Describing the Einstein Lambda in the form of a parametrized Curvature Radius Λ(n)/R_{Hubble}= Λ(n)H_{o}/c = Λ(n)/R_{H }in AdS then enables the scale of the classically geometric Einstein Lambda in squared Hubble units to unitize the quantum geometric Planck-brane-string units by modular mirror duality between the curvature 1/R(n)^{2} and its radius R(n) with the gravitational parameter G(n)(M(n) modulating the mass M in its Schwarzschilded 'eternal' form to the energy content in the universe by E=mc^{2}.

As the magnitude of the Einstein Lambda of the instanton is of the order of 2x10^{85} acceleration units and relates to the baryonic matter (BM) density in the seedling mass M_{o}/2M_{∞} =½Ω_{BMps}; with 'closure' mass M_{∞}=R_{H}.c^{2}/2G_{o}=c^{3}/2G_{o}H_{o} relates to the

wormhole mass m_{ps}=hf_{ps}/c^{2} =2.222..x10^{-20} kg; as quantum eigenstate; the coupling between the Planck density ρ_{P}=l_{P}^{3}/m_{P}=1.85x10^{96} and the 'critical closure' density ρ_{H}=M_{∞}/R_{H}^{3} becomes subject to the overall mass evolution from the wormhole mass to the

'closure mass' over the evolution cycle, ending after for n_{reset}=234.5 for 3.96 Trillion years.

The coupling between the Planck Density and the Inflaton-Instanton Density so is: m_{ps}/R_{H}^{3} = hf_{ps}c^{4}/8G_{o}^{3}M_{H}^{3} = 5.45x10^{-99} .

A modular brane mirror-T duality between the inversion properties of two parts of a heterotic supermembrane (class HE(8x8)) and can be expressed in the unitary dimensions of the Gravitational Constant G_{o} in [m^{3}][kg^{-1}][s^{-2}]=(Volume)(Angular Acceleration)/(Mass)=(Angular Acceleration)/(Density).

Brane mirror duality unifies the electromagnetic and gravitational interactions via the coupling of their fine structures.

The quantization of mass m so indicates the coupling of the Planck Law in the frequency parameter to the Einstein law in the mass parameter.

The postulative basis of M-Theory utilizes the coupling of two energy-momentum eigenstates in the form of the modular duality between so termed 'vibratory' (high energy and short wavelengths) and 'winding' (low energy and long wavelengths) self-states.

The 'vibratory' self state is denoted in: E_{ps}=E_{primary sourcesink}=hf_{ps}=m_{ps}c^{2} and the 'winding' and coupled self state is denoted by: E_{ss}=E_{secondary sinksource}=hf_{ss}=m_{ss}c^{2}

The F-Space Unitary symmetry condition becomes: f_{ps}f_{ss}=r_{ps}r_{ss}=(λ_{ps}/2π)(2πλ_{ss})=1

The coupling constants between the two eigenstates are so: E_{ps}E_{ss}=h^{2} and E_{ps}/E_{ss}=f_{ps}^{2}=1/f_{ss}^{2}

The Super-membrane E_{ps}E_{ss} then denotes the coupled superstrings in their 'vibratory' high energy and 'winded' low energy self states.

The coupling constant for the vibratory high energy describes a MAXIMISED frequency differential over time in df/dt|_{max}=f_{ps}^{2} and the coupling constant for the winded low energy describes its MINIMISED reciprocal in df/dt|_{min}=f_{ss}^{2}.

F-Theory also crystallizes the following string formulations from the E_{ps}E_{ss} superbrane parameters.

**Electromagnetic Fine structure: α _{e} = 2πke^{2}/hc = e^{2}/2ε_{0}hc**

**Gravitational Fine structure (Electron): α**

_{g}= 2πG_{o}m_{e}^{2}/hc = {m_{e}/m_{Planck}}^{2}**Gravitational Fine structure (Primordial Nucleon): α**

_{n}= 2πG_{o}m_{c}^{2}/hc**Gravitational Fine structure (Planck Boson): α**

_{Planck}= 2πG_{o}m_{Planck}^{2}/hc1/E

_{ps}=e*=2R

_{e}c

^{2}=√{4αhce

^{2}/2πG

_{o}m

_{e}

^{2}}=2e√α[m

_{P}/m

_{e}=2e√{α

_{e}/α

_{g}} = {2e

^{2}/m

_{e}}√(k/G

_{o})=2e

^{2}/G

_{o}m

_{e}= e

^{2}/2πε

_{o}m

_{e}for G

_{o}= 1/k = 4πε

_{o}for the cosmological unification of the fine structures.

E

_{ps }= 1/E

_{ss}= 1/e* = √{α

_{g}/α

_{e}}/2e = G

_{o}m

_{e}/2e

^{2}

Here e* is defined as the inverse of the sourcesink vibratory superstring energy quantum E

_{ps}=E* and becomes a

*New Physical Measurement Unit*is the

*StarCoulomb (C*)*and as the physical measurement unit for 'Physical Consciousness'.

R

_{e}is the 'classical electron radius' coupling the 'point electron' of Quantum- Electro-Dynamics (QED) to Quantum Field Theory (QFT) and given in the electric potential energy of Coulomb's Law in: m

_{e}c

^{2}=ke

^{2}/R

_{e}; and for the electronic restmass m

_{e}.

Alpha α is the electromagnetic fine structure coupling constant α=2πke^{2}/hc for the electric charge quantum e, Planck's constant h and light speed constant c.

