A quasi-periodic ~2400-year climate cycle – or not?

Posted: July 29, 2020 by oldbrew in Analysis, climate, Cycles, data


We’ll look here at examples of where a 2400 year period has been identified by researchers in radiocarbon data.
– – –
Part of the abstract below is highlighted for analysis. The original Talkshop post on the paper in question:
S. S. Vasiliev and V. A. Dergachev: 2400-year cycle in atmospheric radiocarbon concentration

Abstract. We have carried out power spectrum, time-spectrum and bispectrum analyses of the long-term series of the radiocarbon concentrations deduced from measurements of the radiocarbon content in tree rings for the last 8000 years. Classical harmonic analysis of this time series shows a number of periods: 2400, 940, 710, 570, 500, 420, 360, 230, 210 and 190 years. A principle feature of the time series is the long period of ~ 2400 years, which is well known. The lines with periods of 710, 420 and 210 years are found to be the primary secular components of power spectrum. The complicated structure of the observed power spectrum is the result of ~ 2400-year modulation of primary secular components. The modulation induces the appearance of two side lines for every primary one, namely lines with periods of 940 and 570 years, of 500 and 360 years, and 230 and 190 years. The bi-spectral analysis shows that the parameters of carbon exchange system varied with the ~ 2400-year period during the last 8000 years. Variations of these parameters appear to be a climate effect on the rate of transfer of 14C between the atmosphere and the the ocean.

Looking at the ‘primary’ numbers:
‘The lines with periods of 710, 420 and 210 years are found to be the primary secular components of power spectrum. The complicated structure of the observed power spectrum is the result of ~ 2400-year modulation of primary secular components.’
[See Figure 3 and Table 1 in the paper]
– – –
Analysing this:
2400-year modulation = Mod-2400
210-year = de Vries cycle
420-year = de Vries pair (21 Jupiter-Saturn conjunctions)

With these periods we can derive as an approximation:
Square root of 2400*210 = 709.93 years
Therefore, the geometric mean of Mod-2400 and de Vries pair = ~710-year

The periods quoted in the abstract are rounded by the authors to the nearest ten years, but the pattern is there.

Substituting more accurate numbers for the de Vries pair:
21 Jupiter-Saturn = 417.16575 years
(via NASA JPL data: https://ssd.jpl.nasa.gov/?planet_phys_par)

Using the formula from this post:
24 * 2503y = 25 * 2402.88y
(24:25 = 2.4:2.5)

Then:
2402.88 * 417.16575 = (1001.1988y)² [geometric mean of the two periods]
2402.88 / 2.4 = 1001.2
417.16575 * 2.4 = 1001.1978

1001.2 * 2.5 = 2503
(2.4 = 12/5)

By the same method for the single de Vries period:
2402.88 * 208.58287 = (707.9545)² [geometric mean of the two periods]
This corresponds to the 710 year period referred to in the paper as a ‘primary secular component’ of the power spectrum.

Cross-check: 707.9545 * √2 = 1001.1988
– – –
Vasiliev and Dergachev cite a NASA-sponsored paper by Hood and Jirikowic:
A Probable Approx. 2400 Year Solar Quasi-cycle in Atmospheric Delta C-14

They say:
The residual record can be modeled to first order as an amplitude modulation of a century-scale periodic forcing function by a approx. 2400 year periodic forcing function.

The caption to their Figure 4 says:
Figure 4. Simple model of amplitude modulation of a 200 year sinusoid (a) by a 2400 year sinusoid (b). The product is shown in (c) and has characteristics that are qualitatively similar to those of the residual 14-C record.

They also refer to:
the empirical evidence discussed here for a dominantly solar origin of both the century-scale and longer-term (~ 2400 year) residual variations in the Conference 14-C record.

Figure 4 is worth a look to get a better idea of what they mean.
– – –
A series of three articles under the title ‘Nature Unbound IV – The 2400-year Bray cycle’ can be found here:
https://judithcurry.com/?s=bray

Finally, Ivanka Charvatova finds a ~2402 year solar inertial motion period in this paper:
Responses of the basic cycles of 178.7 and 2402 yr in solar–terrestrial phenomena during the Holocene

Comments
  1. oldbrew says:

    Another way of looking at it (Fig. 3):
    710² / 210 = 2400 (2400.476)
    (210*2 = 420)
    – – –

  2. Paul Vaughan says:

    Starting Point for Discussion

    J: 11.__________ years
    S: 29.__________ years
    U: 8_.__________ years
    N: 16_.__________ years

    Someone please fill in Charvatova’s model parameters with ALL the decimal places.

  3. Paul Vaughan says:

    clarification:
    in Julian years please

  4. oldbrew says:

    ‘Charvatova (2000) suggests to a possible relation of the ~2400 year period to a similar one discovered in radio-carbon series.’ – Vasiliev & Dergachev.

    See also:
    Recurring variations of probable solar origin in the atmospheric Δ14C time record
    L. L. Hood J. L. Jirikowic
    First published: January 1990
    https://doi.org/10.1029/GL017i001p00085
    Citations: 22

  5. Paul Vaughan says:

    Mayan Queen of Spaades “Beat IT”

    BSfool bell lure in a moon stir US sly helios fear IC OB survey shh! UN:
    No. 1’s eye sees what’s We’11-hood D-UN.

    “No. 1 WW ants-tube. D-feat-D” — MJ

    WHO’s REM main knew Jan. EU wary?
    “EU have too shh!O-theme that Eur.a11y knot[OB]scured…

    IC the climate casino awe weights real ease of just won list honor.

  6. oldbrew says:

    Charvatova says:
    It was also found that very long (nearly 370 yr) intervals of the solely trefoil orbit of the SIM occurred in steps of 2402 yr. Such exceptional intervals occurred in the years 159 BC–208 AD, 2561–2193 BC, 4964–4598 BC, etc. A stable behaviour of ST phenomena during these long segments is documented.

    There is no mathematical model as such in her paper. The dates quoted are derived from the solar inertial motion diagrams, as documented in Figure 4.

  7. oldbrew says:

    V&D: The complicated structure of the observed power spectrum is the result of ~ 2400-year modulation of primary secular components.

    Main frequency: 2400/710 = 710/210
    (They used multiples of 10 years in the graphic).

  8. Paul Vaughan says:

    Tune Knight: Sci11UNs Communication

    OBacknowledge-D:
    “There is no mathematical model as such in her paper”

    IT exists and IT was US-D.
    IT simply was knot ID-D.

    5 Fathom: UN knew SOS eerie US mind set:

    I will twist the knife and bleed my aching heart
    and tear IT apart
    — Garbage #1 CRush

    “Bury center” is political code weather left or right.

    2400 Q-borg: Tim Channon found no. th’ UN ‘ear IT? (too 2 UN 1 seeks 6 hive 5?)

    Mess Age in the Err Craft: UN Dare Grown-D Miss $Sell Vac Scene

    Bet ER: DO the review. WHO’s BRI11ant ID was IT to knot have IC report numb brrs and their SORCE err or CRaft? Maybe some UN WHO want-D to cause more D-tech-dive “work”.

    Share ring? Knot incline-D UN dare “classified” “intellectual property” of IT‘s “trade secrets”.
    IT has US tie-D in knots. Sea We-D knot free 2’s peek.

    UN of verse AI: trains later. Comp PR UN-D ?
    If knot: tough shh!IT for 5 moor. Then AI of US in no cent loom $ UN eerie.

    With the numbers listed precisely: Green light’s instant (copy/paste/post).
    May Be. butter right to IC with Anon name US. WHO’s Quest UNs AB out IT.

  9. oldbrew says:

    The Vasiliev & Dergachev paper doesn’t rely on Charvatova at all. Neither do any of the other studies they cite, AFAIK.

    But Charvatova’s ~370 year period either exists and repeats or it doesn’t. Obviously she says it does, with supporting SIM diagrams (Fig. 4 in the PRP paper).

    This is 2403 years (53 Saturn-Uranus = 2402.934 years).
    Left and right are about 120 degrees different.

  10. oldbrew says:

    More discussion with Bill Howell:

    An Independent Verification of Ivanka Charvátová’s Solar Inertial Motion (SIM) Curves
    William Neil Howell

    Click to access Howell%20-%20solar%20inertial%20motion%20-%20NASA-JPL%20versus%20Charvatova.pdf


    (version: incomplete, uncorrected second draft 18Jun08)

    Some other Bill Howell stuff here – includes various links:

    Charvatova’s Solar Inertial Motion (SIM) hypothesis – similar SIM curves yield similar solar activity
    http://www.billhowell.ca/Charvatova%20solar%20inertial%20motion%20&%20activity/_Charvatova%20-%20solar%20inertial%20motion%20&%20activity.html

  11. Paul Vaughan says:

    OB: You leave the impression you think Charvatova used the Seidelmann (1992) model.

    I advise that someone write to her and ask for precise verification. I want to run comparative diagnostics at this stage as I have the whole thing cracked and all that’s left is sensitivity testing.

    I’ve already given all of the puzzle pieces. I just haven’t assembled them all in one place. The main reason is I get so incredibly annoyed with that stupid filter that favors paragraphs over blocks of numbers and algebra.

    Lots of things fall out clearly. I’ll list a few now:

    1. 96 & 104 combine to give 100 and this is clear in aa index diagnostics (solves a puzzle I discovered in 2008).
    2. V&D 940 & 570 do not arise from 710 & 2400 but rather 710 and 2900. If you study the figures in V&D carefully there’s no getting around this. They threw a curve-ball in their paper and I’ve figured out what they missed and/or misrepresented: DO (which Tim Channon found).
    3. Landscheidt Cycle (980 years) shows up yet another way: as a base-level UN slip cycle. (This is a new result I turned up while reviewing with analogous extensions.)
    4. There are 4 noteworthy jovian slip cycles bundled around 2300-2400. One of them fits V&D like a glove. Luminaries will find clear scope for further exploration and diagnostics here.

    It’s all algebraically analogous to JEV.
    I’ll share some insights in “climate casino” format, which was chosen to deliberately accent parallels.

  12. Paul Vaughan says:

    Jovial Game of Celestial Hearts

    J = 1 / 11.8626151546089 ____ Cards Featured
    S = 1 / 29.4474984673838

    C = 1 / 835.546575435631 ____ Card Players Shuffled with Deck
    C = Challenger of Harmonic Means = Queen of Spades
    = Contestant Rocking Hearts 4 Calendar Mayan-D Count Roll

    B = 1 / 19.8650360864628 = +1J-1S = beat __________________ Dealer’s Time Table
    R = 1 / 16.9122914926352 = +0.5J+0.5S = harmonic mean
    I = 1 / 8.4561457463176 = +1J+1S = axial period

    R+C = 1 / 16.5767613988929 = +0.5J+0.5S+1C ____ Splitting the Deck
    R-C = 1 / 17.2616851219298 = +0.5J+0.5S-1C

    1B = ⌊1(R-C)/B⌉B = harmonic of B slipping nearest 1(R-C) ____1 Whole
    |1(R-C)-1B| = |W-| = 1 / 131.716392653884 = -0.5J+1.5S-1C __|Solar Core|
    2B = ⌊2(R-C)/B⌉B = harmonic of B slipping nearest 2(R-C) ____2 Halves
    |2(R-C)-2B| = |H-| = 1 / 65.8581963269422 = -1J+3S-2C __|Solar Opposition|
    5B = ⌊4(R-C)/B⌉B = harmonic of B slipping nearest 4(R-C) ____4 Quarters
    |5B-4(R-C)| = |Q-| = 1 / 50.071541259993 = +3J-7S+4C __|Solar Rights|

    1B = ⌊1(R+C)/B⌉B = harmonic of B slipping nearest 1(R+C) ____1 Whole
    |1(R+C)-1B| = |W+| = 1 / 100.143082519986 = -0.5J+1.5S+1C __|Galactic Core|
    2B = ⌊2(R+C)/B⌉B = harmonic of B slipping nearest 2(R+C) ____2 Halves
    |2(R+C)-2B| = |H+| = 1 / 50.0715412599931 = -1J+3S+2C __|Galactic Opposition|
    5B = ⌊4(R+C)/B⌉B = harmonic of B slipping nearest 4(R+C) ____4 Quarters
    |5B-4(R+C)| = |Q+| = 1 / 96.1829470900284 = +3J-7S-4C __|Galactic Rights|

  13. Paul Vaughan says:

    19.8650360864628 = (29.4474984673838)*(11.8626151546089) / (29.4474984673838 – 11.8626151546089)
    8.4561457463176 = (29.4474984673838)*(11.8626151546089) / (29.4474984673838 + 11.8626151546089)
    16.9122914926352 = (29.4474984673838)*(11.8626151546089)/((29.4474984673838+11.8626151546089)/2)

    ⌊ 29.4474984673838 / 11.8626151546089 ⌉ = ⌊2.4823783022197⌉ = 2
    29.4474984673838 / 2 = 14.7237492336919
    61.0464822565173 = (14.7237492336919)*(11.8626151546089) / (14.7237492336919 – 11.8626151546089)

    ⌊ 61.0464822565173 / 19.8650360864628 ⌉ = ⌊3.07306173473895⌉ = 3
    61.0464822565173 / 3 = 20.3488274188391
    835.546575435631 = (20.3488274188391)*(19.8650360864628) / (20.3488274188391 – 19.8650360864628)

  14. Paul Vaughan says:

    ⌊ 131.716392653884 / 19.8650360864628 ⌉ = ⌊6.63056397585119⌉ = 7
    131.716392653884 / 7 = 18.8166275219835
    356.533700137559 = (19.8650360864628)*(18.8166275219835) / (19.8650360864628 – 18.8166275219835)

    ⌊ 65.8581963269421 / 19.8650360864628 ⌉ = ⌊3.31528198792559⌉ = 3
    65.8581963269421 / 3 = 21.9527321089807
    208.886643858908 = (21.9527321089807)*(19.8650360864628) / (21.9527321089807 – 19.8650360864628)

