On Constrained Functional Extremization as the Unified Pricing Grammar of Gravity and Electromagnetism: A Meta-Unification Framework Based on Variational Isomorphism

This paper demonstrates that the Einstein field equations and Maxwell's equations do not need to be unified as the same kind of force—they are already two independent instantiations of the same unified pricing algorithm. This pricing algorithm is defined as "Li" (the price at which a transaction is finally reached under given constraints): under given constraints, a system automatically converges to the unique state that extremizes a certain functional among all possible states. The core argument proceeds in four steps: (i) gravity and electromagnetism cannot be unified at the level of "force"—their symmetry groups, field variable types, and action forms exhibit fundamental differences; (ii) however, at the meta-level, they share the same pricing grammar: both of their field equations are Euler-Lagrange equations derived from action extremization, sharing the identical extremum criterion δS = 0; (iii) more importantly, they exhibit precise structural isomorphism at the gauge field theory level—both gravity and electromagnetism are curvature equations of a connection, sharing the pricing grammar of "gauge connection → curvature → extremum criterion → field equation," with the former acting on the spacetime diffeomorphism group and the latter on the U(1) gauge group—this difference represents distinct implementations of the same algorithm under different constraints, not a difference in the algorithm itself; (iv) this grammar is not a coincidence internal to physics—classical mechanics, quantum mechanics, cosmology, chemical equilibrium and non-equilibrium thermodynamics, market clearing and financial instability, channel capacity and complexity threshold, and entropy increase—the core pricing rules of these disciplines were independently derived throughout history, by researchers unaware of each other's work, using different mathematical tools, yet ultimately converging on the same constrained extremization skeleton in mathematical structure. Independently derived isomorphism is more convincing than intentionally inherited isomorphism. This paper precisely situates Li's unification framework within the history of ideas through systematic comparison with previous unification attempts, and honestly marks the boundary of the theory: in the arena of science, the theory remains at "all things conform to the description of Li"; under the stars of philosophy, the theory retains the ontological conviction that "all things are themselves the unfolding of Li"—but conviction is not a proven theorem. The highest is not conquest; the highest is honesty. Keywords: unified field theory; variational principle; constrained extremization; functional analysis; gauge field theory; interdisciplinary unification; incompleteness theorem; complexity threshold