**Topic013: Quantum Periodicity Uniqueness, Completeness, Bill Tifft, 11/15/15**

With the recognition of how to precisely predict redshift periodicities the underlying basis of Quantum Temporal Cosmology is essentially complete. One important task remains. How complete and unique is the periodicity pattern? Are there other decay processes and redshift patterns hidden in the known data? Three studies have been made to look for power inconsistent with the Lehto-Tifft equations. The first test looked for any power excess distributed between periods predicted by the set of ninth-root periods which are spaced by the factor1.08. Locations scaled by factors ranging from 0.95 to 1.04 in steps of 0.01, to bridge the 1.08 interval, were searched for evidence of excess power. The Virgo data set was examined in the range from 23 to 126 km/s, which includes the range of well known periods consistent with the quality and redshift range available. A lower limit is set by the point where noise dominates the search interval which is set by the degrees of freedom in the redshift interval. The average power distribution over several profile width intervals in the Virgo data is shown in the left frame of the lead figure for this topic. The figure is number 3.22 in my book (for book information or acquisition see Post001 and Post002) and the lower left frame in the ASP paper Figure 8. The mean power of a power spectrum over the sample range is 1.0. If period matches were accidents power and scatter (scatter in the power is shown in book figure 3.23) should be distributed. In fact both power and dispersion peak symmetrically and sharply at the predicted set. Although only a partial subset of the ninth root periods are actually present, the predicted locations account uniquely for all the excess power detectable. A Student’s t comparison of the average power about the three central and three maximum off values returns a probability of only .0001 that the two populations come from the same distribution. Not only do we have a precise model for redshift periods, but it appears to be unique and complete in the range studied.

Power in a power spectrum is distributed exponentially. At very high power the exponential interpretation of the probability of a specific power peak is questionable although the mean over a set as discussed above should be quite reliable. A test, described later, will show that the actual signal from a set of predicted related periods can be extracted from a spectrum. However, to completely avoid using peak power to evaluate properties of a signal one can use binomial statistics to examine the distribution of peak locations above a cutoff power level. The right hand frame of the lead figure in this topic (the upper frame of book figure 3.24) shows the distribution of peaks above power 4 in the Virgo sample. Pk/P – 1 is the distribution of a peak location, divided by the predicted period, about zero. Vertical lines mark fractional deviations of peaks within the ninth root of two intervals. The curved line at right marks the spacing of the degrees of freedom (independent peak spacings) determined by the redshift and period range involved. Individual predicted periods and their T values are listed at the left. Large circles mark the average of the individual peak deviations found in subsamples as selection parameters are varied.

T values detected, as previously noted, are non random. T = 0, 1, 5, and 6 are dominant and concentrated, not randomly spread as noise should be. Three of the five most deviant detections are weak and in rare T classes, quite likely just noise fluctuations. Eight of the 14 (averages) have at least 1 detection within the central 1/8 of the interval, unlikely at the 0.00008 level. More conservatively, using only the mean values, nine of the 14 trials fall within the central 1/4 of the range, unlikely at the 0.002 level. Allowing for the reality of preferred T values and redshift variation in small steps which will spread measures, a realistic assessment of the likelihood of the lack of randomness is around 0.0005, quite consistent with the power analysis. A slight possible systematic trend with period is consistent with redshift variation in small steps within observed cascades between levels. As developed later in particle physics, particles including galaxies appear to be built on basic vector models with radial or azimuthal forms. Certain patterns, such as the proton and electron, are very stable. Related particles have slight shifts within the same pattern. In a decay chain between the most stable configurations the greater the shift, past or approaching a stable state, the more unstable the configuration. A slightly deviant concentration near the more stable values is not unexpected. Note, for example, spread near 24 km/s already shown to occur in Topic009. Such shifts will also be seen later in comparisons between galaxy and quasar redshifts. Deviations may be related to varying stability as galaxies transform between levels of vector configurations within a complex energy level pattern. Such a pattern, with remarkably frequent transitions, is consistently seen.

As a third test I will demonstrate the presence of a signal distinct from background noise by showing that the power distribution in the spectrum is rendered non-exponential. The sample discussed, from section 3.9.3 of my book, involves the distribution of local galaxy redshifts used in a phase-deviation diagram (figure 3.41 in my book), with profile widths between 150 and 300 km/s for which both Fisher-Tully (FT) and Tifft-Cocke (TC) redshifts were available. The periodicity range examined, from 3 to 20 km/s, demonstrates the presence and properties of short periods involving spectra with high degrees of freedom, 770 in this case, thus lots of noise. Further details of this sample, not related to the process of signal extraction, will be discussed in a later topic. The exponential probability (in parentheses) of the occurrence of peaks with power above 4 (0.018), 5 (0.0067), 6 (0.0025) or 8 (0.0003), times the number of degrees of freedom in a spectrum, yields the expected number of noise peaks which should be present. The OBSERVED/EXPECTED (O/E) ratios actually present are 20/14, 12/5, 8/2 and 4/0. The EXCESS/MISFIT (EX/M) ratio is 6/12. 7/5, 6/2 and 4/0 There is a consistent non-exponential signal excess of 6, all of which fit predicted periods, and misfit numbers are completely consistent with expected noise. Nearly all the high power excess peaks associate with the dominant T = 0 periods. When the short period range limit moved to 4 km/s, dramatically reducing the degrees of freedom to 500, noise drops precipitously showing how critical the low period limit is. The O/E ratios drop to 15/9, 9/3, 7/1, 4/0 with EX/M at 6/8, 6/3, 6/2. 4/0. The signal is virtually unchanged and misfits are consistent with noise reduction. The pattern confirms the exponential power distribution of noise structure.

Using the expected period distribution for detected peaks in the above sample we can carry out a binomial uniqueness test as was done for the Virgo sample. In the period range 2.8 to 20 km/s there are 63 possible periods in 7 location intervals scaled away from 9 predicted dominant cube-root Lehto-Tifft periods in the redshift range. Only 1 contains a match in excess of power 4. The probability of 1 success in 63 trials is 0.7. For the 9 trials at the single set of predicted periods 3 contain a match in excess of power 4. which has a probability of 0.0005 of occurring by accident. A Student’s t test comparing the mean power and power dispersion for the on and shifted matches returns a likelihood less than 0.001 that the patterns have the same parent distribution. Neither power nor power distribution yields any evidence of power significantly away from the Lehto-Tifft predicted period set.

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