Modular Dynamics, Sigma Term, and Nuclear Magic Numbers from the Rendering Algebra R12

Abstract

We construct the modular Hamiltonian H = C2(su(3)) + C2(su(4)) on the rendering algebra R12 = M3(C) ⊗ M4(C) and derive its spectrum {0, 3, 4, 7}. Color confinement projects physical states onto M4 (dim = 16), placing the nucleon in the (1, 15) sector at energy Econf = 4. We prove the color blocking theorem: Tr(λa) = 0 forbids all finite-order perturbative transitions between
(1, 15) and (8, 15), establishing the chiral effect on the nucleon as purely non-perturbative. Applying
the Cauchy duality theorem (Paper XXXIX) and Casimir-dimension duality (Paper XL), we derive σπN /mN = exp(−C2(adj, SU(3))) = e −Nc = e −3 = 0.0498 (1.5%). This A-tier result closes the derivation chain to nuclear physics with zero free parameters: σπN → c1, c3, c4 (NLO ChEFT, 7%); aV = σπN ΩΛ/Tw = 15.82 MeV (volume energy, 0.9%); the shortrange repulsion is identified with the (¯3, 6) cross-sector projection; the Nilsson parameters κ = Ωb = Z/Tw (1.8%) and µ = 1/Tw = 1/2 (exact, Fermi function) reproduce all seven magic numbers 2, 8, 20, 28, 50, 82, 126 and predict 184. We present 25 results including the sigma meson mass mσ = 2mN /(3√φ) and the nuclear effective mass m∗/mN = 1 − 8Ωb.