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Many of these "proofs" are either circular in logic, or assume unproven (and often implausible) premises.  

Many of these "proofs" are either circular in logic, or assume unproven (and often implausible) premises.  

The typical defect in these proofs is an unstated premise that the [[Theory of Relativity]] must be consistent with the conservation of energy, and then implicitly redefine mass as a function of energy so that energy is then conserved under Relativity.

The typical defect in these proofs is an unstated premise that the [[Theory of Relativity]] must be consistent with the conservation of energy.  The "proof" implicitly redefine mass as a function of energy (and the speed of light) so that energy is then conserved under Relativity.

Proponents of the formula rarely state their assumptions clearly.  In fact, they assume what they purport to prove:  that Relativity can make sense only if energy is somehow directly related to mass based on the speed of light.  By assuming that Relativity is true and that energy is conserved, the formula itself is assumed also.

Proponents of the formula rarely state their assumptions clearly.  In fact, they assume what they purport to prove:  that Relativity can make sense only if energy is somehow directly related to mass based on the speed of light.  By assuming that Relativity is true and that energy is conserved, the formula itself is thereby also assumed.

Many of the "proofs" use contrived thought experiments to try to bootstrap the kinetic energy of a moving particle into an assertion about the energy of a resting mass.  Aside from changes in potential energy, there is no direct connection between the kinetic energy of a moving mass and its rest energy.

Many of the "proofs" use contrived thought experiments to try to bootstrap the kinetic energy of a moving particle into an assertion about the energy of a resting mass.  Aside from changes in potential energy, there is no direct connection between the kinetic energy of a moving mass and its rest energy.