2.3.1 it is seen that the Cooper CPT Charge 1 mv2 = 425. tdx conjugated pairs travel over distance 2 eA ± ()3323BB− Secondarily, the electric field(E) is calculated in terms of the forces()mtd v/ time exerted by the wave vectors()K . in each quarter cycle or resonant CPT time±/4. 10

evidence can be taken as another measure of superconductivity ⎧ ⎛ 2 mvtdx ⎞ ⎛ 1 ⎞ ⎫ ⎪10 gs cos()θ ⎪ equivalent to the Meissner effect. ⎜ time⎟ ⎜ e ⎟ ⎪ ⎝ ±±⎠ ⎝ ⎠ ⎪ ⎪ ⎛ 1 mv ⎞ ⎛ 1 ⎞ ⎪ Table 2.5.1 with dielectric plot ⎪9 tdy ⎪ 2 0 ⎜ ⎟ ⎜ ⎟ TK 8.95E+13 2,135 1,190 447 93 8.11E-16 ⎪ ⎝ gs time±±⎠ ⎝ e ⎠ ⎪ c ()

E = 0 ⎨ ⎬ (2.5.2) TK 9.07E+14 3.38E+09 3.31E+09 2.03E+09 9.24E+08 2.729 ⎛ 11m v ⎞ ⎛ 1 ⎞ b () ⎪2 t dz ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎪ D (erg/e/cm) 3.63E+17 8.65E+06 4.82E+06 1.81E+06 3.77E+05 3.29E-12 ⎝ gs cos()θ time⎠ ⎝ e ⎠ ⎪ ±±⎪ E (erg/e/cm) 1.96E+21 2.28E+05 9.49E+04 2.18E+04 2.07E+03 5.34E-23 ⎪ ⎛ mvtdC ⎞ ⎛ 1 ⎞ ⎪ ⎪6⎜ cos()θ ⎟ ⎜ ⎟ ⎪ E (volt/cm) 5.87E+23 6.84E+07 2.85E+07 6.55E+06 6.22E+05 1.60E-20 ⎩ ⎝ time± ⎠ ⎝ e± ⎠ ⎭ ε 1.85E-04 37.95 50.83 82.93 181.82 6.16E+10

The trisine cell surface is (section/cos()θ ) and the volume is vε (cm/sec) 2.20E+12 4.87E+09 4.21E+09 3.29E+09 2.22E+09 1.21E+05 expressed as cavity. Note that the trisine cavity, although 2 Φ (gauss cm ) 1.52E-05 3.36E-08 2.90E-08 2.27E-08 1.53E-08 8.33E-13 bounded by (2section/cos()θ ), still has the passageways for ε Cooper CPT Charge conjugated pairs e to move from cell () Using the computed dielectric, the energy associated with cavity to cell cavity. superconductivity can be calculated in terms of the standard The displacement()D as a measure of Gaussian surface Coulomb's law electrostatic relationship e2/()ε B as presented containing free charges e is computed by taking the 4π ()± ()in equation 2.5.7. solid angle divided by the characteristic trisine area 2 ⎧ chain e± ⎫ ()section cos()θ with the two(2) charges () e± contained ⎪cavity B ⎪ therein and in accordance with Gauss's law as expressed in ⎪ ε ⎪ equation 2.1.1. ⎪ 2 ⎪ ⎪ 1 chain e± 3 ⎪ kTbc= ⎨g sΜ y ⎬ (2.5.7) cos()θ cos()θ cavity ε B De= 4π ± (2.5.3) ⎪ ⎪ 2 section ⎪ 1 chain e2 ⎪ Now a dielectric coefficient ()ε can be calculated from the ⎪ tan()θ ± ⎪ g cavity ε A electric field()E and displacement field()D [61]. Note that ⎩⎪ s ⎭⎪ ε < 1 for superluminal velocities as justified by resonant where Μa is the Madelung constant in y direction as computed condition as numerically indicated in table 2.5.1. in equation 2.4.9. D Significantly, equation 2.5.7 is dimensionally congruent with ε = (2.5.4) E equation 2.5.7a. This implies that mass ()mt is created and Assuming the trisine geometry has the relative magnetic destroyed in each time± cycle. permeability of a vacuum k 1 then a modified velocity of 2 ()m = 1 ⎛ m ⎞ cavity 1 light v can be computed from the dielectric coefficient kT t ()ε ()ε bc= ⎜ ⎟ (2.5.7a) 3 ⎝ time± ⎠ mAt and the speed of light()c where ()ε0 = 1. c c A conversion between superconducting temperature and black vε == body temperature is calculated in equation 2.5.9. Two primary ε ε (2.5.5) km energy related factors are involved, the first being the 2 2 ε0 superconducting velocity vdx to light velocity vε and the () 2 () Trisine incident/reflective angle (Figure 2.3.1) of 30 degrees is second is normal Debye wave vector()K Dn to superconducting -1 1/2 0 2 less than Brewster angle of tan ()εε =45 assuring total Debye wave vector()K Ds all of this followed by a minor reflectivity as particles travel from trisine lattice cell to trisine rotational factor for each factor involving ()me and()mp . For lattice cell. reference, a value of 2.729 degrees Kelvin is used for the universe black body temperature T as indicated by Now the fluxoid ()Φε can be computed quantized according to ()b () e as experimentally observed in superconductors. experimentally observed microwave radiation by the Cosmic Background Explorer (COBE) and later satellites. Although 2πv Φ = ε the observed minor fluctuations (1 part in 100,000) in this ε e (2.5.6) ± universe background radiation indicative of clumps of matter What is generally called the Tao effect [65,66,67], wherein it forming shortly after the big bang, for the purposes of this is found that superconducting micro particles in the presence of report we will assume that the experimentally observed an electrostatic field aggregate into balls of macroscopic uniform radiation is indicative of present universe that is dimensions, is modeled by the calculated values for E in table isotropic and homogeneous. 2.5.1. using equation 2.5.2. When the electrostatic field Verification of equation 2.5.6 is indicated by the calculation exceeds a critical value coincident with the critical temperature that superconducting density ()mt cavity (table 2.9.1) and ()Tb , the microscopic balls dissipate. This experimental 11 present universe density (equation 2.11.4) are equal at this ⎧ 2 ⎫ ⎧ 2 o Dvxxε Ex vεx Debye black body temperature T of 2.729 K . ⎧k ⎫ ⎪ 2 ⎪ ⎪ 2 ()b mx E D v v 2 ⎪ ⎪ ⎪ x x dx ⎪ ⎪ dx 2 2 1 ⎛ vε ⎞ 3 ⎪ ⎪ ⎪ ⎪ ⎪ T = gT ⎪ ⎪ ⎪ Dvyyε Ey ⎪ ⎪vεy b ⎜ ⎟ sc (2.5.8) k = k = = ⎝ vdx ⎠ m ⎨ my ⎬ ⎨ 2 ⎬ ⎨ 2 (2.5.12) ⎪ ⎪ ⎪ Ey Dvydy⎪ ⎪vdy ⎪ ⎪ ⎪ 2 ⎪ ⎪ 2 c ⎛ 18 ⎞ Dvzzε Ez vεz kT 2 Kc ⎪kmz ⎪ ⎪ ⎪ ⎪ bb==π⎜ 2 ⎟ A (2.5.9) ⎩ ⎭ 2 2 λ ⎝ g ⎠ ⎪ Ez Dvzdz⎪ ⎪vdz bs ⎩ ⎭ ⎩ Also note that the dielectric ε and permeability k are The black body temperatures ()Tb appear to be high relative to () ()m related as follows: superconducting()Tc , but when the corresponding wave 3 length()λb is calculated in accordance with equation 2.5.7, it is 2 ⎛ mt ⎞ 2 nearly the same and just within the Heisenberg ()∆z parameter km = εθcos () (2.5.12a) ±±⎝⎜ m ⎠⎟ for all Tc and Tb as calculated from equation 2.1.18 and e presented in table 2.5.2. It is suggested that a black body Then the de Broglie velocities (vvdx, dy and v dz ) as per oscillator exists within such a volume as defined by ()∆∆∆xyz equations 2.4.1, 2.4.2, 2.4.3 can be considered a sub- and and is the source of the microwave radiation at 2.729 o K . superluminal speeds of light internal to the superconductor Further, it is suggested that the trisine geometry when scaled to resonant medium as per equation 2.5.13. These sub- and nuclear dimensions is congruent with the experimentally superluminal resonant de Broglie velocities as presented in observed Standard Model energy and mass (quark, gluon, weak equation 2.5.13 logically relate to wave vectors K A , K B and strong force) parameters. and KP . Table 2.5.2 based on equations 2.1.20 and 2.5.9 ⎧ c ⎫ ⎧ 1 E v ⎫ ⎪ ⎪ x c2 TK0 8.95E+13 2,135 1,190 447 93 8.11E-16 dx ⎪ c () k D ⎪ ⎪ kmxε ⎪ ⎪ m x TK0 9.07E+14 3.38E+09 3.31E+09 2.03E+09 9.24E+08 2.729 ⎪ ⎪ ⎪ ⎪ b () ⎪ ⎪ c ⎪ ⎪ 1 Ey 2 vdy ⎬ = ⎨ ⎬ = ⎨ c (2.5.13) ∆x (cm) 1.06E-14 2.17E-09 2.90E-09 4.74E-09 1.04E-08 3.52 k D ⎪ ⎪ kmyε ⎪ ⎪ m y 9.17E-15 1.88E-09 2.52E-09 4.10E-09 9.00E-09 3.05 ⎪ ⎪ ⎪ ⎪ ∆y (cm) c 1 Ez 2 vdz ⎪ ⎪ ⎪ ⎪ c ∆z (cm) 2.23E-15 4.56E-10 6.10E-10 9.96E-10 2.18E-09 0.74 ⎭ k D ⎩⎪ kmzε ⎭⎪ ⎩⎪ m z 1.59E-15 3.25E-10 4.35E-10 7.10E-10 1.56E-09 0.53 λb (cm) Combining equations 2.5.12 and 2.5.13, the Cartesian

dielectric sub- and superluminal velocities ( vεx , vεy & vεz ) can Based on the same approach as presented in equations 2.5.1, be computed and are presented in Table 2.5.3. 2.5.2 and 2.5.3, Cartesian x, y, and z values for electric and Table 2.5.3 Dielectric Velocities ( v , v & v ) displacement fields are presented in equations 2.5.10 and εx εy εz 2.5.11. TK0 c () 8.95E+13 2,135 1,190 447 93 8.11E-16

0 ⎧ F ⎫ ⎧mv 1 TKb ⎫ x tdx () 9.07E+14 4.43E+09 3.31E+09 2.03E+09 9.24E+08 2.729 Ex ⎪ ⎪ ⎪ e time e k 1.88E-01 3.86E+04 5.16E+04 8.43E+04 1.85E+05 6.26E+13 ⎪ ⎪ ± ⎪ ⎪ ±± m ⎪ F mv ⎪ ⎪ y ⎪ ⎪ tdy 1 vεx (cm/sec) 2.15E+11 4.76E+08 4.11E+08 3.22E+08 2.17E+08 1.18E+04 Ey ⎬ = ⎨ ⎬ = ⎨ (2.5.10) e time e 2.49E+11 5.50E+08 4.75E+08 3.72E+08 2.51E+08 1.36E+04 ⎪ ⎪ ± ⎪ ⎪ ±± vεy (cm/sec) ⎪ ⎪ F ⎪ ⎪ m v 1 z t dz vεz (cm/sec) 1.02E+12 2.27E+09 1.96E+09 1.53E+09 1.03E+09 5.62E+04 Ez ⎭⎪ ⎪ ⎪ ⎪ ⎩ e± ⎭ ⎩time±± e We note with special interest that the Lorentz-Einstein resonant 4π relationship expressed in equation 2.5.14 equals 2, which we D ⎫ ⎧ x e± define as Cooper for all T . ⎪ ⎪approach () c ⎪ ⎪ 1 ⎪ ⎪ 4π == 2 Dy ⎬ = ⎨ e± (2.5.11) 2 side vdx (2.5.14) ⎪ ⎪ 1− 2 ⎪ ⎪ 4π vdy e± Dz ⎭⎪ ⎪seection As a check on these dielectric Cartesian velocities, note that ⎩ Now assume that the superconductor material magnetic they are vectorially related to vε from equation 2.5.5 as indicated in equation 2.5.15a. As indicated, the factor '2' is permeability()km is defined as per equation 2.5.12 noting related to the ratio of Cartesian surfaces approach, section and that kkmmxmymz=== k k. side to trisine area cos()θ section and cavity/chain. 12

⎛ cos()θ ⎞ ⎛ chain ⎞ ()approach ⎝⎜ section⎠⎟ ⎝⎜ cavity⎠⎟ 2.6 Fluxoid And Critical Fields Based on the material fluxoid Φ (equation 2.5.5), the critical ⎛ cos θ ⎞ ⎛ chain ⎞ ()ε () fields H (equation 2.6.1), H (equation 2.6.2) & H + ()section ⎜ ⎟ ⎜ ⎟ (2.5.15) ()c1 ()c2 ()c ⎝ section⎠ ⎝ cavitty⎠ (equation 2.6.3) as well as penetration depth()λ (equation ⎛ cos()θ ⎞ ⎛ chaain ⎞ 2.6.5) and Ginzburg-Landau coherence length ()ξ (equation + ()side ==2 2.6.6) are computed. Also the critical field H is alternately ⎝⎜ section⎠⎟ ⎝⎜ cavity⎠⎟ ()c computed from a variation on the Biot-Savart law.

222 222 Φε (2.6.1) vvvvvvvεεεεεεε=++=++2 xyz xyz (2.5.15a) H = c1 πλ 2 Based on the Cartesian dielectric sub- and superluminal t v , v & v ) resonant justified velocities ( εx εy εz , corresponding 4π Cartesian fluxoids can be computed as per equation 2.5.16. H c2 = Φε (2.6.2) section ⎧2πv ⎫ Φ εx ⎧ εx ⎫ ⎪ ⎪ e m g vey e HHH ± t s e ⎪ ⎪ ⎪ ± ⎪ ccc==12 = ±λ (2.6.3) ⎪ ⎪ vεx ⋅⋅ A time me 6 cavity c ⎪ ⎪ ⎪2πvεy ⎪ ⎨Φεy ⎬ = ⎨ ⎬ (2.5.16) ⎪ ⎪ ⎪ e± ⎪ (2.6.4) nc = ⎪ ⎪ ⎪2πv ⎪ cavity Φ εz ⎩⎪ εz ⎭⎪ ⎪ ⎪ It was observed by Huxley [70] that an increasing magnetic ⎩ e± ⎭ field quenched the URhGe superconductor at 2 tesla (20,000 The trisine residence resonant CPT time is confirmed in terms ± gauss) initiated the URhGe superconductor at 8 tesla (20,000 of conventional capacitance()C and inductance()L resonant gauss) quenched the URhGe superconductor at 13 tesla circuit relationships in x, y and z as well as trisine dimensions. (130,000 gauss) ⎧ LC⎫ xx The trisine model H correlates well with the reported [70] ⎪ ⎪ ()c2 time± = ⎨ LCyy⎬ = LC (2.5.17a) 2 (20,000 gauss) with a calculated modeled superconductor ⎪ ⎪ magnetic critical field value ()H c2 of 1.8 tesla (18,000 gauss) LCzz ⎩⎪ ⎭⎪ for critical temperature ()Tc of .28 K. Where relationships between Cartesian and trisine capacitance At critical temperature ()Tc of .4 Kelvin, the trisine model and inductance is as follows: predicts superconductor magnetic critical field ()H c2 of 2.9 tesla (29,000 gauss), which is substantially lower than the 2 CC===xyz C C (2.5.18) observed 13 tesla (130,000 gauss). This could be explained in LL===222 xyz L L (2.5.19) terms of the ferromagnetic properties of URhGe, which essentially mask the superconductor critical magnetic field. LLLxyz== (2.5.20) But it still remarkable how well the model correlates to the CCCxyz== (2.5.21) observed superconducting magnetic critical field()H c2 in this Table 2.5.4 Capacitance and Inductive Density case.