G_{o} is the Newtonian gravitational constant as applicable in the Planck-Mass m_{P}=√(hc/2πG_{o}).

Alternatively expressed, the mass seedling M_{o} of the QBB manifests in an emerging 'Black Hole' evolution and is bounded by the 'closure' mass M_{∞}=M_{Hubble}. There so must be an energy gradient between the Hubble mass and the seedling mass in direct proportion to the de Broglie inflaton/instanton event.

The universe begins with a baryon matter seed of 2.813% in AdS spacetime, which allows the emergence of a first generation family of 'eternal' Black Holes to seed 'eternal' White Holes manifesting as quasars which seed galaxies from a first generation of proto-stars as a function of temperature and independent of mass. Those 'ylem' stars can be shown to be naturally degenerate 'proto' magnetars and neutron stars, whose gravitational inward pressure is balanced by their thermal heat content deriving from the Cosmic radiation temperature.

The general Jeans formulation for ylem stars is R_{ylem} = √{kTR_{e}^{3}/G_{o}m_{c}^{2}}, R_{e}=e^{2}/4πε_{o}m_{e}c^{2} and m_{c} is a prototypical nucleon mass.

The Dark Energy Interaction Goldstone Gauge Boson is the Graviton manifesting from its 11/5D AdS membrane space Dirichlet 'open string' attachment from AdS space-time into dS space-time.

*"** Spacetime tells matter how to move; matter tells spacetime how to curve."*

Wheeler's succinct summary of Einstein's theory of general relativity, in *Geons, Black Holes, and Quantum Foam*, p. 235. - 1998 by John Archibald Wheeler and Kenneth Ford; W.W.Norton and Company; New York, London

The quote of John Archibald Wheeler can be extended in a question.

*"Is the presence of matter required for spacetime to curve or is the presence of spacetime sufficient for a matter dynamic to emerge and to eventuate?"*

**The Birth of Space and Time with Space-time Inflation Curvature preceding Big Banged Wormhole Matter**

The problem of the singularity regarding the creation event, known as the Quantum Big Bang in General Relativity is well addressed in the literature of science and an appropriate solution to the infinite continuous regression of space-time parameters is resolved in a disengagement and an unnecessity of and for a prior existing space-time background for matter to act within.

It is the nature of the string itself, which in particular initial- and boundary conditions allows the concepts of space and time to emerge from the qualitative and quantitative nature of the string definition itself.

In particular a discretization of dynamical parameters of an describing cosmology in defined Planckian parameters serves to replace the infinite mathematical point like singularity to a so called Planck string vibration formulated as the Planck length l_{Planck} and which can be considered to be a minimum displacement as a wormhole radius.

Any displacement scale below the Planck length then is rendered unphysical and so becomes mathematically inapplicable to describe the dynamics in a physical universe, described by a cosmology, which defines a string epoch characterized by a cosmological inflationary Inflaton from a defined Planck-Length-Quantum-Oscillation/Fluctuation to a so labeled Quantum Big Bang Instanton.

The Big Bang Instanton becomes defined in a final manifestation of the Planckian Supermembrane (Class I at Planck Time) from the Inflaton to the Instanton (Class HE 8x8 at Weyl-Wormhole Time) and transforming the string energies across string classes from I to IIB, HO 32 and IIA to HE 8x8.

The Inflaton defined a 11-dimensional supermembraned de Sitter closed - and positively curved cosmology with a defined Hubble Event horizon for a positive spheroidal curvature information bound.

This closed universe contained no matter seed, but was defined in its curvature through a unification condition relating the electromagnetic fine structure alpha {α=2πke^{2}/hc} to the gravitational finestructure omega {ω = 2πG_{o}m_{Planck}^{2}/hc}

via ke^{2} = e^{2}/4πε_{o} = G_{o}M_{unification}^{2} for e^{2} = {G_{o}/k}M_{unification}^{2} requiring dimensional mensuration identity in inflaton space [C^{2}/Jm]=[Farad/meter] = [Jm/kg^{2}] for [C] = [C*] = [Star-Charge Coulomb] = [m^{3}/s^{2}] = [VolumexAngular Acceleration] and for the Maxwell

Constant 1/c^{2} = μ_{o}.ε_{o} = {120π/c}{1/120πc} and 'Free Space' Inflaton Impedance Z_{o}= electric field strength E/magnetic field strength H = √(μ_{o}/ε_{o}) = cμ_{o} = 1/cε_{o} = 120π}.

Therefore,

[G_{o}]_{mod} =[4πε_{o}]_{mod} = [1/k]_{mod} and the inflaton dimensionless string modular unity is G_{o}k=1 for e^{2} = G_{o}^{2}.M_{unification}^{2} = M_{unification}^{2}/k^{2} and M_{unification} = 30[ec] that is 30 modulated magnetic monopole masses. It is this 30[ec]modular inflaton mass, which

represents the initial breaking of the inherent super-symmetry of the Planck superstring class I at the Planck energy level to the monopole superstring energy of superstring class IIB with the Planck mass being replaced by 30 monopole masses as the

integration of 30 [ec] monopole masses at the [ec^{3}]_{mod} = 2.7x10^{16} GeV energy level of the Grand Unification Energy separating the quantum gravity from the GUT symbolized as SEW.G.

Setting ω=1 defines the Planck-Mass and setting a proto-nucleon seed m_{c}=m_{Planck}.α^{9} allows the breaking of the inflaton super-symmetry in the superstring classes.

