Cross-Layer Closure Equations: From Euler's Formula to the Event Horizon
The Physical Foundation of the SAE Framework

Writing Declaration: This paper was independently authored by Han Qin. All intellectual decisions, framework design, and editorial judgments were made by the author.

Abstract

ZFCρ Paper II (DOI: 10.5281/zenodo.18927658) established the mathematical structure of inter-layer transitions: two remainders + one act → closure. It provided a complete closure-equation table from L₀ through L₃. This paper takes over that prediction and extends the structure to the physical layers L₃→L₄ and L₄→L₅.

Core identifications: (1) In the static, spherically symmetric, vacuum sector of 4D general relativity, the preferred closure representative for L₃→L₄ can be identified as the Schwarzschild condition rc² − 2GM = 0, with ct (the spatial distance of causal unfolding) and G (spacetime curvature coupling) as two remainders. (2) In the equilibrium / microcanonical context, the preferred closure representative for L₄→L₅ can be identified as the Boltzmann relation S − k_B ln W = 0, with S (entropy) and ln W (logarithm of microstates) as two remainders; macroscopically closed but microscopically non-closed.

The complete table exhibits three meta-observations: a bookend structure (non-closure at both ends, closure in the middle three layers), decreasing closure tightness (from exact arithmetic points to conditional macroscopic closure), and the birth of physical dimensions (L₃→L₄ is the first closure equation carrying physical units).

This paper also reveals the unity of the two foundational SAE axioms: remainder must develop (P1) = a single remainder has no dual, no closure equation, and can only unfold; remainder conservation (P2) = the remainder finds its dual, E₁+E₂=0, closure. The two axioms are two phases of the same thing.

Claim strength layering. Inherited mathematical structure comes from ZFCρ Paper II. The L₃→L₄ and L₄→L₅ physical candidates are identificatory conjectures. P1/P2 unification and the three meta-observations are meta-theses. The extrapolation that no closure equation exists above L₅ belongs to the philosophical afterword and is not part of the physics claim proper.

Keywords: Self-as-an-End, remainder, closure equation, layer transition, event horizon, Boltzmann relation, dualization, black hole, speed of light, physical dimensions