Toward the Unification of Physics and Number Theory

This paper introduces the notion of simplex-integers and shows how, in contrast to digital numbers, they are the most powerful numerical symbols that implicitly express the information of an integer and its set theoretic substructure. A geometric analogue to the primality test is introduced: when [Formula: see text] is prime, it divides [Formula: see text] for all [Formula: see text]. The geometric form provokes a novel hypothesis about the distribution of prime-simplexes that, if solved, may lead to a proof of the Riemann hypothesis. Specifically, if a geometric algorithm predicting the number of prime simplexes within any bound [Formula: see text]-simplex or associated [Formula: see text] lattice is discovered, a deep understanding of the error factor of the prime number theorem would be realized — the error factor corresponding to the distribution of the non-trivial zeta zeros, which might be the mysterious link between physics and the Riemann hypothesis [D. Schumayer and D. A. W. Hutchinson, Colloquium: Physics of the Riemann hypothesis, Rev. Mod. Phys. 83 (2011) 307]. It suggests how quantum gravity and particle physicists might benefit from a simplex-integer-based quasicrystal code formalism. An argument is put forth that the unifying idea between number theory and physics is code theory, where reality is information theoretic and 3-simplex integers form physically realistic aperiodic dynamic patterns from which space, time and particles emerge from the evolution of the code syntax.

If we are successful, it will be the only microscopic first-principles theory of everything, providing analytical expressions for values such as the Planck constant -a first-principles theory of everything.
What clues does nature offer for what a first-principles theory of everything should look like?
Reality is geometric.
And both classic and quantum theory indicate reality is made of information.
Geometric symbolism may explain how a geometric reality can be made of information. Perhaps a mosaic-like language of 3D geometric symbols could express the meaning of a geometric reality like ours.

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Quasicrystalline codes are exactly such languages. And they are inherently non-local and described by noncommutative geometry.

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Information Causality Loops Non-determinism Consciousness Pixelation E8 Crystal Golden Ratio In a pending paper, we introduce the notion of simplex-integers and show how simplexes are the most powerful numeric symbol expressing the counting function and set theoretic substructure of an integer. We introduce a purely geometric primality test for simplex integers, which extends also to the A n lattice series: We introduce a purely geometric primality test for simplex integers, which extends also to the A n lattice series: A simplex-integer is prime if and only if its 0-simplexes divide evenly into each sum of its sub-simplexes. Prime simplexes have a certain divisional symmetry in their geometry, for lack of a better term.
For example, the 5 vertices of the simplex-integer associated with the prime number 5 divides evenly into its ten 1-simplexes, ten 2-simplexes and five 3-simplexes. For example, the prime number theorem incorrectly predicts 21.7 primes within the same bound of numbers.
And it is precisely this error value of 25 -21.7 = 3.3 (or the error in any bound) that correlates to the non-trivial zeta zeros. An interesting difference between simplex-integer number theory and digital number theory is algebraic versus transcendental numbers.

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Specifically, to force the additive series of digital numbers to converge, we must invert them in the zeta function and place a power on the denominator.

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Information Causality Loops Non-determinism Consciousness Pixelation E8 Crystal Golden Ratio When s = 2, we get the integers to converge relative to the transcendental expression: 1/(1 2 ) + 1/(2 2 ) + 1/(3 2 )… = π 2 /6 Information Information Causality Loops Non-determinism Consciousness Pixelation E8 Crystal Golden Ratio When s = 2, we get the integers to converge relative to the transcendental expression: 1/(1 2 ) + 1/(2 2 ) + 1/(3 2 )… = π 2 /6 By contrast, the circumradii of the simplex-integer additive series converges without the zeta function to the algebraic expression 1/√2. The philosophy of discreteness versus smoothness that is currently debated among quantum gravity theorists is related to the fundamental differences between "smooth" transcendental numbers like π and e versus algebraically "crisp" numbers like the golden ratio or √2. The golden ratio is the precise point where a black hole's modified specific heat changes from positive to negative. 4

= ϕ
The golden ratio is the precise point where a black hole's modified specific heat changes from positive to negative.
And it is part of the equation for the lower bound on black hole entropy.
The golden ratio is the precise point where a black hole's modified specific heat changes from positive to negative.
And it is part of the equation for the lower bound on black hole entropy.
The golden ratio even relates the loop quantum gravity parameter to black hole entropy. The metric unit is simply the distance from the Equator to a pole, the minimally distorted ¼ circumference of Earth.

Golden Ratio
And the dimensionless ratio of the Earth to Moon pole through pole diameters is .

Golden Ratio
So the French had good intuition about this even though they did not know about the golden ratio relationship.
In other words, we have an Earth-Moon "clock" with two spherical "gears" that have a size ratio of with a ratio to ¼ of the Earth circumference of about 1, which we use as the unit value for our scientific metric system. Consider that the rotation cycles of these two golden ratio based spherical gears might accordingly be based on their dimensionless ratios. Consider that the rotation cycles of these two golden ratio based spherical gears might accordingly be based on their dimensionless ratios.
The Earth day itself is a geometric system based on the rotation of Earth's mass, as acted upon by a golden ratio relationship with the Moon's mass. In other words, the 2 5 3 3 . based division of the golden ratio based Earth Moon "clock" may be no more man-made than the golden ratio related metric system based. The nine recent determinations of the Planck constant cover five separate methods. Where there is more than one recent determination for a given method, the value of h given here is a weighted mean of the results, as calculated by CODATA.
If the quasicrystalline spin network approach to quantum gravity is correct, why is the Planck length only about 99.9% of the golden ratio? The nine recent determinations of the Planck constant cover five separate methods. Where there is more than one recent determination for a given method, the value of h given here is a weighted mean of the results, as calculated by CODATA.
To answer that, one should understand that our quantum gravity framework will predict that systems like the Earth-Moon clock settle into spatial and time based patterns that approximate the golden ratio substructure of spacetime because these will be energetically favorable. The nine recent determinations of the Planck constant cover five separate methods. Where there is more than one recent determination for a given method, the value of h given here is a weighted mean of the results, as calculated by CODATA.
Our research program is focused on projecting the E8 crystal to 3D and 4D, which creates a golden ratio based binary code of pixelated space and causality loops requiring emergent consciousness.