Replacing the proto-nucleon mass m_{c }= √{hc/2πG_{o}}.{60πe^{2}/h}^{9} = 9.924724523x 10^{-28} kg by the effective electron mass m_{e} = ke^{2}/2G_{o}(1.125x10^{12}) =9.290527148x10^{-31 }kg sets the Electromagnetic Interaction/Gravitational Interaction ratio

**EMI/GI = e ^{2}/G_{o}^{2}m_{e}^{2} = {e/G_{o}m_{e}}^{2} = 2.421821677x10^{42 }**using string units.

The Instanton following the Inflaton then defines a 10-dimensional superstringed Anti de Sitter open - and negatively hyperbolically curved cosmology, bounded in the 11-dimensional asymptotic Hubble Event Horizon.

**Modular String Duality and the Wormhole Curvature Boundary **

The concept of modular (Mirror/T) duality in super-membrane theory relates a maximum or large spacial displacement radius R as a low frequency and low energy named as a 'winding mode' to its inverse minimum or small spacial displacement radius 1/R as a high frequency and high energy as a 'vibratory mode'.

The utility of either 'string mode' would then result in an identical physical description either using a 'macro quantum' radius R or a 'micro quantum' radius 1/R.

c = _{λmin}.f_{max} = 2πR_{min}.f_{max} as light speed invariance for the vibratory string mode R_{curvature} = R_{min} = λ_{min} /2π = Wormhole Perimeter/2π

1/c = 1/(λ_{min}.f_{max}) = λ_{max}.f_{min} = λ_{max}/f_{max} = R_{max}.f_{min}/2π as lightspeed invariance for the winding string mode R_{curvature} = R_{max} = 2π._{λmax} = 2π/Wormhole Perimeter

For the Harmonic Planck Energy Oscillator Energy E^{o} = ½hf_{o} = ½m_{o}c^{2} = ½kT_{Planck}

Planck Mass = m_{Planck} = √{hc/2πG_{o}}

Planck Energy = E_{Planck} = m_{Planck}.c^{2} = √{hc^{5}/2πG_{o}} = hf_{Planck} = hc/l_{Planck} = kT_{Planck}

Planck Length =l_{Planck} = λ_{Planck}/2π = √{hG_{o}/2πc^{3}}

Planck Temperature = T_{Planck} = E_{Planck}/k = √{hc^{5}/2πk^{2}G_{o}}

Planck Density = ρ_{Planck}= m_{Planck}/V_{Planck} = √{4π^{2}c^{10}/h^{2}G_{o}^{4}}/2π^{2} = c^{5}/πhG_{o}^{2}= 9.40x10^{94} kg/m^{3} for 9x10^{60} permutation vibratory string eigenstates by |f_{ps}^{2}|_{mod}.

Energy Density Inflaton/Energy Density Instanton = E_{Planck}.V_{BigBang}/E_{BigBang}V_{Planck }with minimum Inflaton Planck Oscillator: E^{o}_{Planck} = ½m_{Planck}c^{2}

= m_{Planck}.r_{wormhole}^{3}/m_{wormhole}.l_{Planck}^{3} = √{(hc/2πG_{o})(2πc^{3}/hG_{o})}(2πc^{3}/hG_{o})(r_{wormhole}^{3}/m_{wormhole}} = (c^{2}/G_{o})(2πc^{3}/hG_{o}){r_{wormhole}^{3}/m_{wormhole}}

= (2πc^{5}/hG_{o}^{2}){r_{wormhole}^{3}/m_{wormhole}} = {4π^{2}k^{2}/h^{2}G_{o}} (hc^{5}/2πG_{o}k^{2}){r_{wormhole}^{3}/m_{wormhole}} = {kT_{Planck}}^{2}(2πr_{wormhole})^{2}{1/h^{2}G_{o}}{r_{wormhole}/m_{wormhole}}

= {E_{Planck}}^{2}.{c/hf_{wormhole}}^{2}.{1/G_{o}}}{r_{wormhole}/m_{wormhole}} = {E_{Planck}/E_{BigBang}}^{2}.{r_{wormhole}}{c^{2}/G_{o}m_{wormhole}} for the minimum Instanton Planck Oscillator: E^{o}_{Planck} = ½m_{wormhole}c^{2}

V_{BigBang}/V_{Planck }= {E_{Planck}/E_{BigBang}}.{r_{wormhole}}{c^{2}/G_{o}m_{wormhole}} for EV_{BigBang}/EV_{Planck }= EV_{Instanton}/EV_{Inflaton }= r_{wormhole}/R^{o}_{wormhole} = N_{Avagadro-Instanton} and counting the amount of wormhole string transformed from the Inflaton as the Instanton of the Quantum

Big Bang and as the constant 5.801197676..x10^{23} in string units.

**The Coupling of the Energy Laws by the Self-Frequency of the Quantum for Mass**

It has been discovered, that the universe contains an intrinsic coupling-parameter between its inertial mass content and its non-inertial energy content.

The matter in the universe is described by the physical parameter termed Mass (M), say as proportional to Energy (E) in Einstein's famous equation Mass M=E/c^{2}.

This mass M then reappears in Newtonian mechanics as the change in momentum (p) defining the Inertial Mass (M_{i}) as being proportional to some applied Force (F) or the 'work done' for a particular displacement {F=dp/dt for p=mv and v a kinematic velocity as the ratio of displacement over time generalised in the light path X=cT}.

