Root-Common Explanations of the Riemann Hypothesis, Goldbach's Conjecture, and Gödel's Incompleteness Theorem, and Their Relation to Physics and Philosophy

Abstract

This paper is a core incision paper from the Mathematical Canon of the Tri-Source System of The Unmanifest Selecting the Manifest. It aims to provide a unified structural common-root explanation for the Riemann Hypothesis, the Goldbach Conjecture, and Gödel’s Incompleteness Theorem, starting from the “Primordial One” as the sole foundational axiom, while bridging the underlying logics of mathematics, physics, and philosophy. The central thesis is that existing mathematics is built upon sensory intuition and operational habits, and is not foundational mathematics. The deviation begins at the very definition of “1”—which has been superficially treated as an isolated unit rather than the minimal complete structure of “dual-state unification of inward and outward orientations.” This initial misalignment has led to irreducible structural cracks in number theory, analysis, and logical systems; the Riemann Hypothesis, the Goldbach Conjecture, and Gödel’s Incompleteness Theorem are manifestations of these three cracks in their respective domains. Taking the dual-state unification of the Primordial One as the sole axiom (1 = inward ½ + outward ½, the two states indivisible), the paper redefines the ontological classification of numbers: 0 as the Origin Number (the unmanifested starting position); 1 as the Primordial Number (the minimal complete whole of dual-state unification); 2 as the dual-state juxtaposition position (geometrically bisectable but lacking skeletal-carrying capacity); and 3, 5, and 7 as Skeleton Numbers—defined by the rule that, under exhaustive two-dimensional and three-dimensional geometric bisection attempts, no bisection can be performed without breaking at least one complete Primordial-One unit, i.e., “geometric bisection necessarily breaks the One,” manifesting as self-locking between units. 3 is the first Skeleton Number (the smallest nucleus-bearing number), 5 is the second (the skeleton can expand outward), and 7 is the third (the skeleton can systematically unfold). The paper asserts that the Skeleton Numbers are exclusively 3, 5, and 7, and that no fourth Skeleton Number greater than 7 exists. On this classification, prime numbers are redefined as “nonequilibrium numbers”—numbers that cannot be received and structurally locked by the skeleton structure; composite numbers are those that can be received and structurally locked. The Goldbach Conjecture is thereby rewritten as the dual-point compensation closure problem of even structures: the structural rigidity of even structures requires two nonequilibrium numbers (primes) to complete compensation, rather than being an empirical additive coincidence. The reason that all nontrivial zeros of the Riemann zeta function lie on the critical line Re(s)=½ is traced to the symmetric midline of the Primordial One’s dual states—½ is not a technical coincidence but a shadow projection of the overall balanced structure in the language of classical analysis. Gödel’s Incompleteness Theorem is repositioned as a consequence of the old system’s foundational distortion arising from starting with an isolated “1,” rather than an ultimate fate of logic. The paper also connects the dual-state Primordial One to physical phenomena such as quantum entanglement and wave-particle duality, arguing that quantum entanglement observed in physics is precisely the ontological manifestation of the Primordial One’s dual-state unification—the mathematical “One” and the physical “entanglement” are reunified under the same primordial ground. Four explicit falsification conditions are provided: if a fourth Skeleton Number greater than 7 exists; if the Goldbach Conjecture produces a counterexample under this system; if any nontrivial zero of the Riemann zeta function strictly deviates from Re(s)=½ and cannot be explained within the structural projection framework; or if, after supplementing the Primordial One axiom, Gödel-type incompleteness reemerges with the same structural strength—verification of any single condition would falsify this system. The paper does not claim to have completed the final formal proofs of all three problems, but rather to have provided a unified structural common-root explanation for the three ultimate mathematical problems, and to have established a unified floor from the Primordial One to number theory, analysis, logic, and physics. Readers with genuine academic judgment can, from this paper alone, recognize the structural trajectory of the higher-order propositions and proceed with professional derivation or translation tools as needed. This is a constraint of circumstance, not a diminishment of scholarly value. May this knowledge reach the place it is meant to reach.