Paper U: Magnetic Interpretation of the Gap Strip: Time-Reversal Symmetry and Kreĭn Geometry

Abstract. This paper develops a magnetic interpretation of the gap strip σ ∈ (1/2, 17/30) within the MNZI programme, unifying three existing programme threads — Krĕın geometry (Papers C and D), geostrophic rigidity (Paper O), and the Zero Density Conjecture (Paper S) — through the common framework of time-reversal symmetry. The J-involution (Jf)(z) = f(−1 / z̄) on L²(Γ\H) is interpreted as the mathematical expression of time-reversal symmetry. A system preserves time-reversal symmetry when its Hamiltonian commutes with J. A magnetic field breaks this symmetry by introducing a term that is odd under J. The effective magnetic field of the Eisenstein scattering system is defined by B(σ, t) = −∂σ log |φ(σ + it)|, where φ(s) = ξ(2s − 1) / ξ(2s) is the Eisenstein scattering coefficient. Main results: (i) Field vanishing on the critical line (Theorem 2.1) The effective magnetic field satisfies B(1/2, t) = 0 for all t. This is proved unconditionally from the functional equation φ(1 − s) · φ(s) = 1 (Lean-verified as MNZI.scatteringPhase_involutive). (ii) Field non-vanishing and the gap (Theorem 4.2) The condition B(σ, t) ≠ 0 for σ ∈ (1/2, 17/30) is implied by the existence of zeros in that strip. Under the Riemann Hypothesis, one has B = 0 throughout the region σ > 1/2, via the Taylor Column Theorem from Paper O. (iii) Flux quantisation (Theorem 3.1) The Buchanan–Seeley anomaly βJ(x) ∈ 2ℤ (Paper D [2], Lean-verified as MNZI.betaJ_mod2_vanishing) is interpreted as a magnetic flux quantisation law: the total effective magnetic flux through any closed contour in ℂ is quantised in units of 2π. This gives the topological meaning of the mod-2 constraint. (iv) RH as field extinction (Conjecture 7.1) The Riemann Hypothesis is equivalent to extinction of the effective magnetic field throughout the half-plane Re(s) > 1/2. Equivalently: all non-trivial zeros lie on the critical line if and only if the effective magnetic field of the Eisenstein scattering system vanishes off the critical line. The gap strip (1/2, 17/30) is therefore the region where B is currently known to be non-zero at most O(T^(30(1−σ)/13)) exceptional heights (from the Guth–Maynard bound [12]). The Zero Density Conjecture would reduce this to zero exceptional heights, extinguishing B throughout the strip. Closing the gap is therefore interpreted as proving field extinction — equivalently, proving that the dielectric of the Eisenstein capacitor never breaks down. (v) The critical line as insulator and capacitor (§5) The critical line simultaneously exhibits two apparently contradictory electrical properties: • it behaves as a topological insulator (field-free, topologically protected, with no leakage), and • it behaves as a perfect capacitor (spectral charge stored with dielectric efficiency π²/12, exceeding the Landauer bound). These are not independent phenomena. Both are forced by the single identity φ(1 − s) · φ(s) = 1. The gap strip is interpreted as the potential dielectric breakdown region, while the Zero Density Conjecture is the statement that dielectric breakdown cannot occur. (vi) The Perfect Duality Principle (§6) The insulator–capacitor duality is presented as the twelfth “perfect duality” identified by the MNZI programme. A unified classification table records all twelve dualities, including: • arithmetic duality• Krĕın duality• Wästlund duality• odd-symmetry duality• fluid duality• magnetic duality• electrical duality• information-theoretic duality• n-bonacci duality and related structures. In every case, the critical line is the unique locus at which two apparently contradictory properties are simultaneously maximised. The Riemann Hypothesis is interpreted as the statement that this perfect duality persists for all non-trivial zeros. All proved results in this paper are unconditional, or conditional only on explicitly named hypotheses. No proof of the Riemann Hypothesis is claimed.