Frame Duality on the Number Line
Year: 2026
Type: preprint
Abstract: EARLY/ROUGH DRAFTS BELOW. SEE VERSION 9 REORGANIZATION : Inspired by Dimensional Mechanics, Riemann I constructs a new observable D_C measuring how well the finite Euler product satisfies the functional equation. It equals 1 exactly on the critical line, prime by prime, and deviates off it. The correction identity R_C D_C = R_C(1−s) shows any off-line zero requires R_C to absorb a growing discrepancy. A non-circular proto-zero filtration and the variance asymmetry theorem establish the critical line as the unique locus where zeros can form. Independently, a curvature reformulation proves RH equivalent to positivity of u = F_σσ/F throughout the strip, with boundary positivity established via the Hadamard product. RH is reduced to noncompensation of R_C. The Realizability Principle, stated as an axiom, closes the gap by denying R_C independent tail freedom. Riemann II, also inspired by DM, develops the Realizability Principle (axiom used in Riemann I) as standalone foundations. Iterated prime-counting enters {0,1,2,3}; the parity recursion stabilizes at x* = (3+√13)/2 with floor 3. The same point is forced by an S³ shell equilibrium with Spin(3) texture and locked topological core, selecting d = 3 uniquely through parity survival and π₃(S³) ≅ ℤ. The principle classifies mathematical statements by frame, absorbs Gödel incompleteness as frame-dependent, and identifies noncompensation as the structural consequence of closure.Yang-Mills, also inspired by DM, applies the identical reduction architecture to the mass gap. The theory is built on S³_R where compactness gives analytic control. The Casimir floor δ_E ≥ c_E g² C_{2,min}(G) > 0 is algebraic and cannot vanish. It is the Yang-Mills analog of the zero-free finite Euler product. The magnetic correction δ_B is the only channel through which the gap can die, which is the analog of R_C. The entire problem reduces to one inequality: |δ_B| ≤ (1−ε) δ_E. A truncated SU(2) lattice diagnostic confirms |δ_B/δ_E| < 0.16 at every tested finite system size in the strong-coupling regime, with the ratio decelerating as the system grows. The same Realizability Principle idea closes the Yang-Mills gap by the same mechanism that closes the Riemann Hypothesis: the infinite-volume correction channel is denied independent existence by arithmetic closure at finite radius.
Author: Sulea, Eliza
Institution: Widener University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7206174331