The Generalized No-Chaos Theorem

The Generalized No-Chaos TheoremBounded Complexity Growth at Any Fixed Coarse-GrainingWithin the Quantum Gravity Consistency Framework (QGCF) - any finite-dimensional, closed, unitarily evolving system satisfying (QGCF1)-(QGCF4) - three logically distinct results are established. First, ontic chaos is impossible in three senses: spectral quasiperiodicity, Poincaré recurrence, and exact vanishing of the asymptotic quantum Lyapunov exponent λL (§5). Second, dynamical (Lyapunov/0-1-test) chaos at any FIXED coarse-graining scale ℓ is also impossible, unconditionally - a direct corollary of the same almost-periodicity that gives λL=0 (§7); the one established quantum-chaos diagnostic that survives this obstruction is level (spectral) statistics, checked preliminarily against a non-integrable/integrable benchmark pair (§7.3). Third, and logically independent of the first two: no observer realized as a proper subsystem can simultaneously reconstruct the complete global state from its own accessible data and evolve under an autonomous unitary law, once genuinely (entangling-ly) coupled to the rest of the system (§8) - a consequence of purification non-uniqueness (Hughston-Jozsa-Wootters) and the non-unitary character of open marginal dynamics, requiring no assumption beyond a generic (non-fine-tuned) boundary coupling. The two failures named in that theorem arise from the same physical mechanism - entanglement across the observer's own boundary - but are shown not to be strictly the same fact, only generically co-occurring. Thermalization of coarse blocks (Calabrese-Cardy growth to a Page-value plateau) is real, established physics, distinct from all of the above, and is presented (§9) together with the timescale τ*(ℓ) it requires, and its role as the only available justification for treating the system's effective Hamiltonian Ĥeff as possessing a meaningful temperature Tc, given that the axioms admit no literal external bath. A note on generality. QGCF does not require a literal discrete graph. (QGCF3) is stated as a disjunction: a discrete, graph-based realization (QGCF3a), giving the sharper results of §8.2 and §8.6.2 exactly; or a continuum realization with only a locality/metric structure (QGCF3b), giving the standard, approximately-causal Lieb-Robinson bound instead. Everything else in this document §5-§7, §8.3-§8.6.1, §9 - depends only on finite-dimensionality and unitarity (QGCF1-QGCF2), or on (QGCF4)'s boundedness/genericity, and holds identically under either branch of (QGCF3).