TK0 c () 8.95E+13 2,135 1,190 447 93 8.11E-16 Figure 2.6.1 Experimental (Harshman Data) and Trisine TK0 Model Cooper CPT Charge conjugated Pair Three b () 9.07E+14 4.43E+09 3.31E+09 2.03E+09 9.24E+08 2.729 Dimensional Concentration as per equation 2.6.4 3 C/v (fd/cm ) 7.70E+09 3.76E+04 2.81E+04 1.72E+04 7.85E+03 2.32E-05 1.0E+23 L/v (h/cm3 ) 4.85E+19 2.37E+14 1.77E+14 1.08E+14 4.94E+13 1.46E+05 1.0E+22

In terms of an extended Thompson cross-section()σ T , it is 2 1.0E+21 noted that the (1/R ) factor [62] is analogous to a dielectric()ε equation 2.5.4. Dimensionally the dielectric ()ε is 1.0E+20 proportional to trisine cell length dimension. Harshman Data Trisine Model Cooper Pair Concentraton (/cm3) The extended Thompson scattering cross section ()σ T [43 1.0E+19 equation 78.5, 62 equation 33] then becomes as in equation 2.5.22. 1.0E+18 0 20 40 60 80 100 120 140 2 2 Critical Temperature Tc ⎛ K ⎞ ⎛ 8π ⎞ ⎛ e2 ⎞ σ = B ± ≈ sside (2.5.22) T ⎜ K ⎟ ⎝⎜ 3 ⎠⎟ ⎜ mv2 ⎟ ⎝ Ds⎠ ⎝ ε t dy ⎠ Also it interesting to note that the following relationship holds 13

22 e±±e 22 (2.6.4a) 2π = mvedxe mc Figure 2.6.3 Experimental (Homes, Pratt & Harshman) A B and Trisine Model Penetration Depth

1.0E-03 Figure 2.6.2 Experimental (Harshman Data) and Trisine Harshman's Data Model Cooper CPT Charge conjugated Pair Two Trisine Theory Dimensional Area Concentration or section Homes' Data Pratt's Data 1E+16 1.0E-04

1E+15 1.0E-05 Penetration Depth (cm) 1E+14

1.0E-06 Harshman Data 1E+13 0.1 1 10 100 Trisine Model Critical Temperature Tc (K) Cooper Pair Area Concentration (/cm2)

1E+12 0 20 40 60 80 100 120 140 See derivation of Ginzberg-Landau coherence length ξ in Critical Temperature Tc equation 2.6.6 in Appendix D. It is interesting to note that

B ξ = 273.e≈ for all Tc , which makes the trisine ⎛ 1 ⎞ ⎛ mv2 ⎞ 11 superconductor mode fit the conventional description of 2 t ε λ = ⎜ 2 ⎟ ⎜ ⎟ 2 operating in the “dirty” limit. [54] ⎝ gs ⎠ ⎝ 2 ⎠ nc () e 2 (2.6.5) m 1 chain (2.6.6) ⎛ 111⎞ ⎛ ⎞ ⎛ mc ⎞ 11 ξ = t = t 2 ⎜ 2 ⎟ ⎜ ⎟ ⎜ ⎟ 2 mKeBcavity ⎝ gs ⎠ ⎝ ε ⎠ ⎝ 2 ⎠ nc e ()± From equation 2.6.5, the trisine penetration depth λ is Figure 2.6.4 Trisine Geometry in the Fluxoid Mode calculated. This is plotted in the above figure along with (Interference Pattern Ψ2 ) Harshman[17] and Homes[51, 59] penetration data. The 1/2 Homes data is multiplied by a factor of (ne chain/ ) to get 2 nc in equation 2.6.5. This is consistent with λ ∝1 ()εnc in 2 equation 2.6.5 and consistent with Uemura Plot λ ∝1 nc over short increments of penetration depth λ . It appears to be a good fit and better than the fit with data compiled by Harshman[17].

Penetration depth and gap data for magnesium diboride MgB2

[55], which has a critical temperature Tc of 39 K indicates a fit to the trisine model when a critical temperature of 39/3 or 13 K The geometry presented in Figure 2.6.4 is produced by three is assumed (see Table 2.6.1). This is interpreted as an intersecting coherent polarized standing waves, which can be called the trisine wave function. This can be compared to indication that MgB2 is a three dimensional superconductor. figure 2.6.5, which presents the dimensional trisine geometry. Table 2.6.1 Experimental ( MgB2 Data [55]) and the related Trisine Model Prediction Equation 2.6.7 represents the trisine wave function()Ψ . This wave function is the addition of three sin functions that are 120 Trisine Data at T c Observed Data [55] 39/3 K or 13 K degrees from each other in the x y plane and form an angle 22.8 degrees with this same x y plane. Also note that 22.8 Penetration 276 260 ±20 Depth (nm) degrees is related to the ()B A ratio Gap (meV) 1.12 3.3/3 or 1.1 ±.3 by tan()9000− 22 . 8 = BA. As indicated in equation 2.5.5, the material medium dielectric Ψ =++exp() sin()2()axxxx10 by 10 cz 10 sub- and superluminal speed of light ()vε is a function of medium dielectric ε constant, which is calculated in general, () + exp() sin()2()aaxxxx20++ by 20 cz 20 (2.6.7) and specifically for trisine by ε = DE (equation 2.5.4). The fact that chain rather than cavity (related by 2/3 factor) must be ++ exp() sin()2()axxx30 b330ycz+ x 30 used in obtaining a trisine model fit to Homes’ data indicates Where: 00 0 0 that the stripe concept may be valid in as much as chains in the abxx1 = cos()()030 cos 22 . 8 1 = sin() 030 cos() 2 228. trisine lattice visually form a striped pattern as in Figure 2.3.1. 0 cx1 = −sin()030 14

00 0 0 abxx2 = cos()()150 cos 22 . 8 2 = sin() 150 cos() 2 228. Figure 2.6.5 Critical Magnetic Field Data (gauss) c = −sin 1500 x2 () compared to Trisine Model as a function of T 00 0 0 c abxx3 = cos()()270 cos 22 . 8 3 = sin() 270 cos() 2 228. 100000000 0 10000000 cx3 = −sin()270 1000000 100000 Figure 2.6.5 Trisine Geometry in the Fluxoid Mode 10000 1000

100 C Hc1 Data Hc2 Data 10 A y Hc Data x 2 + 2 = 2 1 Hc1 Trisine B A B C Hc2 Trisine 0.1 Hc Trisine Critical Fields Hc1, Hc, and Hc2 0.01

0.001 0.1 1 10 100 1000 2B Critical Temperature Tc 3 Cell Cavity 2 B 2B 2.7 Superconductor Internal Pressure, Casimir Force 3 And Deep Space Thrust Force (Pioneer 10 & 11) The following pressure (P) values in Table 2.7.1 as based on

2B equations 2.7.1 indicate rather high values as superconducting 3 ()Tc increases. This is an indication of the forces that must be contained in these anticipated materials or lattices. These forces are rather large for conventional materials especially at 2B 3 nuclear dimensions, but range to very small to the virtual lattice existing in outer space. 2 B

⎧ K B ⎫ ⎧ Fx ⎫ ⎪time approach ⎪ ⎪approach ⎪ ⎪ ± ⎪ ⎪ ⎪ 2B B ⎪ K ⎪ ⎪ Fy ⎪ kT P P bc Indicates Trisine Peak Each Peak is raised by height A = ⎨ ⎬ = ⎨ ⎬ = (2.7.1a) ⎪time± side ⎪ ⎪side ⎪ cavity F ⎪ K A ⎪ ⎪ z ⎪ In table 2.6.2, note that the ratio of London penetration ⎪ ⎪ ⎪ ⎪ time secction ssection length()λ to Ginzburg-Landau correlation distance()ξ is the ⎩ ± ⎭ ⎩ ⎭ constant()κ of 58. The trisine values compares favorably with 22 2 Casimir London penetration length λ and constant κ of 1155(3) Å AB+ cavity π vdx ⎛ ⎞ () () P = (2.7.1b) and 68 respectively as reported in reference [11, 17] based on B chain 240 A4 ⎝⎜ ppressure ⎠⎟ experimental data.

Table 2.6.2 Critical Fields and Lengths ⎧ mvtdx ⎫

0 ⎪ ⎪ TK time approach c () 8.95E+13 2,135 1,190 447 93 8.11E-16 ⎪ ± ⎪

0 TK ⎪ mvtdy ⎪ kTbc b () 9.07E+14 4.43E+09 3.31E+09 2.03E+09 9.24E+08 2.729 P = ⎨ ⎬ = (2.7.1c) 3.29E+19 1.74E+06 8.36E+05 2.46E+05 3.46E+04 1.64E-17 time± side caviity H c (gauss) ⎪ ⎪ ⎪ ⎪ H (gauss) 4.36E+16 2.30E+03 1.11E+03 3.26E+02 4.57E+01 2.17E-20 mvtdz c1 ⎪ ⎪ ttime± section c me 2 ⎩ ⎭ H c 1.20E+16 1.30E+08 8.39E+07 4.02E+07 1.24E+07 1.99E-06 vdx mt 3 To put the calculated pressures in perspective, the C-C bond has a reported energy of 88 kcal/mole and a bond length of 2.49E+22 1.31E+09 6.32E+08 1.86E+08 2.61E+07 1.24E-14 H c2 (gauss) 1.54 Å [24]. Given these parameters, the internal chemical 3 4.66E+39 5.43E+23 2.26E+23 5.20E+22 4.94E+21 1.27E-04 nc (/cm ) bond pressure ()CBP for this bond is estimated to be: 1.05E-11 2.16E-06 2.89E-06 4.71E-06 1.03E-05 3.50E+03 kcal erg 1 mole 1 bond λ (cm) 88 4. 18x 1010 mole kcal60210. x 23 bond −8 3 cm3 ξ (cm) 1.81E-13 3.72E-08 4.98E-08 8.12E-08 1.78E-07 60.30 ()15410. x 12 erg Figure 2.6.5 is graph of generally available critical field data as = 16710. x 3 compile in Weast [16]. cm Within the context of the superconductor internal pressure 15

numbers in table 2.7.1, the internal pressure requirement for ()mt cavity is the mass equivalent energy density (pressure) the superconductor is less than the C-C chemical bond pressure 2 0 ()mt c cavity value at TEKc = 811. − 16 at the order of design Tc of a few thousand degrees Kelvin indicating that there is the possibility of using materials with representing conditions in space, which is 100 percent, the bond strength of carbon to chemically engineer high transferred to an object passing through it. performance resonant superconducting materials. mt 22 (2.7.4) Force= − AREA c= − AREA()ρU c In the context of interstellar space vacuum, the total pressure cavity 2 mt c cavity may be available (potential energy) and will be of Note that equation 2.7.4 is similar to the conventional drag such a magnitude as to provide for universe expansion as equation as used in design of aircraft with v2 replaced with c2 . astronomically observed. (see equation 2.11.21). Indeed, there The use of c2 vs. v2 means that the entire resonant energy 2 is evidence in the Cosmic Gamma Ray Background Radiation content (mtc /cavity) is swept out of volume as the spacecraft 2 for energies on the order of mct or 56 Mev or (log10(Hz) = passes through it. Of course Force= Ma , so the Pioneer 22.13) or ( 2.20E-12 cm) [73]. In the available spectrum [73], spacecraft M/ AREA , is an important factor in establishing there is a hint of a black body peak at 56 Mev or (log10(Hz) = deceleration, which was observed to be on the order of 8.74E-8 22.13) or ( 2.20E-12 cm). This radiation has not had an cm2/sec. identifiable source until now. The force equation is repeated below as the conventional drag Table 2.7.1 with pressure (P) plot equation in terms of Pioneer spacecraft and the assumption that

0 space density ρ (2.11.16) is essentially what has been TK ()U c () 8.95E+13 2,135 1,190 447 93 8.11E-16 determined to be a potential candidate for dark matter: TK0 b () 9.07E+14 4.43E+09 3.31E+09 2.03E+09 9.24E+08 2.729 2 v pressure 5.76E+37 1.60E+11 3.71E+10 3.21E+09 6.34E+07 1.42E-35 FMaCA==dcUρ erg/cm3 2 pressure (2.7.4) 8.35E+32 2.32E+06 5.39E+05 4.66E+04 9.20E+02 2.06E-40 24 6 ()psi where:. Cd =+ + 4 [660] empirical R pressure e 1+ Re 5.76E+36 1.60E+10 3.71E+09 3.21E+08 6.34E+06 1.42E-36 ()pascal and where: Total F = force pressure 1.36E+24 2.45E+19 1.02E+19 2.35E+18 2.23E+17 5.73E-09 erg/cm3 M = Pioneer 10 & 11 Mass 1E+12 a = acceleration 1E+10 Ac = Pioneer 10 & 11 cross section 1E+08

1E+06 When the Reynolds’ number (Re) is low (laminar flow 1E+04 condition), then Cde= ()24 R and the drag equation reduces to

Pressure (pascal) 1E+02 the Stokes’ equation FvD= 3πµ . [60]. When the Pioneer 1E+00 spacecraft data is fitted to this drag equation, the computed drag force is 6 orders of magnitude lower than observed 1 10 100 1000 10000 Critical Temperature Tc (K) values.µ (fluid absolute viscosity) Assuming the superconducting current moves as a longitudinal In other parts of this report, a model is developed as to what wave with an adiabatic character, then equation 2.7.2 for this dark matter actually is. The model dark matter is keyed to a 6.38E-30 g/cc value ()ρU (2.11.16) in trisine model (NASA current or de Broglie velocity ()vdx holds where the ratio of heat capacity at constant pressure to the heat capacity at observed value of 6E-30 g/cc). The proposed dark matter constant volume ()δ equals 1. consists of mass units of mt (110.12275343 x electron mass) per volume (Table 2.2.2 cavity- 15733 cm3). These mass units cavity are virtual particles that exist in a coordinated lattice at base vdx = δ pressure (2.7.2) mt energy ω /2under the condition that momentum and energy The Casimir force is related to the trisine geometry by equation are conserved - in other words complete elastic character. This 2.7.3. dark matter lattice would have internal pressure (table 2.7.1) to withstand collapse into gravitational clumps. section π 2v Casimir Force= g3 dz (2.7.3) Under these circumstances, the traditional drag equation form s 3 240A4 is valid but velocity (v) should be replaced by speed of light In a recent report[53] by Jet Propulsion Laboratory, an (c). Conceptually, the equation 2.7.4 becomes a thrust unmodeled deceleration was observed in regards to Pioneer 10 equation rather than a drag equation. Correspondingly, the and 11 spacecraft as they exited the solar system. This 2 space viscosity is computed as momentum/area or (m c/(2∆x )) deceleration can be related empirically to the following t where ∆x is uncertainty dimension in table 2.5.1 at T = 8.1E- dimensional correct force expression (2.7.4) where c 16