    ⌊ 50.0715412599931 / 19.8650360864628 ⌉ = ⌊2.52058647374493⌉ = 3
    50.0715412599931 / 3 = 16.690513753331
    104.443321929454 = (19.8650360864628)*(16.690513753331) / (19.8650360864628 – 16.690513753331)

  15. Paul Vaughan says:

    ⌊ 131.716392653884 / 9.93251804323141 ⌉ = ⌊13.2611279517024⌉ = 13
    131.716392653884 / 13 = 10.1320302041449
    504.413226524327 = (10.1320302041449)*(9.93251804323141) / (10.1320302041449 – 9.93251804323141)

    ⌊ 65.8581963269421 / 9.93251804323141 ⌉ = ⌊6.63056397585119⌉ = 7
    65.8581963269421 / 7 = 9.40831376099173
    178.266850068779 = (9.93251804323141)*(9.40831376099173) / (9.93251804323141 – 9.40831376099173)

    ⌊ 100.143082519986 / 19.8650360864628 ⌉ = ⌊5.04117294748986⌉ = 5
    100.143082519986 / 5 = 20.0286165039972
    2432.25439579341 = (20.0286165039972)*(19.8650360864628) / (20.0286165039972 – 19.8650360864628)

    ⌊ 356.533700137559 / 50.0715412599931 ⌉ = ⌊7.12048583218722⌉ = 7
    356.533700137559 / 7 = 50.9333857339369
    2959.13381403679 = (50.9333857339369)*(50.0715412599931) / (50.9333857339369 – 50.0715412599931)

    ⌊ 504.413226524327 / 131.716392653884 ⌉ = ⌊3.82954024446897⌉ = 4
    504.413226524327 / 4 = 126.103306631082
    2959.13381403688 = (131.716392653884)*(126.103306631082) / (131.716392653884 – 126.103306631082)

  16. Paul Vaughan says:

    ⌊ 504.413226524327 / 65.8581963269421 ⌉ = ⌊7.65908048893794⌉ = 8
    504.413226524327 / 8 = 63.0516533155409
    1479.56690701844 = (65.8581963269421)*(63.0516533155409) / (65.8581963269421 – 63.0516533155409)

    939.447668560927 = (2959.13381403679)*(713.067400275117) / (2959.13381403679 – 713.067400275117)
    574.604095118134 = (2959.13381403679)*(713.067400275117) / (2959.13381403679 + 713.067400275117)

    228.511660883048 = (2432.25439579341)*(208.886643858908) / (2432.25439579341 – 208.886643858908)
    192.365894180056 = (2432.25439579341)*(208.886643858908) / (2432.25439579341 + 208.886643858908)
    504.413226524332 = (2432.25439579341)*(417.773287717816) / (2432.25439579341 – 417.773287717816)
    356.533700137555 = (2432.25439579341)*(417.773287717816) / (2432.25439579341 + 417.773287717816)

  17. Paul Vaughan says:

    ⌊ 208.886643858908 / 100.143082519986 ⌉ = ⌊2.08588190619376⌉ = 2
    208.886643858908 / 2 = 104.443321929454
    2432.25439579362 = (104.443321929454)*(100.143082519986) / (104.443321929454 – 100.143082519986)

    8.63084256096492 = 17.2616851219298 / 2
    4.31542128048246 = 17.2616851219298 / 4

    ⌊ 19.8650360864628 / 17.2616851219298 ⌉ = ⌊1.15081673348482⌉ = 1
    19.8650360864628 / 1 = 19.8650360864628
    131.716392653884 = (19.8650360864628)*(17.2616851219298) / (19.8650360864628 – 17.2616851219298)

    ⌊ 19.8650360864628 / 8.63084256096492 ⌉ = ⌊2.30163346696964⌉ = 2
    19.8650360864628 / 2 = 9.93251804323141
    65.8581963269421 = (9.93251804323141)*(8.63084256096492) / (9.93251804323141 – 8.63084256096492)

    ⌊ 19.8650360864628 / 4.31542128048246 ⌉ = ⌊4.60326693393928⌉ = 5
    19.8650360864628 / 5 = 3.97300721729256
    50.0715412599931 = (4.31542128048246)*(3.97300721729256) / (4.31542128048246 – 3.97300721729256)

    8.28838069944647 = 16.5767613988929 / 2
    4.14419034972324 = 16.5767613988929 / 4

    ⌊ 19.8650360864628 / 16.5767613988929 ⌉ = ⌊1.19836653303036⌉ = 1
    19.8650360864628 / 1 = 19.8650360864628
    100.143082519986 = (19.8650360864628)*(16.5767613988929) / (19.8650360864628 – 16.5767613988929)

    ⌊ 19.8650360864628 / 8.28838069944647 ⌉ = ⌊2.39673306606072⌉ = 2
    19.8650360864628 / 2 = 9.93251804323141
    50.0715412599931 = (9.93251804323141)*(8.28838069944647) / (9.93251804323141 – 8.28838069944647)

    ⌊ 19.8650360864628 / 4.14419034972324 ⌉ = ⌊4.79346613212144⌉ = 5
    19.8650360864628 / 5 = 3.97300721729256
    96.1829470900285 = (4.14419034972324)*(3.97300721729256) / (4.14419034972324 – 3.97300721729256)

  18. Paul Vaughan says:

    Jovial Game of Celestial Hearts

    U = 1 / 84.016845922161 ____ Cards Featured
    N = 1 / 164.791315640078
    C = 1 / 48590.8284812209 ____ Card Players Shuffled with Deck
    C = Challenger of Harmonic Means = Queen of Spades
    = Contestant Rocking Hearts 4 Calendar Mayan-D Count Roll

    B = 1 / 171.406220601552 = +1U-1N = beat __________________ Dealer’s Time Table
    R = 1 / 111.292543528394 = +0.5U+0.5N = harmonic mean
    I = 1 / 55.6462717641972 = +1U+1N = axial period

    R+C = 1 / 111.038221334937 = +0.5U+0.5N+1C ____ Splitting the Deck
    R-C = 1 / 111.548033396551 = +0.5U+0.5N-1C

    2B = ⌊1(R-C)/B⌉B = harmonic of B slipping nearest 1(R-C) ____1 Whole
    |2B-1(R-C)| = |W-| = 1 / 369.89908519333 = +1.5U-2.5N+1C __|Solar Core|
    3B = ⌊2(R-C)/B⌉B = harmonic of B slipping nearest 2(R-C) ____2 Halves
    |2(R-C)-3B| = |H-| = 1 / 2340.74786894369 = -2U+4N-2C __|Solar Opposition|
    6B = ⌊4(R-C)/B⌉B = harmonic of B slipping nearest 4(R-C) ____4 Quarters
    |4(R-C)-6B| = |Q-| = 1 / 1170.37393447185 = -4U+8N-4C __|Solar Rights|

    2B = ⌊1(R+C)/B⌉B = harmonic of B slipping nearest 1(R+C) ____1 Whole
    |2B-1(R+C)| = |W+| = 1 / 375.617889254102 = +1.5U-2.5N-1C __|Galactic Core|
    3B = ⌊2(R+C)/B⌉B = harmonic of B slipping nearest 2(R+C) ____2 Halves
    |2(R+C)-3B| = |H+| = 1 / 1962.57776108677 = -2U+4N+2C __|Galactic Opposition|
    6B = ⌊4(R+C)/B⌉B = harmonic of B slipping nearest 4(R+C) ____4 Quarters
    |4(R+C)-6B| = |Q+| = 1 / 981.288880543385 = -4U+8N+4C __|Galactic Rights|

  19. Paul Vaughan says:

    171.406220601552 = (164.791315640078)*(84.016845922161) / (164.791315640078 – 84.016845922161)
    55.6462717641972 = (164.791315640078)*(84.016845922161) / (164.791315640078 + 84.016845922161)
    111.292543528394 = (164.791315640078)*(84.016845922161)/((164.791315640078+84.016845922161)/2)

    ⌊ 164.791315640078 / 84.016845922161 ⌉ = ⌊1.96140802277619⌉ = 2
    164.791315640078 / 2 = 82.395657820039
    4270.09258127429 = (84.016845922161)*(82.395657820039) / (84.016845922161 – 82.395657820039)

    ⌊ 4270.09258127429 / 171.406220601552 ⌉ = ⌊24.9121214287811⌉ = 25
    4270.09258127429 / 25 = 170.803703250972
    48590.8284812209 = (171.406220601552)*(170.803703250972) / (171.406220601552 – 170.803703250972)

    55.7740166982756 = 111.548033396551 / 2
    27.8870083491378 = 111.548033396551 / 4

    55.5191106674687 = 111.038221334937 / 2
    27.7595553337344 = 111.038221334937 / 4

    85.7031103007758 = 171.406220601552 / 2
    42.8515551503879 = 171.406220601552 / 4

    27.8231358820986 = 55.6462717641972 / 2
    13.9115679410493 = 55.6462717641972 / 4

    9.93251804323141 = 19.8650360864628 / 2
    4.9662590216157 = 19.8650360864628 / 4

    4.2280728731588 = 8.4561457463176 / 2
    2.1140364365794 = 8.4561457463176 / 4

  20. Paul Vaughan says:

    ⌊ 171.406220601552 / 111.548033396551 ⌉ = ⌊1.53661355904147⌉ = 2
    171.406220601552 / 2 = 85.7031103007758
    369.89908519333 = (111.548033396551)*(85.7031103007758) / (111.548033396551 – 85.7031103007758)

    ⌊ 171.406220601552 / 55.7740166982756 ⌉ = ⌊3.07322711808295⌉ = 3
    171.406220601552 / 3 = 57.1354068671839
    2340.74786894369 = (57.1354068671839)*(55.7740166982756) / (57.1354068671839 – 55.7740166982756)

    ⌊ 171.406220601552 / 27.7595553337344 ⌉ = ⌊6.17467457748694⌉ = 6
    171.406220601552 / 6 = 28.5677034335919
    981.288880543382 = (28.5677034335919)*(27.7595553337344) / (28.5677034335919 – 27.7595553337344)

    ⌊ 369.89908519333 / 171.406220601552 ⌉ = ⌊2.15802602668191⌉ = 2
    369.89908519333 / 2 = 184.949542596665
    2340.74786894369 = (184.949542596665)*(171.406220601552) / (184.949542596665 – 171.406220601552)

    ⌊ 171.406220601552 / 111.038221334937 ⌉ = ⌊1.54366864437174⌉ = 2
    171.406220601552 / 2 = 85.7031103007758
    375.617889254102 = (111.038221334937)*(85.7031103007758) / (111.038221334937 – 85.7031103007758)

    ⌊ 375.617889254102 / 85.7031103007758 ⌉ = ⌊4.38278013406827⌉ = 4
    375.617889254102 / 4 = 93.9044723135256
    981.288880543382 = (93.9044723135256)*(85.7031103007758) / (93.9044723135256 – 85.7031103007758)

    ⌊ 171.406220601552 / 27.8870083491378 ⌉ = ⌊6.1464542361659⌉ = 6
    171.406220601552 / 6 = 28.5677034335919
    1170.37393447185 = (28.5677034335919)*(27.8870083491378) / (28.5677034335919 – 27.8870083491378)

    ⌊ 1170.37393447185 / 111.292543528394 ⌉ = ⌊10.5161936044102⌉ = 11
    1170.37393447185 / 11 = 106.397630406532
    2419.09562407727 = (111.292543528394)*(106.397630406532) / (111.292543528394 – 106.397630406532)

    2385.623964918 = (2432.25439579341)*(2340.74786894369)/((2432.25439579341+2340.74786894369)/2)
    2402.24320580332 = (2419.09562407727)*(2385.623964918)/((2419.09562407727+2385.623964918)/2)

  21. Paul Vaughan says:

    ⌊ 104.443321929454 / 19.8650360864628 ⌉ = ⌊5.25764571858129⌉ = 5
    104.443321929454 / 5 = 20.8886643858908
    405.375732632252 = (20.8886643858908)*(19.8650360864628) / (20.8886643858908 – 19.8650360864628)

    ⌊ 96.1829470900285 / 19.8650360864628 ⌉ = ⌊4.84182090943057⌉ = 5
    96.1829470900285 / 5 = 19.2365894180057
    608.063598948375 = (19.8650360864628)*(19.2365894180057) / (19.8650360864628 – 19.2365894180057)

    1216.12719789677 = (608.063598948375)*(405.375732632252) / (608.063598948375 – 405.375732632252)

    2432.25439579355 = (208.886643858908)*(192.365894180057) / (208.886643858908 – 192.365894180057)

    ⌊ 208.886643858908 / 19.8650360864628 ⌉ = ⌊10.5152914371626⌉ = 11
    208.886643858908 / 11 = 18.9896948962643
    430.953071338593 = (19.8650360864628)*(18.9896948962643) / (19.8650360864628 – 18.9896948962643)

    ⌊ 192.365894180057 / 19.8650360864628 ⌉ = ⌊9.68364181886114⌉ = 10
    192.365894180057 / 10 = 19.2365894180057
    608.063598948375 = (19.8650360864628)*(19.2365894180057) / (19.8650360864628 – 19.2365894180057)

    1479.5669070184 = (608.063598948375)*(430.953071338593) / (608.063598948375 – 430.953071338593)

  22. Paul Vaughan says:

    Mods: Something like 4 calculation-laden comments are caught in the filter.

  23. Paul Vaughan says:

    Alternating Strategy

    Warning: On the last DO thread I flooded the discussion with a lot of water.

