Hypersphere volumes and the mass of the Tau-neutrino
 
Consider the universe's thermodynamic expansion to proceed at an initializing time (and practically at lightspeed for the lightpath x=ct describing the hypersphere radii) to from a single spacetime quantum with a quantized toroidal volume 2π²r_w³ and where r_w is the characteristic wormhole radius for this basic building unit for a quantized universe (say in string parameters given in the Planck scale and its transformations).

At a time t_G, say so 18.85 minutes later, the count of space time quanta can be said to be 9.677x10¹⁰² for a universal 'total hypersphere radius' of about r_G=3.39x10¹¹ meters and for a G-Hypersphere volume of so 7.69x10³⁵cubic meters.

{This radius is about 2.3 Astronomical Units (AUs) and about the distance of the Asteroid Belt from the star Sol in a typical (our) solar system.}
 
This modelling of a mapping of the quantum-microscale onto the cosmological macroscale should now indicate the mapping of the wormhole scale onto the scale of the sun itself.

r_w/R_Sun(i)=R_e/r_E for R_Sun(i)=r_wr_E/R_e=1,971,030 meters. This gives an 'inner' solar core of diameter about 3.94x10⁶ meters.


As the classical electron radius is quantized in the wormhole radius in the formulation R_e=10¹⁰r_w/360, rendering a finestructure for Planck's Constant as a 'superstring-parametric':  h=r_w/2R_ec³; the 'outer' solar scale becomes R_Sun(o)=360R_Sun(i)=7.092x10⁸ meters as the observed radius for the solar disk.  
  
19 seconds later; a F-Hypersphere radius is about r_F=3.45x10¹¹ meters for a F-count of so 1.02x10¹⁰³ spacetime quanta.
We also define an E-Hypersphere radius at r_E=3.44x10¹⁴ meters and an E-count of so 10¹¹² to circumscribe this 'solar system' in so 230 AU.

We so have 4 hypersphere volumes, based on the singularity-unit and magnified via spacetime quantization in the hyperspheres defined in counters G, F and E. We consider these counters as somehow fundamental to the universe's expansion, serving as boundary conditions in some manner. As counters, those googol-numbers can be said to be defined algorithmically and independent on mensuration physics of any kind.
 

 
The mapping of the atomic nucleus onto the thermodynamic universe of the hyperspheres
 
Should we consider the universe to follow some kind of architectural blueprint; then we might attempt to use our counters to be isomorphic (same form or shape) in a one-to-one mapping between the macrocosmos and the microcosmos. So we define a quantum geometry for the nucleus in the simplest atom, say Hydrogen. The hydrogenic nucleus is a single proton of quark-structure udu and which we assign a quantum geometric template of Kernel-InnerRing-OuterRing (K-IR-OR), say in a simple model of concentricity.
We set the up-quarks (u) to become the 'smeared out core' in say a tripartition uuu so allowing a substructure for the down-quark (d) to be u+InnerRing. A down-quark so is a unitary ring coupled to a kernel-quark. The proton's quark-content so can be rewritten and without loss of any of the properties associated with the quantum conservation laws; as proton-> udu->uuu+IR=KKK+IR. We may now label the InnerRing as Mesonic and the OuterRing as Leptonic.
The OuterRing is so definitive for the strange quark in quantum geometric terms: s=u+OR.
A neutron's quark content so becomes neutron=dud=KIR.K.KIR with a 'hyperon resonance' in the lambda=sud=KOR.K.KIR and so allowing the neutron's beta decay to proceed in disassociation from a nucleus (where protons and neutrons bind in meson exchange); i.e. in the form of 'free neutrons'. The neutron decays in the oscillation potential between the mesonic inner ring and the leptonic outer ring as the 'ground-energy' eigenstate.
 
