Can I call it a Principal? More vigorous than a theory, a principle is more of a foundational concept, yet not quite a Law. I have, perhaps incorrectly, referred to this theory as the "Horizon Principal".
In Quantum Mechanics, I think of how we describe a particle with no mass. An "effective mass" calculated as an equivalent to its energy. The uniqueness of 1 is where this fallacy of the mind becomes apparent. I imagine the uniqueness of 1 as a panspermia of neurotoxin, contaminating the foundations of Physics and Mathematics.
If there exists an experiment where the dualist properties of uniqueness are falsifiable, and the experiment doesn't falsify, we then have a viable theory. I need to enlist the help if Mathematics to set up this theory. Does it stand or fall? The implications cascade beyond the number 1 to all natural numbers.
On the subject of Quantum Physics. One doesn't signify uniqueness, only a trend. 2 is the first Quantum integral in the QFM Horizon Principle.
"I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me." - Isaac Newton
From Newton's perspective, he can see an ocean of possibilities yet no shoulder standing will change his horizon for the pepples he ponders are also a horizon. This is relativity.
Remember it's just about the mathematics and nothing metaphysical. Well, mathematics is a form of metaphysics. We can all agree that the number 1 is one. Right? Well, let's re-consider the mathematical representation of integer numero uno.
At the end of all this you say, "No, it's not a viable theory". I say, test the math out, what does this tell us by re-imagining the number and concept of the number one. What does it tell us when we apply the math to the equations of Quantum Physics? Again, I am not a mathematician yet I think that mathematically the clarification for this theory is within calculus, specifically limits as the quantum locality approaches uniqueness (but can never achieve it). I am hopeful, that this will further explain continuity and deterministic outcomes for fundamental particles.
This idea, or synapse, presupposes a theory that unifies the wave/particle duality.
The Theory states, "God does not play dice" as it disproves the Parallel Postulate, unifies the wave/particle duality from the dual slit experiment.
In Euclidean geometry, a point is a primitive notion from Euclidean geometry. Equating our numeral representation of "one" to this element of geometry limits humanity's understanding of the universe. Until Max Plank ushered in a new age of physics called Quantum Mechanics the need for axiomatic duality that exists within the concept of one. What if non-existence of uniqueness (no ones) in nature is a fundamental law. Proving something doesn't exist is a tough thing to do. I think we have wasted more than five millennia on that task already. We could call it the inverse law of nature and show it as a 1 or 1/1, infinitely reducing to infinity without entropy or randomness. Complete and utter oblivion.
Why was this "overlooked"? My best guess is that it predates human consciousness. This is worthy of some further thinking not included within the scope of this book.
Let's for a minute talk about elements and space. Earlier I said that this conversation is about math, only math and nothing but math. You must avert your wandering mind. Stay away from the temptations of physics. For now, don't think of elements and space in the terms of physics. Purely mathematics (for now). The point, as a fundamental element, has the property of uniqueness. I disagree. The point has a duality. Euclid defined a point as axioms of uniqueness. Changing the axiom, or properties of one, does not change the truth within Euclidean mathematics. This we did already once with the field theories of Einstein and the works of Newton. Instead of applying distances between two points in the geodesic and extrapolating the space-time implications, the origin and properties of the point connects distances and speaks to deterministic outcomes.
The current estimate for the size of a proton was shown by Rutherford in his scattering experiment in 1911. Rutherford showed that the nucleus of the atom is very small and dense compared to the rest of the atom. Its diameter was later found to be on the order of 10−10 m. If you compare this to the size of a Planck length of 1.6e-26 nanometers. To give some perspective, the width of a human hair is 17 to 181 µm and there are 1,000 µms in a nm (nanometer). The difference is one millionth of a meter (µm) and one billionth of a meter (nm). The scale is staggering.
Back to Euclid and his geometry. Regarding the properties of the line, for now, I will concede with the no thickness, “breadthless” postulate as an unprovable property of Euclidean geometry. Leaving the line alone, my focus is the point. I am proposing a theory that gives dimension to the point but not the line, unless the introduction of line breadth resolves some incongruence in the theory, I will reserve this as a possibility. In Quantum Mechanics, the line is a path, a trajectory whereas the point is determined, “unique” position in space, a predetermined and predictable location. I will build this theory from Euclidean geometry and then on to the four dimensions of space-time while reminding everyone that I am not a physicist nor a mathematician. I welcome your assistance with the theory, to make it more robust and rigorous. Ultimately the theory needs to be “built up” to represent the concepts explained and then, of course, made falsifiable in an experiment. According to Popper, anything else would be pseudo-science. Consider two points and the line segment that connects the two. The points have a diameter equal to some distance greater than the breadth of the line. For conversational purposes, let’s say the diameter of the point is the diameter of the Planck length. This theory is only applicable to fundamental particles, like the ones found in Quantum Mechanics. The reason I am arbitrarily saying the diameter of the point is the Planck length is because the Planck length is already agreed as the smallest unit of length found in the Universe. I think some theories have gone even smaller than the Planck length but I will use the ℓ P as our starting “point” or diameter. Perhaps the diameter is twice the length of the Planck length or a changing variable? The important constituent of this theory is that the diameter of the point is definable. Now, how does the “breadthless” line fall within the point?
We know from the famous double slit experiment that individual electrons appear to interfere with itself and mathematically goes through both slits while going through neither, goes through a single slit, while at the same time going through just the other?? These possibilities are all mathematically possible! At the same time!! Which is a mashup of contradiction. Or so our current understanding and interpretation goes. If, as in my theory, the two-dimensional point has a duality, a diameter with a spectrum of possible “end-point” locations, this would provide a range of possible destinations. Obviously prior knowledge of which slit the electron goes through will yield a deterministic outcome. Having a deeper understanding of fundamental properties a point will add greater understanding to this mystery. At this point I am not certain on how to proceed, scientifically or mathematically. Mathematicians and Physicists please write, comment and engage. I do know that from Euclid the two dimensional geometry needs to be extrapolated into the four dimensions of spacetime. I imagine a spherical boundary condition with “flat spots” equal to the Planck length (variable) or it’s variable increment.
What are the alternatives to Euclid's geometry? Non-Euclidean geometry introduces curvature whereas Projective and Affine Geometry differentiates from Euclidean geometry through projections, distance and angles.
Is the geometry symmetrical? Yes I would imagine. Again only the mathematics and physics will offer further understanding.
On a coordinate, two-dimensional XY graph you increase from zero towards 1 at the same rate as you decrease from 2 towards 1.