    (Aside: That was deliberate. I’ve been choked about intolerable Western — and especially Five Eyes moves — e.g. lockdowns, double-standards on racism that attack beautiful Chinese people, aggressive campaigns of financial terror now not only on “rogue” foreign nations but on millions of homeland citizens, Boris back-stabbing hardcore volunteer supporters and thus reminding us to remain absolutely cynical about lending trust, incomprehensibly Orwellian climate psy-ops, the creepy shift in our society towards supporting big tech monopolies, the list goes on – the times are running savage interference on better ways.)

    Some of the stuff buried in that DO flood is pure signal. Other stuff is a flood of subharmonics and harmonics. Those with key intuition will know the difference.

    HERE I have NOT watered anything down. Until better avenues for the greater good open up I’m balancing protest of “democratic” psy-ops with solid contribution.

    After the moderator clears the backlog, 2 steps remain.

    Above JS & UN were given separate treatment.

    So the next step is to review how JS & UN slip past one another.
    We’ve covered this before. It’s review with sharpening perspective.

    Then a review of JEV will underscore parallel structure.

    Each of the recipes has 3 ingredients. The procedure stays the same.
    There’s only 1 theme: how the meeting points slip on an axially-locked frame.

    Alert readers see that I give this hierarchical treatment to measure the rate at which nested geometric features slip past one another.

  24. Paul Vaughan says:

    Jovial Game of Celestial Hearts

    J = 1 / 11.8626151546089 ____ Cards Featured
    S = 1 / 29.4474984673838

    C = 1 / 171.406220601552 ____ Card Players Shuffled with Deck
    C = Challenger of Harmonic Means = Queen of Spades
    = Contestant Rocking Hearts 4 Calendar Mayan-D Count Roll

    B = 1 / 19.8650360864628 = +1J-1S = beat __________________ Dealer’s Time Table
    R = 1 / 16.9122914926352 = +0.5J+0.5S = harmonic mean
    I = 1 / 8.4561457463176 = +1J+1S = axial period

    R+C = 1 / 15.3934519460015 = +0.5J+0.5S+1C ____ Splitting the Deck
    R-C = 1 / 18.7636626447678 = +0.5J+0.5S-1C

    1B = ⌊1(R-C)/B⌉B = harmonic of B slipping nearest 1(R-C) ____1 Whole
    |1(R-C)-1B| = |W-| = 1 / 338.432743555957 = -0.5J+1.5S-1C __|Solar Core|
    2B = ⌊2(R-C)/B⌉B = harmonic of B slipping nearest 2(R-C) ____2 Halves
    |2(R-C)-2B| = |H-| = 1 / 169.216371777979 = -1J+3S-2C __|Solar Opposition|
    4B = ⌊4(R-C)/B⌉B = harmonic of B slipping nearest 4(R-C) ____4 Quarters
    |4(R-C)-4B| = |Q-| = 1 / 84.6081858889894 = -2J+6S-4C __|Solar Rights|

    1B = ⌊1(R+C)/B⌉B = harmonic of B slipping nearest 1(R+C) ____1 Whole
    |1(R+C)-1B| = |W+| = 1 / 68.3854913151666 = -0.5J+1.5S+1C __|Galactic Core|
    3B = ⌊2(R+C)/B⌉B = harmonic of B slipping nearest 2(R+C) ____2 Halves
    |3B-2(R+C)| = |H+| = 1 / 47.4074465992974 = +2J-4S-2C __|Galactic Opposition|
    5B = ⌊4(R+C)/B⌉B = harmonic of B slipping nearest 4(R+C) ____4 Quarters
    |4(R+C)-5B| = |Q+| = 1 / 122.665716840277 = -3J+7S+4C __|Galactic Rights|

  25. Paul Vaughan says:

    9.38183132238388 = 18.7636626447678 / 2
    4.69091566119194 = 18.7636626447678 / 4

    7.69672597300077 = 15.3934519460015 / 2
    3.84836298650038 = 15.3934519460015 / 4

    ⌊ 122.665716840276 / 4.9662590216157 ⌉ = ⌊24.6998226041719⌉ = 25
    122.665716840276 / 25 = 4.90662867361105
    408.644083615556 = (4.9662590216157)*(4.90662867361105) / (4.9662590216157 – 4.90662867361105)

    ⌊ 408.644083615556 / 84.6081858889895 ⌉ = ⌊4.82984098195556⌉ = 5
    408.644083615556 / 5 = 81.7288167231111
    2401.54232383281 = (84.6081858889895)*(81.7288167231111) / (84.6081858889895 – 81.7288167231111)

  26. Paul Vaughan says:

    ⌊ 169.216371777979 / 19.8650360864628 ⌉ = ⌊8.51830175598286⌉ = 9
    169.216371777979 / 9 = 18.8018190864421
    351.291236535953 = (19.8650360864628)*(18.8018190864421) / (19.8650360864628 – 18.8018190864421)

    ⌊ 338.432743555958 / 9.93251804323141 ⌉ = ⌊34.0732070239314⌉ = 34
    338.432743555958 / 34 = 9.95390422223406
    4622.9545388013 = (9.95390422223406)*(9.93251804323141) / (9.95390422223406 – 9.93251804323141)

    ⌊ 338.432743555958 / 4.9662590216157 ⌉ = ⌊68.1464140478629⌉ = 68
    338.432743555958 / 68 = 4.97695211111703
    2311.47726940065 = (4.97695211111703)*(4.9662590216157) / (4.97695211111703 – 4.9662590216157)

    ⌊ 338.432743555958 / 16.9122914926352 ⌉ = ⌊20.0110519442818⌉ = 20
    338.432743555958 / 20 = 16.9216371777979
    30622.009569219 = (16.9216371777979)*(16.9122914926352) / (16.9216371777979 – 16.9122914926352)

  27. Paul Vaughan says:

    Humility, not Hubris

    Here’s a fun one. Some insights are really satisfying. Most of this is review, but here I wrap it up and tie on the ribbon.

    2500.20304503681 = (30622.009569219)*(2311.47726940065) / (30622.009569219 – 2311.47726940065)
    2149.24339501203 = (30622.009569219)*(2311.47726940065) / (30622.009569219 + 2311.47726940065)
    2402.13901582301 = (2500.20304503681)*(2311.47726940065)/((2500.20304503681+2311.47726940065)/2)

    2450.19021822768 = (2500.20304503681)*(2402.13901582301)/((2500.20304503681+2402.13901582301)/2)

    OB: Note that 15 * 2450 = 36750.

    Note the parallel structure with our old primorial (as in number theory) mnemonic device, which doubles here as a suggestion that event series and slip cycle aggregation criteria differ fundamentally.

    2502.5 = (30030)*(2310) / (30030 – 2310)
    2145 = (30030)*(2310) / (30030 + 2310)
    2402.4 = (2502.5)*(2310)/((2502.5+2310)/2)

    2451.42857142857 = (2502.5)*(2402.4)/((2502.5+2402.4)/2)

    Also recall that simple sporadic groups tie into monstrous moonshine via primorials.

  28. Paul Vaughan says:

    Moderators: 2 more calc.-laden comments have been trapped in the filter, bringing the total to 6.

    Supplementary note for readers working their way through the V&D calculations:
    2959 = 2 * 1479.5
    713 = 2 * 356.5

    V&D fits the JS slip-cycle frame flawlessly. Note that UN and JS-slip-on-UN aren’t even needed — which really underscores the dominance of JS in the strongest level of windowing.

    I’m very curious about the 1 misrepresentation V&D made. There’s probably some delectable history there, but it may be difficult to expose.

    They’ve done such beautiful work I would choose to let them save face. They carried the ball quite far and the one loose end they left was easy to figure out.

    Getting really technical: Look at the peak just to the right of 0.001 in Figure 3. Take the reciprocal. It’s a number less than 1000. They say it’s 940 and that is consistent with that peak in Figure 3.

    If anyone tries to tell you 940 should be 1000 or 1010 (6 or 7% off) remind them that we often hit below 0.00001% and tend to not even look at things with errors above 0.01%.

    Something around 1000 or 1010 may be wonderfully meaningful in some other way, but it isn’t consistent with V&D’s well-illustrated summary of 940.

    I showed in detail above what accounts for 940 and 570. That suggests a nice diagnostic avenue for further exploration. The world doesn’t have enough people I would trust with that sort of work.

    The last thing I plan to do here is review JEV so readers have a more familiar parallel structure that might help them comparatively ground their orientation in JS & UN slip cycles.

  29. Paul Vaughan says:

    96

    I bring this up because readers not comparing the decimal digits may conflate their thinking about 96 year cycles of different origins.

    Never forget the 96 year lunisolar cycle to which I’ve had to point repeatedly. It is VERY close to the the JS slip cycle outlined above (which means challenging — and interesting — diagnostics).

    There are 2 other 96 year JS cycles of entirely different natures.
    All of these 96 year cycles tie into a monstrous bundle in such a way that they differentiate precisely.
    The number 96^3 is a building block for all of them. Don’t worry about that, but take this seriously:

    Minimalism:

    I wouldn’t trust any climatologist who can’t at least follow and deeply understand the simple derivation of 96 in the lunisolar context. If this isn’t ringing a bell, you’ve missed something important — maybe repeatedly.

  30. Paul Vaughan says:

    Mods: There are still 2 calculation-laden comments caught in the filter.

  31. Paul Vaughan says:

    Lunisolar 96 Review

    The 96 year lunisolar slip-cycle was derived years ago from time-height sections of quasibiennial, annual, and semi-annual oscillations.

    http://ugamp.nerc.ac.uk/hot/ajh/qbo.htm — includes animation

    Here’s a clean, simple outline I shared with talkshop reader Ed:
    https://tallbloke.wordpress.com/suggestions-18/#comment-116526

    ⌊ 0.999978614647502 / 0.0745030006844627 ⌉ = ⌊13.421991134057⌉ = 13
    0.999978614647502 / 13 = 0.0769214318959617
    2.36966735541038 = (0.0769214318959617)*(0.0745030006844627) / (0.0769214318959617 – 0.0745030006844627)

    ⌊ 2.36966735541038 / 0.499989307323751 ⌉ = ⌊4.73943606533166⌉ = 5
    2.36966735541038 / 5 = 0.473933471082076
    9.0943796900619 = (0.499989307323751)*(0.473933471082076) / (0.499989307323751 – 0.473933471082076)

    ⌊ 9.0943796900619 / 0.999978614647502 ⌉ = ⌊9.09457418073658⌉ = 9
    9.0943796900619 / 9 = 1.0104866322291
    96.1613372617316 = (1.0104866322291)*(0.999978614647502) / (1.0104866322291 – 0.999978614647502)

  32. Paul Vaughan says:

    Let’s throw this here right beside that:

    100.143082519986 = (104.443321929454)*(96.1829470900285)/((104.443321929454+96.1829470900285)/2)

    2432.25439579362 = (104.443321929454)*(100.143082519986) / (104.443321929454 – 100.143082519986)
    2432.25439579349 = (100.143082519986)*(96.1829470900285) / (100.143082519986 – 96.1829470900285)
    2432.25439579355 = (208.886643858908)*(192.365894180057) / (208.886643858908 – 192.365894180057)

    In 2008 I estimated 105 and 97 from the drift of seasonal patterns in annual aa index. At the time I didn’t know 96 so well.

    Some years later Vukcevic and I had a different take on an ENSO clustering pattern.
    He suggested 104, but 96 was a better fit.

  33. Paul Vaughan says:

    96: Review of Chandler, QBO, & Polar Motion Definitions

    Lunar Draconic = Nodal Month
    0.0745030006844627 = 27.212221 / 365.25
    0.0372515003422313 = 13.6061105 / 365.25

    Terrestrial Tropical Year
    0.999978614647502 = 365.242189 / 365.25
    0.499989307323751 = 182.6210945 / 365.25

    Chandler Wobble
    ⌊ 0.499989307323751 / 0.0372515003422313 ⌉ = ⌊13.421991134057⌉ = 13
    0.499989307323751 / 13 = 0.0384607159479808
    1.18483367770519 = (0.0384607159479808)*(0.0372515003422313) / (0.0384607159479808 – 0.0372515003422313)

    QBO = Quasibiennial Oscillation
    ⌊ 0.999978614647502 / 0.0745030006844627 ⌉ = ⌊13.421991134057⌉ = 13
    0.999978614647502 / 13 = 0.0769214318959617
    2.36966735541038 = (0.0769214318959617)*(0.0745030006844627) / (0.0769214318959617 – 0.0745030006844627)

    Polar Motion (Group Wave)
    ⌊ 0.999978614647502 / 0.0372515003422313 ⌉ = ⌊26.843982268114⌉ = 27
    0.999978614647502 / 27 = 0.0370362449869445
    6.40939079526111 = (0.0372515003422313)*(0.0370362449869445) / (0.0372515003422313 – 0.0370362449869445)

    Also note:
    ⌊ 835.546575435636 / 6.40939079526111 ⌉ = ⌊130.36286944048⌉ = 130
    835.546575435636 / 130 = 6.42728134950489
    2302.60937468698 = (6.42728134950489)*(6.40939079526111) / (6.42728134950489 – 6.40939079526111)

  34. Paul Vaughan says:

    Moderators: 2 sets of calculations stuck in the filter derive quantities used in other calculations listed above. Are you able to find them? Or do I need to post them again?

  35. Paul Vaughan says:

    Q:SST UN on Corporate Seas of ITees Mono Pole Lie?

    If few are puttin’ the V&D truffle with IC IT’s a 50 year slide across theme agen[t]a bridge.

    X$marks the control verse see AI Luke Duke slide across the hood in Hazard County.

    CC: Causun Caughter clean-SJ!VE!UN Trains Later

    “First pass-D the post”:
    Then SCroll-D up past “16 Figures“.

    This is cleans h!ave UN sci11UNs comm. on C[ENSO]Rship.