It is also well understood, that the inertial mass M_{i} has a gravitational counterpart described not by the change in momentum of inertia carrying matter agglomerations; but by the geometric curvature of space containing matter conglomerations. This Gravitational Mass M_{g} is measured to be equivalent to the Inertial Mass M_{i} and is formulated in the 'Principle of Equivalence' in Einstein's Theory of General Relativity.

F-Theory then has shown, that this Inertial Mass M_{i} is coupled inherently to a 'mass-eigen' frequency via the following formulation:

(1) Energy E=hf=mc^{2} (The Combined Planck-Einstein Law)

(2) E=hf iff m=0 (The Planck Quantum Law E=hf for light speed invariance c=λf)

(3) E=mc^{2} iff f=f_{o}=f_{ss} (The Einstein Law E=mc^{2} for the light speed upper limit)

(1) Whenever there is mass (M=M_{i}=M_{g}) occupying space; this mass can be assigned either as a photonic mass {by the Energy-Momentum relation of Special Relativity: E^{2}=E_{o}^{2}+(pc)^{2}} by the photonic momentum p=h/λ=hf/c} OR a 'rest mass' m_{o}=m/√[1-(v/c)^{2}] for 'rest energy' E_{o}=m_{o}c^{2}.

The 'total' energy for the occupied space so contains a 'variable' mass in the 'combined' law; but allows particularisation for electromagnetic radiation (always moving at the Maxwell light speed constant c in Planck's Law and for the 'Newtonian' mass M in the Einstein Law.

(2) If M=0, then the Einstein Law is suppressed in favour of the Planck Law and the space contained energy E is photonic, i.e. electromagnetic, always dynamically described by the constancy of light speed c.

(3) If M>0, then there exists a mass-eigen frequency f_{ss}=f_{o}=E_{ss}/h=m_{ss}c^{2}/h, which QUANTIZES all mass agglomerations m=Σm_{ss} in the mass quantum m_{ss}=E_{ss}/c^{2}.

**The Wave Matter of de Broglie:** **λ _{deBroglie} = h/p **

The Wave matter of de Broglie from the Energy-Momentum Relation is applied in a (a) nonrelativistic, a (b) relativistic and a (c) superluminal form in the matter wavelength: λ_{deBroglie} = h/p = hc/pc for (pc) = √{E^{2} - E_{o}^{2}}= m_{o}c^{2}.√{[v/c]^{2}/(1-[v/c]^{2})}

(a) Example:

A pellet of 10g moves at 10 m/s for a de Broglie wavelength v_{dB} = h/mv = h/0.1 = 6.7x10^{-33} m*

This matter wavelength requires diffraction interference pattern of the order of λ_{dB} to be observable and subject to measurement

(b) Example:

An electron, moving at 80% of light speed 'c' requires relativistic development

E_{o }= m_{o}c^{2} with E = mc^{2} = m_{o}c^{2}/√{1-[v/c]^{2}}, a 66.66% increase in the electron's energy describing the Kinetic Energy E - E_{o} = {m - m_{o}}c^{2} for a relativistic momentum p = m_{o}c.√{[0.8]^{2}/(1-[0.8]^{2})} = (1.333..) m_{o}c = h/λ_{deBroglie} and for a relativistic de Broglie

wavelength, 60% smaller, than for the non-relativistic electron in λ_{deBroglie }= h/1.333..m_{o}c < h/0.8m_{o}c =λ_{deBroglie} (1.83x10^{-12} m relativistic and 3.05x10^{-12} m* non-relativistic for an electron 'restmass' of 9.11x10^{-31} kg* and measurable in diffraction interference

patterns with apertures comparable to this wavematter scale).

(c) The de Broglie matter wave speed in its 'group integrated' form derives from the postulates of Special Relativity and is defined in the invariance of light speed 'c' as a classical upper boundary for the acceleration of any mass M.

In its 'phase-individuated' form, the de Broglie matter wave is 'hyper accelerated' or tachyonic, the de Broglie wave speed being lower bounded by light speed 'c'

v_{phase} = wavelength.frequency = (h/mv_{group})(mc^{2}/h) = c^{2}/v_{group} > c for all v_{group} < c

m = Energy/c^{2} = hf/c^{2} = hc/λ_{deBroglie}c^{2} = h/λ_{deBroglie}c = m_{deBroglie} = [Action as Charge^{2}]_{mod}/c(Planck-Length Oscillation) = [e^{2}]_{mod}/cl_{Planck}√alpha = [e^{2}c^{2}/ce]_{mod} = [ec]_{modular}

as monopole mass of GUT-string IIB and as string displacement current mass equivalent for the classical electron displacement 2R_{e} = e*/c^{2} = [ec]_{modular} as Wormhole minimum spacetime configuration for the Big Bang Instanton of Big Bang wormhole energy

quantum E_{ps}=hf_{ps}=m_{ps}c^{2}=kT_{ps} as a function of e*=1/E_{ps} of Heterotic superstring class HE 8x8 and relating the Classical Electron Diameter {2R_{e}} as Monopole Mass [ec]_{mod} in Curvature Radius r_{ps}=λ_{ps}/2π = G_{o}m_{ps}/c^{2}.

The factor 2G_{o}/c^{2} multiplied by the factor 4π becomes Einstein's Constant k = 8πG_{o}/c^{2} = 3.102776531x10^{-26 } m/kg describing how spacetime curvature relates to the mass embedded in that spacetime in the theory of General Relativity.