16 K and as per equation 2.1.20. This is in general agreement The sun referenced Pioneer velocity of about 12 km/s with gaseous kinetic theory [19]. or earth (around sun) orbital velocity of 29.8 km/sec The general derivation is in one linear dimensional path (s) and or sun (circa milky way center) rotation velocity of 400 km/sec in accordance with Newton’s Second Law, Resonant Mass and or the CMBR dipole of 620 km/s Work-Energy Principles in conjunction with the concept of or any other velocity 'v' magnitude (0> 1 () ⎝⎜ c2 ⎠⎟ established within trisine model and used. = 0 at vc= (photon has zero mass) Pioneer (P) Translational Calculations Clearly, an adiabatic process is described wherein the change P mass 241,000 gram in internal energy of the system (universe) is equal in absolute P diameter 274 cm (effective) magnitude to the work (related to objects moving through the P cross section 58,965 cm2 (effective) universe) or Work = Energy. Momentum is transferred from P area/mass 0.24 cm2/g the trisine elastic space lattice to the object moving through it. P velocity 1,117,600 cm/sec The hypothesis can be stated as: space kinematic viscosity 1.90E+13 cm2/sec 3 An object moving through momentum space space density 6.38E-30 g/cm will slow down. Pioneer Reynold's number 4.31E-01 unitless thrust coefficient 59.67 It is understood that the above development assumes that P deceleration 8.37E-08 cm/sec2 dv/ dt ≠ 0 , an approach which is not covered in standard one year thrust distance 258.51 mile texts[61], but is assumed to be valid here because of it laminar thrust force 9.40E-03 dyne describes in part the actual physical phenomenon taking place when an object impacts or collides with an trisine elastic space Time of object to stop 1.34E+13 sec lattice cell. The elastic space lattice cell essentially collapses JPL, NASA raised a question concerning the universal application of the observed deceleration on Pioneer 10 & 11. B → 0 and the entire cell momentum mc is transferred to () ()t In other words, why do not the planets and their satellites the object with cell permittivity ()ε and permeability ()km proceeds to a value of one(1). experience such an deceleration? The answer is in the AMc ratio in the modified thrust relationship. In terms of an object passing through the trisine elastic space 2 mct Ac 22Actm CPT lattice at some velocity vo , the factor Fs/ mt is no () () aC≈− t = −Ct ρUtcC= − c (2.7.6) longer dependent of object velocity ()vo as defined in s M M cavity 2 ()vvdx<<< o cbut is a constant relative to c which reflects Using the earth as an example, the earth differential movement the fact that the speed of light is a constant in the universe. per year due to modeled deceleration of 4.94E-19 cm sec2 2 Inspection would indicate that as ()vo is in range would be 0.000246 centimeters (calculated as at 2 , an ()vvdx<<< o c then the observed object deceleration "a" is a unobservable distance amount). Also assuming the trisine constant (not tied to any particular reference frame either superconductor model, the reported 6 nanoTesla (nT) (6E-5 translational or rotational): gauss) interplanetary magnetic field (IMF) in the vicinity of F earth, (as compared to .2 nanoTesla (nT) (2E-6 gauss [77]) a = M interstellar magnetic field congruent with CMBR) will not allow the formation of a superconductor CPT lattice due to the and specifically for object and trisine elastic space lattice well known properties of a superconductor in that magnetic relative velocities v as follows: ()o fields above critical fields will destroy it. In this case, the 17

space superconductor at Tc = 8.1E-16 K will be destroyed by a are statistically equal to each other and the two space craft magnetic field above exited the solar system essentially 190 degrees from each other implies that the supporting fluid space density through which Hcdxet()()( c v m m chain cavity )= 2.E-6 gauss they are traveling is co-moving with the solar system. But this as in table 2.6.2. It is conceivable that the IMF decreases by may not be the case. Assuming that the supporting fluid some power law with distance from the sun, and perhaps at density is actually related to the cosmic background microwave some distance the IMF diminishes to an extent wherein the radiation (CMBR), it is known that the solar system is moving space superconductor CPT lattice is allowed to form. This at 500 km/sec relative to CMBR. Equivalent decelerations in may be an explanation for the observed “kick in” of the irrespective to spacecraft heading relative to the CMBR would Pioneer spacecraft deceleration phenomenon at 10 Astronomical Units (AU). further support the hypothesis that deceleration is independent of spacecraft velocity. Earth Translational Calculations Earth mass 5.98E+27 gram Figure 2.7.1 Pioneer Trajectories (NASA JPL [53]) Earth diameter 1.27E+09 cm with delineated solar satellite radial component r. Earth cross section 1.28E+18 cm2 Earth area/mass 2.13E-10 cm2/g Earth velocity 2,980,010 cm/sec space kinematic viscosity 1.90E+13 cm2/sec space density 6.38E-30 g/cm3 Earth Reynold's number 2.01E+06 unitless thrust coefficient 0.40 unitless

thrust force (with Ct ) 2.954E+09 dyne Earth deceleration 4.94E-19 cm/sec2 One year thrust distance 0.000246 cm laminar thrust force 4.61E-01 dyne Time for Earth to stop rotating 6.03E+24 sec Now to test the universal applicability of the unmodeled Pioneer 10 & 11 decelerations for other man made satellites, Executing a web based [58] computerized radial rate dr/dt one can use the dimensions and mass of the Hubble space (where r is distance of spacecraft to sun) of Pioneer 10 & 11 telescope. One arrives at a smaller deceleration of 6.79E-09 Figure 2.7.1), it is apparent that the solar systems exiting 2 cm/sec because of its smaller area/mass ratio. This low velocities are not equal. Based on this graphical method, the deceleration is swamped by other decelerations in the vicinity spacecraft’s velocities were in a range of about 9% (28,600 – of earth, which would be multiples of those well detailed in 27,500 mph for Pioneer 10 “3Jan87 - 22July88” and 26,145 – reference 1, Table II Pioneer Deceleration Budget. Also the 26,339 mph for Pioneer 11 ”5Jan87 - 01Oct90”). This trisine elastic space CPT lattice probably does not exist in the reaffirms the basic thrust equation used for computation in that, immediate vicinity of the earth because of the earth’s velocity magnitude of the spacecraft is not a factor, only its milligauss magnetic field, which would destroy the coherence direction with the thrust force opposite to that direction. The of the trisine elastic space CPT lattice. observed equal deceleration at different velocities would Hubble Satellite Translational Calculations appear to rule out deceleration due to Kuiper belt dust as a Hubble mass 1.11E+07 gram source of the deceleration for a variation at 1.092 or 1.19 would Hubble diameter 7.45E+02 centimeter be expected. Hubble cross section 5.54E+05 cm2 Also it is observed that the Pioneer spacecraft spin is slowing Hubble area/mass 0.0499 cm2/g down from 7.32 rpm in 1987 to 7.23 rpm in 1991. This Hubble velocity 790,183 cm/sec equates to an angular deceleration of .0225 rpm/year or 1.19E- 2 space kinematic viscosity 1.90E+13 cm2/sec 11 rotation/sec . JPL[57] has acknowledged some systematic space density 6.38E-30 g/cm3 forces that may contribute to this deceleration rate and suggests Hubble Reynolds' number 1.17E+00 unitless an unmodeled deceleration rate of .0067 rpm/year or 3.54E-12 thrust coefficient 23.76 rotations/sec. [57] Now using the standard torque formula: thrust force (with Ct ) 7.549E-02 dyne 2 Hubble deceleration 6.79E-09 cm/sec Γ = Iω (2.7.7) One year thrust distance 3,378,634 cm The deceleration value of 3.54E-12 rotation/sec2 can be laminar thrust force 7.15E-08 dyne replicated assuming a Pioneer Moment of Inertia about spin Time for Hubble to stop rotating 1.16E+14 sec axis(I) of 5.88E+09 g cm2 and a 'paddle' cross section area of The fact that the unmodeled Pioneer 10 & 11 deceleration data 3,874 cm2 which is 6.5 % spacecraft 'frontal" cross section 18

which seems reasonable. The gross spin deceleration rate of launch a series of spacecraft of varying AMc ratios and .0225 rpm/year or 1.19E-11 rotation/sec2 results in a pseudo exiting the solar system at various angles while measuring their paddle area of 13,000 cm2 or 22.1% of spacecraft 'frontal" rotational and translational decelerations. This would give a cross section which again seems reasonable. more precise measure of the spatial orientation of dark matter Also it is important to note that the angular deceleration rate in the immediate vicinity of our solar system. An equal for each spacecraft (Pioneer 10, 11) is the same even though calculation of space energy density by translational and they are spinning at the two angular rates 4 and 7 rpm rotational observation would add credibility to accumulated respectfully. This would further confirm that velocity is not a data. factor in measuring the space mass or energy density. The calculations below are based on NASA JPL problem set Figure 2.7.2 Space craft general design to quantify values[57]. Also, the sun is moving through the CMBR at 600 translational and rotary thrust effect. Brown is km/sec. According to this theory, this velocity would not be a parabolic antenna, green is translational capture factor in the spacecraft deceleration. element and blue is rotational capture element. Pioneer (P) Rotational Calculations P mass 241,000 g P moment of inertia 5.88E+09 g cm2 P diameter 274 cm P translational cross section 58,965 cm2 P radius of gyration k 99 cm P radius r 137 cm paddle cross section 3,874 cm2 paddle area/mass 0.02 cm2/g P rotation speed at k 4,517 cm/sec P rotation rate change 0.0067 rpm/year P rotation rate change 3.54E-12 rotation/sec2 P rotation deceleration at k 2.20E-09 cm/sec2 P force slowing it down 1.32E-03 dyne space kinematic viscosity 1.90E+13 cm2/sec The assumption that the Pioneer spacecraft will continue on space density 6.38E-30 g/cm3 forever and eventually reach the next star constellation may be P Reynolds' number 4.31E-01 unitless wrong. Calculations indicate the Pioneers will come to a stop relative to space fluid in about 400,000 years. We may be more thrust coefficient 59.67 isolated in our celestial position than previously thought. thrust force 1.32E-03 dyne One year rotational thrust When this thrust model is applied to a generalized design for a 1,093,344 cm 2 distance Solar Sail [63], a deceleration of 3.49E-05 cm/sec is P rotational laminar thrust calculated which is 3 orders higher than experienced by 7.32E-23 dyne force Pioneer spacecraft. Consideration should be given to spin Time for P to stop rotating 2.05E+12 sec paddles for these solar sails in order to accumulate spin These calculations suggest a future spacecraft with general deceleration data also. In general, a solar sail with its design below to test this hypothesis. This design incorporates characteristic high AMc ratio such as this example should be general features to measure translational and rotational thrust. very sensitive to the trisine elastic space CPT lattice. The A wide spectrum of variations on this theme is envisioned. integrity of the sail should be maintained beyond its projected General Space Craft dimensions should be on the order of solar wind usefulness as the space craft travels into space beyond Jupiter. Trisine geometry B at TKTEKbc==2.. 711 8 12− 16 or 22 cm (Table 2.2.1) or larger. The Space Craft Reynold’s number Sail Deceleration Calculations would be nearly 1 at this dimension and the resulting thrust the Sail mass 300,000 gram greatest. The coefficient of thrust would decrease towards one Sail diameter 40,000 centimeter with larger Space Craft dimensions. It is difficult to foresee Sail cross section 1,256,637,000 cm2 what will happen with Space Craft dimensions below 22 cm. Sail area/mass 4,189 cm2/g The spacecraft (Figure 2.7.2) would rotate on green bar axis Sail velocity 1,117,600 cm/sec while traveling in the direction of the green bar. The red space kinematic viscosity 1.90E+13 cm2/sec 2 paddles would interact with the space matter as mc slowing space density 6.38E-30 g/cm3 the rotation with time in conjunction with the overall Sail Reynold's number 6.30E+01 unitless spacecraft slowing decelerating. thrust coefficient 1.45 A basic experiment cries out here to be executed and that is to 19