  36. oldbrew says:

    Dergachev and Vasiliev claim to have found a ~2300 year dipole period in their 2018 paper…

    Long-term changes in the concentration of radiocarbon and the nature of the Hallstatt cycle

    In the second case (AM-2), the modulation of ∼210-year variations of the SMP is absent. But the existence of ∼2300-year oscillations of the dipole moment is allowed. In the second case, unlike AM-1, the amplitude of the 210-year oscillations in the production rate depends on the phase of the 2300-year variations in the dipole moment, which agrees with the observations.

    Based on the comparison of AM-1 and AM-2 models of amplitude modulation, it can be concluded that changes in the dipole moment of the geomagnetic field are responsible for the observed ∼2300-year variations in the concentration of radiocarbon.

    https://www.sciencedirect.com/science/article/abs/pii/S1364682617306168

    But before that:
    Highlights

    • Analysis of the radiocarbon series shows that there is an amplitude modulation of the ∼210-year cycle.

    • Two models of amplitude modulation of radiocarbon variations are considered: solar and geomagnetic.

    • The geomagnetic field are responsible for the observed ∼2400-year variations in the concentration of radiocarbon.
    – – –
    09 November 2019
    One Hundred Thousand Years of Geomagnetic Field Evolution – open access
    https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2019RG000656

  37. Paul Vaughan says:

    96 is simple with QBO, annual, and semi-annual, but years later ERSSTv3b2 has not been restored.

    The primorials 2310 & 210 come into focus when jovian harmonic means are aggregated.

    I haven’t shared that yet.

    There is no community where these things can be discussed sensibly. The climate propaganda has saturated the minds of highly intelligent elite. It’s remarkable how unsuspecting they are that they’ve been duped. Worse: They interpret propaganda as “unbiased news”. I declare them a lost cause. It is not reasonable to think they can be reached. Together they constitute a truly monstrous obstacle to justice and integrity.

    With humility we know the removal or persistence of these justice and integrity barriers is in God’s hands. We are not here to be regarded as members of a fringe cult. Shifts in “climate” power structures would be most welcome, should they occur.

    What would it take to engineer a truly productive climate dialogue? Many things that don’t presently exist and many things that may not exist anytime soon. For sure God is the only negotiator who could make the needed arrangements.

  38. oldbrew says:

    See climate cycles graphic below.

  39. oldbrew says:

    Major Climate Change Occurred 5,200 Years Ago: Evidence Suggests That History Could Repeat Itself
    Date: December 24, 2004
    Source: Ohio State University

    Evidence shows that around 5,200 years ago, solar output first dropped precipitously and then surged over a short period. It is this huge solar energy oscillation that Thompson believes may have triggered the climate change he sees in all those records.

    https://www.sciencedaily.com/releases/2004/12/041219142907.htm

    Also: Major Climate Change Occurred 5,200 Years Ago
    https://phys.org/news/2004-12-major-climate-years-evidence-history.html
    – – –
    2400 years before the Homeric Minimum

  40. oldmanK says:

    oldbrew: see oldmanK says:
    July 28, 2020 at 6:48 am

  41. oldbrew says:

    Javier says:
    The 208-year de Vries cycle is only apparent in cosmogenic records as prominent peaks of isotope generation with that spacing during one thousand-year period windows centered 2500 years apart.

    http://euanmearns.com/periodicities-in-solar-variability-and-climate-change-a-simple-model/

    The de Vries cycle is 1/12th of 2503 years.
    2503*24 = 2402.88*25, as per the post above.

    The de Vries pair is 21 Jupiter-Saturn conjunctions.
    The geometric mean of the de Vries pair and 2402.88 years is 1001.2 years (‘one thousand-year period window’).
    1001.2 * 2.4 = 2402.88
    1001.2 / 2.4 = 417.166 = 21 J-S

  42. Paul Vaughan says:

    Monde Stir US…

    104.212132286568 = ⌊(e^√10π)^(1/4)⌉^4 – e^√10π
    -103.947369666712 = ⌊(e^√13π)^(1/2)⌉^2 – e^√13π
    104.007114381762 = ⌊(e^√18π)^(1/4)⌉^4 – e^√18π
    104.001742574386 = ⌊(e^√22π)^(1/2)⌉^2 – e^√22π
    -103.999977946281 = ⌊(e^√37π)^(1/2)⌉^2 – e^√37π
    104.000034332275 = ⌊(e^√58π)^(1/2)⌉^2 – e^√58π — Srinivasa Ramanujan

    “I’m USA Eur. aura is ink CR red a bull” — Rihanna

    152.000658869743 = ⌊(e^√25π)^(1/2)⌉^2 – e^√25π
    152.000658869743 = ⌊(e^5π)^(1/2)⌉^2 – e^5π

    …Double Vision

    152 = 104 + 48
    304 = 208 + 96
    608 = 416 + 192
    1216 = 832 + 384
    2432 = 1664 + 768

    2432 = 16*152 = 16*(104 + 48) = 8*(208 + 96)

    ⌊2432.25439579357⌋ = 8 * ( ⌊208.886643858908⌋ + ⌊96.1829470900285⌋ )

    “now We’re ROC kin on the ⌊D-ants floor⌋ act UN naughty” — R

  43. Paul Vaughan says:

    mod 24 Mono Pole Lie on the C[ENSO]Rship

    “WHO knew?…” — Rihanna “Please Don’t Stop the Music

    178.266850068779 = (208.886643858908)*(96.1829470900285) / (208.886643858908 – 96.1829470900285)
    65.8581963269421 = (208.886643858908)*(96.1829470900285) / (208.886643858908 + 96.1829470900285)
    131.716392653884 = (208.886643858908)*(96.1829470900285)/((208.886643858908+96.1829470900285)/2)

    “I wanna take Q away — R

    96.1829470900285 = (208.886643858908)*(178.266850068779) / (208.886643858908 + 178.266850068779)
    1216.12719789678 = (208.886643858908)*(178.266850068779) / (208.886643858908 – 178.266850068779)

    2432.25439579357 = (417.773287717816)*(356.533700137559) / (417.773287717816 – 356.533700137559)

    what goes on: bet wean US
    No. 1 has tune O: this is a private show” — R

  44. Paul Vaughan says:

    On the last theme, also recall from far above:

    2432.25439579355 = (208.886643858908)*(192.365894180057) / (208.886643858908 – 192.365894180057)

    Scalewise note halving lands on components all equal to themselves mod 24 in 3 steps:

    152 = 104 + 48
    76 = 52 + 24
    38 = 26 + 12
    19 = 13 + 6

    Leaving it there for now.

    _ _ _

    Mediating “New” Brew

    Until now we’ve only looked at axial PAIRS — for example locking on an E+V, J+S , or U+N axis. With some creative naming to mirror the challenges of present human times, what structures axially lock to The Jovian Collective?

    Let’s begin by OBserving how J-S slips on the harmonic mean of ALL the Jovians, before casting “symphonic” light on something truly primorial.

    Definining the axis:

    29.3625733662892 = harmean(11.8626151546089,29.4474984673838,84.016845922161,164.791315640078)

    Defining how it’s split by the long JS cycle:

    30.4320075307947 = (835.546575435631)*(29.3625733662892) / (835.546575435631 – 29.3625733662892)

    28.3657510805208 = (835.546575435631)*(29.3625733662892) / (835.546575435631 + 29.3625733662892)

    Q: With what period does the cycle-pair thus derived (“65” & “66” below) slip collectively an on axially-locked frame?

    ⌊ 30.4320075307947 / 19.8650360864628 ⌉ = ⌊1.53193819524611⌉ = 2
    30.4320075307947 / 2 = 15.2160037653974
    65.0170708690834 = (19.8650360864628)*(15.2160037653974) / (19.8650360864628 – 15.2160037653974)

    ⌊ 28.3657510805208 / 19.8650360864628 ⌉ = ⌊1.42792346095212⌉ = 1
    28.3657510805208 / 1 = 28.3657510805208
    66.2869734167119 = (28.3657510805208)*(19.8650360864628) / (28.3657510805208 – 19.8650360864628)

    ⌊ 66.2869734167119 / 16.9122914926352 ⌉ = ⌊3.91945547092645⌉ = 4
    66.2869734167119 / 4 = 16.571743354178
    822.985424077451 = (16.9122914926352)*(16.571743354178) / (16.9122914926352 – 16.571743354178)

    ⌊ 822.985424077451 / 65.0170708690834 ⌉ = ⌊12.6579898644556⌉ = 13
    822.985424077451 / 13 = 63.3065710828809
    2406.31881498933 = (65.0170708690834)*(63.3065710828809) / (65.0170708690834 – 63.3065710828809)

    Reason Ants

    Varying aggregation criteria for comparative study, a sharp observer won’t take long to notice nearly-resonant parallel structure. Slip cycles hinge on geometric limits. Discrete-continuous relations define boundary conditions limiting dynamics.

    It’s remarkable that in years of discussion no one ever algebraically defined or systematically cataloged the wide variety of geometric structures that can be precisely defined with varying aggregation criteria. It doesn’t make any sense that there would be such a striking level of ignorance of something so tractable.

    Hubristic C[ENSO]Rship

    Weather ignorance or deception (systematic dominance by elite prosperity plan), IT’s dark either way. Certainly we aren’t going to take an IT must fear IC “climate” less sun from those WHO can’t or won’t make explicit truly monstrous boundaries limiting heliospheric geometry.

    The Surf ACE

    Naturally fleeting mono pole lie yields PR O fit ably in green land to monster US competition, surf(UN) invigorating cycles of rebirth.

    Here we studied full-circle orbits.

    Next we’ll look at something “node-worthy”. Soon you may understand precisely how JS monstrously pins UN in primorial luck D-own.

    Populism won’t look so geometric ally O-min US if just a few more simply real eyes:

    “Mathematics is the queen of the sciences—and number theory is the queen of mathematics.” — Gauss

    The D-tails may be in torn ally sCRambled, bot IT is a belle law in aggregate.

  45. Paul Vaughan says:

    Mods: Mediation is trapped in the filter. Morse queued 4 real ease soon. Watch 4 IT.

  46. Paul Vaughan says:

    2406.7904774739 = 100 * 24.067904774739

    Review

    29.4530098610638 = 2 * 14.7265049305319
    14.7265049305319 = 1 / 0.0679047747389632
    The tower in the mod 24 spectrum.

    0.0679047747389632 = ⌊(e^√7π)^(1/1)⌉^1 – e^√7π
    24.067904774739 = ⌊(e^√7π)^(1/2)⌉^2 – e^√7π
    24.067904774739 = ⌊(e^√7π)^(1/3)⌉^3 – e^√7π
    24.067904774739 = ⌊(e^√7π)^(1/4)⌉^4 – e^√7π
    24.067904774739 = ⌊(e^√7π)^(1/6)⌉^6 – e^√7π
    24.067904774739 = ⌊(e^√7π)^(1/12)⌉^12 – e^√7π

    Leech let US.

  47. Paul Vaughan says:

    Q:Con’s peer ACE sea?

    BRI10nese: “Eur. talks IC comms.lippin’ UN dr.”

    1793_)
    1826 (
    1859_) SCD care ring 10? “I’m USA Eur. aura is UN CR edible – if EU don’t have to go: don’t” – Rihanna
    1892 (
    1925_)
    1958 (
    1991_) Q: 132 years later (peer score bon luke back) what can be noticed?
    2024 ( “Ink CR red ABull” rich “helped” those they muzzled and locked out of jobs and housing?
    2057_)
    2090 (

    “Therefore, our modern technological society needs take this seriously, study extreme space weather events, and also understand all the subtleties of the interactions between the Sun and the Earth.”

    Miss UN dr. cure $rent of mess age?

    =
    [Obama] drew particular attention during the 2020 Democratic primary when he said many of the world’s problems have been due to “old people, usually old men, not getting out of the way.”
    =

    Weather rich or just naive left’s tricked off-track: bot Q-wrap as DCoy “conspiracy”.

    =
    One of their political weapons is ‘cancel culture’ — driving people from their jobs, shaming dissenters and demanding total submission from anyone who disagrees. This is the very definition of totalitarianism, and it is completely alien to our culture and our values, and it has absolutely no place in the United States of America,” Trump said in a July 3 speech at Mount Rushmore.
    =

    (to be continued)

  48. Paul Vaughan says:

    Q: Seize the Day Light?

    The next 2 comments rock naive D’Cept shh! UN, but rewards fall O 4bearing with icon IC We’ave.

    =
    Every highly composite number is a product of primorials (e.g. 360 = 2 × 6 × 30).
    […]
    Any completely multiplicative function is defined by its values at primorials, since it is defined by its values at primes, which can be recovered by division of adjacent values.
    […]
    The n-compositorial of a composite number n is the product of all composite numbers up to and including n. The n-compositorial is equal to the n-factorial divided by the primorial n#. The compositorials are 1, 4, 24, 192, 1728, […]
    =

    Q: Pole ice brutal IT seek cures free doom of speech?

    Note that 30030 is the first primorial with primorial+1 = prime.
    30031 came up in precisely monstrous Suggestions without explanation.

    I wanna take q-away” — Rihanna “pleas Don’s top the music”

    IT’s now dawn: Artificial “Intelligence” reads comments and writes presscriptions.

    “Logic” Based in False Assumptions:shh!hacks Constitution

    The cue is weather C[ENSO]Rship’s caste.in caring 10.

    Also note that primorial 2310 + 1 = 2311 is a prime. When a few readers get really serious about review, they’ll encounter 2311 ….for commparissun (wool eye soon in.off) with 2310 D-arrived below “Just Dance”.

    Skip the Disco

    West turn C[ENSO]Rship D-sends:
    Canoe XI past in.comprehensible disruption and corruption fatally land Don US.in UN O-Thor hell.

    Real climate discussion hasn’t eve UN Bay gone. At this stage ( see 22 a gust of 2020 win-D at 11:22 AM ) we’re only test UN on the C[ENSO]Rship weather IT will be all O-wed to beg.in.

