A professor inquired in October 2026 about simultaneously having discrete and continuous.
My reply:

You say: [H]ow it is possible for space to be discrete for one type of particle and continuous for another?

The paper is regarding the motion of mass and light through space. That motion is through a single, continuous space. Space itself is not fundamentally different for them; rather, the concept of ‘discrete’ arises strictly from the motion of mass. Because mass and light are distinct physical entities, it could well be that they have distinct modes of motion within the same continuous space. We already have a fundamental kinematic divide: mass is constrained to speeds strictly less than c, whereas light propagates at exactly c. The key argument of the paper is that their underlying difference in motion – discrete versus continuous – is physically the foundation of this very distinction in their velocities.

You say: How are the points of the discrete space selected from those of the continuum?

There is no selection of “points of discrete space” because there is no discrete space, in our theory. Space remains entirely continuous. Instead, the distinction lies in the dynamics: mass exhibits discrete motion, while light exhibits continuous motion. Our model does not rely on a spatial lattice with a fixed minimum length scale. As mass moves faster, it will approach (but not actually reach) an infinite number of jumps per time, with jump length approaching zero. This shows that a discrete space with a minimum length scale is not only absent from our framework, but is physically incompatible even with the discrete motion of mass through space.
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You say: How do you formulate a theory in which light and massive particles interact if the space they feel is so different?

There is only one continuous space and thus “the space they feel” is one and the same space. In classical physics, mass and light “interact” by the Galilean velocity addition formulas. What reason is there that both motions have to be continuous for them to interact through some form of velocity addition formulas? We introduce new formulas that add velocities by computing jumps per time and jump lengths.

The old common-sense physics question was: how can the velocity of light be constant for moving observers and how can the motion of these observers not matter? Our velocity addition equations have the key success that they – through the mathematics of actual infinity – answer this. Jump lengths addition, which modifies continuous-motion Galilean velocity addition, allows the mathematics of actual infinity to play a key role.

A professor inquired in August 2026 about “jumps” being used when referring to continuous motion and about the use of actual infinity.
My reply:

You say: If the motion of light is continuous, then is it really true to say that it makes an actual infinity of discrete jumps as opposed to saying it makes no jumps? To me, continuity means zero jumps.

It is said that mathematics is the language of physics.
In the language of mathematics, where the concepts of actual infinity and zero are included, continuous motion can be mapped to being an infinite number of jumps per time of zero length.
Without applying mathematics, or without including the concepts of actual infinity and zero in the mathematics, one could argue that considering continuous motion to have no jumps or zero jumps would be the final physics of such motion. Having the needed mathematics available allows one to examine what continuous motion can mathematically and physically mean in terms of jumps per time and jump length.
Discrete motion, which is a property of mass, has finite number of jumps per time and the jump lengths get smaller as jumps per time increase (as given by the formula in Section 2 of paper) such that the limit of their product approaches c. Thus, as mass moves faster, it will approach (but not actually reach) an infinite number of jumps per time, with jump length approaching zero. On the other hand, continuous motion (which is a property of the massless) has an actual infinite number of jumps per time and these jumps are of zero length, with the product of these two being c. In this mathematical framework of motion, no jumps or zero jumps would mean number of jumps per time being zero (which could only happen for mass).
(Of course, all motion is spoken of in terms of inertial frames.)

You say: With regard to the concept of actual infinity itself, I believe that Cantor in constructing his actual infinities had to make some assumptions … So I’m skeptical about applying the concept of actual infinity to the physical world.

Actual infinity has been argued in proposals regarding properties of the physical world since ancient times, long before Cantor and his actual infinities.
(We mention ancient references to actual infinity in the physical world in Sections 3 and 4 of paper.)
Is there any argument that mathematically rules out the proposal, being made since ancient times, that the universe is actually infinite? Is Cantor’s mathematics needed for one to agree with this ancient proposal about the physical world?
Is there any argument that mathematically rules out the infinite divisibility of space, which has been a common view since ancient times? Is Cantor’s mathematics needed to hold this ancient and common view?

You say: With regard to the concept of actual infinity … difficult to apply to the physical world itself — the Axiom of Choice, for instance.

Our use of actual infinity does not depend on such additional matters regarding actual infinity and the physical world. To suggest that we cannot use actual infinity in the manner that we have, one must provide specific mathematical reasons. That some other matters regarding actual infinity can be further debated does not mean that such additional discussions would affect all other proposals and mathematical use of actual infinity in the physical world.

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