_{1}

^{*}

Fermion mass is modeled as an analogue of Weyl curvature, which by hypothesis emerges when closed, spin-2 strings sweep out closed world tubes. Second order curvature classes result when closed world tubes circulate and themselves sweep out closed tubes etc. Gauge invariance distinguishes admissible curvature classes from the larger set that would constitute an infinite continuum of possibilities. Admissible curvature classes account for known quark masses and predict a new quark of mass 30 GeV/c
^{2}. Super-symmetric interactions among prescribed fermions and super-partners conserve electrical charge,
I
_{3}, color and generation and are therefore regarded by hypothesis as preserving a minimal irreducible representation of a super-symmetric SU(5).

The theory of H. Weyl associates increments of vector magnitude with curvature states

and gauge transformations:

[

This analogue of Weyl curvature will be known as W-curvature and will be regarded as physically admissible (as time and position independent) if and only if the implicit gauge is preserved in the sense introduced by F. London:

This second order W-curvature is physically admissible if and only if

By hypothesis the proposed model associates the W-curvature classes (7) with the masses of composite states

The proposed model is calibrated by associating a specific W-curvature state

quark state

that

[

Interpretation of the mass (10) will be deferred until the massive states described by expressions (11) through (14) have been interpreted. The theoretical mass represented by (11) motivates the association of (11) with the spin-^{2}), where ^{2}) and where, again,

The theoretical masses that are described by expressions (12) through (14) are also established by observation [

To interpret the mass of the composite that is described by expression (10), it is first observed that the left- handed muon ^{2}.

Generation and electrical charge are preserved by the pure SUGRA interactions that are described by vertices

and

Moreover,

Composites such as

and

If the generic

It should be observed that identical leptons as well as identical quarks can emerge within composite, spin-

It is observed that the interaction (16) conserves four quantum numbers: electrical charge,

where

where

where “7” designates the predicted quark and “

where A is the “average quark” introduced above. (The average “A” is obtained by averaging the masses, ^{2} and the

The proposed symmetry models an energy interval between the Planck energy (10^{19} GeV) and the energy level of grand unification (10^{15} GeV) and as such models a marginal domain in which interactions may involve the composites “A” or may involve up-down pairs. By hypothesis, “A” and the up-down pair are never active at the same energy level (never involved in interactions at the same energy level). If “A” is active, then up-down is inactive and conversely. In this context Equation (19) and Equation (20) together model quarks in the entire marginal domain between the Planck level and the grand unification level.

Fermion mass is attributed to an analogue of Weyl curvature which, by hypothesis, occurs when closed, spin-2 strings sweep out closed world tubes. Additional classes of massive fermions are attributed to higher orders of curvature that occur as world tubes circulate and, they themselves sweep out closed tubes etc. The proposed hypothesis clearly parallels Wheeler’s ideal, which attributes mass to space-time curvature. The composites _{3} neutral). Leptons are converted into quarks by absorptions of such fields. Inverse fields are also postulated.

As an analogue of the Weyl theory, the proposed model is subject to gauge transformations and a requirement of gauge invariance distinguishes admissible curvature classes and consequent mass classes from the larger set that will constitute an infinite continuum of possibilities. The proposed model correctly approximates the masses of observed quarks and of the tauon and predicts a new quark of (approximate) mass 30 GeV/c^{2}.

Super-symmetric interactions that conserve electrical charge, I_{3}, color and generation are regarded as preserving a minimal irreducible representation of a super-symmetric SU(5).