**Authors:** J. S. Markovitch

The fine structure constant and the quark and lepton mixing angles are shown to arise naturally in the
course of altering the symmetry of two algebraic identities. Specifically, the symmetry of the
identity x^{2} = *x*^{2} is "broken" by making the substitution
*x*^{n} → *x*^{n} - *y*^{p} on its left side, and the substitution
*x* → *x *- *z* on its right side,
where *p* equals the order of the identity; these substitutions convert the above identity into the
equation *x*^{2} - *y*^{2} = (*x* - *z*)^{2}. These same substitutions are also applied
to the only slightly more complicated identity (*x*/*a*)^{3} + *x*^{2} = (*x*/*a*)^{3} +
*x*^{2} to produce this second equation (*x*^{3} - *y*^{3}) / *a*^{3} + *x*^{2} - *y*^{3} =
(*x* - *z*)^{3} / *a*^{3} + (*x* - *z*)^{2}. These two equations are then shown to share
a mathematical property relating to *dz*/*dy*, where, on the second equation's left side, this
property helps define the special case (*x*^{3} - *y*^{3}) / *a*^{3} + *x*^{2} - *y*^{3} =
(10^{3} - 0.1^{3}) / 3^{3} + 10^{2} - 0.1^{3} = 137.036, an equation which incorporates a value close to the
experimental fine structure constant inverse. Moreover, on the second equation's right side, this same
special case simultaneously produces values for the sines squared of the mixing angles. Specifically, the
sines squared of the leptonic angles *φ*_{12}, *φ*_{23}, and
*φ*_{13} appear as 0.3, 0.5, and not larger than roughly 1/30 000, respectively; and the
sines squared of the quark mixing angles *θ*_{12} and *θ*_{13} appear
as 0.05, and close to 1/90 000, respectively. Despite closely mirroring so many experimental values, including
the precisely-known fine structure constant, the above mathematical model requires no free parameters adjusted
to fit experiment.

**Comments:** 23 pages

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[v1] 11 Oct 2010

[v2] 26 Oct 2010

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