We could not find any experiment that directly proves inertial mass increase, though it has been inferred from some experiments. But there are alternative explanations for those observations. Like other forces, we experience mass through the effect it produces on others. This is the difference between the body (dimension) and it’s mass. While energy creates temperature variation, mass creates pressure variation. Both are related to the body. Inertial mass is a mass parameter giving the inertial resistance to acceleration of the body as determined by its momentum, when it is subjected to a force that is not due to gravity. It is found by applying a known force to an unknown mass, measuring the acceleration, and applying Newton's Second Law, m = F/a. If the same body with the same mass is displaced with application of different energy, its effect (a) would appear as different in proportion to the applied force (F), i.e., with mass constant, F varies with a. Thus, in principle, it need not be mass variation, but can appear so due to energy variation.
Switching over from Cyclotron to Synchrotron is not based on observational necessity, but based on difference between relativistic and non-relativistic theories. They are two different theories independent of each other. The variation factor γ proposed in SR is a mathematical structure apparent to the observer and not real to the physical structure being observed. This difference is generally not recognized - mostly by the brane-world proponents. Unlike three dimensional physical structures, mathematical structures are two dimensional. The graph may represent space, but it is not space itself. The drawings of a circle, a square, a vector or any other physical representation, are similar abstractions. The circle represents only a two dimensional cross section of a three dimensional sphere. The square represents a surface of a cube. Without the cube or similar structure (including the paper), it has no physical existence. An ellipse may represent an orbit (a circle with a moving center), but it is not the dynamical orbit itself. The vector is a fixed representation of velocity; it is not the dynamical velocity itself, and so on.
The so-called simplification or scaling up or down of the drawing does not make these structures abstract. The basic abstraction is due to the fact that the mathematics that is applied to solve physical problems actually applies to the two dimensional diagram, and not to the three dimensional space. The numbers are assigned to points on the piece of paper or in the Cartesian graph, and not to points in space. If one assigns a number to a point in space, what one really means is that it is at a certain distance from an arbitrarily chosen origin. Thus, by assigning a number to a point in space, what one really does is assign an origin, which is another point in space leading to circular logic. The point in space can exist by itself as the equilibrium position of various forces. But a point on a paper exists only with reference to the arbitrarily assigned origin. If additional force is applied, the locus of the point in space resolves into two equal but oppositely directed field lines. But the locus of a point on a graph is always unidirectional and depicts distance – linear or non-linear, but not force. Thus, a physical structure is different from its mathematical representation. Hence, the basic assumptions of all topologies, including symplectic topology, linear and vector algebra and the tensor calculus, all representations of vector spaces, whether they are abstract or physical, real or complex, composed of whatever combination of scalars, vectors, quaternions, or tensors, and the current definition of the point, line, and derivative are necessarily at least one dimension less from physical space.
Perception is related to physical structures. The field created by the content of our eye interacts with similar fields created by other objects measuring those in the process. They are not two dimensional like the impressions on a photographic plate, but three dimensional like a mould, i.e., a shaped cavity that is used to give a definite form to fluid or plastic material including radiation that behave like fluids. Thus, we see three dimensional objects. The photographic plate on the other hand represents the mathematical structure in two dimensions. Thus, it is an abstraction, which is not physical.
While there is no direct evidence of increase in inertial mass, there are many experiments to conclusively show that the ratio of gravitational mass to inertial mass is constant. Gravity is said to be an attractive force, i.e., the force applied by one body brings the other body towards it, i.e., towards the direction of the applied force. Inertia displaces the body in the direction opposite to the direction of the applied force. Since both the displacements in opposite directions are known to be equivalent, this implies that the concept of inertial mass is only apparent like a mirage and can not be real.