Λ-EG — Emergent Gravitational Coherence: Theory, Ontology, Cosmology (T.O.C.)

Abstract

Λ-EG THEOREM-MU: The manuscript proposes an exact geometric foundation for the standard MOND interpolation function, $\mu(x) = \frac{x}{\sqrt{1+x^2}}$, proving that it corresponds to the cumulative probability of the projection ratio $\cot\theta$ under a three-dimensional isotropic distribution. Lambda-EGv10: This release updates the de Sitter derivation of the critical acceleration scale and clarifies the separation between the primary scalar construction and the Dirac–Lichnerowicz consistency route. Axioms and Main Theorem: * Axiom 1 (Isotropy): Microscopic channel directions $\hat{n}$ are symmetric under inversion ($\hat{n} \to -\hat{n}$) and distributed invariantly under $SO(3)$ with uniform measure $dP = \frac{1}{2\pi} \sin\theta \, d\theta \, d\phi$ over the hemisphere. * Axiom 2 (Tilt Activation): A channel activates when misalignment with the local gravitational field $\mathbf{g}$ exceeds a critical threshold: $|\mathbf{g} \times \hat{n}| > a_0 |\hat{n} \cdot \hat{g}|$, or equivalently, $\theta > \theta_0(x) = \arctan(1/x)$ where $x = g/a_0$. Exact Result: The activated solid angle fraction integrates directly to $\mu(x) = \cos(\arctan(1/x)) = \frac{x}{\sqrt{1+x^2}}$. Uniqueness Properties and Structure * Bijection: Any monotonic function is the solid angle fraction of some gate; the rigidity of the theorem lies in the fact that the standard function isolates the threshold variable $\cot\theta$ (longitudinal over transverse component). * Gate Comparison: Among simple gates generated by projections of $\mathbf{g}$ onto $\hat{n}$ (Magnitude, Longitudinal, Perpendicular, and Tilt), only the Tilt gate yields an analytic, strictly increasing $\mu$ function with correct MOND limits at both extremes. * Empirical Family: In the family $\mu_\alpha(x) = \frac{x}{(1+x^\alpha)^{1/\alpha}}$, the threshold parameter matches the projection ratio if and only if $\alpha = 2$. Dynamic Corollaries and BTFR Attractor * Flow Function: The derivative of the standard $\mu$ function yields the exact flow function $\beta(\Phi) = -\frac{\Phi(2+\Phi)(1+\Phi)}{2+\Phi(2+\Phi)}$. * Stability: It guarantees a linear attractor with $\beta'(0) = -1$ and curvature $\kappa = 1/2$, converting the Baryonic Tully-Fisher Relation (BTFR) into a stable dynamical attractor. * Numerical Verification: A Monte Carlo analysis with $10^7$ gates confirmed the activation rate and the shape of $\beta$ without parameter tuning. Scope and Postulate * Postulate Independence: The geometric theorem is independent of physics. To connect geometry with dynamics, the Response Postulate is formally established: the gravitational sector response is directly proportional to the geometric fraction of active channels.