L(6+7) as an Entropic Hamilton–Jacobi Wave Map: The Quantum Wave Function as a Projection of Extremal Histories in an Extended State Space

This paper formulates an entropic Hamilton–Jacobi wave map within the L(6+7) framework, in which the quantum wave function is not treated as a primary isolated object but as the observable projection of an ensemble of extremal histories in an extended state space. The starting point is the Lohmiller–Slotine/MIT bridge: a quantum wave function can be reconstructed from a multi-valued classical action and the density of the associated classical flow. The proposed extension embeds this bridge into the canonical L(6+7) corpus: the unified variational root, the internal seven-sector, entropic time, stable-history selection, and meso–micro reductions. The central result is a wave-map formula in which the phase of a branch is determined by the full action, the amplitude is determined by a density-entropic weight, and the observable wave arises by projection over the internal seven-sector. In the frozen-sector limit the construction returns to the standard Hamilton–Jacobi action-density map; when the internal sector is active, phase and contrast corrections arise and can be tested in a minimal double-slit model.