  49. Paul Vaughan says:

    This Ain’t Bridge IT App.

    “IT’s get tune late – I’m make kin my way over to my favor IT plays…” — Rihanna

    385 = (13*11*7*5*3)/(13+11+7+5+3)
    210 = 7*5*3*2
    96.25 = 385 / 4

    66 = (210)*(96.25) / (210 + 96.25)
    132 = (210)*(96.25)/((210+96.25)/2) — app.ears carbon luke back above and below

    96.25 = (210)*(66) / (210 – 66)
    385 = 4 * 96.25 = 396 – 11 = (4*9*11) – 11

    hear’s mnemonic D-vice sov. 1859
    te[a]ch UN O-crazy ware O as sin “or we’ll” tote AI IT eerie UN O-Thor-IT

    8 = 3 + 5 = 11 – 3
    385 = 35 * 11

    210 = 35 * 6
    66 = 11 * 6

    2310 = 11*7*5*3*2 = 35 * 66

    Few will get this, but TB might. Hear’z-a6[LOD]s twist know win XI=6 in rome UN no. morals:

    165 = 2.5 * 66
    66 = 2/5 * 165

    key no. Q-win saw dawns *’s’ ABove 8.
    Q: WHO’s lucky star’s key UN hitch 70s class [SIM phone] IC?

    Fort Win tee acu-punk cheer ring: IT’s the Great Chimerican Reef IT!

    Caring 10 is a big deal for stay Be/11/e IT.
    Q: WHO sat? UN. Knock, knock. WHO’s there? Sat. Sat WHO? Sat UN.

    “Don’t fee like$SatUN bot ayaM to them” — kneel young “key[*8]pawn rock in$in the fee world”

    She seas the need, IV-without better cooperation in the buy area, We dr.own in luck D-own 4 muzzled evicition by sci11UNs.

    No. wororries: 6’ll save US from 16[.5]figures pen-job Bay-err rihanna grande.

    16 figures pen job buy don mess age:

  50. Paul Vaughan says:

    Q: IC’s now dawn?

    “I still can’t believe they’re IS ally
    I never believe them and I never assume
    I think EU already know how far I’d go not to say
    Leap of faith – DO EU doubt?
    We’ve got somethin’ to reveal
    No. 1 Can know how we feel
    Whatever EU DO: don’t tell anyone”

    Queens of the Stone Age “The Lost Art of Keeping a Secret”

    For the longest time we never dared to say a word about the 1859 & 1991 anomalies.

    Anyone can easily note the dates are 132 years apart, even if they don’t think to anchor the frame to the Carrington Event.

    How fast does 132 slip past other structures we’ve already noted?
    The answers are familiar.

    ⌊ 208.886643858908 / 131.716392653884 ⌉ = ⌊1.58588190619376⌉ = 2
    208.886643858908 / 2 = 104.443321929454
    504.413226524325 = (131.716392653884)*(104.443321929454) / (131.716392653884 – 104.443321929454)

    ⌊ 178.266850068779 / 131.716392653884 ⌉ = ⌊1.35341430536454⌉ = 1
    178.266850068779 / 1 = 178.266850068779
    504.413226524326 = (178.266850068779)*(131.716392653884) / (178.266850068779 – 131.716392653884)

    ⌊ 131.716392653884 / 96.1829470900285 ⌉ = ⌊1.36943602414881⌉ = 1
    131.716392653884 / 1 = 131.716392653884
    356.533700137558 = (131.716392653884)*(96.1829470900285) / (131.716392653884 – 96.1829470900285)

    ⌊ 131.716392653884 / 50.0715412599931 ⌉ = ⌊2.63056397585119⌉ = 3
    131.716392653884 / 3 = 43.9054642179614
    356.533700137558 = (50.0715412599931)*(43.9054642179614) / (50.0715412599931 – 43.9054642179614)

    ⌊ 131.716392653884 / 100.143082519986 ⌉ = ⌊1.31528198792559⌉ = 1
    131.716392653884 / 1 = 131.716392653884
    417.773287717817 = (131.716392653884)*(100.143082519986) / (131.716392653884 – 100.143082519986)

    Easy stuff.
    With 2020 hindsight? You could say too easy — way too easy.

    Aside

    To within less than 0.1% the following tune OB’s suggestions (about products, squares, square-roots, and geometric means):

    ⌊ 504.413226524327 / 356.533700137559 ⌉ = ⌊1.41477012223448⌉ = 1
    504.413226524327 / 1 = 504.413226524327
    1216.12719789678 = (504.413226524327)*(356.533700137559) / (504.413226524327 – 356.533700137559)

    ⌊ 1008.82645304865 / 713.067400275117 ⌉ = ⌊1.41477012223448⌉ = 1
    1008.82645304865 / 1 = 1008.82645304865
    2432.25439579355 = (1008.82645304865)*(713.067400275117) / (1008.82645304865 – 713.067400275117)

    What happened in 1991 is now a full order of magnitude less of a mystery.

    Trusting naive analogs leads to slip psych culls with truly monstrous consequences. A good defenceman can be neither ignorant nor naive, especially when facing abusively deceptive opponents.

    If someone thinks they’re going to line up 66 and 132 year cycles and predict the future, there are some big less UNs.

    For example nonlinear aliasing implies slip cycles, with discrete costs (call them slip psych culls) for failing awareness.

    Another big challenge: Where shock dissipation has more than 1 possible avenue, sensible data interpretation demands integration of coupled multivariate bundles. This goes far beyond the skill and judgement of almost all data analysts. The pool of capable candidates is vanishingly small.

    We already knew deficient western math education systems need a major overhaul, but this underscores the need for decades of mentoring beyond that. Brutal political interference is sure to obstruct anyone working on these problems.

    Board D-UN the Half-Pipe

    =
    “I realized that I was crazy to have imagined that the Supreme Court, or Congress, or President Obama, seeking to distance his administration from President George W. Bush’s, would ever hold the IC legally responsible — for anything,” he wrote, using an abbreviation for the intelligence community.
    =

    https://txtify.it/ctvnews.ca/world/william-barr-vehemently-opposed-to-pardoning-snowden-ap-exclusive-1.5073936

    Eddy Grant’s Canoe

    Eye sea at least 3 avenues bundled to ROC-piercing enlightenment:

    IC = Intensive Care
    IC = Ivanka Charvatova
    IC = “Intelligence” (what IT’s call-D) Comm. UN IT

    “If EU could flip a switch and open Eur. 3rd eye, EU’d sea that _______” — Muse

  51. Paul Vaughan says:

    Distill UN Monster US Moonshine

    Remember Uncle Jesse Duke’s plea bargain?

    ⌊ 131.716392653884 / 104.443321929454 ⌉ = ⌊1.26112795170238⌉ = 1
    131.716392653884 / 1 = 131.716392653884
    504.413226524329 = (131.716392653884)*(104.443321929454) / (131.716392653884 – 104.443321929454)

    and Laskar’s p = 25685?

    ⌊ 417.773287717816 / 131.716392653884 ⌉ = ⌊3.17176381238752⌉ = 3
    417.773287717816 / 3 = 139.257762572605
    2432.25439579359 = (139.257762572605)*(131.716392653884) / (139.257762572605 – 131.716392653884)

    ⌊ 2432.25439579341 / 131.716392653884 ⌉ = ⌊18.4658442794188⌉ = 18
    2432.25439579341 / 18 = 135.125244210745
    5221.17476429629 = (135.125244210745)*(131.716392653884) / (135.125244210745 – 131.716392653884)

    Well, “the good ole boys” got the general airborn.

    20884.6990571851 = 4 * 5221.17476429629

    0.999978499752023 = (20884.6990571851)*(1.00002638193018) / (20884.6990571851 + 1.00002638193018)
    0.999978614647502 (-0.000000089121 = % error)
    25684.4120517011 = (1.00001743371442)*(0.999978499752023) / (1.00001743371442 – 0.999978499752023)

    0.999978500643215 = (25685)*(1.00001743371442) / (25685 + 1.00001743371442)
    1.00002638282146 = (20884.6990571851)*(0.999978500643215) / (20884.6990571851 – 0.999978500643215)
    1.00002638193018 (-0.000000089125 = % error)

    error = 2.8 seconds per century

    Q: WHO seize simple implicationsand D-ants colony whatnotice?

  52. oldmanK says:

    PV thanks again, good tips though maybe not as intended. Anyway stay safe and don’t trust; its wicked.

    Two comments deriving from the above:

    Quote: “Every highly composite number is a product of primorials (e.g. 360 = 2 × 6 × 30).” Wiki says “A highly composite number is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan (1915). However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato,—“. Not quite. 360 = 3*3*2*2*2*5 was known minimally 3000 years before Plato. It derives from the days in a season which vary about 90, where 90 = 3*3*2*5 is the best scale to use. Greeks inherited from Babylonian – base60 (but not Sumerian since that was a quinary system).
    Used here : https://melitamegalithic.wordpress.com/2017/04/17/melitamegalithic/

    Re the chart on ‘Eccentricity, obliquity, precession’, to my mind that does not hold for millions of years. Since the earth sees abrupt changes in gravitational forces (at times of Kepler trigons + moon), then it is not a steady cycle, even without a change of obliquity. The latter may also be subject to abrupt change for the same reason given for precession.

  53. Paul Vaughan says:

    oldmanK: The question I would hope you would be left asking yourself (focusing on my last comment as an aside in the context of the bigger message) is about models.

    The bigger picture: Seidelmann’s (1992) parameters precisely tie an algebraic generalization of Bollinger’s (1944) graphical method directly to Ramanujan and monstrous moonshine.

    In Bulgarian they have an expression “nogo slatka”. (I have no idea if I’m spelling that right.)
    It means: “too sweet”. It’s expression is often accompanied by giggling.

    The Jesse Duke aside: Why should the correction of a single highly suspicious error in V&D tie all of the lines in that classic paper so precisely to all of the lines in Laskar’s classic paper??

    This isn’t a question I expect anyone to answer. (We ink, we yank.)

    Node-worthy harmonic means: Perhaps not everyone will pick up The Jovian Collective subtle tees of 2310 & 210 (next stop on the scheduled route), presented alongside your fav-O-red 980 pet layer-red cake.

  54. Paul Vaughan says:

    Before we move on to nodes, here’s something I left out above — see August 21, 2020 at 10:19 pm — when I introduced axial lock to The Jovian Collective:

    ⌊ 65.0170708690834 / 16.9122914926352 ⌉ = ⌊3.8443679200655⌉ = 4
    65.0170708690834 / 4 = 16.2542677172708
    417.761369612522 = (16.9122914926352)*(16.2542677172708) / (16.9122914926352 – 16.2542677172708)

    ⌊ 417.761369612522 / 65.0170708690834 ⌉ = ⌊6.42541049647892⌉ = 6
    417.761369612522 / 6 = 69.6268949354204
    982.019421406588 = (69.6268949354204)*(65.0170708690834) / (69.6268949354204 – 65.0170708690834)

    See JEV slip-cycle previously derived (491.049532939153 with 982 phi notes) for comparison. That was back in the clumsier days before I condensed Bollinger’s method into the terse “climate casino” algorithm. Everything became orders of magnitude simpler conceptually when framed around locked axes and harmonic means.

    Also see: _1_ , _2_ , & _3_

    Alert readers will have noticed a logical reversal needed in a statement I made in haste about 30031 above.

    30031 = 59 * 509
    How’s that for a monstrous centerpiece?

    The first primorial+1 that’s not prime ties simply to monster group representation.

    Recall:
    196883 = (59-12)*(59)*(59+12)
    Thus:
    196883 = (30031/509-12)*(30031/509)*(30031/509+12)

    That’s the missing link tying lunisolar 96^3 (see previous Suggestions) to the present discussion of jovian slip-cycles. Near an exchange I had with TB about John Conway I gave all the needed ingredients.

    Attentive readers should soon see (at least vaguely while striving for deeper grasp) Piers Corbyn’s 132 year lookback rationale in monstrous context.

    132 is an integrated multivariate bundle. 66 alone doesn’t tell the story. 132 has variable expression, affording diagnostic avenues by which to explore multivariate observations holistically.

    Hammer-wielders will focus sequentially on 1 variable at a time.

  55. Paul Vaughan says:

    Sea Arch Tar Gets

    Quick note to tie 509 to the next primorial:

    1001 = 13*11*7

    510510 = 17*13*11*7*5*3*2
    509 = ( 17*13*11*7*5*3*2 – 13*11*7 ) / ( 13*11*7 )
    509 = ( 17*5*3*2 – 1 )

    Comparatively note parallel structure:
    509 = (17)*(5*3*2) – 1
    30031 = (13*11*7)*(5*3*2) + 1

    59 = ( (13*11*7)*(5*3*2) + 1 ) / ( (17)*(5*3*2) – 1 )

    Leaving it there for now.

    Humility: We should have seen the monstrous tie-in years ago. Has anyone ever seen it brought up in (public) climate discussion? Or in orbital stability discussions? More generally: Where is the knowledge applied in design? Is that public or secret (trade secret, intellectual property, classified, etc.)?

    If we had the precise model parameters used in IC’s study, that would pinpoint database sea arch tar gets.

    Next: 2310, 210, & 982 from The Nodes of The Jovian Collective.

  56. Paul Vaughan says:

    Ramanujan Dialed Leadership into Monstrous Cycles

    We’ve seen JEV 104 & 208 many times.
    Let’s take a quick look at JV & JE.