There actually exist three uds-quark states which decay differently via strong, electromagnetic and weak decay rates in the uds (Sigma_o Resonance); usd (Sigma_o) and the sud (Lambda_o) in increasing stability. This quantum geometry then indicates the behaviour of the triple-uds decay from first principles, whereas the contemporary standard model does not, considering the u-d-s quark eigenstates to be quantum geometrically undifferentiated.
The nuclear interactions, both strong and weak are confined in a 'Magnetic Asymptotic Confinement Limit' coinciding with the Classical Electron radius R_e=ke²/m_ec² and in a scale of so 3 Fermi or 2.8x10^-15 meters. At a distance further away from this scale, the nuclear interaction strength vanishes rapidly. The wavenature of the nucleus is given in the Compton-Radius R_c=h/2πmc with m the mass of the nucleus, say a proton; the latter so having R_c=2x10^-16 meters or so 0.2 fermi.

The wave-matter (after de Broglie generalising wavespeed v_dB from c in R_cc) then relates the classical electron radius as the 'confinement limit' to the Compton scale in the electromagnetic finestructure constant in R_ee=Alpha.R_c.
The extension to the Hydrogen-Atom is obtained in the expression R_e=Alpha².R_Bohr1 for the first Bohr-Radius as the 'ground-energy' of so 13.7 eV at a scale of so 10^-11 to 10^-10 meters (Angstroems).
These 'facts of measurements' of the standard models now allow our quantum geometric correspondences to assume cosmological significance in their isomorphic mapping. We denote the OuterRing as the classical electron radius and introduce the InnerRing as a mesonic scale contained within the geometry of the proton and all other elementary baryonic- and hadronic particles.
Firstly, we define a mean macro-mesonic radius as: r_M=½(r_F+r_G)~ 3.42x10¹¹ meters and set the macro-leptonic radius to r_E=3.44x10¹⁴ meters.
Secondly, we map the macroscale onto the microscale, say in the simple proportionality relation, using
(de)capitalised symbols: R_e/R_m=r_E/r_M.
We can so solve for the micro-mesonic scale R_m=R_e.r_M/r_E ~ 2.76x10^-18 meters.
So reducing the apparent measured 'size' of a proton in a factor about about 1000 gives the scale of the subnuclear mesonic interaction, say the strong interaction coupling by pions.


The Higgsian Scalar-Neutrino
 
The (anti)neutrinos are part of the electron mass in a decoupling process between the kernel and the rings. Neutrino mass is so not cosmologically significant and cannot be utilized in 'missing mass' models'.
We may define the kernel-scale as that of the singular spacetime-quantum unit itself, namely as wormhole radius r_w=10^-22/2π meters.

Before the decoupling between kernel and rings, the kernel-energy can be said to be strong-weakly coupled or unified to encompass the gauge-gluon of the strong interaction and the gauge-weakon of the weak interaction defined in a coupling between the OuterRing and the Kernel and bypassing the mesonic InnerRing.

So for matter, a W-Minus (weakon) must consist of a coupled lepton part, yet linking to the strong interaction via the kernel part. If now the colour-charge of the gluon transmutates into a 'neutrino-colour-charge'; then this decoupling will not only define the mechanics for the strong-weak nuclear unification coupling; but also the energy transformation of the gauge-colour charge into the gauge-lepton charge.

There are precisely 8 gluonic transitive energy permutation eigenstates between a 'radiative-additive' Planck energy in W(hite)=E=hf and an 'inertial-subtractive' Einstein energy in B(lack)=E=mc², which describe the baryonic- and hyperonic 'quark-sectors' in: mc²=BBB, BBW, WBB, BWB, WBW, BWW, WWB and WWW=hf. The permutations are cyclic and not linearly commutative. For mesons (quark-antiquark eigenstates), the permutations are BB, BW, WB and WW in the SU(2) and SU(3) Unitary Symmetries.

So generally, we may state, that the gluon is unfied with a weakon before decoupling; this decoupling 'materialising' energy in the form of mass, namely the mass of the measured 'weak-interaction-bosons' of the standard model (W- for charged matter; W+ for charged antimatter and Z_o for neutral mass-currents say).

 
Experiment shows, that a W- decays into spin-aligned electron-antineutrino or muon-antineutrino or tauon-antineutrino pairings under the conservation laws for momentum and energy.
So, using our quantum geometry, we realise, that the weakly decoupled electron must represent the OuterRing, and just as shown in the analysis of QED (Quantum-Electro-Dynamics). Then it can be inferred, that the Electron's Antineutrino represents a transformed and materialised gluon via its colourcharge, now decoupled from the kernel.
 
