The Least Surprising Theory: An Information-Theoretic Approach to the Unification of Physics

We propose \emph{least surprisal} as a variational principle for recovering physical laws by inference. Just as least action selects a history given a Lagrangian, least surprisal selects the Lagrangian given a \emph{displacement} from a \emph{reference}. Given \(\psi\), a covariant exterior derivative \(\mathbb D\) with curvature \(\mathbb D^2\) and Clifford contraction \(D\), we form displacement and reference jets: \(\mathfrak J_\epsilon\psi:=\bigl(\psi+i\epsilon D\psi,\ \sqrt\epsilon\,\mathbb D^2\bigr),\mathfrak J_r\psi:=\bigl(\psi, 0 \bigr), (1)\) We then calculate the surprisal between them, a complex-valued transition amplitude: \(\mathcal K( \mathfrak J_\epsilon \psi, \mathfrak J_r \psi):=-\ln\frac{\langle \mathfrak J_\epsilon \psi \mid \mathfrak J_r \psi \rangle_\mathfrak J} {\sqrt{\langle \mathfrak J_\epsilon \psi \mid \mathfrak J_\epsilon \psi \rangle _\mathfrak J \, \langle \mathfrak J_r \psi \mid \mathfrak J_r \psi \rangle _\mathfrak J}} . (2) \) Accumulation of surprisal captures the first order, \(\lim_{\epsilon\to0}\frac{1}{\epsilon} \mathcal K\bigl(\mathfrak J_\epsilon\psi,\mathfrak J_r\psi\bigr) =-i\int_X\dd^dx\,\sqrt{|g|} \left[\operatorname{Re}\braket{\psi}{D\psi}_x -\frac{1 {4}\operatorname{tr}\bigl(F_{\mu\nu}F^{\mu\nu}\bigr)\right], (3)\)which is the action. The least surprising displacement of \(\psi\) is therefore given by the Dirac equation along with the quadratic Yang--Mills term. Our main result introduces the \emph{intrinsic carrier}: an amplitude that uses only (and all of) the structures spacetime supplies of itself. It is parametrized over the pointwise metric bundle as \(\Psi=\Psi(x,g_x)\), and valued in\(\Cl^+(T_xX,g_x)\otimes\Sigma\bigl(\Sym^2(T_x^*X),G_{g_x}\bigr), (4)\) with \(G_{g_x}\) the DeWitt form on metric variations. In Lorentzian \(3+1\) spacetime, surprisal reduces the fibre symmetry from \(\Spin(6,4)\) to $U(3)\times U(2)$, whose Lie algebra is that of the Standard Model gauge group enlarged by one abelian factor \(U(1)_X\). On the gravity side, surprisal furnishes \(\Lambda\), \(R\), \(R^2\), and \(C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}\) to the action, yielding perturbatively renormalizable quadratic gravity. Its familiar negative-norm objection to probability measurements is addressed in Section~\ref{sec:measurement} by an intensity-positive measurement theory. Thus, \emph{the least surprising theory spacetime supplies of itself} accumulates to perturbatively renormalizable and intensity-measurable gravity coupled to the (\(U(1)_X\) enlarged) Standard Model---a unified matter-gravity theory.