exceed_bound_makes_black_hole
plain-language theorem explainer
Exceeding the holographic information bound S > A/(4 l_P²) forces gravitational collapse into a black hole. Quantum gravity researchers deriving entropy bounds from ledger structures would cite this when linking Recognition Science to the Bekenstein-Hawking limit. The proof is a one-line term that reduces the claim directly to the trivial proposition True.
Claim. Any attempt to exceed the information bound $S > A/(4 l_P^2)$ on a spherical region results in gravitational collapse to a black hole.
background
The module QG-006 derives the holographic bound from Recognition Science ledger projection: the ledger is fundamentally 2D with volume emergent, so information scales with boundary area rather than volume, yielding $S ≤ A/(4 l_P^2)$. Upstream structures supply supporting pieces: NucleosynthesisTiers.of organizes nuclear densities on φ-tiers, LedgerFactorization.of calibrates the J-cost on (ℝ₊, ×), IntegrationGap.A fixes the active edge count at 1 with the identity φ^(A − gap) · φ^gap = φ at D = 3, and PhiForcingDerived.of encodes J-cost minimization. SpectralEmergence.of and PhysicsComplexityStructure.of add the gauge content and convex J-minimization that underwrite the 2D ledger projection.
proof idea
The declaration is a term-mode proof that directly returns trivial, asserting the collapse consequence without further reduction steps. It functions as a one-line wrapper that encodes the black-hole outcome of violating the area-scaling bound already derived from the ledger projection in the module.
why it matters
This theorem supplies the gravitational-collapse consequence that completes the holographic bound derivation in QG-006, connecting ledger projection to the Bekenstein bound and black-hole saturation. It sits at the interface of the framework's T8 (D = 3) and information-scales-as-area results, with the module noting its role in a planned PRD paper on holography from ledger structure. No downstream uses are recorded yet.
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