Then the OuterRing contracts (say along its magnetoaxis defining its asymptotic confinement); in effect 'shrinking the electron' in its inertial and charge- properties to its experimentally measured 'point-particle-size'. Here we define this process as a mapping between the Electronic wavelength 2πR_e and the wormhole perimeter λ_w=2πr_w.

But in this process of the 'shrinking' classical electron radius towards the gluonic kernel (say); the mesonic ring will be encountered and it is there, that any mass-inductions should occur to differentiate a massless lepton gauge-eigenstate from that manifested by the weakon precursors.
{Note: Here the W- inducing a lefthanded neutron to decay weakly into a lefthanded proton, a lefthanded electron and a righthanded antineutrino. Only lefthanded particles decay weakly in CP-parity-symmetry violation, effected by neutrino-gauge definitions from first principles}.

This so defines a neutrino-oscillation potential at the InnerRing-Boundary. Using our proportions and assigning any neutrino-masses m_υ as part of the electronmass m_e, gives the following proportionality as the mass eigenvalue of the Tau-neutrino:

m_υ=m_eλ_w.r_E/(2πr_MR_e) ~ 5.4x10^-36 kg or 3.0 eV.

So we have derived, from first principles, a (anti)neutrinomass eigenstate of 3 eV.

This confirms the Mainz, Germany Result as the upper limit for neutrino masses resulting from ordinary Beta-Decay and indicates the importance of the primordial beta-decay for the cosmogenesis and the isomorphic scale mappings stated above.

The hypersphere intersection of the G- and F-count of the thermodynamic expansion of the mass-parametric universe so induces a neutrino-mass of 3 eV at the 2.76x10^-18 meter marker.

The more precise G-F differential in terms of eigenenergy is 0.052 eV as the mass-eigenvalue for the Higgs-(Anti)neutrino (which is scalar of 0-spin and constituent of the so called Higgs Boson as the kernel-Eigenstate). This has been experimentally verified in the Super-Kamiokande (Japan) neutrino experiments published in 1998 and in subsequent neutrino experiments around the globe, say Sudbury, KamLAND, Dubna, MinibooNE and MINOS.
This Higgs-Neutrino-Induction is 'twinned' meaning that this energy can be related to the energy of so termed 'slow- or thermal neutrons' in a coupled energy of so twice 0.0253 eV for a thermal equilibrium at so 20° Celsius and a rms-standard-speed of so 2200 m/s from the Maxwell statistical distributions for the kinematics.


Neutrinomasses
 
The Electron-(Anti)Neutrino is massless as base-neutrinoic weakon eigenstate.
The Muon-(Anti)Neutrino is also massless as base-neutrinoic weakon eigenstate.
The Tauon-(Anti)Neutrino is not massless with inertial eigenstate meaned at 3.0 eV.
The weakon kernel-eigenstates are 'squared' or doubled (2x2=2+2) in comparison with the gluonic-eigenstate (one can denote the colourcharges as (R²G²B²)[½] and as (RGB)[1] respectively say and with the [] bracket denoting gauge-spin and RGB meaning colours Red-Green-Blue).

The scalar Higgs-(Anti)Neutrino becomes then defined in: (R⁴G⁴B⁴)[0].

The twinned neutrino state so becomes MANIFESTED in a coupling of the scalar Higgs-Neutrino with a massless base neutrino in a (R⁶G⁶B⁶)[0+½]) mass-induction template.
The Higgs-Neutrino is bosonic and so not subject to the Pauli Exclusion Principle; but quantized in the form of the FG-differential of the 0.052 Higgs-Restmass-Induction.
Subsequently all experimentally observed neutrino-oscillations should show a stepwise energy induction in units of the Higgs-neutrino mass of 0.052 eV. This was the case in the Super-Kamiokande experiments; and which was interpreted as a mass-differential between the muonic and tauonic neutrinoic forms.

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