Frame Duality on the Number Line

Inspired by Dimensional Mechanics, Riemann I constructs the observable D_C measuring how well the finite Euler product satisfies the functional equation. Its modulus 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 that any off-line zero requires R_C to absorb the finite-product discrepancy through paired leading-coefficient compensation. A non-circular proto-zero filtration and the variance asymmetry theorem establish the critical line as the unique fixed line with both exact finite-stage coupling and unbounded fluctuation budget. A curvature reformulation proves RH equivalent to positivity of u = F_σσ/F throughout the strip, with boundary positivity established via the Hadamard product. An exact operator realization constructs the bronze toral automorphism A₃ ∈ SL(2,ℤ) with contracting eigenvalue n* and dynamical zeta ζ_{A₃}(z) = (1−z)²/(1−11z+z²), whose complete zero set lies on the critical line. The arithmetic completion identifies x* as the fundamental unit of Q(√13), giving n* = exp[−√13 L(1,χ₁₃)] via the class-number formula and embedding the bronze residual in genuine prime arithmetic through ζ_{Q(√13)} = ζ·L(s,χ₁₃). A contour winding theorem shows the off-line zero count equals the cutoff-independent winding index of R_C, giving a topologically quantized obstruction. The completed spectral development unfolds the unit-period bronze Gaussian into Hecke–Bessel L-function towers, realizes the full Riemann zero divisor as a canonical quotient of the Q(√13) divisor, and derives an explicit prime-loop Dirichlet energy form of the Guinand–Weil formula. RH is reduced to noncompensation of R_C; the Realizability Principle closes that bridge by denying R_C an independent tail channel, while the contour and spectral constructions state the precise remaining analytic estimate.

Riemann II, also inspired by DM, develops the Realizability Principle 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 shell factorization P(x) = −2(x²+1)(x²−3x−1) is identified as the joint characteristic polynomial of the metallic transfer M₃ and the compact inversion J ∈ SU(2), with J reversing A₃ = M₃². A bronze-Frobenius system combines unit transport with rational-prime actions: split primes preserve the squared-unit transport while inert primes reverse it, and the squared commutator recovers exactly the missing odd prime-power repetitions, giving the ordinary prime-loop trace as a half-sum. A completed spectral construction distinguishes periodic and antiperiodic Gaussian sectors on the closed bronze circle, producing integer and half-integer Hecke–Bessel towers with exact completed reflection, an antiperiodic spectral gap π²/(4 log² α), and strictly positive periodic Toeplitz matrices. The principle classifies mathematical statements by frame, absorbs Gödel incompleteness as frame-dependent, identifies noncompensation as the structural consequence of closure, and states a conditional completed-Weil operator criterion: if a faithful realization satisfies the bronze contraction bound T*T ≼ bI, then Weil positivity and classical RH follow.

Yang–Mills, also inspired by DM, applies the same reduction architecture to the mass gap on S³_R, where compactness provides analytic control. The algebraic Casimir floor δ_E ≥ c_E g² C₂,min(G) > 0 guarantees a positive finite-stage contribution, analogous to the zero-free finite Euler product in Riemann, while the magnetic correction δ_B is isolated as the only potential cancellation channel, analogous to R_C. The open-frame problem thus reduces to the explicit noncompensation bound |δ_B| ≤ (1−ε)δ_E. A truncated SU(2) strong-coupling lattice diagnostic supports this bound, finding |δ_B/δ_E| < 0.16 at every tested finite system size, with growth decelerating as system size increases. The closed-frame route then invokes the same Realizability Principle architecture used in Riemann, with arithmetic closure removing the independent infinite-volume correction channel and the finite-R* noncompensation bound supported by the lattice diagnostic.