The ‘sinc’ function [sin(X)/X] for the mean energy
level is derived by integrating power over space and mode number. It is
equivalent to a beam-formed sum of signals in the above equation summed by
Fourier Transform along a continuous line array. I would suggest the reader look
at a standard book on Sonar e.g. Urick "Principles of Underwater Sound" or
"Sonar for Practising Engineers". c Thomson Marconi Sonar Ltd, 2^{nd}
Edition. 1998. A. D. Waite.

Correlation processing are used in sonar and radar systems. Active pulses that are returned by scatterers are compared with the known replica to work out time delays and therefore distances of objects. Other differential information can be gained from phase, the simplest being Target Doppler.

In fact, it appears the GUE (Gaussion Unitary Ensemble) approach (and its
other variates e.g. CUEs Circular Unitary Ensembles) is actually equivalent to
Replica Correlation Beam forming used in sonar, where integration of
cross-correlated terms annihilate each other with coherency at the same mode.
GUE is a generator of all Hermitian matrices. We say there is 10 logN processing
gain^{5} for N
independent signals (viz. N degrees of Freedom), and N can be very large. In
fact they are infinite matrices. In fact, the GUEs sum up to form larger GUEs in
self-similar fashion. It is important to remember, the Rayleigh^2 has been
normalised with the local mean level spacing up the Critical Line so it does not
‘spike up’ at the lower level, but represents a variance in scale. The reasoning
is as follows:

A x = λ x for some eigenvalue of matrix A. Let A be any Hermitian, then

x^{H} A^{H} A x = λ^{H} λ. x^{H} x

If we cross correlate then sum residual of all terms (Expectation operator
‘E’) in each entry of A^{H} A . They mutually form zero cross
correlation

E< a_{ij} a_{km} > = 0 if i≠ k or j≠ m;

but E< a_{ij} a_{km} > ≠ 0 non-zero if i=k & j=m

Within some inner product (e.g. integration between 0 and 2π)

For the sum (generator) of all possible Hermitian matrices, the only values
left over exist in the trace^{6} of Gen {A^{H} A}

A relationship between the size of the matrix N and the height T along the
line has been used in empirical fits in random matrix formulae (Keating/Snaith)
with high accuracy moments calculated (up to 5?) that have been shown to be
remarkable. These are discrete distribution of large, finite N, rather like
t-distributions that are over finite sample sizes rather than infinite
populations. My modulated Rice equation describes the limiting case N = ∞,
rather than any large finite N so my Rayleigh^2 curve shown in Fig 1 is the
noise limited case (T→∞). In fact the spike that I propose is a growing
partition ^{7}
number, as in Ramanujan’s work and in the density of energy levels in certain
simple quantum systems.

By further Sonar analogy, Zeta encodes all frequencies in a manner of an
up-sweep Linear period Modulated (LPM pulse). Such Wideband pulses contain all
the frequencies with a large BT (Bandwidth Time product) ^{8}. The frequency bands are summed
together in the level spacing correlation.

This has something to do with simulating the prime Number Theorem (PNT) through the Riemann-Von Mangoldt relationship of the mean density of spacings between zeros along the imaginary axis, which rises logarithmically (over a complete cycle 2π), where Number of zeros up to height T is Nzeros ≈ (T/2π) ln (T/2π) - T/2π. The regularity of spacing is proportional to the local or instantaneous frequency dΦ/dT up the line.

This expresses all fundamental informational units (‘infons’), so there exists a relationship between primes and new/additional information albeit coupled in a somewhat complicated manner though the Zeta function.

The Montgomery-Dyson curve and modulated RICE exhibit side lobe discontinuity as already mentioned. Furthermore they represent sampling across all variations in scale (in particular all scales with the latter). An analogy from Sonar would be transmitter and Receiver in the ocean. If there was no relative motion and the configuration was completely static, then the variation would not be very interesting. If we move the Transmitter or Receiver away from (or alternatively towards) each other, which could be at varying relative speed, and therefore at unequally spaced sampling. We would see the interference profile (Lloyd’s Mirror) of peaks and nulls due to bottom and surface bounce, where the integration is independent of step size but outlines a profile. This is the analysis of sampling.

**4 Fractal Details within the ‘bit'**

The RICE statistics can go one better than these in that they modulate the waves with very large wavelengths that are not seen within the usual (Euclidean) trigonometric Fourier series. In fact, cosmological (gravity) waves may have wavelengths of cosmological scale that we could simply never directly experience. By modal analysis and defining a complete Fourier set of modes at this larger scale we can include a random variation at greater than the Nyquist sampling at the small-scale. In contrast, the other methods are defined over a complete spectrum only (Dirichlet), and so do not include these modulations in their descriptions.

Normal mode theory does not pick up the fractal details, my equation does. This is seen as decoherence in non-isolated quantum systems. In fact the Montgomery-Dyson ‘sinc’ curve can only describe Euclidean space and breaks down at the smaller scale as shown below:

This breakdown at small scale is analogous to Mercator projections at the Earth’s Poles through the map of scale.

Notice the discrete ‘teeth’ in the Noise and Signal curves in the smaller scale, and all converge to become noise limited as x→ 0.

Whereas higher up the scale the Montgomery-Dyson and Modulated RICE graphs are conjoined as shown in the following.

In fact, in the literature (Planat et al) large modulating wavelengths are analysed by the use of Ramanujan sums where conventional Discrete Fourier Transform is not suited to represent the low frequency regime.

This is where Chaos Theory comes in, because chaotic systems by definition do not display regularity of periodicity (asymptotically large periods T→ ∞) and so do not lend themselves to conventional Fourier analysis.

In fact, the high frequency components due to Casimir Effect will not be seen in conventional computers. Computers are deterministic devices because the ‘bit’ cannot be penetrated, whereas the modulated RICE description is true for all scales including the micro-scale beyond the threshold level of observable reality.

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