Bounded Curvature and Singularity Resolution via Combinatorial Synergy Saturation in Emergent Gravity

Context: This manuscript represents the cosmological and astrophysical application track of the DAGI research program at Whytics, addressing Penrose-Hawking singularities through the lens of graph-theoretic information capacity limits. Abstract: Classical General Relativity (GR) permits curvature singularities because the continuum field equations allow unbounded focusing and arbitrarily large local stress–energy. This paper reframes the issue in the Directed Acyclic Graph Interpretation (DAGI) program, where geometry is not fundamental: it is an emergent continuum description of a discrete causal graph. In the DAGI emergent-gravity dictionary developed in the master paper (Paper I), the operational curvature source is not an arbitrarily localizable fluid density on a manifold, but a bounded combinatorial resource: the density of irreducible higher-order informational constraints (Möbius synergies) supported by finite graph degrees of freedom. We define a controllable curvature-source functional (band-limited cumulative positive synergy) and prove a capacity theorem: for finite local state dimension and bounded causal fan-in, the synergy density—and therefore the effective geometric stress–energy—is strictly bounded. This yields an admissible class of saturation laws that recover GR at low densities while enforcing limiting curvature at high densities. The de Sitter-like equation of state in the saturated core is derived (not assumed): once the geometric response obeys an admissible saturation law ₄₅₅ = F (ₘ), the Bianchi identity forces the effective pressure to approach p₄₅₅ -₄₅₅ c² as F' 0. We also show how the information-to-energy conversion scale is not a tunable dial: it is jointly constrained by the weak-field (Poisson) limit and by horizon thermodynamics in the Jacobson/Bekenstein-Hawking sense. As primary phenomenological applications of the framework, we show (i) a non-singular de Sitter core is forced in spherical collapse with curvature scale fixed by the capacity bound, and regular metrics (e. g. Hayward) are merely convenient interpolants, and (ii) in cosmology the same capacity bound replaces the Big Bang divergence by an automatic high-density de Sitter phase ("inflation for free") without postulating an inflaton. We explicitly demarcate scope: the paper derives the background inflationary dynamics, while CMB perturbations and reheating require a separate fluctuation theory for the synergy field. Finally, the capacity bound implies a falsifiable evaporative trajectory for small black holes: as the outer horizon approaches the capacity-fixed core, the surface gravity tends to zero, the Hawking temperature turns over and cools to T 0, and stable "synergy remnants" remain as cold dark-matter candidates. Key Highlights: Rigorous Capacity Theorem: Proves mathematically that infinite spacetime curvature is impossible in a discrete causal substrate due to strict combinatorial bounds on causal fan-in and local state dimensions. Derived de Sitter Cores: Demonstrates that a vacuum-like equation of state (w -1) is a forced, derived consequence of the Bianchi identity under geometric saturation, reversing geodesic focusing naturally. Inflation Without an Inflaton: Provides a natural, background-level mechanism for early-universe inflation driven entirely by the graph's informational capacity limits, featuring an automatic graceful exit. Falsifiable Black Hole Thermodynamics: Predicts a turnover in Hawking evaporation that forbids terminal gamma-ray flashes, leaving stable, collisionless "synergy remnants" as highly viable cold dark matter candidates.