thrust force 10 dyne 2 2 GM 2 ⎛ Area 2 ⎞ 2 Sail acceleration 3.49E-05 cm/sec v 0 +=+vvC⎜ 0 − t ρct⎟ R0 ⎝ M ⎠ (2.7.8b) one year thrust distance 1.73E+10 cm where v v laminar thrust force 1.37E+00 dyne = δ 0 Time of Sail to stop 1015 year Assuming the sun orbital velocity perturbing factor δ to be .9 The following calculation establishes a distance from the sun at and the time (t) at the estimated age of the solar system which the elastic space CPT lattice kicks in. There is a energy (5,000,000,000 years), then the smallest size objects that could density competition between the solar wind and the elastic remain in sun orbit are listed below as a function of distance space CPT lattice. Assuming the charged particle density at (AU) from the sun. earth is 5 / cm3 [57] and varying as 1/R2 and comparing with Space object upper size elastic space density of kTc/cavity, then an equivalent energy AU diameter (meter) density is achieved at 3.64 AU. This of course is a dynamic Mercury 0.39 9.04 situation with solar wind changing with time, which would Venus 0.72 10.67 result in the boundary of the elastic space CPT lattice changing Earth 1.00 11.63 with time. This is monitored by a space craft (Advanced Mars 1.53 13.04 Composition Explorer (ACE)) at the L1 position (1% distance Jupiter 5.22 18.20 to sun) and can be viewed at Saturn 9.57 21.50 http://space.rice.edu/ISTP/dials.html. A more descriptive term Uranus 19.26 26.10 would be making the elastic space CPT lattice decoherent or Neptune 30.17 29.60 coherent. It is interesting to note that the calculated boundary is close to that occupied by the Astroids and is considered the Pluto 39.60 31.96 Astroid Belt. Kuiper Belt 50 35.95 60 37.57 Elastic Space CPT lattice intersecting 70 39.03 with solar wind Oort Cloud? 80 40.36 AU 100 41.60 Mercury 0.39 150 46.75 Venus 0.72 Earth 1.00 200 50.82 Mars 1.53 300 57.20 400 62.24 Jupiter 5.22 This calculation would indicate that there is not small particle Saturn 9.57 dust remaining in the solar system, which would include the Uranus 19.26 Kuiper Belt and hypothesized Oort Cloud. Also, the calculated Neptune 30.17 small object size would seem to be consistent with Asteroid Pluto 39.60 Belt objects, which exist at the coherence/decoherence AU 3.64 boundary for the elastic space CPT lattice. n at AU 0.38 cm-3 1/3 This analysis has implications for large-scale galactic r 1/n 1.38 cm structures as well. Indeed, there is evidence in the Cosmic coordination # 6 unitless Gamma Ray Background Radiation for energies on the order volume 0.44 cm3 of m c2 or 56 Mev or (log (Hz) = 22.13) or (2.20E-12 cm) e2 1 T energy t 10 c [73]. In the available spectrum [73], there is a hint of a black 3 r volume Tb volume 1.13E-34 erg / cm body peak at 56 Mev or (log10(Hz) = 22.13) or ( 2.20E-12 cm). kT energy This radiation has not had an identifiable source until now. It Trisine bc cavity volume 1.13E-34 erg / cm3 is conceivable that energy densities may vary spatially within a Now assuming the basic conservation of total kinetic and galaxy with generally increased energy density towards the potential energy for an object in circular orbit of the sun with center. This would imply that a generally radial boundary velocity v and at distance R equation 2.7.8 provides an condition would exist within a galaxy and may be responsible ()0 ()0 for observed anomalous dark matter galactic rotation. indication of the time required to perturb the object out of the circular orbit in the direction of the sun. Apparent clumping of galactic matter may be due to localities where elastic space CPT lattice does not exist due to GM 2 2 2 decoherent effects of various energy sources existing at that v 0 +=+vvat()0 − (2.7.8a) R0 particular locality. Dark energy and Dark matter may be one and the same. Reference [76] indicates the very existence of dark matter and just such clumping thereof as well as a thrust effect on the dark matter on galactic material. Reference [79] 20 substantiates the independence of dark and visible matter space kinematic viscosity 1.90E+13 cm2/sec distribution. According to this correlation, the observed dark space density 6.38E-30 g/cm3 matter distributions are due to discrete quantum mechanical New Horizon Reynold's number 3.31E-01 unitless states of the ubiquitous dark matter which is uniform and drag coefficient 76.82 currently unobservable in most of space but clumps in the drag force (with drag coefficient) 1.525E-02 dyne vicinity of massive objects such as galaxies or residual effects New Horizon deceleration 3.24E-08 cm/sec2 of historical massive collision events. one year drag distance 16,132,184 cm Reference 64 presents an experimental protocol in which a laminar drag force 7.21E-03 dyne comoving optical and sodium atom lattice is studied at Time of New Horizon to stop 5.00E+13 sec nanoKelvin a temperature. When the moving lattice is stopped (generating laser beams extinguished), the sodium atoms stop New Horizon (NH) Predictive Rotational Calculations rather than proceeding inertially. This unexplained anomaly NH mass 470,000 g may be explained in a similar manner as Pioneer deceleration. NH moment of inertia NASA JPL 4.01E+09 g cm2 The following calculation indicates a sodium atom deceleration large enough to indicate essentially an instantaneous stop. NH diameter 315 cm NH translational cross-section 77,931 cm2 sodium temperature 1.00E-06 Kelvin NH radius of gyration k 131 cm sodium mass 3.82E-23 gram NH radius r 158 cm sodium van der Waals diameter 4.54E-08 centimeter 2 cm 2 paddle cross-section 60,210 sodium cross-section 1.62E-15 cm paddle area/mass 1.28E-01 cm2/g 2 sodium area/mass 4.24E+07 cm /g NH rotation speed at k 5,971 cm/sec optical lattice velocity 3.0 cm/sec NH rotation rate change 0.2000 rpm/year 2 space kinematic viscosity 1.90E+13 cm /sec NH rotation rate change 1.06E-10rotation/sec2 2 space density 6.38E-30 g/ cm NH rotation deceleration at k 8.68E-08 cm/sec2 sodium Reynold's number 7.15E-11 unitless NH force slowing it down 2.04E-02 dyne thrust coefficient 3.36E+11 unitless 2 space kinematic viscosity 1.90E+13 cm /sec thrust force (with thrust space density 6.38E-30 g/cm3 coefficient) 3.116E-12 dyne NH Reynold's number 4.36E-01 unitless sodium deceleration 8.16E+10 cm/sec2 thrust coefficient 59.09 time to stop sodium atom 3.68E-11 sec thrust force (with thrust 2.04E-02 dyne distance to stop sodium atom 5.51E-11 cm coefficient) An experiment is envisioned whereby larger atomic clusters One year rotational thrust distance 43,139,338 cm are decelerated in a more observable manner because of their NH rotational laminar thrust force 1.90E-02 dyne increased mass. It assumed that such an anomalous Time for NH to stop rotating 6.88E+10 sec deceleration will occur only at nanoKelvin temperatures where This New Horizon deceleration characteristics will be the kT energies are essentially zero. monitored in the vicinity of Jupiter passage on February 28, It is understood that the New Horizon spacecraft launched on 2007 and thereafter. January 19, 2006 presents an opportunity to replicate the Pioneer 10 & 11 deceleration. The tables below provide a 2.8 Superconducting Resonant Current, Voltage And predictive estimate of this deceleration. Because of the New Conductance Horizon smaller area to mass ratio than Pioneer, it is The superconducting electrical current()Ie is expressed as in anticipated that the New Horizon anomalous deceleration will terms of a volume chain containing a Cooper CPT Charge be smaller than the Pioneer anomalous deceleration by a factor conjugated pair moving through the trisine CPT lattice at a de of about .4. Also, because the New Horizon Moment of Inertia Broglie velocity()vdx . about the spin axis is smaller than Pioneer, it is predicted that I Cooper e v the rate of spin deceleration will be greater for the New edx= ± (2.8.1) Horizon than observed for the Pioneer space crafts. area chain The superconducting mass current I is expressed as in terms New Horizon (NH) Predictive Translation Calculations ()m of a trisine resonant transformed mass ()mt per cavity New Horizon mass 470,000 gram containing a Cooper CPT Charge conjugated pair and the New Horizon diameter 210 cm (effective) cavity Cooper CPT Charge conjugated pair velocity (vdx). New Horizon cross section 34,636 cm2 (effective) 2 m New Horizon area/mass 0.07 cm /g ⎛ t ⎞ Im = ⎜ ⎟ vdx (2.8.2) trisine area/mass 5.58E+30 cm2/g ⎝ cavity⎠ New Horizon velocity 1,622,755 cm/sec It is recognized that under strict CPT symmetry, there would 21 be zero current flow. It must be assumed that pinning of one the sub- and superluminal nuclear conditions side of CPT will result in observed current flow. Pinning ()vcvcvcvcdx<< & εε x& () dx>> & x . (beyond that which naturally occurs) of course has been ()()group velocity phase velocity = c2 observed in real superconductors to enhance actual ⎛ 1 v2 ⎞ 1 ⎛ chain v2 ⎞ (2.8.6) supercurrents. This is usually performed by inserting unlike ()v εx cv () εx cc2 dx ⎜ 2 ⋅ ⎟ ≅ dx ⎜ 2 ⋅ ⎟ ≅ atoms into the superconducting molecular lattice or ⎝ π vdx ⎠ 2 ⎝ ccavity vdx ⎠ mechanically distorting the lattice at localized points [72]. In the Nature reference [51] and in [59], Homes reports Also, there are reports that superconductor surface roughness experimental data conforming to the empirical relationship enhances super currents by 30% [72]. This could be due to the Homes’ Law relating superconducting density ρs at just inherent “roughness” of the CPT symmetry better expressing aboveT , superconductor conductivity σ and T . itself at a rough or corrugated surface rather than a smooth one. c dc c ρσsdcc∝ T Homes' Law ref: 51 (2.8.7) Secondarily, this reasoning would provide reasoning why a net charge or current is not observed in a vacuum such as trisine The Trisine model is developed forTc , but using the Tanner’s model extended to critical space density in section 2.11. law observation that nSN = n 4 (ref 52), we can assume that E = E 4. This is verified by equation 2.5.1. Now Table 2.8.1 with super current ()amp/ cm2 plot. fN fS substituting IEefN()area for superconductor conductivity σ dc , 0 TK and using cavity for super current density ρs , the following c () 8.95E+13 2,135 1,190 447 93 8.11E-16 relationship is obtained and is given the name Homes’ TK0 b () 9.07E+14 4.43E+09 3.31E+09 2.03E+09 9.24E+08 2.729 constant. 2 Ie ()amp/ cm 5.56E+32 3.17E+11 9.83E+10 1.39E+10 6.01E+08 4.56E-26 Note: σσss()cgs = 8.99377374E+11 x ( S/cm ) 2 Im ()gcm//sec1.16E+26 6.61E+04 2.05E+04 2.90E+03 1.25E+02 9.52E-33 22 σ kT3π e A Homes' constant ==dc b c ± 1E+14 2 m2 B ρs t 5 1E+12 cm = 191, 537 (2.8.8) sec3 1E+10 or σ kT 1E+08 ρ = dc b c (ccgs units) s 191, 537 1E+06 Equation 2.8.8 is consistent with equation 2.8.3. Supercurrent (amp/cm^2) 1E+04 The results for both c axis and a-b plane (ref 51) are 1 10 100 1000 10000 Critical Temperature Tc (K) graphically presented in the following figure 2.8.1. On average the Homes’ constant calculated from data exceeds the Homes’ A standard Ohm's law can be expressed as follows with the constant derived from Trisine geometry by 1.72. resistance expressed as the Hall resistance with a value of 25,815.62 ohms (von Klitzing constant). Figure 2.8.1 Homes’ data compared to Trisine Model voltage = () current () resistance Homes’ Constant 3.50 ⎛ ⎞ kTbc ⎛ e± ⎞ 2π (2.8.3) = approach v 3.00 ⎝⎜ dx ⎠⎟ ⎜ 2 ⎟ e± chain ⎝ e ⎠ ()± Equation 2.8.4 represents the Poynting vector relationship or in 2.50 other words the power per area per resonant CPT time± being 2.00 transmitted from each superconducting cavity:

1.50 me 1 kTbc= Exc H()2 side time± vε (2.8.4) Homes' Data/Homes' Constant mt 2 1.00 Equation 2.8.4 can be rearranged in terms of a relationship Homes' Data 0.50 Homes' Constant between the de Broglie velocity ()vdx and modified speed of light velocity ()vεx as expressed in equation 2.8.5. 0.00 2 2 0 102030405060708090100 1 Superconductor Critical Temperature Tc 2 3 ⎛ mt ⎞ () e± B B v v 34 c vvcv εx = 2 ⎜ ⎟ dx = ⋅⋅dx= π ⋅⋅ dx 48π ⎝ me ⎠ A A (2.8.5) There may be some degrees of freedom considerations12⋅ kTbc/ , 22⋅ kTbc/ , 32⋅ kTbc/ that warrant This leads to the group/phase velocity relationship (as looking into that may account for this divergence although previously stated in 2.1.30) in equation 2.8.6, which holds for conformity is noteworthy. Also 1.72 is close to sqrt(3), a value 22

with Plot of Cooper Pair density that is inherent to Trisine geometry. This correspondence () warrants a closer look. In general, the conventional and high TK0 c () 8.95E+13 2,135 1,190 447 93 8.11E-16 temperature superconductor data falls in the same pattern. It is TK0 concluded that Homes’ Law can be extrapolated to any critical b () 9.07E+14 4.43E+09 3.31E+09 2.03E+09 9.24E+08 2.729 3 2.34E+14 2.73E-02 1.13E-02 2.61E-03 2.48E-04 6.38E-30 temperatureTc . ρC (g/cm )

Θdx 1.44E+25 4.00E-02 9.28E-03 8.03E-04 1.58E-05 3.55E-48 2.9 Superconductor Apparent Weight Reduction In A Θ 3.59E+26 1.00 2.32E-01 2.00E-02 3.96E-04 8.88E-47 Gravitational Field dT Based on the superconducting helical or tangential velocity 1E+01

()vdT calculated from equation 2.4.5, the superconducting 1E-01 gravitational shielding effect()Θ is computed based on Newton's gravitational law and the geometry as presented in 1E-03 (g/cm^3) Figure 2.9.1. 1E-05 Cooper Pair Density Figure 2.9.1 Superconductor Gravitational Model Based 1E-07 1 10 100 1000 10000 on Super current mass flow with Mass Velocities()vdx Critical Temperature Tc (K) and()vdT . Experiments have been conducted to see if a rotating body's m v t dT weight is changed from that of a non-rotating body [21, 22, mtvdz 23]. The results of these experiments were negative as they m v t dx Earth clearly should be because the center of rotation equaled the center mass. A variety of gyroscopes turned at various angular velocities were tested. Typically a gyroscope with effective mt vdx

mt vdy radius 1.5 cm was turned at a maximum angular frequency of M 22,000 rpm. This equates to a tangential velocity of 3,502 R F cm/sec. At the surface of the earth, the tangential velocity is calculated to be:

gRE or 791,310 cm/sec. The expected weight reduction if center of mass was different then center of rotation()r according to equations 2.9.2 and 2.9.3 would be: 2 ()3,, 502 791 310 or 1.964E-5 With a material density()ρ of 6.39 g/cc [17], the gravitational of the gyroscope weight. The experiments were done with shielding effect()Θ is .05% as observed by Podkletnov and gyroscopes with weights of about 142 grams. This would Nieminen (see Table 2.9.1). indicate a required balance sensitivity of 1.964E-5 x 142 g or m nm (2.9.1) 2.8 mg, which approaches the sensitivity of laboratory ρ ==tt cavity 2 balances. Both positive[20] and negative results [22, 23] are experimentally observed in this area although it has been 2 vdtGm M (2.9.2) generally accepted that the positive results were spurious. Fm== F ↑ t R ↓ R2 Under the geometric scenario for superconducting materials presented herein, the Cooper CPT Charge conjugated pair Superconducting Force F ()↑dx centers of mass and centers of rotation r are different making Θ = () dx Earth's Gravvitational Force F the numerical results in table 2.9.1 valid. Under this scenario, ()↓ (2.9.3) it is the superconducting material that looses weight due to ρ Earth Radius v2 = dx transverse Cooper pair movement relative to a center of ρ Eaarth Mass G gravity. The experimentally observed superconducting shielding effect cannot be explained in these terms. Superconducting Force F ()↑dT Θ = dT Earth's Gravvitational Force F ()↓ 2.10 BCS Verifying Constants (2.9.4) 2 Equations 2.9.1 - 2.9.8 represent constants as derived from ρ Earth Radius vdT = BCS theory [2] and considered by Little [5] to be primary ρ EEarth Mass G constraints on any model depicting the superconducting Table 2.9.1 Cooper Pair Concentration and Shielding ()Θ phenomenon. It can be seen that the trisine model constants 23 compare very favorably with BCS constants in brackets {}. 2 γπTc cavity ⎧2 ⎫ = ⎨ ⎬ (2.10.7) 2 (2.10.1) kb 3 CDkTDTbcb= 10.. 2()∈ = 10 2k ⎩ ⎭ cavity cavity BCS 2 2 2 k 2∆o −Euler 22π b (2.10.2) = 2πe {} 3. 527754 (2.10.8) γπε= Dk()Tb = 3 cavity 3 T cavity kTbc c

2 Within the context of the trisine model, it is important to note 1 γπTc ⎧ ⎫ (2.10.3) = ..178 ⎨ or 170⎬ that the Bohr radius is related to the proton radius Bp by this H 2 18 () c ⎩ ⎭ BCS gap as defined in 2.10.8

Euler CD 3⋅ jump cavity B m e = 152...= 155152 (2.10.4) p t 2 {} {} (2.10.9) γπTc 2 ∆∆∆xyzBohr radius m 2π e 2 k cavity 2 ⎛ K A 2 ⎞ Table 2.10.1 Electronic Heat Jump b at Tc − K Dn ⎜ ⎟ 0 TK ⎝ ⎠ (2.10.5) c () = jump{}10....2 2= 10 12{} 10 8.95E+13 2,135 1,190 447 93 8.11E-16 0 2mkTtbc TK b () 9.07E+14 4.43E+093.31E+09 2.03E+09 9.24E+08 2.729

