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| Formato: | Recurso digital |
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Zenodo
2026
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| Acceso en línea: | https://doi.org/10.5281/zenodo.19447126 |
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- <p>We present a limit consistency analysis for the variational formulation based on bilocal unitary propagation kernels already established in the Action--Phase program. We start from the Euler--Lagrange equation of the bilocal kernel,<br>\[ (\nabla_x^2+\nabla_y^2)W(x,y) = -\kappa\,\rho_W(x,y), \]<br>obtained without ad hoc introductions and without circular derivation.<br>It is shown that, under controlled hypotheses of quasi-locality, diagonal dominance, and small modulation, the restriction of the bilocal equation to the $x=y$ sector leads to a local scalar theory parameterized by a field $\Theta(x)$. In particular, it is demonstrated that this projection satisfies an equation of the type <br>\[ \nabla^2 \Theta(x) = -\kappa_{\mathrm{eff}}\,\rho_\phi(x). \]<br>From the relationship between the local unitary kernel and the effective phase, it is further shown that, in the first-order regime, the phase satisfies an equation of the form <br>\[ \nabla^2 \Phi_{\mathrm{eff}}(x) = 2\kappa_{\mathrm{eff}}\,\rho_\phi(x), \] <br>which allows establishing a structural matching with the Poisson equation of the classical gravitational field.<br>The obtained result does not constitute a complete derivation of general relativity, nor does it establish a final covariant formulation. Its role is more restricted and precise: to demonstrate that the formalism of bilocal unitary filters contains, as a controlled limit, the structure of the gravitational field in the weak regime.</p>