Index Theory, Modular Hamiltonians, and de Sitter Entropy: The Topological Origin of the Bekenstein-Hawking Coefficient (ქვესათაური: Algebraic Derivation of the Factor of Two via Modular Splitting)

Published March 9, 2026 | Version v1

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Index Fixation (Proved): We prove that the projection \tilde{P}_{ker} onto the kernel of the twisted Dirac operator D_{\chi} on the bifurcation sphere S^{2m-2} satisfies the identity \tilde{\tau}_{can}(\tilde{P}_{ker}) = \chi(S^{2m-2}) = 2. As an element of the Type II_{\infty} crossed product, this topological invariant fixes the dimensionality of the modular kernel independently of the de Sitter radius or Newton’s constant.
Modular Splitting (V \oplus JV): We provide a rigorous spectral decomposition of the full modular Hamiltonian K. We show that the Bekenstein-Hawking entropy corresponds to the sum of two isomorphic sectors related by the modular conjugation J.
Derivation of the BH Coefficient: By identifying the single-sector contribution with the local Noether charge density (A/8G_N), we show that the global algebraic trace—which accounts for both V and JV sectors—identically recovers the A/4G_N coefficient.
Interaction Stability: We prove that the index \text{Index}(D_{\chi}) = 2 is stable under Yukawa-type interactions, establishing that the topological factor is a robust invariant of the interacting theory rather than an artifact of free-field approximations.

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