The selfduality of the superstring IIB aka the Magnetic Monopole selfstate in GUT Unification 2R_{e}/30[ec]_{mod} = 2R_{e}c^{2}/30[ec^{3}]_{mod} = e*/30[ec^{3}]_{mod} ∝ κ for a proportionality constant {κ*}=2R_{e}/30k[ec]_{mod} = 2R_{e}.c^{2}/8πe = e*/8πe =1.2384..x10^{20} kg*/m* in string units for Star Charge in Star Colomb C*/Electro Charge in Coulomb C unified.

The monopolar Grand Unification (SEWG gravitational decoupling SEW.G) has a Planck string energy reduced at the IIB string level of e*=[ec^{3}]_{modular} for m_{ps}c^{2}c/[ec]_{modular} = [c^{3}]_{modular} = 2.7x10^{25} electron volt or 4.3362x10^{6} J for a monopole mass

[ec]_{modular }= m_{monopole }= 4.818x10^{-11} kg* .

Mass M = n.m_{ss} = Σm_{ss} = n.{h/2πr_{deBroglie}c} .[E_{ss}.e*]_{mod} = n.m_{ps}.[E_{ss}.{9x10^{60}}.2π^{2}R_{rmp}^{3}]_{mod} = n.m_{ps}.[E_{ss}.{2R_{e}.c^{2}}]_{mod} = n.[E_{ps}.E_{ss}]_{mod}.[2R_{e}]_{mod} for λ_{deBroglie}=λ_{ps}=h/m_{ps}c and [E_{ps}.e*]_{mod} =1

{2R_{e}c^{2}} = 4G_{o}M_{Hyper} for the classical electron radius R_{e}=ke^{2}/m_{e}c^{2} and describes its Hyper-Mass M_{Hyper-electron} = R_{e}c^{2}/2G_{o} = ke^{2}/2G_{o}m_{e} = 1.125x10^{12} kg* for an effective electron mass of m_{e} = ke^{2}/2G_{o}(1.125x10^{12}) =9.290527148x10^{-31 }kg* in string units

and where k=1/4πε_{o }= 9x10^{9} (Nm^{2}/C^{2})*.

The curvature radius for the electron mass m_{e} = r_{electron}c^{2}/2G_{o} then becomes r_{electron} = 2G_{o}m_{e}/c^{2} = 2.293957...x10^{-57} m* in string-membrane inflaton space as 1.44133588x10^{-34 }r_{ps} in the wormhole instanton space.

R_{e}/r_{inflaton-electron} = M_{Hyper-electron}/m_{e} = 1.2109108..x10^{42} = ½(EMI/GI) = ½(e^{2}/G_{o}^{2}m_{e}^{2}) =½ {e/G_{o}m_{e}}^{2} = ½(2.421821677x10^{42 }) for the classical electron radius R_{e} halved from the classical electron diameter 2R_{e} from the definition for the modulated

supermembrane coupled in E_{ps}E_{ss}=h^{2 }and E_{ps}/E_{ss}=f_{ps}^{2}=1/f_{ss}^{2}.

Mass M = n.m_{ss} = Σm_{ss} = n.{m_{ps}} .[E_{ss}.e*]_{mod} = n.{m_{ps}}[{hf_{ss}}{f_{ps}/f_{ss}}.2π^{2}R_{rmp}^{3}]_{mod} = n.m_{ps}.[E_{ps}.e*/f_{ss}^{2}]_{mod} = n.m_{ps}/|f_{ss}^{2}|_{mod}

**Hyper-Mass and the Hawking Modulus in Curvature of Space-time **

A general solution for the Curvature Radius R

_{Curvature}embedded in a spacetime and as a static boundary condition for a Black Hole is given as the Schwarzschild metric from the field equations of General Relativity:

Curvature Radius: ------------R

_{Curvature}= 2G

_{o}M/c

^{2}

for Hyper-Mass:----------------M_{Hyper} = hc^{3}.e*/4πG_{o} = ½N_{Avagadro-Instanton}.m_{wormhole}

Hyper-Mass M_{Hyper} describes a higher dimensional Inflaton mass for a lower dimensional Instanton curvature radius and becomes the relationship between the beginning and the end of the string epoch in the Planck Radius of the Inflaton and the physicalized wormhole of the Quantum Big Bang as the Instanton.

The wormhole of the Instanton r_{wormhole}=r_{ps}=r_{min} then forms the displacement quantum for the expanding cosmology in both the classical geometry of General Relativity (GR) and the quantum geometry of Quantum Relativity (QR).

Using the Schwarzschild metric for a mass of 70 kg would calculate a Curvature Radius for a mass equivalent Black Hole of (2.22..x10^{-10})(70)/c^{2} = 1.728..x10^{-25} meters*.

This is below the boundary condition of r_{min} = 10^{-22}/2π m = 1.591549..x10^{-23} m* for which the minimum mass requirement is found to be 6445.775.. kg*.