    JV

    0.648846557532906 = (11.8626151546089)*(0.615197263396975) / (11.8626151546089 – 0.615197263396975)
    0.584866011394422 = (11.8626151546089)*(0.615197263396975) / (11.8626151546089 + 0.615197263396975)
    1.16973202278884 = (11.8626151546089)*(0.615197263396975)/((11.8626151546089+0.615197263396975)/2)

    ⌊ 11.8626151546089 / 0.615197263396975 ⌉ = ⌊19.2826201617126⌉ = 19
    11.8626151546089 / 19 = 0.624348166032049
    41.9737045040379 = (0.624348166032049)*(0.615197263396975) / (0.624348166032049 – 0.615197263396975)

    ⌊ 41.9737045040379 / 0.648846557532906 ⌉ = ⌊64.6897236592169⌉ = 65
    41.9737045040379 / 65 = 0.645749300062122
    135.278456610988 = (0.648846557532906)*(0.645749300062122) / (0.648846557532906 – 0.645749300062122)

    1.17993473711141 = (135.278456610988)*(1.16973202278884) / (135.278456610988 – 1.16973202278884)

    ⌊ 1.17993473711141 / 0.648846557532906 ⌉ = ⌊1.81851120794699⌉ = 2
    1.17993473711141 / 2 = 0.589967368555707
    6.50141930950091 = (0.648846557532906)*(0.589967368555707) / (0.648846557532906 – 0.589967368555707)

    -6.49165552476461 = ⌊(e^2π)^(1/2)⌉^2 – e^2π
    -6.49165552476461 = ⌊(e^√4π)^(1/2)⌉^2 – e^√4π

    104 = 16 * 6.5

    JE

    1.0920796543202 = (11.8626151546089)*(1.00001743371442) / (11.8626151546089 – 1.00001743371442)
    0.922270140470534 = (11.8626151546089)*(1.00001743371442) / (11.8626151546089 + 1.00001743371442)
    1.84454028094107 = (11.8626151546089)*(1.00001743371442)/((11.8626151546089+1.00001743371442)/2)

    0.546039827160099 = 1.0920796543202 / 2

    ⌊ 11.8626151546089 / 1.00001743371442 ⌉ = ⌊11.8624083487694⌉ = 12
    11.8626151546089 / 12 = 0.988551262884078
    86.2160970415933 = (1.00001743371442)*(0.988551262884078) / (1.00001743371442 – 0.988551262884078)

    ⌊ 86.2160970415933 / 1.0920796543202 ⌉ = ⌊78.9467111675672⌉ = 79
    86.2160970415933 / 79 = 1.0913430005265
    1617.90178364771 = (1.0920796543202)*(1.0913430005265) / (1.0920796543202 – 1.0913430005265)

    1.84664560787919 = (1617.90178364771)*(1.84454028094107) / (1617.90178364771 – 1.84454028094107)

    ⌊ 1.84664560787919 / 1.0920796543202 ⌉ = ⌊1.69094406307633⌉ = 2
    1.84664560787919 / 2 = 0.923322803939597
    5.97511772872138 = (1.0920796543202)*(0.923322803939597) / (1.0920796543202 – 0.923322803939597)

    ⌊ 5.97511772872138 / 0.546039827160099 ⌉ = ⌊10.9426408688857⌉ = 11
    5.97511772872138 / 11 = 0.543192520792853
    104.17029708512 = (0.546039827160099)*(0.543192520792853) / (0.546039827160099 – 0.543192520792853)

    104.212132286568 = ⌊(e^√10π)^(1/2)⌉^2 – e^√10π
    104.212132286568 = ⌊(e^√10π)^(1/4)⌉^4 – e^√10π

  57. Paul Vaughan says:

    Mods: You’ll find a PC Mayan guide in the anti-math filter.

  58. Paul Vaughan says:

    Memorreryless Relay Channels

    Here’s a simpler one.

    JMa

    2.23525255532127 = (11.8626151546089)*(1.88084761346252) / (11.8626151546089 – 1.88084761346252)
    1.62344612704193 = (11.8626151546089)*(1.88084761346252) / (11.8626151546089 + 1.88084761346252)
    3.24689225408385 = (11.8626151546089)*(1.88084761346252)/((11.8626151546089+1.88084761346252)/2)

    ⌊ 11.8626151546089 / 1.88084761346252 ⌉ = ⌊6.30705808897014⌉ = 6
    11.8626151546089 / 6 = 1.97710252576816
    38.6331302796673 = (1.97710252576816)*(1.88084761346252) / (1.97710252576816 – 1.88084761346252)

    ⌊ 38.6331302796673 / 2.23525255532127 ⌉ = ⌊17.2835638584538⌉ = 17
    38.6331302796673 / 17 = 2.27253707527455
    136.24137607071 = (2.27253707527455)*(2.23525255532127) / (2.27253707527455 – 2.23525255532127)

    3.32616102529218 = (136.24137607071)*(3.24689225408385) / (136.24137607071 – 3.24689225408385)

    1.66308051264609 = 3.32616102529218 / 2

    ⌊ 2.23525255532127 / 1.66308051264609 ⌉ = ⌊1.34404350139658⌉ = 1
    2.23525255532127 / 1 = 2.23525255532127
    6.49700560030246 = (2.23525255532127)*(1.66308051264609) / (2.23525255532127 – 1.66308051264609)

    104 = 16 * 6.5

    John Conway herd. Monstrous Moonshine’s the voice of God.

  59. oldmanK says:

    PV thanks for the tip. The Seidelmann 1992, a long read but putting much in perspective.

    The tip ‘models’ I found at ch5, in which I tried to understand the analyses. I get the impression it was all secular consideration.
    That in spite of at 1.432 ‘Forces on Extended Bodies’ that say “A simple example is that of a two-body system (such as the Sun and a planet in which one body is an oblate spheroid that is rotating about its principal axis. The torque which may be considered to act on the equatorial bulge, has a gyroscopic effect and causes the principal axis to precess around the normal to the orbit plane at a constant inclination. In more complex systems a nutational motion is superimposed on the main precessional motion and an irregular body may tumble as it moves around its orbit”.

    This is a point I have been concerned with. Apart from the secular forces there are also abrupt short duration torques, which act on the oblate earth when in particular orientations (solstice or equinox times?). One would be an S,J,E,moon,Sun alignment. Apparently no such conditions are considered. It also could mean that the inclination to the orbit plane is no longer constant. Maths may give an indication (what Dodwell questioned). Other proxies -read archaeology and ancient texts- are more vocal there.

    The Astro Almanac makes a review of the historical aspects (a good read). Ch12 on calendars. Historical info can and is informative, providing data that the Almanac does not provide. Example early Akkadian text of the ‘Flood from the West’ that say ‘the gods planned the destruction on the darkest night of the month when there was no moon’ (monstrous moonlight or darkness??). It is good to remember the Akkadian text still remembered events of late forth/early third millennium. Ancient texts have some to say about abrupt events, and the celestial factor that might be to blame.

    Re the Bulgarian expression, I think much is lost there in repeated translation. Whatever, the Bulgarians retain much that others have suffered and lost from ‘cancel culture’ (to use a modern term but which applies perfectly). And in connection with Ch12 – Calendars, there were two technical inventions that changed the future of the naked ape, both agrarian. That is, the calendar and the plough. Both go back to fourth millennium bce and earlier. The Bulgarians never quite lost their ancient folklore. The agrarian Almanac has a very long history; so has the calendar. See https://mythandmegalithica.blogspot.com/2019/08/normal-0-microsoftinternetexplorer4_25.html

  60. Paul Vaughan says:

    2432.42097975989 = 2^5 * φ^9
    2432 = 32 * 76 = 16 * 152

    No doubt OB seize 76 = 4 * 19 = 152 / 2 is a Lucas number.
    The relevance of the following will become apparent soon:

    2605.00959689363 = 5 * φ^13
    2605 = 5 * 521 = 25 * 104.2

    OB will note:
    5 & 13 are both Fibonacci numbers.
    521 = 5 * 104.2 is a Lucas number.
    521 is prime.
    520 (not a prime) = 521 – 1 = 5 * 104

  61. Paul Vaughan says:

    FYI I’ve cracked this whole thing wide open.
    Mayans, Ramanujan, Bollinger, Seidelmann — it all ties in tightly with number theory.

    Not for a second do I believe this is not already deeply understood by some.
    I am beyond certain that I am simply reinventing knowledge of the wheel.

    Maybe the difference is I didn’t work from theory but rather generalized methods based on Bollinger & Ramanujan to take precise measurements of discrete-continuous relations in Siedelmann’s model to arrive at deeper appreciation of Mayan wisdom.

  62. oldbrew says:

    PV wrote: 521 = 5 * 104.2 is a Lucas number.

    All Lucas numbers are the sum of two Fibonacci numbers, in this case: 377+144.
    The ratio of the two Fib. nos. approaches Phi² ever closer as the series goes on.

  63. Paul Vaughan says:

    Mods: There’s an important calculation-laden comment stuck in the filter on the most recent XR thread. For sound reasons it’s best to have IT out in the open (and hopefully widely well-understood) before the derivation of almost-integer 2310 is posted here.

  64. Paul Vaughan says:

    Moderators?

  65. Paul Vaughan says:

    Moderators: You will find key calculations (a repeat of a previous comment stripped of entertainment content) caught in the filter, along with completely benign commentary.

  66. Paul Vaughan says:

    Normally I wait until previous comments have been released from the filter, but we are already a full month behind schedule so I’m readily altering course.

    24.0685740890598 = 22.1384769776823 + 11.8626151546089 – 9.93251804323141
    24.067904774739 = ⌊(e^√7π)^(1/x)⌉^x – e^√7π with x = 2, 3, 4, 6, 12
    0.002780941370 = % error

    1.0000278094137 = 24.0685740890598 / 24.067904774739
    1.00002638193018 = 365.259636 / 365.25 = terrestrial anomalistic year period
    0.000142744586 = % error

  67. Paul Vaughan says:

    Oldbrew, I need to check that you are aware of this:

    1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 8^2 = 10/(1/2-836/2432)

    split sum
    not prime
    1 + 9 + 15 = 25 = 5^2
    prime
    3 + 5 + 7 + 11 + 13 = 39 = 64 – 25 = 8^2 – 5^2

    (8^2 – 5^2)*(792 – 22) = 13*11*7*5*3*2 = 30030

    Recall that 30031 is the first primorial+1 that’s composite:
    (8^2 – 5^2)*(792 – 22)+1 = 59 * 509
    (8^2 – 5^2)*(7920 – 22*5*2)+8^2*(1/2-836/2432) = 59 * 5090

    ⌊ 10181.3753891113 / 2 ⌋ = 5090 (check moderation filter)

    11 = (((8^2 – 5^2)*(7920 – 22*5*2)+8^2*(1/2-836/2432))/5090) mod 24

    196883 = (((8^2-5^2)*(7920-22*5*2)+8^2*(1/2-836/2432))/5090)*((((8^2-5^2)*(7920-22*5*2)+8^2*(1/2-836/2432))/5090)^2-(0+1+1+1+2+3+5+8+13+21+34+55))

    Recall key emphasized many times in the past:
    13*11*7*5*3 / (13+11+7+5+3) = 11 * 210 / 6 = 2310 / 6

    As a learning exercise readers can check if this holds for other primorials:
    13*11*7*5*3 / (13+11+7+5+3) is an integer.

    We’ve a lot more ground to cover.

  68. Paul Vaughan says:

    Found on Conway Triangle base:
    181=(2^15-7)^(1/2)
    11=(2^7-7)^(1/2)
    5=(2^5-7)^(1/2)
    3=(2^4-7)^(1/2)
    1=(2^3-7)^(1/2)

    256=(181^2+7)/(11^2+7)
    2=√(181^2+7)/√(11^2+7)=(181^2+7)/(11^2+7)=(181^2+7)/(11^2+7)

    260=(181^2+7)/(11^2+7)+(11^2+7)/(5^2+7)

    104=2*((181^2+7)/(11^2+7)+(11^2+7)/(5^2+7))/5
    152=(181^2+7)/(11^2+7)/2+(11^2+7)/(5^2+7)/2+2*(2^7-7)^(1/2)
    744=2^2*(2^15-7)^(1/2)+2^2*(2^5-7)^(1/2)

    96=(2^15-7)^(1/2)/2+(2^7-7)^(1/2)/2
    178=152+26
    100=48+52

    2310=√((2^7-7)*(70^2)*(2^5-7)*(2^4-7))/5
    210=√((70^2)*(2^5-7)*(2^4-7))/5

    30=2√((2^5-7)*(2^4-7))
    6=2√(2^4-7)

    7=2+5
    10=2*5
    70=2*5*(2+5)
    70^2=1^2+2^2+3^2+….+22^2+23^2+24^2

  69. Paul Vaughan says:

    29.3625733662893 = harmean(11.8626151546089,29.4474984673838,84.016845922161,164.791315640078)
    29.3622914957372 = 4*(59^(1/2)-MOD(59^(1/2),1)/2)
    0.000959974640 = % error

    59.0021651086552 = (29.3625733662893/4+MOD(29.3625733662893/4,1))^2
    59
    0.003669675687 = % error

    5256 = 7920-2400-240-24 = 7920-2664

    44.2780679879602 = 29.3625733662893 * 2^3*5^(1/2) / 11.8626151546089
    22.1390339939801 = 44.2780679879602 / 2

  70. Paul Vaughan says:

    ⌊ 100.143082519986 / 22.1392314983837 ⌉ = ⌊4.52333146827148⌉ = 5
    100.143082519986 / 5 = 20.0286165039972
    210.08956088803 = (22.1392314983837)*(20.0286165039972) / (22.1392314983837 – 20.0286165039972)

    ⌊ 131.716392653884 / 22.1392314983837 ⌉ = ⌊5.94945640563451⌉ = 6
    131.716392653884 / 6 = 21.9527321089807
    2605.99576083602 = (22.1392314983837)*(21.9527321089807) / (22.1392314983837 – 21.9527321089807)

    104.239830433441 = 2605.99576083602 / 25

  71. Paul Vaughan says:

    1/5 = (1/11)*(1/16) / (1/11 – 1/16)
    1/6 = (1/5)*(1/11) / (1/5 – 1/11)
    1/16 = (1/5)*(1/11) / (1/5 + 1/11)

