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The most difficult math problem of all time - solved in a perfunctory, concise way, an accepted way, consistently; and without error.
Exposition And Sub-Proofs of Landau's Pole Prime Number Theory; Group Coordinates Magnitude Theory
The Rho Function - Measure of density of prime number (and then general exponential) distribution; in the reals.
A prime number is a foundational unit of arithmetic distribution (multiplication), and thus comprises the building-block root of Categorical Set Theory. Prime numbers are arbitrarily and relatively dipolar. Upon set-theoretic recombination (or substitution), this proves Goldbach's Conjecture for any set of combinatorially-rearranged odd addends, and the problems may be rephrased after such precedent:
1) Goldbach's Theorem: Prime numbers are subject to arbitrary substitution in the real integers, meeting the condition of relative dipolarity as progressive coordinates. Restatement of: Prime Non-Predictability 2) The Twin Prime Theorem: The arbitrariness of prime substitution has no limit, thus there is always at most a median gap between their sequenced complementarity bridge. Restatement of: Infinite Number of Relative Primes 3) Legendre's Conjecture: There is no bound-grouping distribution which constricts the substitution of primes - in the place of general, relative dipolarity, or logarithmic-exponential asymptotic tangency of continuation evolution. Restatement of: The Definition Proper of "Arbitrariness" 4) The Near-Square Primes Theorem: A corollary of (2) and (3), upon analytic statistical distribution. Restatement of: Arbitrary Distributive Dimensions of Freedom (the Proper Definition of the Prime Number [as the base principle of the coordinate system])
Rephrased collectively, these four former "problems" comprise the basis of Modal Coordinate Theory, a sub-branch of classical mechanics which is comprised of what may be termed "Aljaddou's System" (for velocity analysis):
1. The Stochastic Distribution Theorem: Randomization exists in nature, and is a co-condition of predictable phenomena. [Validation of Absolute Newtonian Space] 2. The Infinite Coordinate System Theorem: The origin of a coordinate system is arbitrary as it is infinitely extendable, applicable, and replaceable. [Validation of Absolute Newtonian Time] 3. The Substitute-System Theorem: Algebraic bijective functions (of multiple variables) form a consistent physical representation. [Validation of Newton's Three Laws of Motion] 4. The Static Origin Theorem: A coordinate system is always ultimately objective. [Validation of Newton's Light Corpuscle Theory]
Landau's Problems - Aljaddou's Solutions (1).pdf