This result shows, that no physical microquantum Black Holes can exist, but that the minimum unitary wormhole quantum of the Instanton is given by a 'wormhole substance' or Inflaton Black Hole Molarity count for a new minimum Planck Oscillator at the HE

8x8 Instanton energy scale E^{o}_{BigBang}=½m_{ps}c^{2}=½hf_{ps}=½kT_{ps}=½E_{ps}=1/2e*

M_{Hyper}/½m_{ps} = 2hc^{3}.e*/4πG_{o}.m_{ps} = N_{Avagadro-Instanton }= r_{ps}/R_{ps}

N_{Avagadro-Instanton} = 5.8012x10^{23 }for hypermass

M_{Hyper} = hc^{3}.e*/4πG_{o} = ½N_{Avagadro-Instanton}.m_{wormhole}

for R_{ps }= G_{o}m_{ps}/c^{2} = r_{ps}/N_{Avagadro-Instanton }= 2.743...x10^{-47} m* in Inflaton membrane space of 11D and string space of 10D

M_{Hyper}/m_{min} = M_{min}/m_{min} = {n.hc^{3}e*/4πG_{o}}/m_{min} = {n.r_{min}c^{2}/2G_{o}}/m_{min} = {hc^{3}.c^{2}/4πG_{o}E_{min}}{ne*} = {hc^{3}.c^{2}/4πG_{o}hf_{min}}{ne*} = {2πr_{min}.hc^{3}.c^{2}/4πG_{o}hc}{ne*} = {r_{min}.c^{2}.E/2G_{o}M}{ne*} = r_{min}/R.{Ene*} for R = 2G_{o}M/c^{2 }

for a generalized energy-mass proportionality c^{2}=E/M in modular membrane duality with ne* = 1/E and ne*E = 1 (Modular Unification) for n.e*hc = n.λ_{min}.

m_{Hyper}/r_{min} = {r_{min}c^{2}/2G_{o}}/r_{min} = c^{2}/2G_{o} = n.m_{min}.Ee*/R = M/R_{curv} for de Broglie wave matter m_{min} = hf_{min}/c^{2} = h/cλ_{min}

Utility of the Schwarzschild metric allows calculation of Black Hole matter equivalents for any mass M>r_{min}c^{2}/2G_{o} say for a planetary mass M_{Earth} = 6x10^{24} kg for a r_{curv} = 0.015 meters and for a solar mass M_{Sun} = 2x10^{30} kg* for a r_{curv} = 4938.3 meters*.

The curvature of the Inflaton calculates as R_{Hubble-11D} = c/H_{o} = 2G_{o}M_{BigBang-Seed}/c^{2} = 1.59..x10^{26} meters* for the Inflaton Mass of 6.47..x10^{52} kg*.

For any mass M<r_{min}c^{2}/2G_{o} say for mass conglomerations smaller than 6445.775 kg* as the characteristic HyperMass for the Instanton, the corresponding curvature radius forms in the Inflaton space preceding the Quantum Big Bang at the time instanton of

t_{min}=t_{ps}=1/f_{ps}=[f_{ss}]_{mod}

The Standard Gravitational Parameter μ= GM = constant = G_{o}M(X^{n}Y^{n})= G_{o}X^{n}.MY^{n} and for (XY)^{n}=1 can be finestructured in a decreasing gravitational constant G(n)=G_{o}X^{n} with a corresponding increase in the mass parameter M as M(n)=M_{o}Y^{n} as say for the

proto-nucleonic mass of the Instanton m_{c}(n_{ps}) as m_{c}(n_{present}) = m_{c}.Y^{npresent} = m_{neutron }< m_{c}Y^{npresent} = 1.711752..x10^{-27} kg* and 958.99 MeV* upper limited

For a changing Gravitational constant G(n_{present}) .m_{neutron}(n_{present})^{2} = G_{o}m_{c}^{2}.Y^{npresent} and is modulated say in A micro-macro Black Hole perturbation M_{o}^{2}/2M_{∞}.M_{MaxHawking }= 1.000543 ~ 1

The Black Holed mass equivalence for astrophysical bodies is well formulated in the application of the basic Schwarzschild metric derived from General Relativity.

Stephen Hawking developed the inverse proportionality between the mass of a Black Hole M and its Temperature T in the form of the Hawking Modulus:

HM = m_{Planck}.E^{o}_{Planck}/k = √{hc/2πG_{o}}{½m_{Planck}.c^{2}/k} = hc^{3}/4πG_{o}k = {M_{Smin}.T_{Smax}} = {m_{ps}.T_{ps}.½N_{Avagadro-Instanton}} = [c^{2}/4π^{2}]_{mod}.{M_{MaxHawking} .T_{Smin} } = 9.131793821x10^{23} kg*K* with (m_{ps}T_{ps} = E_{ps}^{2}/kc^{2 }= 1.002117..π)

The Hawking Modulus so has mensuration units [Mass][Temperature] in [kg][K(elvin)], which reduce to [Mass]{Energy] in [kg][J(oules)] for ignoring the Stefan-Boltzmann entropy constant k.

And so M_{min}.T_{max }= hc³/4πG_{o}k = [c^{2}/4π^{2}]_{mod}.M_{max}.T_{min }= ½m_{Planck}.T_{Planck }= M_{MaxHawking}. [c^{2}/4π^{2}]_{mod}.T_{ss} and the Hawking Mass is determined as M_{MaxHawking} = λ_{max}πc²/G_{o} = 2.54469..x10^{49} kg*.

Hyper-Mass M_{Hyper} (n_{ps}) = hc^{3}.e*/4πG_{o} = ½N_{Avagadro-Instanton}.m_{wormhole} = 6445.775 kg at the Instanton boundary n=n_{ps} so increases to M_{Hyper}(n_{present})Y^{npresent} =hc^{3}.e*/4πG_{o}X^{npresent }~ 11,117.26 kg as the projected Instanton boundary mass for the wormhole

radius r_{wormhole} = r_{ps} = N_{Avagadro-Instanton}.R_{ps} modulating the Inflaton curvature with the Instanton curvature and utilizing n_{present}=1.1327... for a decreased perturbed G(n_{present}) = 6.442x10^{-11} G-string units for the Standard Gravitational Parameter

G(n)m_{i}Y^{k}(n).m_{j}Y^{n-k} = G_{o}m_{c}^{2} = constant for G(n)=G_{o}X^{n}.