3 33 kb ⎛ erg ⎞ 3π 3 6.52E+24 7.59E+08 3.16E+08 7.28E+07 6.90E+06 1.78E-19 2 ⎝⎜ ⎠⎟ Tc cavity = cavity cm 1 3 3 2 2 ⎛ B⎞ 2 6 mt ⎜ ⎟ kb (2.10.6) Harshman [17] has compiled and reported an extended listing ⎝ A⎠ t of electronic specific heat jump at Tc data for a number of = 3. 6196E − 19 (Saam's Constant) superconducting materials. This data is reported in units of Equation 2.10.6 basically assumes the ideal gas law for mJ/mole K 2 . With the volume per formula weight data for the superconducting charge carriers. superconductors also reported by Harshman and after 2 multiplying byTc , the specific heat jump at Tc is expressed in Figure 2.10.1 Homes’ Data Compared to Saam’s terms of erg/cm3 and graphed in figure 2.10.2. Constant Superconductor Critical Temperature Tc Figure 2.10.2 Electronic Specific Heat Jump()Cd at Tc 1.00E-14 comparing Harshman compiled experimental data and 1.00E-15 predicted trisine values based on equation 2.10.1 1.00E-16 1.00E+08 1.00E+07 1.00E-17 1.00E+06

1.00E-18 1.00E+05

1.00E-19 1.00E+04 1.00E+03 Experimental Data Homes' Data & Saam's Constant 1.00E-20 Trisine Model 1.00E+02 Homes' Data 1.00E-21 Saam's Constant Electronic Specific Heat Jump at Tc 1.00E+01 1.00E-22 1.00E+00 0 102030405060708090100 0 20 40 60 80 100 120 140 Critical Temperature Tc

When the 1/ρs data from reference [51, 59] is used as a In addition it is important to note a universal resonant measure of cavity volume in the equation 2.10.6 , the following superconductor diffusion coefficient Dc based on Einstein’s plot is achieved. It is a good fit for some of the data and not 1905 publication [75] which adds to the universality of the for others (figure 2.10.1). One can only conclude that the trisine model. number of charge carriers in the cavity volume varies 1 considerably from one superconductor to another with perhaps ()23B 2 2 kT c −2 (2.10.10) only the Cooper CPT Charge conjugated pair being a small Dc == =×6.6052607910 time 3πµU ()x fraction of the total. Also, degrees of freedom may be a factor as in the kinetic theory of gas or other words – is a particular superconductor one-, two- or three- dimensional 2.11 Superconducting Resonant Cosmological Constant

(/,/,/)122232⋅⋅⋅kTbc kT bc kT bc as Table 2.6.1 data for The following relations of radius()RU , mass()MU , MgB2 would imply? Also it is important to remember that density()ρU and time()TU in terms of the superconducting Homes’ data is from just above Tc and not at or below Tc . cosmological constant are in general agreement with known data. This could be significant. The cosmological constant as 24 presented in equation 2.11.1 is based on energy/area or density of the universe calculated from observed celestial universe surface tension()σU , as well as energy/volume or objects. As reported in reference [27], standard cosmology universe pressure()pU . The universe surface tension()σU is model allows accurate determination of the universe baryon 3 independent of ()Tc while the universe pressure()pU is the density of between 1.7E-31 to 4.1E-31 g/cm . These values are value at current CMBR temperature of 2.729 o K as presented 2.7 - 6.4% of the universe density reported herein based the in table 2.7.1 and observed by COBE [26] and later satellites. background universe as superconductor. 2 The cosmological constant ()CU also reduces to an expression RU π relating trisine resonant velocity transformed mass m the Universe time() TU ==3 ()t 4 mcG gravitational constant G and Planck's constant . Also a 3c t (2.11.6) () () reasonable value of 71.2 km/sec-million parsec for the Hubble = 432. EE 17 sec ( 1 .37 10 yeears) constant()HU from equation 2.11.13 is achieved based on a universe absolute viscosity()µU and energy dissipation Figure 2.11.2 Shear forces along parallel planes of an rate()PU . elemental volume of fluid 3 16π G HUt4 mG CU ===4 σU 2 (2.11.1) v dx 3c 3π kT σ = bc (2.11.2) U section

1 (2.11.3) RU ==225.E 28 cm (47.5 billion light years)) CU

Figure 2.11.1 Closed Universe with Mass MU and

Radius RU

This is an indication that the universe has been expanding at the speed of light (c) since the beginning accounting for the early universe inflation and present universe expansion. As illustrated in Figure 2.11.2 equation 2.11.13 for the Hubble

constant()HU is derived by equating the forces acting on a cube of fluid in shear in one direction to those in the opposite direction, or ⎡ dτ ⎤ pyzU ∆∆ ++τ ∆zxy ∆∆ ⎣⎢ dz ⎦⎥ (2.11.7) ⎡ dpU ⎤ =++τ ∆∆xy pU ∆∆∆xyz ⎣⎢ ddx ⎦⎥

Here pU is the universe pressure intensity, τ the shear intensity, and ∆x , ∆y , and ∆z are the dimensions of the cube. The mass of the universe()MU is established by assuming a It follows that: spherical isotropic universe (center of mass in the universe dτ dpU (2.11.8) center) of radius()RU such that a particle with the speed of = light cannot escape from it as expressed in equation 2.11.4 and dz dx graphically described in figure 2.11.1. The power expended, or the rate at which the couple does work()τ ∆∆xy , equals the torque ()τ ∆∆xy ∆ z times the c2223π c (2.11.4) angular velocitydv dz . Hence the universe power()PU MRUU= ⋅ = 32 G 4 mGt consumption per universe volume()VU of fluid is PV x y zdvdz x yz dvdz . = 3.02E+56 g UU= ⎣⎡()ττ∆∆ ∆⎦⎤ () ∆∆∆ = (() Defining τ = µU dv dz 3 14mcG62 m tt and Hdvdz= ρUU= M ⋅ 33==4 U 2 4ππRU cavity TEc =811.1−16 2 Tb =2. 729 (2.11.5) then PUU V= µµ U() dv dz= UU H 3 2 638. E 30 g/cm or HPUUUU= µ V = − () This value for Universe density is much greater than the The universe absolute viscosity()µU is based on 25 momentum/surface area as an outcome of classical kinetic Now from equation 2.11.4 and 2.11.13 the universe mass theory of gas viscosity [19]. In terms of our development, the ()MU is defined in terms of the Hubble constant ()HU in momentum changes over sectional area in each cavity such equation 2.11.15. that: 3222 4ππ3 33c c RcU MRUU====ρ U mvtdx (2.11.9) 32 µ = 3 HGUt4mG G (2.11.15) U section = 302. Eg 56 Derivation of the kinematic viscosity υ in terms of absolute ()U Now from equation 2.11.13 and 2.11.5 the universe density viscosity as typically presented in fluid mechanics [18], a ()ρU is defined in terms of the Hubble constant ()HU in constant universe kinematic viscosity ()υU is achieved in equation 2.11.16. This differs from the universe critical terms of Planck's constant() and Cooper CPT Charge density equation in reference [27] by a factor of 2/3, which indicates an inflationary universe. conjugated pair resonant transformed mass()mt . 22 2 HUU3 chain H cavity µU A ρU == υU ===µU π 8ππG 8 cavity G mmt ρUtB (2.11.16) (2.11.10) g cm2 = 638. E − 30 = 139. E − 02 cm3 ssec Now from equation 2.11.13 and 2.11.6 the universe time ()TU Derivation of the Universe Energy Dissipation Rate P ()U is defined in terms of the Hubble constant ()HU in equation assumes that all of the universe mass ()MU in a universe 2.11.17. volume V is in the superconducting state. In other words, ()U 1 our visible universe contributes an insignificant amount of TU ==4.32E+17 or 137. E 10 years (2.11.17) mass to the universe mass. This assumption appears to be HU valid due to the subsequent calculation of the Hubble constant P U (2.11.18) ()HU of 71.2km/sec-million parsec (equation 2.11.13) which Universe mass creation rate() MU = 2 falls within the experimentally observed value[20] on the order c of 70-80 km/sec-million parsec and the recently measured 70 Table 2.11.1 Vacuum Space parameters km/sec-million parsec within 10% as measured by a NASA TK0 c () Hubble Space Telescope Key Project Team on May 25, 1999 8.95E+13 2,135 1,190 447 93 8.11E-16 TK0 after analysis of 800 Cepheid stars, in 18 galaxies, as far away b () 9.07E+14 4.43E+093.31E+09 2.03E+09 9.24E+08 2.729 as 65 million light years. µU (g/cm/sec) 2.09E+01 3.78E-04 1.57E-04 3.62E-05 3.44E-06 8.85E-32

2 2 MvvUdxdx (2.11.11) σU (erg/cm ) 8.05E+23 4.58E+02 1.42E+02 2.01E+01 8.70E-01 6.61E-35 PU = 2 RU PU (erg/sec) 8.23E+62 9.59E+46 3.99E+46 9.19E+45 8.72E+44 2.24E+19 46 MU (g/sec) 9.16E+41 1.07E+26 4.44E+25 1.02E+25 9.70E+23 2.50E-02 4 3 3π VRUU==π 93 This development would allow the galactic collision rate 3 16mGt (2.11.12) calculation in terms of the Hubble constant()HU (which is = 4 .774Ecm+ 85 3 considered a shear rate in our fluid mechanical concept of the

3 universe) in accordance with the Smolukowski relationship PU 4 mGct H == [12, 13, 14] where JU represents the collision rate of galaxies U µ V π 2 UU (2.11.13) of two number concentrations ()nnij, and diameters()ddij, : = 231./secE − 18 1 3 JnnHdd=+ (2.11.19) or in more conventionally reported units: UijUij6 () km 1 The superconducting Reynolds' number R , which is defined H = 71. 23 ()e U sec million parsec as the ratio of inertial forces to viscous forces is presented in Now from equation 2.11.1 and 2.11.13 the cosmological equation 2.11.20 and has the value of one(1). In terms of conventional fluid mechanics this would indicate a condition of constant ()CU is defined in terms of the Hubble constant ()HU in equation 2.11.14. laminar flow [18]. H 1 mt 11 mvtdx U −1 Re ==vdx cavity = 1 (2.11.20) CU ===4.45E-29 cm (2.11.14) cavity section µµ secction 3c RU

And the Universe Escape Velocity (vU)

HRUU GM U vcU == = (2.11.14a) 3 RU 26

2 2 ⎡ 2 ⎛ T ⎞ ⎤ T −Euler ⎛ vdy ⎞ 22 2 HHTo1 = ⎢1− π −Euler ⎥ at ~0 (2.12.7) aU = πe ⎜ ⎟ 3 ⎜ eT⎟ T 2P ⎢ ⎝ π c ⎠ ⎥ c ⎝ ⎠ TEc =811. − 16 ⎣ ⎦ Tb =2. 729 (2.11.21) T 22 HHk22= at near 1 RHUU Pc U −8 cm Tos2 (2.12.8) Tc ==cHU = = 6.93x10 2 3 µUUV sec 2 2 2 ⎡if HHcT1 > cT 2 ⎤ The numerical value of the universal deceleration()aU AvgH cT = ⎢ 2 2 ⎥ (2.12.9) expressed in equation 2.11.21 is nearly equal to that observed ⎣then HHcT1 else cT 2 ⎦ with Pioneer 10 & 11 [53], but this near equality is viewed as consequence of equation 2.11.22 for the particular dimensional 2.13 Superconductor Resonant Energy Content characteristics (cross section()Ac , Mass()M and thrust coefficient()Ct ) of the Pioneer 10 & 11 space crafts. Table 2.13.1 presents predictions of superconductivity energy content in various units that may be useful in anticipating the HU 228Ac − cm acHUU==cC≈ t ρU cx= 6.93 10 2 (2.11.22) uses of these materials. c M sec This acceleration component of the universal dark Table 2.13.1 Superconductor Resonant Energy in Various matter/energy may provide the mechanism for observed Units. universe expansion and primarily at the boundary of the elastic TK0 c () 8.95E+13 2,135 1,190 447 93 8.11E-16 space CPT lattice and Baryonic matter (star systems, galaxies, 0 TK etc) and under the universal gravity and quantum mechanical b () 9.07E+14 4.43E+093.31E+09 2.03E+09 9.24E+08 2.729 relationship as follows: erg/cm3 7.19E+38 2.00E+12 4.64E+11 4.01E+10 7.92E+08 1.78E-34 2 7.19E+34 2.00E+08 4.64E+07 4.01E+06 7.92E+04 1.78E-38 2 mMtU 3 MU h 1 joule/liter mct == G 2 3 1.93E+33 5.37E+06 1.25E+06 1.08E+05 2.13E+03 4.77E-40 RU 82π mttUmR BTU/ft 2 2 2 (2.11.23) kwhr 2 ⎛ 3 ⎞ h 1 h 1 3 5.66E+29 1.57E+03 3.65E+02 3.16E+01 6.23E-01 1.40E-43 3 kg2 ft = ⎜ ⎟ 2 = πεms 2 3 ⎝ 8π ⎠ 2mBtproton 2mAt hp-hr 1.01E+29 2.82E+02 6.54E+01 5.66E+00 1.12E-01 2.51E-44 Which fits well into the relativistic derivations in Appendix F. gallon gasoline 1.55E+27 4.30E+00 9.98E-01 8.63E-02 1.70E-03 3.82E-46 equivalent 2.12 Superconducting Resonant Variance At T/Tc Table 2.13.2 represents energy level of trisine energy states Establish Gap and Critical Fields as a Function of TTusing c and would be key indicators material superconductor Statistical Mechanics [19] characteristics at a particular critical temperature. The BCS 22 − KB chain 2 − gap is presented as a reference energy. ze==2mkTebccavity e3 (2.12.1) s Table 2.13.2 Energy (mev) associated with Various Trisine 22 33 Wave Vectors. The energy is computed by 2m K and 3π ()t cavity = 1 3 3 3 expressing as electron volts(ev). The BCS gap is computed B 22 2 2 ⎛ ⎞ 2 2 (2.12.2) as exp Euler2 m K . 6 mt kTbc ()π ()() tB ⎝⎜ A⎠⎟ t TK0 c () 8.95E+13 2,135 1,190 447 93 8.11E-16 3 3 TK0 b () 2 m k T2 cavity 13.% 2 9.07E+14 4.43E+093.31E+09 2.03E+09 9.24E+08 2.729 ()π tbc ⎛ T ⎞ 2 z = ≈ (2.12.3) K 267,138 183.98 102.55 38.52 8.01 6.99E-17 t 8 33 ⎜ T ⎟ B π ⎝ c ⎠ 2 KC 896,191 617.22 344.02 129.22 26.89 2.34E-16 3 3 2 kT T bc 2 Tc ⎛ 2 c ⎞ K Ds 831,836 572.89 319.32 119.95 24.96 2.18E-16 z ⎛ T ⎞ 2 − ⎛ T ⎞ 2 ⎜ − ⎟ t kTb 3 T ⎝ 3 T ⎠ 2 kt ==e ee = e (2.12.4) K 320,119 220.47 122.88 46.16 9.60 8.37E-17 z ⎝⎜ T ⎠⎟ ⎝⎜ T ⎠⎟ Dn s c c 2 KP 356,184 245.31 136.73 51.36 10.69 9.31E-17 2 chain 22 (2.12.5) KKDs+ B 1,098,974 756.87 421.86 158.46 32.97 2.87E-16 kkst= − = − k t 3 cavity 22 KKDn+ B 587,257 404.45 225.43 84.68 17.62 1.54E-16 22 1 T 1 1 T 1 KKDs+ C 1,728,027 1190.11 663.34 249.17 51.84 4.52E-16 ∆T = 1− =+1 22 KK1,216,310 837.68 466.91 175.38 36.49 3.18E-16 3 Tc ⎛ 1 ⎞ 3 Tkctln() (2.12.6) Dn+ C ln⎜ ⎟ 22 ⎝ kt ⎠ KKDs+ P 1,188,020 818.20 456.05 171.30 35.64 3.11E-16 27

