Coupled Gerbes, Transgression, Explicit Theta Sectors, and N-ality on T 3

This paper consolidates in one place the complete bundle-gerbe and transgression calculations carried out for the three-torus M = T³ = S¹_t × T²_{x,y}. Starting from the degree-N Hermitian line bundle L_N → T², we construct explicitly the bundle gerbe 𝒢_N = (Y, ℒ_N, μ), Y = ℝ × T² → T³, equip it with connection and curving, compute its 3-curvature H_N = 2πiN dt ∧ dx ∧ dy, and prove that its Dixmier-Douady class is DD(𝒢_N) = N[dt] ⌣ [dx ∧ dy] ∈ H³(T³, ℤ). We then construct an explicit direct Hilbert bundle-gerbe module over 𝒢_N, compute the full unsimplified transgression-curvature formula on loop space, identify the restriction of the transgression to the winding-n loop sector with L_N^{⊗ n}, and prove the coupled classification of winding blocks by curvature and flat parameters (nN, [φ], [ψ]) ∈ ℤ × (ℝ/ℤ)². We derive the global convolution algebra 𝒜_N = C_c^∞(Y^[2], ℒ_N), prove its associativity, compute the universal-cover deck-group 3-cocycle and the local associator, and prove the twisted K-groups K⁰(S¹ × Σ_g, h) ≅ H²(S¹ × Σ_g, ℤ),K¹(S¹ × Σ_g, h) ≅ H¹(S¹ × Σ_g, ℤ) ⊕ ℤ_|N|, for h = Nu ⌣ η. The second half of the paper gives the explicit transgressed groupoid-gerbe over T², the explicit transgressed sector modules (E_{φ,ψ})_n = L_N^{⊗ n} ⊗ F_{φ,ψ}, their matrix-valued twisted convolution algebra, the magnetic-translation operators, the exact coupled sector equation (φ, ψ) ↦ (φ + nNb, ψ + nNa), and the full holomorphic derivation of the theta sectors. In particular, for d = nN > 0 we derive the Fourier recursion for holomorphic coefficients, prove directly that dim H⁰(E_τ, ℒ_{n,φ,ψ}^{hol}) = d, construct an explicit theta basis, compute the Heisenberg action on that basis, and prove the N-ality decomposition ΘSpace_{n,φ,ψ} = ⊕_{r=0}^{N-1} ΘSpace^{(r)}_{n,φ,ψ},dim ΘSpace^{(r)}_{n,φ,ψ} = n (n > 0). We conclude with detailed tables in the style of Freed–Hopkins–Lurie–Teleman bookkeeping, including the distinguished ℤ₃ × ℤ₂ six-sector slice at N = 3 and n = 1.