PAPER–Σ: Symplectic Foundations of Action, Global Structure, and Observable Readout

Abstract

This paper develops a pre-metric symplectic foundation for action, global structure, and observable readout in the Fracture–Berry–Tension framework. Its central claim is deliberately narrower than a direct derivation of matter, gravity, dark sectors, black holes, or Hawking radiation. A symplectic form is available before a Lorentzian metric or a physical clock: it supplies action periods, capacities, Hamiltonian transport, and prequantisation data without presupposing an already emergent spacetime. Four results organise the paper. First, under an explicit discreteness hypothesis, the admissible period group of the symplectic form has a minimal positive generator Amin. Prequantisation relates the period lattice to a chosen quantum of action, while symplectic capacity and Gromov non-squeezing protect nonzero action scales against arbitrary symplectic compression. They do not independently derive the numerical value of Planck’s constant, an operator uncertainty relation, metric completeness, or the absence of curvature singularities. Second, Darboux local standardness is separated from global geometric triviality. FBT16A is imported only through the typed obstruction package Oup = (ρ, cρ, ˆ︁cB, HolArel ). Nontrivial obstruction data rule out a globally uniform passive phase atlas. This is a global bundle-and-transport statement; it does not force pointwise curvature variation, matter localisation, sector number, entropy production, or projection failure. Third, for a realised compactification carrier equipped with a descended line and a specified finite-dimensional readout map, we introduce V(Σb) =(︃∫︂ΣbΩ, c1(Lb), rankRobsb)︃.This action–topology–readout signature is invariant under admissible symplectic, bundle, and readout equivalences. Fourth, a smooth family of readout maps admits determinantal rank strata. Rank-drop loci are closed, and in the analytic category they are analytic determinantal sets. This gives a rigorous language for full, partial, and failed readout without identifying a stratum automatically with visible matter, dark matter, an event horizon, or a black-hole interior. The resulting foundational order is symplectic action −→ global topology and transport⇓typed observable readout −→ conditional spacetime interpretation