22 KKDn+ P 676,303 465.78 259.61 97.52 20.29 1.77E-16 equation 2.11.13 implies the following: 2 chain K A 6,049,289 4166.20 2322.15 872.27 181.48 1.58E-15 K B TEc =895. 13 = 8π mct (2.14.7b) BCS gap 471,198 324.52 180.88 67.94 14.14 1.23E-16 TEb =907. 14 cavity also:

⎧ 1 2 ⎫ mv 2.14 Superconductor Resonant Gravitational Energy ⎪ tdx ⎪ ⎪2ε xmk ⎪ 1 mv2 As per reference[25], a spinning nonaxisymmetric object ⎨ 2 ⎬ = rdx ⎪ chain 1 1 e ⎪ 2 Tc =8.111E − 16 rotating about its minor axis with angular velocity ω will Tb =2. 729 ⎪cavity2εε k B ⎪TE=811. − 16 radiate gravitational energy in accordance with equation 2.14.1 ⎩ xm CMBR⎭ c Tb =2. 729 . 32G 22 6 (2.14.1) 2c2 E g = 5 I3ςω 5c where: 2ε xm k = 2 (2.14.7c) vdx Tc =8.111E − 16 This expression has been derived based on weak field theory Tb =2. 729 assuming that the object has three principal moments of inertia Where mt and mr are related as in Equation 2.1.25 and I1, I2, I3, respectively, about three principal axes abc>>and ∆x , ∆y , ∆z are Heisenberg Uncertainties as expressed in ς is the ellipticity in the equatorial plane. equation 2.1.20. ab− In the case of a superconductor, the radius R is the radius of ς = (2.14.2) ab curvature for pseudo particles described in this report as the Equations 2.14.1 with 2.14.2 were developed from first variable B. All of the superconductor pseudo particles move in principles (based on a quadrapole configuration) for predicting asymmetric unison, making this virial equation applicable to gravitational energy emitted from pulsars [25], but we use this particular engineered superconducting situation. This is them to calculate the gravitational energy emitted from rotating consistent with Misner Gravitation [35] page 978 where it is Cooper CPT Charge conjugated pairs with resonant trisine indicated that gravitational power output is related to “power flowing from one side of a system to the other” carefully mass (mt) in the trisine resonant superconducting mode. incorporating “only those internal power flows with a time- Assuming the three primary trisine axes C>B>A corresponding changing quadrupole moment”. The assumption in this report to a>b>c in reference [25] as presented above, then the trisine is that a super current flowing through a trisine ellipticity (or measure of trisine moment of inertia) would be: superconducting CPT lattice has the appropriate quadrupole CB− chain A moment at the dimensional level of B. The key maybe a linear ς = ==.2795 (2.14.3) CB cavity B back and forth trisine system. Circular superconducting

Assuming a mass ()mt rotating around a nonaxisymmetric systems such as represented by Gravity Probe B gyroscopic axis of resonant radius B to establish moment of inertia()I3 elements indicate the non gravitational radiation characteristic then equation 2.14.1 becomes a resonant gravitational radiation of such circular systems reinforcing the null results of emitter with no net energy emission outside of resonant cavity: mechanical gyroscopes [21, 22, 23].

2 2 2 6 Eg 32G ()ς ()mBt ⎛ 2 ⎞ = 5 ⎜ ⎟ (2.14.4a) 2.15 Critical Optical Volume Energy (COVE). time±±5 vdx ⎝ time ⎠ This section considers the concept of critical volume, which where angular velocity ω = 2 time± , and replacing the speed of light ()c with resonant consequential sub- and super- would be analogous to critical mass in atomic fission. The critical volume would establish a parameter beyond which the luminal de Broglie speed of light ()vdx , then equation 2.14.4 in terms of dimension ‘B’ becomes the Newtonian gravitational output energy would exceed the input energy. It is anticipated energy relationship in equation 2.14.5: that this energy would have a benign character and not be 2 usable in explosive applications. The basic principle is that a mt coherent beam length is not related to input power creating that EGg = − (2.14.5) B beam while virtual particle generation (in accordance with the All of this is consistent with the Virial Theorem. CPT theorem) is a function of length. Given that volume is Gravitational Potential Energy/2 = Kinetic Energy length cubed, then there must be a critical volume at which the energy of created virtual particles exceeds the input power to 1 31 2 2 the volume. The following is a derivation of this concept. Gmt = mvrdx TTE=811. − 16 (2.14.7) 2 ∆∆∆xyz++TEc =811. − 16 2 c 2 T =2. 729 Tb =2. 729 b ⎛ 1⎞ ⎛ mvtxε ⎞ and Poytning Vector() S= c⎜ ⎟ (2.15.1) ⎝ 2⎠ ⎝⎜ cavity⎠⎟ 2 cavity mt then hH TE=811. − 16 = G U c (2.14.7a) 2 Tb =2. 729 chain B p Tc =8895. E 13 2 ⎛ cmv⎞ ⎛ txε ⎞ TEb =907. 14 Input Power23 B S = ⎜ ⎟ (2.15.2) ()⎝ 2⎠ ⎝⎜ A ⎠⎟ Which is the same form as 2.14.5 and in conjunction with 28

Now consider the out put power. 1 mt e± 2 3 gges = cavity H c ⎛ 1⎞ ⎛ mvtdx⎞ ⎛ 1⎞ ⎛ mvtdx⎞ 22m mv Output Power = ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ (2.15.3) eeε ⎝ 2⎠ ⎝ time ⎠ ⎝ 2⎠ ⎝ 2B ⎠ with equation 3.1 containing the constant ()ce2 and relating Now calculate output energy. electron mass()me , trisine resonant transformed mass ()mt 3 and the characteristic trisine geometric ratio()BA. ⎛ 1⎞ ⎛ mvtdx⎞ (2.15.4) Output Energy () Length = ()Length 2 1 2 2 ⎜ ⎟ ⎜ ⎟ − ⎝ 22⎠ ⎝ B ⎠ 1 mt 4 ⎛ c⎞ ⎛ A B ⎞ 2 gges= 43π 2 72 ++ 12 2 119 (3.1) 2 m ⎝⎜ e ⎠⎟ ⎝⎜ B A ⎠⎟ The input energy is calculated as follows: e ± or ⎛ cmv⎞ ⎛ 2 ⎞ (2.15.5) Input Energy ()2 B txε ()2B 2 2 2 = ⎜ ⎟ ⎜ ⎟ mt ⎛ c ⎞ ⎛ A B ⎞ ⎝ 2⎠ ⎝ A ⎠ 22= 3 72 ++ 122 19 (3.1a) m ⎝⎜ gge ⎠⎟ ⎝⎜ B A ⎟⎠ Now define the critical condition: ees± and Energy Out (2.15.6) 2 let = 1 mt ⎛ 8πGMUU H ⎞ ⎛ A B⎞ Energy In 2322= ⎜ + ⎟ (3.1b) m ⎝⎜ gec ⎠⎟ ⎝ B A⎠ e s ± Now a critical Length (L) can be established. and also ⎛ cmv⎞ ⎛ 2 ⎞ 22⋅ B ⎛ c ⎞ ⎛ v2 ⎞ ⎛ B⎞ GM H L = txε 2B = 4 εx B UU 12/ ⎜ ⎟ ⎜ ⎟ 3 ⎜ ⎟ ⎜ 2 ⎟ ⎜ ⎟ (2.15.7) 3 = 3 (3.1c) ⎝ 2⎠ ⎝ A ⎠ mvtdx⎝ v dx⎠ ⎝ vdx ⎠ ⎝ A⎠ c It is envisioned that this length would be a measure of trisine The extension to traditional superconductivity discussion is lattice size necessary for criticality and in particular a defined postulated wherein the underlying principle is resonance Critical Optical Volume defined at the of intersection of three expressed as a conservation of energy and momentum (elastic lengths. To retrieve the power, one side (+ or -) of the virtual character). The mechanics of this approach are in terms of a pair would have to be pinned (perhaps optically). In this case cell (cavity) defined by momentum states: power could be retrieved by electrically coupling lattice to a K123,, K K and K 4 pick up coil or arranging to intercept 56 Mev radiation. and corresponding energy states: 2 2 2 2 K1 ,, K2 K3 and K4 3. Discussion by a required new mass called trisine mass ()mt , which is This effort began as an effort towards formulating a theoretical intermediate to the electron and proton masses. The cell frame work towards explaining the Podkletnov's and dimensions can be scaled from nuclear to universe dimensions Nieminen's [7] results but in the process, relationships between with the trisine mass remaining constant. The triangular electron/proton mass, charge and gravitational forces were hexagonal cell character is generally accepted by conventional established which appear to be unified under this correlation solid-state mechanics to be equivalent in momentum and real with links to the cosmological constant, the dark matter of the space making the energy (~K2) and momentum (~K) cell universe, and the universe black body radiation as a (cavity) identity possible. Also, the hexagonal or triangulation continuation of its big bang origins and nuclear forces. energy (~K2) and momentum (~K) have a 1/3, 2/3 character The overall correlation is an extension to what Sir Arthur inherent to the standard model 1/3, 2/3 quark definitions. Eddington proposed during the early part of the 20th century. Observation of the universe from the smallest (nuclear He attempted to relate proton and electron mass to the number particles) to the largest (galaxies) indicates a particularly static of particles in the universe and the universe radius. Central to condition making our elastic assumption reasonable and his approach was the fine structure constant: applicable at all dimensional scales. Indeed, this is the case. ()ce2 This simple cellular model is congruent with proton density and radius as well as interstellar space density and energy which he knew to be approximately equal to 137. This density. relationship is also found as a consequence of our development in equation 2.1.19, the difference being that a material In order to scale this cellular model to these dimensions varying by 15 orders of magnitude, special relativity Lorentz dielectric()ε modified velocity of light()vε is used. To transforms are justifiably suspended due to resonant determine this dielectric()vε , displacement()D and electric fields()E are determined for Cooper CPT Charge conjugated cancellation in the context of the de Broglie hypothesis. pairs moving through the trisine CPT lattice under conditions Superluminous velocities, which occur above 3,100,000 K and -10 of the trisine CPT lattice diamagnetism equal to −14π (the less than 10 cm or within nuclear cell dimensions are then Meissner effect). Standard principles involving Gaussian theoretically acceptable. surfaces are employed and in terms of this model development Most importantly, the gravity force is linked to the Hubble the following relationship (3.1) is deduced from equation constant through Planck’s constant. This is a remarkable 2.1.19 repeated here result. It indicates that gravity is linked to other forces at the proton scale and not the Planck scale. Indeed, when the model 29 trisine model is scaled to the Planck scale (B= 1.62E-33 cm), a Other verifying evidence is the trisine model correlation to the mass of 1.31E+40 Gev/c2 (2.33E+16 g) is obtained. This value ubiquitous 160-minute resonant oscillation [68,69] in the would appear arbitrary. Indeed the model can be scaled to the universe and also the model’s explanation of the Tao effect

Universe mass (MU) of 3.02E56 grams at a corresponding [65, 66, 67]. dimension (B) of 1.42E-53 cm. Also dimensional guidelines are provided for design of room temperature superconductors and beyond. These dimensional 4. Conclusion scaling guidelines have been verified by Homes’ Law and A momentum and energy conserving (elastic) CPT lattice generally fit within the conventionally described superconductor operating in the “dirty” limit. [54] called trisine and associated superconducting theory is postulated whereby electromagnetic and gravitational forces The deceleration observed by Pioneer 10 & 11 as they exited are mediated by a particle of resonant transformed mass the solar system into deep space appear to verify the existence (110.12275343 x electron mass or 56.2726/c2 Mev) such that of an space energy density consistent with the amount the established electron/proton mass is maintained, electron theoretically presented in this report. This translational and proton charge is maintained and the universe radius as deceleration is independent of the spacecraft velocity. Also, computed from Einstein's cosmological constant is 2.25E28 the same space density explains the Pioneer spacecraft cm, the universe mass is 3.0E56 gram, the universe density is rotational deceleration. This is remarkable in that translational 6.38E-30 g/cm3 and the universe time is 1.37E10 years. and rotational velocities differ by three (3) orders of magnitude. The cosmological constant is based on a universe surface tension directly computed from a superconducting resonant General conclusions from study of Pioneer deceleration data energy over surface area. are itemized as follows: The calculated universe mass and density are based on an 1. Dark Matter and Dark Energy are the same thing and are isotropic homogeneous media filling the vacuum of space represented by a Trisine Elastic Space CPT lattice which is analogous to the 'aether' referred to in the 19th century (but still composed of virtual particles adjacent to Heisenberg in conformance with Einstein's relativity theory) and could be Uncertainty considered a candidate for the 'cold dark matter' in present 2. Dark Matter and Dark Energy are related to Cosmic universe theories and as developed by Primack [80]. Universe Microwave Background Radiation (CMBR) through very large density is 2/3 of conventionally calculated critical density. space dielectric (permittivity) and permeability constants. This is in conformance with currently accepted inflationary 3. A very large Trisine dielectric constant (6.16E10) and CPT state of the universe. equivalence makes the Dark Matter/Energy invisible to The 'cold dark matter', a postulated herein, emits a CMBR of electromagnetic radiation. o 2.729 K as has been observed by COBE and later satellites 4. A velocity independent thrust force acts on all objects although its temperature in terms of superconducting critical passing through the Trisine Space CPT lattice and models the o temperature is an extremely cold 8.11E-16 K . translational and rotational deceleration of Pioneer 10 & 11 Also, the reported results by Podkletnov and Nieminen spacecraft. wherein the weight of an object is lessened by .05% are 5. The Asteroid Belt marks the interface between solar wind theoretically confirmed where the object is the superconductor, and Trisine, the radial extent an indication of the dynamics of although the gravitational shielding phenomenon of the energetic interplay of these two fields YBa Cu O is yet to be explained. It is understood that the 237− x 6. The Trisine Model predicts no Dust in the Kuiper Belt. The Podkletnov and Nieminen experiments have not been Kuiper Belt is made of large Asteroid like objects. replicated at this time. It could be that the very high super currents that are required and theoretically possible have not be 7. The observed Matter Clumping in Galaxies is due to replicated from original experiment. magnetic or electrical energetics in these particular areas at such a level as to destroy the Coherent Trisine. The trisine model provides a basis for considering the phenomenon of superconductivity in terms of an ideal gas with 8. The apparent high observed speed of outer star rotation in 1, 2 or 3 dimensions (degrees of freedom). The trisine model galaxies is due to forced redistribution with R of orbital was developed primarily in terms of 1 dimension, but velocities with time due to disproportionate change in R of inner faster moving stars relative to R of outer slower orbiting experimental data from MgB2 would indicate a direct translation to 3 dimensions. stars in accordance with orbital velocity (v) relationship: GM These trisine structures have an analog in various resonant v2 = structures postulated in the field of chemistry to explain R properties of delocalized π electrons in benzene, ozone and a This is a residual effect left over from the time when galaxies myriad of other molecules with conservation of energy and were primarily dust or small particles when the formula: momentum conditions defining a superconductive (elastic) 2 deceleration= CdU Aρ c condition. 30

