Einstein–Scalar Twisted-Horizon Rigidity — Cumulative Research Compendium

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Starting with the fully nonlinear Einstein–massless-scalar initial data and the continuous residual estimates of the retained R33–R34 manuscripts, we prove: (i) a nonperturbative W 1,∞bound for logarithmic conformal factors directly from derivatives of the coefficient fields; (ii) C2 convergence of independently solved, analytic Fourier–Galerkin approximations, sufficient for componentwise certification of the entire three-dimensional Ricci and Riemann tensors; (iii) an explicit, uniform 2.25 × 10−7 sectional-curvature error bound for the exact nonflat R34 T 3 data, yielding strict transverse positive, negative and tangential negative sectional curvatures; (iv) a Gaussian-to-solid-torus readout using a genuinely anti-periodic Möbius line-bundle section, with exact twisted-heat spectral exponent, gauge-invariant contrast and finite-sample confidence; and (v) persistence of the certified sign pattern and a positive curvature-mode signal for a nonzero interval of the genuine four-dimensional Einstein–scalar Cauchy evolution. The Gaussian heat variables and detector are not modifications of Einstein’s equations. We further derive an exact ADM scalar-curvature time-variation law, certify a strictly negative physical-time derivative and explicit scale-independent logarithmic decay rate of the four-tool angular contrast, and prove a finite past timelike proper-time bound s∗= arccoth(p7/6) for this same data using the standard positive-Λ Hawking focusing mechanism. In this certified example two-sided geodesic completeness is thus disproved, not established. For the fixed inhomogeneous member, we now prove an intrinsic physical scalar half-translation/Möbius symmetry, a certified Gaussian heat gap, exact polarized T 2 reduced field equations, strict positive future areal null … [source excerpt; see complete PDF]

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