    2.90909090909091 = 2432 / 836
    6.4 = (2.90909090909091)*(2) / (2.90909090909091 – 2)
    1.18518518518519 = (2.90909090909091)*(2) / (2.90909090909091 + 2)

    144 = 59^2 – 196883 / 59
    196883 = 59 * ( 59^2 – (0+1+1+1+2+3+5+8+13+21+34+55) )

  72. Paul Vaughan says:

    14.6812866831446 = harmean(11.8626151546089/2,29.4474984673838/2,84.016845922161/2,164.791315640078/2)

    14.9438634806119 = (835.546575435631)*(14.6812866831446) / (835.546575435631 – 14.6812866831446)

    7.47193174030594 = 14.9438634806119 / 2

    ⌊ 19.8650360864628 / 7.47193174030594 ⌉ = ⌊2.65862119420933⌉ = 3
    19.8650360864628 / 3 = 6.62167869548761
    58.190595753165 = (7.47193174030594)*(6.62167869548761) / (7.47193174030594 – 6.62167869548761)

    (to be continued — calculation-laden continuation may get stuck in filter)

  73. Paul Vaughan says:

    (calculation-laden continuation — filter often stops long calculations, especially ones containing round-off bars — a very important one is still stuck in the filter above)

    ⌊ 58.190595753165 / 16.9122914926352 ⌉ = ⌊3.44072805145922⌉ = 3
    58.190595753165 / 3 = 19.396865251055
    132.03288413456 = (19.396865251055)*(16.9122914926352) / (19.396865251055 – 16.9122914926352)

    ⌊ 58.190595753165 / 19.8650360864628 ⌉ = ⌊2.92929725875602⌉ = 3
    58.190595753165 / 3 = 19.396865251055
    823.031677829323 = (19.8650360864628)*(19.396865251055) / (19.8650360864628 – 19.396865251055)

    ⌊ 823.031677829327 / 60.3231000190041 ⌉ = ⌊13.6437231768599⌉ = 14
    823.031677829327 / 14 = 58.7879769878091
    2310.09042512369 = (60.3231000190041)*(58.7879769878091) / (60.3231000190041 – 58.7879769878091)

  74. Paul Vaughan says:

    3.73596587015297 = 14.9438634806119 / 4

    ⌊ 19.8650360864628 / 3.73596587015297 ⌉ = ⌊5.31724238841867⌉ = 5
    19.8650360864628 / 5 = 3.97300721729256
    62.6178493532417 = (3.97300721729256)*(3.73596587015297) / (3.97300721729256 – 3.73596587015297)

    ⌊ 62.6178493532417 / 16.9122914926352 ⌉ = ⌊3.70250532759029⌉ = 4
    62.6178493532417 / 4 = 15.6544623383104
    210.483935211463 = (16.9122914926352)*(15.6544623383104) / (16.9122914926352 – 15.6544623383104)

  75. Paul Vaughan says:

    4.2280728731588 = 16.9122914926352 / 4

    ⌊ 62.6178493532417 / 4.2280728731588 ⌉ = ⌊14.8100213103611⌉ = 15
    62.6178493532417 / 15 = 4.17452329021611
    329.604596559097 = (4.2280728731588)*(4.17452329021611) / (4.2280728731588 – 4.17452329021611)

    ⌊ 329.604596559097 / 58.190595753165 ⌉ = ⌊5.66422447292388⌉ = 6
    329.604596559097 / 6 = 54.9340994265162
    981.621857403477 = (58.190595753165)*(54.9340994265162) / (58.190595753165 – 54.9340994265162)

  76. Paul Vaughan says:

    I have a storm of further material on deck. I hit a main vein a month ago when I finally checked something really simple that was on my list for years while I dealt with other things.

    Moderators: Please release the comment that is stuck in the filter above. It’s a foundation for further discussion which shall remain impaired until the comment appears.

    I have philosophical notes to share on organizing (with the benefit of earned hindsight) aggregation criteria systematically for comparison, contrast, sorting, classification, taxonomy, and awareness of linkages between different but related patterns contained in the greater whole.

  77. Paul Vaughan says:

    a few notes to help readers see structure

    39 = (8^2 – 5^2) = (13+11+7+5+3)

    770 = (792 – 22) = (4900+49000+49000+490000)^(1/2)
    7700 = (7920 – 22*5*2) = (490000+4900000+4900000+49000000)^(1/2)
    can be partitioned with
    70^2=1^2+2^2+3^2+…+22^2+23^2+24^2

  78. Paul Vaughan says:

    31.5822874030921 = (417.773287717815)*(29.3625733662892) / (417.773287717815 – 29.3625733662892)

    ⌊ 31.5822874030921 / 19.8650360864628 ⌉ = ⌊1.58984294141877⌉ = 2
    31.5822874030921 / 2 = 15.791143701546
    77.0004727270529 = (19.8650360864628)*(15.791143701546) / (19.8650360864628 – 15.791143701546)

    Definition

    From now on I may summarize such calculations like this:

    77.0004727270529 = slip(31.5822874030921,19.8650360864628)

  79. Paul Vaughan says:

    Another example:

    27.4343882446567 = (417.773287717815)*(29.3625733662892) / (417.773287717815 + 29.3625733662892)

    71.9989110164714 = slip(27.4343882446567,19.8650360864628) summarizes the following:

    ⌊ 27.4343882446567 / 19.8650360864628 ⌉ = ⌊1.38103893319137⌉ = 1
    27.4343882446567 / 1 = 27.4343882446567
    71.9989110164714 = (27.4343882446567)*(19.8650360864628) / (27.4343882446567 – 19.8650360864628)

  80. Paul Vaughan says:

    Key test of math(?)resistant filter (with condensed notation) :

    I introduced a simple key in 3 comments (including 1 stuck in filter) here.

    The other side of the coin:

    56.573884877055 = (61.0464822565173)*(29.3625733662892) / (61.0464822565173 – 29.3625733662892)

    372 ~= 371.982512516528 = slip(56.573884877055,19.8650360864628)
    372 * 2 = 744

    -743.777680155239 = ⌊(e^√19π)^(1/3)⌉^3 – e^√19π
    -743.999775171279 = ⌊(e^√43π)^(1/3)⌉^3 – e^√43π
    -743.999816894531 = ⌊(e^√67π)^(1/3)⌉^3 – e^√67π

    744 / 5 / 2 / 2 / 2 = 18.6

    Sharp recognition of Fib and Luc appears to be blocking the community (may be only optics?) from recognizing something more general. I’ve refocused expression accordingly, so more ground’s covered collectively.

  81. Paul Vaughan says:

    Introducing a resource to help readers stay organized on the scenic route:

    http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/

    Helpful roadmap and table:

    https://en.wikipedia.org/wiki/Pariah_group

    (Seemingly endless) interconnectivity relating simple math may try readers’ patience, with delightful rewards 4 those who can weather learning.

    744 = 24 * 31
    152 = 8 * 19
    104 = 8 * 13
    58 = 2 * 29

    2^8 = 256 = 152 + 104

  82. oldmanK says:

    PV, help needed. Not all of us (read I) can perceive in the numbers the kernel or grain of salt the mind requires.

    I have stumbled – finally – on a more mundane ‘number salad’ that made sense, and it was awe inspiring. Background: earth’s orbit was near circular 5k yrs ago (it is said). So a season worked out to 91 days. Equals 13*7 weeks (meaning the 7day wk is a pragmatic do not cult, sacred or damned.). Also 13 lunar months.
    But for the ancient calendar, 90 was a better for extended division; 3^2*5*2 and *2^2 for whole circle = 360; the ancient division of the circle, much earlier than thought. Divide circle circumference by radius = 12 (hours per day; night and day measured separately – known from Egyptian water clock, copied from earlier Babylonian.)
    And there is more. ‘Number evidence’ is near incontestable -not by chance- , but it takes a special neuron arrangement at birth to get such insight.

  83. Paul Vaughan says:

    One more link:

    https://en.wikipedia.org/wiki/Sporadic_group

    oldmanK: Thanks. Although there’s no express service by which to express the stream of natural insights from the scenic route, I share what I can when I can.

  84. Paul Vaughan says:

    The present politics is beyond intractable. It’s dangerous to engage in open political discourse.
    I’m venturing off into the purely mathematical for awhile. Role with the punches sort of thing.

    It’s an opportunity to study how things fit together. For example #comment-160817 may look “unphysical” but that’s a reflection.

    From the discrete-continuous relations of nonlinear physical aliasing (coupled sets of periodic nonlinear things) there are simple sets of recognizable symmetries.

    Fib & Luc are in the set of reflections too but that doesn’t need to stop us from seeing to the next level of generality beyond that.

    Don’t expect explanations. Just expect numbers. This has to get really abstract and clean. Think of it as needed cleansing from deadly-toxic politics.

  85. Paul Vaughan says:

    correcting the link from the last comment: hear

  86. oldbrew says:

    PV says:
    Oldbrew, I need to check that you are aware of this:

    1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 = 8^2 = 10/(1/2-836/2432)
    – – –
    Every sum of odd numbers starting from 1 is the square of however many numbers are in the series, in this case 8 odd numbers from 1 = 64.

  87. Paul Vaughan says:

    OK OB, so maybe Paul Pukite seize libration too….
    2.90909090909091 = 2432 / 836

    I know you realize I didn’t express it in a different form to confuse readers.

    The way observed things fit together is pretty much exactly a function of discrete-continuous relations, some can learn. Given all the exponentials inevitably aliased periodically in the system, it can’t stably be any other way.

    We’re here to help so few see symmetry reflected and translated. You know we won’t be reaching a wider audience directly.

    “In God We Trust”

  88. Paul Vaughan says:

    12-step from 3 bases:
    5,17,29,41,53,65,77
    7,19,31,43,55,67
    11,23,35,47,59,71

    3 composites neatly loop back to prime base:
    35=5*7
    55=5*11
    77=7*11

    1 composite points to a more inclusive base:
    65=5*13

    In the days of Bollinger (1952), Seidelmann’s (1992) precision was not available. Maybe that is why we never see Ramanujan’s name publicly tied directly to Mayan astronomy? Maybe its all covered in countless books? I don’t have them, but as a hobby I reinvent wheels.

    53,65,77
    52,64,76

    Symmetry:
    260=(53+77)+(65+65)=4*65=16^2+2^2
    256=(52+76)+(64+64)=4*64=16^2

    104=2*52
    152=2*76

    13=104/8
    19=152/8

  89. oldbrew says:

    2.90909090909091 = 2432 / 836

    2432/836 reduces by a factor of 76 to 32/11.

  90. Paul Vaughan says:

    152.000658869743 = ⌊(e^√25π)^(1/2)⌉^2 – e^√25π
    104.001742574386 = ⌊(e^√22π)^(1/2)⌉^2 – e^√22π
    47 = 25+22
    104.000034332275 = ⌊(e^√58π)^(1/4)⌉^4 – e^√58π
    71 = 58+13
    103.947369666712 = e^√13π – ⌊(e^√13π)^(1/2)⌉^2
    59 = 37+58-25-11
    103.999977946281 = e^√37π – ⌊(e^√37π)^(1/2)⌉^2
    22 = 58-25-11
    11 = 2*29-25-22 = 59 mod 24
    18 = 36/2 = (58-22)/2 = 29-11
    104.007114381762 = ⌊(e^√18π)^(1/4)⌉^4 – e^√18π
    10 = 7920 / 18 / 22 / 2
    104.212132286568 = ⌊(e^√10π)^(1/4)⌉^4 – e^√10π
    7920 = 360*22 = 10*(58-22)*22
    22 = ( 7920 – (2+3+5+7+11+13+17+19) * (2+3+5+7+11+13+17+19+23) ) / (2+3+5)

    256 = (19-3)*(13+3) = (152 / 8 – 3)*(104 / 8 + 3) = ((19+13)/2)^2 = (11+5)^2 = 16^2
    256 = ((152 + 104) mod 24)^2 = 152 + 104

    260 = 16^2 + 2^2 = 152 + 104 + 4

    4.01969522320721 = e^√2π – ⌊(e^√2π)^(1/4)⌉^4

    24 = 19 + 5

    1 / 260.000490812393 = +(Φ/19/2)(J+S)+(ΦΦ/5/2)(J-S) ———- harmonic mean
    1 / 260.000490812393 = +(Φ/19/2+ΦΦ/5/2)J-(Φ/19/2-ΦΦ/5/2)S

    24 = 76 – 52 = (152 – 104) / 2
    24 = 22 + 2 = 13 + 11

    84.0196952232072 = e^√2π – ⌊(e^√2π)^(1/11)⌉^11
    84.0196952232072 = EXP(2^(1/2)*PI())-ROUND(EXP(2^(1/2)*PI())^(1/11),0)^11
    84.016845922161 = 1/U
    0.003391344932 = % error

  91. Paul Vaughan says:

    exactly — I knew you’d notice

    mods: list of calculations (with only 2 words) just got caught in the filter.

  92. Paul Vaughan says:

    Note that 84 (last calculation) is stable not just for 11 but for integers above 11.

    Similary…
    -22.1406926327793 = ⌊(e^√1π)^(1/8)⌉^8 – e^√1π
    …is stable for integers above 8.

    As with Fib, Luc, & Binet, note modularity:
    -0.140692632779267 = ⌊(e^√1π)^(1/1)⌉^1 – e^√1π
    104.859307367221 = ⌊(e^√1π)^(1/7)⌉^7 – e^√1π

    e.g. construct 22 or primorial 210 or 11 = 59 mod 24 or 208 (with complement)

    Then note:
    0.140692640692641 = 2*5/(71+1/13)
    0.000005624583 = % error

    Recall: 11.86, 29.45, 8.45, 19.86, & 22.14 = 8.45*phi^2 can be added and subtracted in combinations to generate Fib & Luc series.