ρU = conventionally accepted space vacuum density passes through our solar system terminal shock and detected 2H 2 what are called anomalous cosmic rays (ACRs) (Figure ρ = U U 8πG 4.1http://imagine.gsfc.nasa.gov/docs/features/bios/christian/an M = object mass omalous.html A = object cross sectional area Figure 4.1 Spectra for Voyager 1 (left column) and Ct = thrust coefficient Voyager 2 (right column) for the time periods 1994 days was primarily applicable and is still applicable in as much as 157-261 (a and b), 1994 days 1 - 105 (c and d) and 1993 galaxies still consist of small particles (< a few meters) days 53 - 157 (e and f). gravitationally coupled to large objects such as stars. 9. The observed Universe Expansion is due to Trisine Space CPT lattice internal pressure and is at rate defined by the speed of light (c) escape velocity. 10. When the Trisine is scaled to molecular dimensions, superconductor resonant parameters such as critical fields, penetration depths, coherence lengths, Cooper CPT Charge conjugated pair densities are modeled. 11. A superconductor is consider electrically neutral with balanced conjugated charge as per CPT theorem. Observed current flow is due to one CPT time frame being pinned relative to observer. 12. The nuclear Delta particles ∆++, ∆+ . ∆0 , ∆- with corresponding quark structure uuu, uud, udd, ddd, with masses of 1232 Mev/c2 are generally congruent with the trisine prediction of (2/3)hc/B or 1242 Mev/c2 where B reflects the proton radius. Also, the experimentally determined mean It is assumed that this terminal shock zone generates ACR 56 lifetime of the delta particles (6E-24 seconds) is consistent and Mev radiation in a mechanism similar to the Pioneer satellites understandably longer than the resonant time for the proton deceleration and expresses itself in a black body curve. (2.68E-25 second). Voyager II will enter our solar system terminal shock zone in

13. It is noted that the difference in neutron (mn) and proton 2008. 56 Mev radiation verification is anticipated (mp) masses about two electron (me) masses (mn-mp~2me). Also there is an indication of a 56 Mev peak in black body 14. The nuclear weak force particles W-W + Z0 are associated curve from the EGRET instrument aboard The Compton with the superconductor specific heat jump (2.10.5) with Gamma Ray Observatory Mission (Comptel) spacecraft, which corresponding masses 78.07 Gev/c2 for W- W + and generally observed the extra galactic diffuse background x-ray 78.07/cos2(θ ) Gev/c2 or 89.3 Gev/c2 for Z0 and at proton radiation. The data graphic is replicated below in Figure 4.2 dimension (B). (experimental mass values for particles W- and by author permission [73] with the 56 Mev resonant energy W+ 80.4 {91.187/ cos2(θ )} and particle Z0 91.187 Gev/c2). superimposed. Verification is anticipated with the launch of This is consistent with observed beta decay related to nuclear The Gamma-ray Large Area Space Telescope (GLAST) weak force wherein parity is broken or other words, the chiral spacecraft platform in 2008 which will provide a much more trisine symmetry is broken. detailed delineation of the extra galactic diffuse X-ray background radiation and potentially the trisine space lattice 56 15. The top quark (Higgs Particle) is associated with energy 2 2 Mev/c dark matter component. (KAKA) at a corresponding mass of 174.62 Gev/c . 16. The up (u), down (d) and charm (c) quarks are associated References [76,79] provide evidence that the dark matter is observed to be spatially separate for visual matter. This can be with energies (2/3 me/mt KBc, 1/3 me/mt KBc & 2/3 KBc) or corresponding masses .00564, .00282 & 1.24 Gev/c2. explained in terms of the correlations presented in this report which indicate that dark matter is essentially a rigid quantum 17. The strange quark (s) is associated with hypotenuse energy state each with its own mass density (KK/c^2) which is (K K & K K ) with a mass of 0.10 Gev/c2. A A A A observed within the boundary of these states which occur in the 18. The bottom (b) quark is associated with BCS vicinity of galaxies or residual historical areas of collision superconducting ‘gap’ energy (2.10.8) with a corresponding events and not generally observed in the vastness of space 2 mass of 4.53 Gev/c . because of its homogeneity congruent with CMBR. 19. The Gluon ratio is 60/25 = 2.4 ≅ B/A=2.379760996. A question logically develops from this line of thinking and Other experimental evidence of the ubiquitous character of the that is “What is the source of the energy or mass density?”. 56 Mev radiation is the NASA Voyager I spacecraft as it After the energy is imparted to the spacecraft (or other objects) 31 passing through space, does the energy regenerate itself locally physicists, chemists and engineers. Each has a major role in or essentially reduce the energy of universal field? And then achieving the many goals that are offered if more complete there is the most vexing question based on the universal scaled understanding of the superconducting resonant phenomenon is superconductivity resonant hypothesis developed in this report attained in order to bring to practical reality and bring a more and that is whether this phenomenon could be engineered at an appropriate scale for beneficial use. complete understanding of our universe.

Figure 4.2 Diffuse X-ray Background radiation Figure 4.3 Conceptual Trisine Generator (Elastic including that from EGRET [73] with 56 Mev Space CPT lattice) imposed.

One concept for realizing this goal, is to engineer a wave crystal reactor. Ideally, such a wave crystal reactor would be made in the technically manageable microwave (cm wave length). Perhaps such microwaves would interact at the ω 2 Figure 4.4 Conceptual Trisine Generator (Elastic Space CPT lattice) Within Articulating Motion energy level in the Trisine CPT lattice pattern. Conceptually a Control Mechanism device as indicated in Figures 4.3 and 4.4, may be appropriate where the green area represents reflecting, generating and polarizing surfaces of coherent electromagnetic radiation creating standing waves which interact in the intersecting zone defined by the trisine characteristic angle of (90 – 22.8) degrees and 120 degrees of each other. A similar approach is made with the National Ignition Facility(NIF) at Livermore, CA. The question immediately arises whether the NIF could be run in a more continuous less powerful mode with selected laser standing wave beams of the trisine characteristic angle of (90 – 22.8) degrees and 120 degrees of each other with no material target and study the resultant interference pattern or lattice for virtual particles. Above and beyond the purely scientific study of such a lattice, it would be of primary importance to investigate the critical properties of such a lattice as the basis of providing useful energy generating mechanisms for the good of society. The major conclusion of this report is that an interdisciplinary team effort is required to continue this effort involving 32

5. Variable And Constant Definitions Fundamental Physical Constants are from National Institute of Standards (NIST). name symbol value units

Attractive Energy V gcm 22sec− 22−− 1 Boltzmann constant kb 1.380658120E-16 gcm sec Kelvin Bohr magneton emc/(2 ) 9.27400949E-21 erg/gauss (gauss cm3) e µB 1 5 gcm2 2 sec−1

Bohr radius ao 5.2917724924E-09 cm −1 de Broglie velocity vvvvdx,,, dy dz dT cm sec Cooper CPT conjugated pair 2 unitless correlation length ξ cm 1 −1 2 2 −1 critical magnetic fields(internal) HHHcc12,, c gcm sec o critical temperature Tc K 21− Diffusion coefficient Dc 6.60526079E-2 cm sec

Dirac’s number gd 1.001159652186 unitless 1 −1 electric field (internal) E gcm2 2 sec−1 1 −1 displacement field (internal) D gcm2 2 sec−1 electron mass me 9.10938975E-28 g earth radius RE 6.371315E+08 cm earth mass M E 5.979E+27 g 3 1 electron charge e 4.803206799E-10 cm2 g 2 sec−1 electron gyromagnetic-factor ge 2.0023193043862 unitless energy Ε gcm22sec− fine structure constant hc e2 1 α 137.03599936 unitless 1 3 fluxoid Φ 2.0678539884E-7 gcm2 2 sec−1 (gauss cm2) 1 3 2 2 −1 2 fluxoid dielectric()ε mod Φε gcm sec (gauss cm ) gravitational constant G 6.6725985E-8 cm321sec−− g London penetration depth λ cm −−11 momentum vectors pppxyz,, gc m sec natural log base e 2.718281828459050 unitless 3 nuclear magneton emc ()2 p µ p 5.05078342E-24 erg/gauss (gauss cm )

gcm12// 52sec− 1 permeability km unitless permittivity (dielectric constant) ε unitless Planck's constant h 2π 1.054572675E-27 gcm 21sec− () 3 12/ Planck length ()Gc 1.61624E-33 cm 5 12/ Planck time ()Gc 5.39121E-44 sec 33

12/ Planck mass ()cG 2.17645E-05 g 12/ Planck energy ()cG5 1.95610E+16 gcm 22sec− () erg 12/ Planck momentum cG3 6.52483E+05 gcm sec−1 () Planck force cG4 1.21027E+49 gcm sec−2 ( dyne ) Planck density cG52 5.15500E+93 gcm −3 () Planck acceleration cG6 () 1.03145E+97 cm sec−2 12/ Planck kinematic viscosity ()cG7 () 5.56077E+53 cm2 sec 12/ Planck absolute viscosity cG93 2.49779E+71 gcm −−11 sec ()() proton mass mp 1.67262311E-24 g proton radius Bp ()2sin()θ 8.59E-14 (8 E-14) cm proton density 2.31E-14 gcm−3 resonant mass mr g resonant CPT time± time± second superconductor gyro-factor gs 1.00971389902 unitless trisine angle θ 22.80 angular degrees trisine areas section, approach, side cm2 trisine unit cell unitless trisine constant trisine 2.010916597 unitless 21−− 2 trisine density of states D()∈t sec gcm trisine dimensions A, B, C, P cm trisine magneton emc 2 µ 8.42151973E-23 erg/gauss (gauss cm3) ()t t gcm12// 52sec− 1 trisine mass mt 1.0031502157E-25 g transformed mass mt 110.122753426 x me g trisine size ratio ()BAt 2.379760996 unitless trisine volume per cell cavity cm3 trisine volume per chain chain cm3 −1 trisine wave vectors KKKKABCDnDs,,, , K cm −−11 universe absolute viscosity µU gc m sec −3 universe density ρU 6.38E-30 gc m −1 universe Hubble constant HU 2.31E-18 sec 21− universe kinematic viscosity νU 1.39E-02 cm sec universe radius RU 2.25E+28 cm universe mass M E 3.02E+56 g −−12 universe pressure pU gc m sec universe time TU 4.32E+17 sec velocity of light c 2.997924580E+10 cm sec−1 −1 ε modified velocity of light vε cm sec 34

Also from table A.1:

Appendix A. Trisine Number mt And B/A Ratio 22 KKBP+ (A.12) Derivation trisine = KK BCS approach [2] PB Table A.1 Trisine Wave Vector Multiples()KK Compared To BCS ω ∆o = KK ⎛ 1 ⎞ BB (A.1) Euler sinh⎜ ⎟ ()KK e ⎝ DV()∈T ⎠ π sinh(trisine ) Kittel approach [4] 2ω ()KK ()KK 1 K trisine ∆o = 1 K (A.2) KKBB KKBBe − 1 K B e DV()∈T − 1 K Let the following relationship where BCS (equation A.1) and Dn 1.1981 0.1852 1.0946 Kittel (equation A.2) approach equal each other define a KP 1.3333 0.2061 1.1547 particular superconducting condition. K Ds 3.1132 0.4812 1.7644 Euler ⎧e ω KC 3.3526 0.5182 1.8310 ⎪ π ⎛ 1 ⎞ K ⎪ sinh A 22.6300 3.4976 4.7571 ⎪ ⎜ ⎟ ⎝ DV()∈T ⎠ But also using the following resonant condition in equation kTbc= ⎨ (A.3) A.13 and by iterating equations A.7, A.8 and A.13. ⎪ ω ⎪ 1 22 ⎧ K B ⎫ ⎪ DV()∈T ⎩e − 1 ⎪ 2m ⎪ ⎪ t ⎪ Define: ⎪ 2 ⎪ kTbc= ⎨ ⎬ 1 (A.13) ⎪ mm ⎪ = trisine (A.4) mmet 22tp DV()∈T ⎪ ()2B + ()A ⎪ ⎩⎪ mmet+ mmtp+ ⎭⎪ Then:

⎛ 2eEuler ⎞ trisine = −−ln 1 This trisine model converges at: ⎝⎜ π ⎠⎟ (A.5) mmte = 110.122753426 x (A.14) = 2. 010991660 BA/ = 2.379760996 Given: Euler ⎧ ()KK ω e ⎫ Appendix B. Debye Model Normal And Trisine ⎪ ⎪ Reciprocal Lattice Wave Vectors 2mkTtbc ⎪π sinh(trisine )⎪ 2 2 = ⎨ ⎬ = K B ()KK (A.6) This appendix parallels the presentation made in ref [4 pages ⎪ ω ⎪ 121 - 122]. ⎪ettrisine 1 ⎪ ⎩ − ⎭ 3 ⎧ L 4π ⎫ Ref: [2,4] ⎛ ⎞ 3 ⎧Debye Spherical⎫ ⎪⎜ ⎟ K Dn ⎪ ⎪⎝ 2π ⎠ 3 ⎪ ⎪ Condition ⎪ And within Conservation of Energy Constraint: N = ⎨ 3 ⎬ = ⎨ ⎬ (B.1) L 3 Debye Triisine ⎛ 1 ⎞ 22 2 2 ⎪⎛ ⎞ ⎪ ⎪ ⎪ KK K K ⎜ ⎟ K Ds Condition ⎜ ⎟ ()B+ C=+() Ds Dn (A.7) ⎪⎝ 2π ⎠ ⎪ ⎩ ⎭ ⎝ gs ⎠ ⎩ ⎭ And Within Conservation of Momentum Constraint: where: 1 gK+ K=+ K K 3 (B.2) ()sB() C() DsDn (A.8) KDs= () KKK A B P And from ()KK from table A.1: cavity dK 22 ⎡ 2 Dn ⎤ ⎧Debye Spherical⎫ ()KK=+ KCDs K (A.9) K dN ⎢ 2πω2 Dn d ⎥ ⎪ Conditioon ⎪ Also from table A.1: = ⎢ ⎥ = ⎨ ⎬ (B.3) BCS .%065 dω ⎢cavity 2 dK Ds ⎥ Debye Trisine 222∆o π K Ds (A.10) 3K ⎪ ⎪ 3. 527754 3 Ds Condition ==Euler ≈ ⎣⎢ 8π dω ⎦⎥ ⎩ ⎭ kTbc e K B Also: ω K = (B.4) ∆ v o = 2 (A.11) kTbc 35

