TCFQ Technical Notes Compendium R56-R79: Subleading Asymptotic Expansions, Symmetry Breaking, and Hamiltonian Reconnections in Symplectic Groupoid Models

The TCFQ (Teoría Constitutiva de Fase Cuántica) project, as detailed in Technical Notes R56 through R79, advances a geometric first-principles derivation of asymptotic expansions for the ground-state energy bₘᵢₙ⁽ⁿ⁾(α/n, β/n) in the large-n limit of SU(n) symplectic groupoid models, yielding bₘᵢₙ⁽ⁿ⁾ = 1 + A(α,β)/n + B(α,β)/n² + C(α,β)/n³ + O(1/n⁴), where coefficients A, B, and C are expressed in closed trigonometric forms distinguishing Regime I (c=α+βπ, non-abelian with symmetry breaking at order 1/n³). Key results include the gap observable Γ∞(α,β) = Aᴵ(c) - Aᴵᴵ(α,β), which measures the Riesz-2 energy deficit between separated and fused configurations on the unit circle. Its Haar-average k₀ = 1 - 2/ζ(2) ≈ -0.2159 arises from the exact geometric invariant Area(Dᴵ) = 3ζ(2), determining the universal coupling λ₀ ≈ 0.3265. The unified variational functional F = Aᴵ + λΓ∞ encapsulates phase transitions and spectral degeneracies through exact bilinear factorizations and kernel decompositions with eigenvalues {+1, 0, -1}, bridging matrix models with 2D quantum gravity through structural invariants independent of external parameters.