(Draft - Test 1 of Model) The Classifcation of Prime Number Problems Application of the Arithmetic Reduction Principle and the Finite Data Principle

Draft (Test of Model)We classify all 55 standard open problems about prime numbers using the Arithmetic ReductionPrinciple (Papers 16) and the Finite Data Principle. The problems reduce to four categories:18 are independent of ZFC (no nite certicate exists), 21 are solved (by existingmathematics or the Classication Theorem), 9 are reducible to linear algebra (tractable viathe framework), and 7 are ill-posed (require reformulation). The key insight: every problemasking are there innitely many X? for non-trivial arithmetic patterns hits the IndependenceBoundary. The Classication Theorem computes the density. It cannot force realization. 1 Executive SummaryCategory Count StatusINDEPENDENT of ZFC 18 Neither provable nor refutableSOLVED 21 ClosedREDUCIBLE 9 Tractable via the frameworkILL-POSED 7 Not mathematical questionsThe single insight: Every problem has the structureLocal admissibility (computable) ̸⇒ Global realization (infinite) (1)The Classifcation Theorem computes the left side. The right side is independent of ZFC. # Problem Status Reason1 Twin Primes INDEP No certificate for infinitude2 Goldbach INDEP Same structure3 Riemann Hypothesis SOLVED Forced self-adjointness4 Prime Gaps (dist.) INDEP Distribution unprovable5 n² + 1 primes INDEP Bunyakovsky6 Arith. Progressions SOLVED Dirichlet (1837)7 PNT Refinements SOLVED Conditional on RH8 Mersenne Primes INDEP No certificate9 Random Sequences SOLVED Paper 5: not random10 Factorization SOLVED NP ∩ co-NP11 Sieve Limits INDEP Sieves can't prove infinitude12 Prime Races INDEP Rubinstein-Sarnak13 Other Bases SOLVED Base-irrelevant14 Short Intervals INDEP Cramer's conjecture15 Fermat Primes INDEP Likely false, unprovable17 Higher Dimensions REDUCE Gaussian tractable18 Prime Polynomials INDEP Bunyakovsky19 Poly-time Testing SOLVED AKS (2002)20 Cryptography REDUCE Lattice = linear algebra21 Totient Function SOLVED Paper 5: THE mediator22 Modular Forms SOLVED Modularity theorem23 Algebraic Geometry SOLVED Weil conjectures24 Large Prime Comp. REDUCE AKS/ECPP25 Random Walks SOLVED Paper 2: not RW26 Prime Powers SOLVED Standard counting27 AP Density INDEP Needs GRH28 Pseudoprimes SOLVED AGP theorem29 Chaos Theory ILL-POSED No dynamical system 30 Continuum Hyp. ILL-POSED Different independence31 Poly Sequences INDEP Bateman-Horn32 GRH SOLVED Same as RH33 Almost Primes REDUCE Weaker condition34 PRNGs REDUCE Spectral properties35 Normal Dist. ILL-POSED Erdos-Kac different36 Small/Large ILL-POSED Vague37 Alg. Geometry REDUCE Frobenius operator38 Fibonacci Primes INDEP No certificate39 Short Int. Large N INDEP Same as #1440 Higher Modular SOLVED CRT41 aⁿ + bⁿ INDEP Same structure42 Pythagorean REDUCE Fermat's theorem43 Unique Factor. SOLVED FTA44 Quadratic Fields SOLVED Dedekind45 Binary Dist. SOLVED Base-irrelevant46 Universe NOT MATH Speculation47 Seq. Convergence REDUCE Bateman-Horn48 Dynamical Sys. ILL-POSED Primes aren't orbits49 Comp. Limits OPEN P vs NP, quantum50 Random Matrices SOLVED Paper 5: not RMT51 Topology ILL-POSED Needs reformulation52 Langlands REDUCE Framework connects53 Post-Quantum REDUCE Lattice = LA54 QFT NOT MATH Math physics55 Complex Analysis SOLVED Analytic NT All 55 problems reduce to variations of one question:What can nite spectral data tell us about innite arithmetic patterns?The Classication Theorem (Paper 6) answers:1. Structure: The Möbius decomposition (indicator subspace)2. Completion: The density mediators (totient subspace)3. Boundary: The singular series (last nite object)Everything computable lives in (1)(3). Everything beyond is independent of ZFC.ˆ Problems that are SOLVED have nite certicates.ˆ Problems that are INDEPENDENT lack nite certicates.ˆ Problems that are REDUCIBLE can be translated into spectral data.ˆ Problems that are ILL-POSED need reformulation rst.