_{min}λ

_{max}=1 wavelength modulation in the T-duality of λ

_{min}=2πR

_{min}=1/λ

_{max}=2π/R

_{max}, we can see, that this modulation closely approximates the geometric mean of the seedling mass in:

_{o}

^{2}/2M

_{∞}.M

_{Max}=M

_{o}

^{2}/8π.M

_{∞}.M

_{Hawking}=3.2895..x10

^{102}/3.2931..x10

^{102}~ 0.998910744...

This also circumscribes the actual to critical density ratio in the omega of the general relativistic treatment of the cosmologies.

Now recall our applied G value in G

_{m}(n)=G

_{o}.and apply our just derived Black Hole Mass modulation coupled to that of the quantum micromasses.

We had: G

_{o}m

_{c}²={G

_{o}X

^{n+k}}.{m

_{c}Y

^{n}}.{m

_{c}Y

^{k}}=G

_{m}(n).m

_{nmax}.m

_{nmin}and where G

_{m}is the actual G value as measured and which has proved difficult to do so in the laboratories.

G

_{m}(n)=G

_{o}.X

^{n+k}=G

_{o}m

_{c}²/m

_{nmax}.m

_{nmin}=G

_{o}m

_{c}²/({m

_{c}Y

^{n}}{m

_{nmin}}) and where we have m

_{nmin}=m

_{c}Y

^{k}} for the unknown value of k with m

_{nmax}=m

_{c}Y

^{n}.

So G

_{m}(n)=G

_{o}.X

^{n+k}=G

_{o}X

^{n}[m

_{c}/m

_{nmin}]=G

_{o}{m

_{c}

^{2}/m

_{c}Y

^{n}}.{M

_{o}

^{2}/8π.M

_{∞}.M

_{Hawking}.m

_{av}} for X

^{k}={m

_{c}/m

_{av}}.{M

_{o}

^{2}/8π.M

_{∞}.M

_{Hawking}}=1.00109044..{m

_{c}/m

_{av}} and where now {m

_{nmin}}={8π.M

_{∞}.M

_{Hawking}.m

_{av}/M

_{o}

^{2}}=1.00109044..m

_{av}.

m

_{av}={M

_{o}²/8π.M

_{∞}.M

_{Hawking}}{m

_{nmin}}={M

_{o}²/8π.M

_{∞}.M

_{Hawking}}{m

_{c}Y

^{k}}=0.9989107..{m

_{c}Y

^{k}} and obviously represents a REDUCED minimum mass m

_{nmin}=m

_{c}Y

^{k}.

But the product of maximum and 'new' minimum now allows an actual finetuning to a MEASURED nucleon mass m

_{N}by: m

_{N}² = m

_{av}Y

^{n}.m

_{c}Y

^{n}=m

_{av}.m

_{nmax}.Y

^{n}.

So substituting for m

_{av}in our G

_{m}expression, will now give the formulation:

G

_{m}(n)=G

_{o}.X

^{n+k}=G

_{o}X

^{n}[m

_{c}/m

_{nmin}]=G

_{o}{m

_{c}

^{2}/m

_{c}Y

^{n}}.{M

_{o}

^{2}/8π.M

_{∞}.M

_{Hawking}.m

_{av}}

_{m}(n)=G

_{o}.X

^{n+k}=G

_{o}X

^{n}[m

_{c}/m

_{nmin}]=G

_{o}{m

_{c}

^{2}/m

_{c}Y

^{n}}.{M

_{o}

^{2}/8π.M

_{∞}.M

_{Hawking}}{m

_{c}Y

^{2n}/m

_{N}

^{2}}

_{m}(n)=G

_{o}.{m

_{c}

^{2}/m

_{N}

^{2}}{M

_{o}

^{2}/8π.M

_{∞}.M

_{Hawking}}Y

^{n}

The average nucleon mass m

_{N}is upper bounded in the neutron mass and lower bounded in the proton mass, their difference being an effect of their nucleonic quark content, differing in the up-down transition and energy level and because of electro charges increasing the intra-quarkian Magneto charge coupling between the two mesonic rings of the neutron and the single mesonic ring in the proton's down- or KIR-quark.

For a Neutron Restmass of: m

_{neutron}=1.6812656x10

^{-27}kg* (941.9111 MeV*) or (1.6749792x10

^{-27}kg and 939.594 MeV) the substitution (and using calibrations m=1.001671358 m*; s=1.000978395 s*; kg=1.003753127 kg* and C=1.002711702 C* gives:

_{p})= G

_{o}{m

_{c}/m

_{neutron}}

^{2}.(0.9989107..)Y

^{np }= 6.670693x10

^{-11}(m

^{3}/kgs

^{2})* or 6.675312x10

^{-11}(m³/kgs²).

For a Proton Restmass of: m

_{proton}=1.6788956x10

^{-27}kg* (940.5833 MeV*) or (1.672618x10

^{-27}kg and 938.270 MeV).

_{p}) = G

_{o}{m

_{c}/m

_{N}}

^{2}.(0.9989107..)Y

^{np }= 6.6895399x10

^{-11}(m

^{3}/kgs

^{2})* or 6.694171x10

^{-11}(m³/kgs²).