3 dK 1 (B.5) ⎧ 1 ⎫ ⎧Trisine ⎫ = 2 ⋅ cavity23 m 2 ⎪ ⎛ t ⎞ 2 ⎪ ⎪Dimensiional⎪ dvω 32⎜ ⎟ ∈ dN ⎪ 8π ⎝ ⎠ 2 ⎪ ⎪Condition ⎪ 2 = ⎨ 1 ⎬ = ⎨ ⎬ (C.5) ⎧cavity ω ⎫ d∈ 1 One ⎧Debye spherical⎫ ⎪ 2 − ⎪ ⎪ ⎪ ⎪ 2 3 ⎪ ⎛ 2mt ⎞ 21B dN ⎪ 2π v ⎪ ⎪ ⎪ ⎪ ∈ 2 ⎪ ⎪Dimensional⎪ = = ⎜ 2 ⎟ Condition ⎨ 2 ⎬ ⎨ ⎬ (B.6) ⎩⎝ ⎠ π 2 ⎭ ⎩ ⎭ dω ⎪3⋅ cavity ω ⎪ ⎪ ⎪ 3 3 ⎩ Debye trisine ⎭ ⎩⎪ 8π v ⎭⎪ 3 1 ⎧ − ⎫ ⎧Trisine ⎫ 3 cavity⎛ 22 m⎞ 2 ⎛ m ⎞ 2 3 ⎪2 ttK ⎪ ⎪Dimensional⎪ ⎧cavity ω ⎫ ⎧ccavity 3 ⎫ 32⎜ ⎟ ⎜ 2 ⎟ C K ⎧Deebye Spherical⎫ dN ⎪ 28π ⎝ ⎠ ⎝ ⎠ ⎪ ⎪Condition ⎪ ⎪ 2 3 ⎪ ⎪ 2 Dn ⎪ Condition ⎪ 6π v ⎪ ⎪ 6π ⎪ ⎪ ⎪ = ⎨ 1 1 ⎬ = ⎨ ⎬ (C.6) N = ⎨ ⎬ = ⎨ ⎬ = ⎨ ⎬ (B.7) d∈ One 3 ⎪⎛ 221mBm⎞ 2 ⎛ 21⎞ 2 ⎪ ⎪ ⎪ ⎪cavity ω ⎪ ⎪cavity 3 ⎪ ⎪Debye Trisine ⎪ tt DDimensional K ⎪⎜ 2 ⎟ ⎜ 2 ⎟ ⎪ ⎪ ⎪ 3 3 ⎪ 3 Ds ⎪ ⎩ Condition ⎭ ⎝ ⎠ π 2 ⎝ ⎠ K Condition ⎩⎪ 8π v ⎭⎪ ⎩ 8π ⎭ ⎩ B ⎭ ⎩ ⎭ Given that N = 1 Cell Trisine 1 ⎧ ⎫ ⎧ ⎫ ⎧ 3 cavity⎛ 2 mt ⎞ ⎫ Dimensional 2 3 K ⎪ ⎪ ⎪⎛ 6π ⎞ ⎪ ⎪ 32⎜ ⎟ C ⎪ Condition ⎧K ⎫ ⎧Debye Spherical⎫ dN ⎪8 π ⎝ ⎠ ⎪ ⎪ ⎪ Dn ⎪⎜ ⎟ ⎪ Condition = D()∈ = ⎨ ⎬ = ⎨ ⎬ (C.7) ⎪ ⎪ ⎪⎝ cavity⎠ ⎪ ⎪ ⎪ d 2m ⎨ ⎬ = ⎨ ⎬ = ⎨ ⎬ (B.8) ∈ ⎪ t ⎪ ⎪Onne ⎪ 1 Debye Trisine 22 Dimensional ⎪ ⎪ ⎪ 3 3 ⎪ ⎪ ⎪ ⎩⎪ K B ⎭⎪ ⎪ ⎪ ⎩K Ds ⎭ ⎛ 8π ⎞ Conditionn ⎩Condition ⎭ ⎪⎜ ⎟ ⎪ ⎩ ⎭ ⎩⎪⎝ cavity⎠ ⎭⎪ Now equating trisine and one dimensional density of states results in equation C.8. Appendix C. One Dimension And Trisine Density Of 8π 3 States KC = 2 3⋅⋅cavity K B (C.8) 4π This appendix parallels the presentation made in ref [4 pages = 144-155]. 33A ∈ = energy Trisine D ∈ = density of states 3 () 2KC 3 = N ⎛ 2π ⎞ (C.1) ⎜ ⎟ Appendix D. Ginzburg-Landau Equation And Trisine ⎝ L ⎠ Structure Relationship This appendix parallels the presentation made in ref [4 One Dimension Appendix I]. 2 2 2 π (C.2) ∈ = N 2 α 2m 2 ψ == (D.1) t ()2B β Cavity

Now do a parallel development of trisine and one dimension density of states. 2222 α kTbc K B H c == = (D.2) ⎡Trisine ⎤ 228βπchain m⋅ chain ⎢ ⎥ t ⎡ 3 ⎤ Dimensional 2KC ⋅ cavity ⎢ ⎥ 2 2 2 ⎢ ⎥ mv mv β m v cavity ⎢ 3 ⎥ ⎢Condition ⎥ tttεεε 8π λ = 2 2 ==2 2 (D.3) ⎢ ⎥ ⎢ ⎥ 4πψ() ⋅ e 4παq 4π ()⋅ e N = ⎢ ⎥ = ⎢ ⎥ (C.3) 1 2 2 ⎢ 1 ⎥ ⎢One ⎥ 2 αα2mt chain chain ⎢⎛ 22mBt ⎞ 2 ⎥ ⎢ ⎥ ∈ β ==22 (D.4) ⎜ 2 ⎟ ⎢Dimensional⎥ 22K kT ⎢⎣⎝ ⎠ π ⎦⎥ Bbc ⎢Condition ⎥ ⎣ ⎦ 2 2 αα2mt chain chain 3 αβ==22 = (D.5) ⎧ 1 ⎫ ⎧Trisine ⎫ cavity 22 K cavity k T cavity 2 ⋅ cavity23 m 2 B bc ⎪ ⎛ t ⎞ 2 ⎪ ⎪Dimensiional⎪ 32⎜ ⎟ ∈ dN ⎪ 8π ⎝ ⎠ 2 ⎪ ⎪Condition ⎪ 22 = = cavity K B cavity ⎨ 1 ⎬ ⎨ ⎬ (C.4) α ==kTbc d∈ 1 One (D.6) ⎪ 2 − ⎪ ⎪ ⎪ chain 2m chain ⎛ 2mt ⎞ 21B t ⎪ ∈ 2 ⎪ ⎪Dimensional⎪ ⎝⎜ 2 ⎠⎟ Condition ⎩ π 2 ⎭ ⎩ ⎭ 36

22 2 21mt ⋅ chain mt chain dp ξ ≡ = = F = (F.1) 22mmα 22K⋅ cavity mK2 cavity dt ee B e B (D.7) and for constant mass dp dmv() dv φπφ⋅⋅24 m cavity F == =m (F.2) εεt dt dt dt H c2 = = 2 (D.8) section πξ me chain then given the 'work energy theorem' dp dmv() dv ds dv F == =m =mv Appendix E. Heisenberg Uncertainty within the Context dt dt dt ds ds of De Broglie Condition ds where = v (F.3) Given Heisenberg uncertainty: dt h and therefore: ∆∆px≥ (E.1) 4π Fds= mvdv And the assumption that mass ‘m’ is constant then: where ds is test particles incremental distance along path ''s h mvx∆∆ ≥ (E.2) Now given Newton's second law in relativistic terms: 4π dp d mv Given de Broglie Condition: F == dt dt v2 (F.4) px h 1− 2 = (E.3) c And again assuming constant mass ‘m’: let Lorentz transform notation as 'b' : 1 mvx= h (E.4) b = 2 Rearranging Heisenberg Uncertainty and de Broglie Condition: v (F.5) 1− 2 h 1 c m ≥ 4π ∆∆vx therefore: 1 (E.5) dp dmvb() mh= F == (F.6) vx dt dt Now transform into a work energy relationshipwith a K factor This implies that: of dimensional units 'length/mass' mm≥ (E.6) ⎛ 1 ⎞ a ds= mbvdv (F.7) but this cannot be (constant mass cannot be greater than itself) ⎝⎜ K ⎠⎟ therefore: acceleration 'a' can vary and the path 's' is arbitrary within the mm= (E.7) dimensional constraints of the equation. therefore: Define acceleration 'a' in terms of change in Volume 'V' as: dV2 1 1 h 1 (E.8) constant h = 2 = (F.8) vx 4π ∆∆vx dt V and incremental path 'ds' as: And: 2 ds==224() area dr (π r ) dr (F.9) vx= 4π∆∆ v x (E.9) 'area' is the surface of the spherical volume 'V' Both the Heisenberg Uncertainty and the de Broglie condition Now work-energy relationship becomes: are satisfied. Uncertainty is dimensionally within the de 2 Broglie condition. Schrödinger in the field of quantum dV11 2 24()πr dr= mbvdv (F.10) mechanics further quantified this idea. dt2 V K Now integrate both sides: 2 Appendix F. Equivalence Principle In The Context Of dV11 4 322 2()πrmcmvb= −− (F.11) Work Energy Theorem dt2 V K 3 () for v << c then b ~ 1 Start with Newton's second law 4 and V = πr3 3 then: 37

2 22 dV12 ⎛ mc mv ⎞ Gm v2 − = − 2 ⎜ ⎟ (F.12) a = − 2 1- 2 (F.22) dt V K ⎝ V V ⎠ r c and: Does this equation correctly reflect the resonant gravitational dV2 1 Kmc⎛ 22mv ⎞ acceleration such as a satellite orbiting the earth?? − = − (F.13) dt2 V 2 ⎝⎜ V V ⎠⎟ Of course this equation reduces to the standard Newtonian gravitational acceleration at 'v << c' and:

2 Gm ⎛ mc mv ⎞ a = − 2 (F.23) − r ⎜ V area time⎟ ⎜ x ⎟ 2 The equation F.7 would appear to be general in nature and be dV1 K ⎜ mv ⎟ an embodiment of the equivalence principle. Various − 2 = ⎜ - ⎟ (F.14) dt V 2 ⎜ areeay time⎟ candidate accelerations 'a' and geometric paths 'ds' could be ⎜ mv ⎟ analyzed within dimensional constraints of this equation. ⎜ - ⎟ ⎝ area time ⎠ A simple case defined by a path ‘mK ds’ may be appropriate z for the modeling the Pioneer deceleration anomaly. This is the same as the Baez narrative interpretation except for ⎛ 1 ⎞ the negative signs on momentum. Since momentum 'mv' is a a⎜ ⎟ mKds=== mads Fds mbvdv (F.24) vector and energy 'mv2 ' is a scalar perhaps the relationship ⎝ K ⎠ should be expressed as: ⎛ mc2 mv ⎞ Appendix G. Koide Constant for Leptons - tau, muon and ± ⎜ V area time⎟ electron ⎜ x ⎟ dV2 1 K ⎜ mv ⎟ Koide Constant for Lepton masses follows naturally from − 2 = ⎜ ± ⎟ (F.15) nuclear condition as defined in paragraph 1.1. dt V 2 ⎜ areayttime⎟ m= 1.777 Gev/c2 ⎜ mv ⎟ τ ⎜ ± ⎟ 2 mµ = .1057 Gev/c ⎝ areaz time⎠ 2 or perhaps in terms of spherical surface area 'arear' me = .000511 Gev/c dV2 1 Kmc⎛ 2 mv ⎞ 2 1 mτ =+cK()CB K 2 − 2 =±⎜ ⎟ (F.16) 3 c dt V 2 ⎝ V arear time⎠ 1 m 2 1 Lets look again at the equation F.7 and define another path ‘s’: m = e KK22+ µ 3 m 2m ()CBc2 dr2 t t ds16 rdr a constant ===π 2 (F.17) π me 1 dt me = ()gcKKsCB− 1 ()+ 2 then 2 mt c (G.1) 2 1 ⎛ 2 ⎞ m =+cK K a 8πrdr= mbvdv (F.18) τ ()Ds Dn 2 ⎝⎜ K ⎠⎟ 3 c 1 m 2 1 Then integrate: e 2 2 mµ =+()KKDs Dn 2 32mmtt c ⎛ 2 ⎞ 222 a⎜ ⎟ 4πrmcmvb= −−() (F.19) m 1 ⎝ K ⎠ π e me = ()gcKKsDsDn− 1 ()+ 2 Given that: 2 mt c mm++ m chain 2 8πG τ µ e == K = 2 (F.20) 2 c mm+m cavity 3 ()τ + µ e and solve equation F.19 for 'a' Gm 1 a = − 2 (F.21) r b or 38

Appendix H. Authority for Expenditure (Cost Estimate) Project Name____ Three Dimensional Laser Energetics Date ___ January 15, 2007 Relating Trisine Geometry to the CPT theorem Location ______Corpus Christi, Texas Description _____ Apparatus and Support Facilities for 10 years Completed Cost PROJECT - INTANGIBLE COSTS Project Physicists______$10,000,000 Project Chemists______$10,000,000 Project Engineers______$10,000,000 Technicians Draftsmen Electronic Electrical Specifications______$10,000,000 Salaries management and office staff $5,000,000 Architectural Services $500,000 Landscaping $250,000 Maintenance $400,000 Office Library Supplies $750,000 Library Reference Searches $750,000 Patent Trademark Attorney Services $1,000,000 Legal Services $500,000 Financial Services $500,000 Travel $1,000,000 Publication Services $500,000 Marketing $500,000 Infrastructure maintainance $500,000 Utilities $400,000 Security $300,000 ______$0 SubTotal ______$52,850,000 Contingencies 50.00% ______$26,425,000 PROJECT - INTANGIBLE COSTS ______$79,275,000

PROJECT - TANGIBLE COSTS Vibration Attenuation foundation______$1,000,000 Laser Reactor Area______2500 sq ft $5,000 per sq ft $12,500,000 Assembly Area______10000 sq ft $500 per sq ft $5,000,000 Office Laboratory Library Space______10000 sq ft $200 per sq ft $2,000,000 Office Laboratory Library Furniture______$500,000 Office Laboratory Computers CAD Software Plotters Displays Printers LAN $500,000 Heating Air Conditioning $1,000,000 Heavy Equipment movement and placement trollleys and lifts $1,000,000 Test Benches and Equipment Storage ______$1,000,000 Test and Measurement Equipment ______$1,000,000 Faraday and Optical Cage______$500,000 Three Dimensional Positioning System______$5,000,000 Laser Generators______$25,000,000 Laser Optics, Polarizers and Rectifiers______$9,000,000 Vacuum Enclosure for Laser and Optics______$5,000,000 Vacuum Pumping Equipment______$5,000,000 Cryogenic Equipment______$5,000,000 Data Sensors and Acquisition Equipment ______$5,000,000 Computers and Servers for Analysing and Storing Data ______$5,000,000 Laser Interference Reactor______$25,000,000 PROJECT - TANGIBLE COSTS ______$115,000,000 TOTAL PROJECT COSTS ______$194,275,000

LEASE EQUIPMENT truck and sedan Vehicle $200,000 Out Sourcing Services $5,000,000 ______$0 ______$0 TOTAL LEASE EQUIPMENT ______$5,200,000

TOTAL COST $199,475,000 39

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