Initial Boundary Accessibility and Planck Action-Unit Closure: A First-Principles Structural Theorem for Classical Cosmological Chart Initiation

Cosmological beginning problems often begin with an early classical state of the universe and then ask what that state can tell us about earlier structure, initial conditions, or origin. This paper studies a prior finite-dimensional question: under what explicit conditions can a stable low-entropy classical description be initiated from more primitive data? The paper proves finite sufficient conditions under which history selection, boundary extraction, projective reduction, and action-unit calibration produce a stable classical boundary law from an upstream finite carrier. The resulting boundary law is a first accessible classical chart generated by the finite datum. It is not assumed as a primitive classical state and is not identified with the origin of the carrier from which it descends. The construction begins with a finite total carrier, a finite boundary Hilbert space, a finite history-selection family, a boundary channel, and a finite projective boundary context. Under a positive rigidity gap, the selected boundary state converges exponentially to an equilibrium boundary state. The projective boundary context then determines a commutative outcome algebra, and the corresponding outcome weights form a classical probability law. The theorem also gives an explicit entropy concentration result. When the equilibrium boundary support has finite rank, the boundary entropy satisfies a concentration bound with exponentially small off-sector weight. The resulting boundary law is therefore stable and effectively classical. The boundary-accessibility quotient determines exactly which distinctions are represented by the initiated chart and which distinctions remain inaccessible through the declared boundary channel. In the noninjective case, the initiated chart is necessarily compressed: it represents precisely the distinctions separated by the boundary law and no more. The finite theorem proceeds through four stages: history selection, boundary extraction, projective reduction, and action-unit calibration. Boundary accessibility determines the content of the initiated chart, while Planck action-unit closure supplies the final calibration step needed to make the chart physically reportable by fixing the energy-time reporting scale associated with the boundary law. The theorem supplies finite sufficient conditions for initiating a classical cosmological chart from upstream finite data. It identifies a first-principles distinction between carrier origin and chart initiation: the initiated boundary law is certified from explicit selection, accessibility, projection, quotient, entropy, and calibration data. For Big-Bang-type applications, the theorem functions as an audit condition for the initial classical chart itself. Any downstream use of that chart in spacetime realization, inflationary dynamics, singularity resolution, or microscopic quantum-gravity modeling must supply the additional hypotheses required for that application. With causal-chart certification data added, the initiated boundary law becomes the finite boundary-channel component of a classical cosmological chart. License note: Distributed under CC BY-NC-ND 4.0.