The first step in constructing this problem is defining a set of concentric spheres that are mapped to another set of concentric spheres. This mapping is from one of the spheres poles to the 2nd spheres opposite pole. This is done for an entire series of concentric spheres to define the fundamental geometry of the kind of relationship that one would expect from one sphere to its mirror image.
sin(2*pi/r) = 0 is the relevant equation. In this equation, we see that the period is equal to exactly one, since the formula for period that you may remember from trigonometry is 2*pi divided by one of the constants in the numerator of the thing you're taking the sin() of.
Each solution r represents an entire circle. On this circle, there is a series of identifiers that represent the angle you're making with the midsection of the circle and a fictitious line extended from the origin.
On this next line, I am going to make a statement that all representations in the brain of the integers are internally consistent. What this means is that, like the mapping between spheres, the brain will use symmetries to reconstruct a representation that can be agreed upon. Universally in the brain, this mapping should occur in order to reconstruct the integers through symmetries. I believe also that this symmetry on the time-frequency-axis is also important in bringing distant regions of the brain together in the perception of music.
Theorem - All representations in the brain of the integers are internally consistant
I use r_1 to denote the series of poles in the original sphere, and r_2 to denote the series of poles in the 2nd sphere.
m : r_1 -> r_2
m is given by the set of lines y=rx, where r_1*x=-r_2*x (it anticommutes)
By chaining together these transformations (figure2), it may be feasible to modify the brain's state of reality by locking onto the reconstruction of a similar state to the integers. What I mean, is that there would be a series of "lenses" converging and diverging on the time-frequency axis, with a frequency of repetition of approximately 40 hz, and varying widths. I have yet to try this yet, I am working on getting the software together. Proof against this may require modification of the theory. What I am saying is, sound can produce a disassociation with reality if it satisfies this equation for the mapping.
By modifying the diagram and allowing the points at poles of a circle in sin(2*pi/r)=0 to trace out a line in R and go backwards/forwards in time, an amazing type of figure can be produced. I believe that this holds the key to a variant of the Feynman-Wheeler retarded/advanced potential concept for electromagnetic radiation, where only the advanced waves are visible.
Memories are recalled over time, and are not instantaneously recalled. In order for them to be recalled, the memory is located in the brain. During this time, it is said to be settling down to a fixed point. Also, during storage the memory is said to be converging to a fixed point.
If you look at 1/2 of the diagram, it's a pretty good representation of a memory converging to a fixed point in a certain attentional set.
If the state of the brain settles down to an appropriate continuum, and sonoluminescence is a quantum system whose state is determined by sound, then perhaps a variant on this can turn a sonoluminescence machine into a telescope for probing the distant past. I am thinking small perturbations relating to this implicit differential equation. In other words, I am saying that if our observation of memory convergence is a dynamical mind-universe phenomenon, then perhaps its underlying nature can be modified by directly modifying quantum fields such as in a sonoluminescence state.
Issues:
Convergence of continuums down to a fixed state isn't an accepted phenomenon.
Proof that synaptic plasticity is a variable in the brain's fluid dynamical equations:
Let t_vector denote a vector representing the flow of time. S_1(1) is a hippocampal state of higher entropy density in the future, S_0(-1) is a state of lower entropy density in the past in the physical universe. {1,-1} are a member of the larger set of events. We use the internal consistency theorem on S_1 and S_0. From this, we know that the representations internally and externally are consistent. Let S_2 denote t_vector isomorphic to integral S_0-S_1 dt. Let L={d_1,d_2,etc} denote the set of lengths of dendrites extending from the neuron, and let this evolve over time. L in dL/dt is then differentiated with respect to t_vector (compensating for our minds bias in the flow of time) then differentiated with respect to t (using the chain rule). However there are some issues
1. Physical significance of the internal consistency theorem isn't universally realized
2. It would require an exact measurement of the state of the hippocampus and a measurement of the overall entropy of the universe, which must only be estimates.