Dear William,
thank you for sharing your ideas. It's an intriguing observation that the emergence of a discrete number of modes in nonlinear dynamics resemble the discretenss of observables in quantum theory. (Although note that the reason why the Tacoma Narrows bridge collapsed was in fact not resonance, but a self-exiciting oscillation.) Also the universality that you mention is present in very much the same way in quantum field theory and statistical field theory, where the process of renormalization leads to a phenomenology at phase transitions which only depends on the field content and the symmetries of the theory, but not on its particular dynamics.
However, it is well-known that quantum theory cannot be an emergent large-scale effective description of an underlying classical system. This is prevented, for example, by the Kochen-Specker theorem. But let me just mention here a few remarks about your essay which should underline this.
(1) You mention that exponential decay can be modelled by an underlying chaotic system with exit states. Under the ergodic hypothesis, this is not surprising since both decays are governed by the differential equation
[math]\dot{N}(t)=-\lambda N(t)\,.[/math]
But now, quantum-mechanical systems can show other types of decay behavior. For example, an exponential decay modulated by an oscillation -- see for example figure 3(a) in this paper.
(2) Concerning Bell inequalities, you speculate that quantum non-locality may possibly be explained in terms of classical conditional. This is incorrect. Classical random correlations also violate the Bell inequalities. A http://en.wikipedia.org/wiki/Local_hidden_variable_theory#Local_hidden_variables_and_the_Bell_tests">local hidden variable theory](https://
http://en.wikipedia.org/wiki/Local_hidden_variable_theory#Local_hidden_variables_and_the_Bell_tests)
is defined in such a way that it does allow classical random correlations. They also satisfy all Bell inequalities due to linearity the latter: suppose you have a model given by some chaotic system which predicts that there is a probability of 3/4 for one set of outcomes, and a probability of 1/4 for another set of outcomes. For each set of outcomes, the CHSH quantity will be +2 or -2; let's say it is +2 in the first case and -2 in the second case. Then due to linearity of expectation values, the overall CHSH inequality for the total ensemble is
[math]\frac{3}{4}\cdot (+2) + \frac{1}{4}\cdot (-2) = +1[/math]
Hence, the CHSH inequality still holds. Of course the same reasoning works with any other numbers defining a statistical ensemble of deterministic hidden variable theories.
(3) Quantum correlations and quantum interference are of a very special kind. For example, quantum probabilities are quadratic in the amplitudes, so that the interference terms are linear in each component. (Try Sorkin's work on quantum measures for detailed accounts of this.) Hence for explaining quantum theory, it is not sufficient to explain why not all correlations are classical; you also need to explain why no correlations are stronger than quantum!