G

_{m}(n)=G

_{o}.X

^{n+k }= 6.670693x10

^{-11}(m

^{3}/kgs

^{2})* then gives k

_{p}=ln{G

_{m}(n

_{p})/G

_{o}}/ln{X} - n

_{p}= 1.0602852 - 1.132711 = -0.0724258

The upper value of the neutron bound so represents an upper limit for the Gravitational Constant as the original quark-lepton bifurcation of the X-Boson precursor given in the KKK kernel. Only the KKK kernel is subject to the mass evolution of the cosmos; the leptonic masses being intrinsically incorporated in the Kernel means.

The m

_{c}.Y

^{n}so serves as an appropriate upper bounded approximation for G(n), subject to leptonic ring IR-OR perturbations.

Basic nucleon restmass is m

_{c}=√Omega.m

_{P}=9.9247245x10

^{-28}kg* or 958.99 MeV*.

(Here Omega is a gauge string factor coupling in the fundamental force interactions as: Cuberoot(Alpha):Alpha:Cuberoot(Omega):Omega and for Omega=G-alpha.)

KKK-Kernelmass=Up/Down-HiggsLevel=3x319.66 MeV*= 958.99 MeV*, using the Kernel-Ring and Family-Coupling Constants.

Subtracting the Ring-VPE (3L) gives the basic nucleonic K-State as 939.776 MeV*. This excludes the electronic perturbation of the IR-OR oscillation.

For the Proton, one adds one (K-IR-Transition energy) and subtracts the electron-mass for the d-quark level and for the Neutron one doubles this to reflect the up-down-quark differential.

An electron perturbation subtracts one 2-2/3=4/3 electron energy as the difference between 2 leptonic rings from the proton's 2 up-quarks and 2-1/3=5/3 electron energy from the neutron' singular up-quark to relate the trisected nucleonic quark geometric template.

Proton m

_{p}=u.d.u=K.KIR.K=(939.776+1.5013-0.5205-0.1735) MeV* = 940.5833 MeV* (938.270 MeV).

Neutron m

_{n}=d.u.d=KIR.K.KIR=(939.776+3.0026-1.0410+0.1735) MeV* = 941.9111 MeV* (939.594 MeV).

This is the groundstate from the Higgs-Restmass-Induction-Mechanismand reflects the quarkian geometry as being responsible for theinertial mass differential between the two elementary nucleons. All groundstate elementary particle masses are computed from theHiggs-Scale and then become subject to various finestructures. Overall, the MEASURED gravitational constant 'G' can be said to be decreasing over time.

The ratio given in k is G

_{m}Y

^{n}/G

_{o }~ 0.600362... and so the present G-constant is about 60% of the one at the Planck Scale. G decreases nonlinearly, but at a present rate of 0.600362/19.12x10

^{9 }per year, which calculates as 3.1400..x10

^{-11}G-units per year.

Generally using the exponential series expansion, one can indicate the change in G.

For X

^{n+k}=z=exp[(n+k)lnX] by (n+k)lnX=lnz for the value u=(n+k)lnX=-0.481212(n+k); z transforms in exponential expansion e

^{u}=1+u+u

^{2}/2!+u

^{3}/3!+u

^{4}/4!+...

For a function f(n)=z=G

_{m}(n)/G

_{o}=X

^{n+k}- f(n)=1-(0.481212.)(n+k)+(0.2316.)(n+k)

^{2}/2-(0.1114.)(n+k)

^{3}/6+(0.0536.)(n+k)

^{4}/24-...+...

_{ps}=λ

_{ps}/R

_{max}=6.2591x10

^{-49 }~ 0

Then G

_{BigBang}=G

_{o}X

^{n}

_{ps}=G

_{o}(to 50 decimal places distinguishing the timeinstanton from the Nulltime as the Planck-Time transform).

G

_{o}represents the quantum gravitational constant applicable for any Black Hole cosmology and can be used to correlate the MOND gravitation with the Newton-Einstein gravitation (previously stated in section 3.8).

For our previously calculated k=ln(G

_{m}Y

^{n}/G

_{o})/lnX and which calculates as k= -0.0724258..

^{2}/2-(0.1114.)(n+k)

^{3}/6+(0.0536.)(n+k)

^{4}/24-(0.0258.)(n+k)

^{5}/120+...-... for f(1.132711)=1-0.51022+0.13016-0.02214+0.00283-0.000288...+...~ 0.6006340 to fifth order approximation to 0.60036246...

Hence, the gravitational constant assumes a value of about 60.0% of its Big Bang initialisation and calculates as 6.675x10

^{-11}G-units for a present cycletime n

_{present}=H

_{o}t

_{present}=1.132711...

The introduction of the mass seed coupling between the macro quantum M

_{o}and the micro quantum m

_{c}=m

_{P}alpha

^{9}(from the gravitational finestructure unification) PERTURBS the 'purely electromagnetic' cosmology in the perturbation factor k and increases the purely electromagnetic G

_{memr }in the black hole physics described.

So gravity appears stronger when one 'looks back in time' or analyses cosmological objects at large distances. The expansion parameter (a) in the Friedmann-Einstein standard cosmology can be rewritten as a curvature ratio R(n)/R

_{max}={n/(n+1)} and

_{o}=½Ω

_{o}=M

_{o}/2M

_{∞}=2G

_{o}H

_{o}M

_{o}/c³ ~0.014015... This open universe is bounded in the 'standing wave' of the Hubble Oscillation of the 11D and 'higher dimensional universe'.