    There are some graphs I haven’t shown that have tell-tale tall spikes (another day).

    An expert on polynomials, number theory, and symmetry groups could approach all of this theoretically (incuding axiomatically) to resolve everything precisely I suspect. Titius-Bode “Law” = BS. Formalized and generalized to weighted ellipses may already be longstanding trade secret I suspect.

    I just casually figured out how to generalize Ramanujan’s & Bollinger’s methods in concert (feels like wheel reinvention) to take measurements from Seidelmann.

    Net search turns up curiously little. Readers: Please provide links if you find any. Thanks …especially for 489426.

  93. Paul Vaughan says:

    Here’s the graph — the quick-&-crude version will have to do for now:

    It’s a simple measure of how close to integer (the lower line).
    Overlaid is a measure of how close to half-integer (the higher line).

    Something more formal another time — but can’t have you thinking the values I highlight repeatedly are pulled out of hat — they are the spikes.

    Lots more to discuss and notice — but there’s a key piece readers may not have put together for themselves (simple calculation in spreadsheet and then plot if they did it way back a year or more ago when I started hinting about this).

    As mentioned before Ramanujan’s method is generalizable to detect Fib & Luc harmonics …but OB already gets Fib & Luc — hence the complementary focus to raise more general community awareness.

    With hindsight we can see (from the numbers) that Mayan astronomy sages knew this stuff. There should be some classic paper(s) and/or book(s) addressing all of this. Can anyone point to it?

    Can be tied in a neat bundle: Mayan astronomy, Ramanujan, Bollinger, & Seidelmann.

  94. Paul Vaughan says:

    clarification: the spike at 6 is 1 ; 24 is at 7

  95. Paul Vaughan says:

    Previously we looked in detail at 5256 year cycles in XEV where XEV = J,S,U,N.

    Now a much quicker look at 5256 in JSUN (without EV).
    The key: generalized Bollinger method generalized 1 level further.

    JS slip on Jovian harmonic mean with “836”/2 gives 72 & 77 at top level (outlined above) and these downstream:

    5258.67315903745 = 2 * 2629.33657951873
    5254.83162698434 = 3 * 1751.61054232811
    5256.70282852086 = 4 * 1314.17570713021
    5256.04637975109 = 6 * 876.007729958515
    5257.51148148286 = 11 * 477.955589225715
    5255.92050420241 = 73 * 71.9989110164714

    JS nodes (1/2 J period & 1/2 S period) slip on everything else the same gives:

    5254.7339739464 = 1 * 5254.7339739464
    5254.7339739464 = 2 * 2627.3669869732
    5254.83162698434 = 6 * 875.805271164057
    5256.70282852086 = 8 * 657.087853565107

    One result appears twice. Using it only once, the grand harmonic mean is 5256.01828111237.

    Sporadic group M11 with order 7920 partitions in noteworthy patterns including 5256 = 72*73. I’ve already noted 7920-2400-240-24 = 5256. I’ll outline one other soon.

    Related:

    Fractals tie Hale and Jupiter to jovian harmonic mean scaled by square root 5 (embedded in calculation shown above).

    1/4 of jovian harmonic mean over J period is ~phi.
    1/4 of jovian harmonic mean over S period is ~1/4.
    These counterbalance with 0.005741242571 = % error.

    Curious hindsight: So-called “experts” attempting to guide discussion of geometric aggregation criteria a decade ago never pointed any of these things out. We’re left wondering: Was that ignorance or deception? The discourse back then was insufficiently enlightening either way and at least some of us had the wisdom to leave that darkness behind and find our own way towards the light.

  96. Paul Vaughan says:

    77 = 2+3+5+7+11+13+17+19
    125970 = (2*3*5*7*11*13*17*19) / (2+3+5+7+11+13+17+19) ——– an integer

    10 = 2+3+5 = (2+3+5+7+11+13+17+19+23) / (2+3) / (2)
    100 = 2+3+5+7+11+13+17+19+23

    770 = (2+3+5+7+11+13+17+19) * (2+3) * (2)
    770 = (2+3+5+7+11+13+17+19) * (2+3+5)
    7700 = (2+3+5+7+11+13+17+19) * (2+3+5) * (2+3) * (2)
    7700 = (2+3+5+7+11+13+17+19) * (2+3+5+7+11+13+17+19+23)

    41 = 2+3+5+7+11+13
    59 = 17+19+23
    sum = 100

    58 = 2+3+5+7+11+13+17
    42 = 19+23
    sum = 100

  97. Paul Vaughan says:

    Noteworthy partition and factorization of M11.

    M top prime = 59+12 = 71
    73 is first prime congruent to 1 mod 24 — i.e. 1 = 73 mod 24
    The sandwiched composite: 72.

    Recipe: Repeatedly halve while staying whole.
    Muliply each by all whole numbers up to the next prime and sum the products.

    Summary:
    7920=(73*72)+(37*36)+(19*18)+(11*10*9)

    5256=73*72
    1332=37*36
    342=19*18
    990=11*10*9

    7920=5256+1332+342+990

    “24 of the 26 terms of A001228 are divisors of Mnr”
    “the first 36 factorials and the first 11 primorials are divisors of Mnr”

    “The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, or is one of 26 sporadic groups that do not follow such a systematic pattern. The monster group contains 20 sporadic groups (including itself) as subquotients. […] called those 20 groups the happy family, and the remaining six exceptions pariahs.” — wiki Monster_group

    Noteworthy factorizations include these numbers:
    7920 = 20*396 = (26-6)*36*11 = (104/4-6)*(18*22)

  98. Paul Vaughan says:

    Key Review

    Noteworthy JS slip on “61” with jovian harmonic mean:

    743.965025033056 = 2 * 371.982512516528
    152.02852505596 = 1216.22820044768 / 8
    104.046309599435 = 104.046309599435 / 1

    396.096251379871 = 1584.38500551949 / 4

    509.086466725612 = 509.086466725612 / 1
    5090.68769455564 = 10181.3753891113 / 2

    489425.999094794 = 978851.998189588 / 2 ——- Sylvester’s Sequence * 271

    This is a real clincher — the link no longer missing.
    All the questions that came up before answered in 1 clean shot.

  99. Paul Vaughan says:

    1/4 Note

    -5.76458831914576 = ⌊(e^√3π)^(1/2)⌉^2 – e^√3π
    *18 ~= 104
    *5*29 ~= 836
    13=18-5
    11=29-18
    24=29-5=13+11

    836 modular construction:
    0.246326271211728 = ⌊(e^√20π)^(1/1)⌉^1 – e^√20π
    -418.753673728788 = ⌊(e^√20π)^(1/2)⌉^2 – e^√20π

    152 / 2 / 5 / 5.76458831914576 ~=
    0.263893127441406 = ⌊(e^√61π)^(1/1)⌉^1 – e^√61π

  100. Paul Vaughan says:

    0.00913045629386033 = ⌊(e^√6π)^(1/1)⌉^1 – e^√6π
    29.4474891061275=1/(AVERAGE(1,5^(1/2))^2*(1/104+1/298))
    11.8626174616523=1/(AVERAGE(1,5^(1/2))^4*(1/104+1/298)^2+1/(84-ROUND(EXP(6^(1/2)*PI()),0)+EXP(6^(1/2)*PI()))/2)^(1/2)

    Recall:
    2(J-S)(J+S)=2(J^2-S^2)
    where
    J = Jupiter frequency = 1 / Jupiter period
    S = Saturn frequency = 1 / Saturn period

    298=104+152+84/2
    298=MOD(104+152,24)^2+42

    70p is weird for all primes p ≥ 149

    5.00156171058154 = 77.0004727270529 – 71.9989110164714
    148.999383743524 = 77.0004727270529 + 71.9989110164714
    149=72+77
    298=2*(72+77)
    196883=59*(59^2-(298-2*77))
    11.0091304562939 = ⌊(e^√6π)^(1/2)⌉^2 – e^√6π
    11=ROUND(EXP(6^(1/2)*PI())^(1/2),0)^2-ROUND(EXP(6^(1/2)*PI())^(1/1),0)^1
    11=(2^7-7)^(1/2)

    77 = 70+7 = 7*11 = 7*(2^7-7)^(1/2) = 2+3+5+7+11+13+17+19

    70 = (1^2+2^2+3^2+…+22^2+23^2+24^2)^(1/2)
    7 = (0.1^2+0.2^2+0.3^2+…+2.2^2+2.3^2+2.4^2)^(1/2)

    24^2 = 576
    2.4^2 = 5.76

  101. Paul Vaughan says:

    31.5822476397426 = (77)*(19.8650360864628)/((77+19.8650360864628)/2)
    27.4342301377126 = (72)*(19.8650360864628) / (72 – 19.8650360864628)
    29.3624656246739 = (31.5822476397426)*(27.4342301377126)/((31.5822476397426+27.4342301377126)/2)
    29.3622914957372= 4*(59^(1/2)-MOD(59^(1/2),1)/2)
    29.3625733662892 = harmean(11.8626151546089,29.4474984673838,84.016845922161,164.791315640078)

  102. Paul Vaughan says:

    76=152/2
    21=42/2
    can derive all luc & fib from there
    55=(152-42)/2
    34=152/2-42
    etc.

  103. Paul Vaughan says:

    9.93251804323141 = 19.8650360864628 / 2
    83.9908202014941 = 9.93251804323141 * 8.4561457463176
    83.9908695437061 = 84 – 0.00913045629386033
    -0.000058747114 = % error
    0.00913045629386033 = ⌊(e^√6π)^(1/1)⌉^1 – e^√6π

  104. Paul Vaughan says:

    298 Review

    1.61803398874989 = φ
    2.61803398874989 = φφ

    56.8900001558329 = (61.0464822565173)*(29.4474984673838) / (61.0464822565173 – 29.4474984673838)

    298 = ⌊297.879908055915⌉ = ⌊113.780000311666φφ⌉ =
    = ⌊φφ(835.546575435631)*(61.0464822565173)/((835.546575435631+61.0464822565173)/2)⌉

    149 = ⌊148.939954027957⌉ = ⌊56.8900001558329φφ⌉
    = ⌊φφ(835.546575435631)*(61.0464822565173)/(835.546575435631+61.0464822565173)⌉

    149 = ⌊148.939954027957⌉ = ⌊56.8900001558329φφ⌉
    = ⌊φφ(61.0464822565173)*(29.4474984673838)/(61.0464822565173-29.4474984673838)⌉

    26 = ⌊26.0036698310514⌉ = ⌊9.93251804323141φφ⌉
    = ⌊φφ(61.0464822565173)*(11.8626151546089)/(61.0464822565173+11.8626151546089)⌉

    52 = ⌊52.0073396621028⌉ = ⌊19.8650360864628φφ⌉
    = ⌊φφ(61.0464822565173)*(29.4474984673838)/(61.0464822565173+29.4474984673838)⌉

    104 = ⌊104.014679324206⌉ = ⌊39.7300721729256φφ⌉
    = ⌊φφ(61.0464822565173)*(29.4474984673838)/((61.0464822565173+29.4474984673838)/2)⌉

    208 = 2*104 = 298 – 61 – 29 = 298 – ⌊61.0464822565173⌋ – ⌊29.4474984673838⌋

    Φ = 1 / φ

    0.618033988749895 = Φ
    0.381966011250105 = ΦΦ

    29.4474891061275 = 77.0945273631841ΦΦ = ΦΦ(298)*(104)/(298+104)

  105. Paul Vaughan says:

    Mods: 298 review is caught in the filter (too many round-off bars exceed filter math tolerance).

  106. Paul Vaughan says:

    22.1384769776823 = 8.4561457463176 * φφ
    44.2769539553646 = 2 * 22.1384769776823 = 16.9122914926352 * φφ

    76.0046051964364 = 44.2769539553646 + 11.8626151546089 + 19.8650360864628
    42.0035130641451 = 19.8650360864628 + 22.1384769776823
    41.9827054421324 = 2 * 29.4474984673838 – 16.9122914926352
    41.9827054421324 = 2*(29.4474984673838 – 8.4561457463176)
    34.0010921322913 = 11.8626151546089 + 22.1384769776823
    21.0017565320726 = 9.93251804323141 + 11.0692384888412
    20.9913527210662 = 29.4474984673838 – 8.4561457463176
    8.00242093185388 = 19.8650360864628 – 11.8626151546089

    more than enough to fill gaps + work up & down

    fib
    55=34+21
    34 above
    21 above
    13=34-21=21-8
    5=13-8
    etc.

    luc
    123=76+47
    76 above
    47=76-29
    29=21+8
    18=47-29
    etc.

    “a connection is made” — elastica

    76 = 2432 / 32 = 836 / 11 = 152 / 2
    13 = 104 / 8

    6.4 = (2.90909090909091)*(2) / (2.90909090909091 – 2)
    1.18518518518519 = (2.90909090909091)*(2) / (2.90909090909091 + 2)
    2.37037037037037 = (2.90909090909091)*(2)/((2.90909090909091+2)/2)
    18.6 = 744 / 40

    See end of Suggestions-42 and beginning of Suggestions-43 for precise lunisolar tie-in.

    42 = 84 / 2 = 298 – 104 – 152 = 298 – 4*64

  107. Paul Vaughan says:

    Review — with extension

    5.93183625861279 = log(61.0464822565173,2)
    11.8636725172256 = 2 * 5.93183625861279
    0.008913402339 = % error

    5.93130757730447 = 11.8626151546089 / 2
    1.00008913402339 = 5.93183625861279 / 5.93130757730447
    2.00012356981122 = 2^1.00008913402339
    2 = 2
    0.006178490561 = % error

    Recall that several cookbook recipes discretely give Saturn’s orbital period from 104 & 298.

    Also note this over there (and elsewhere):

    22.1348630064402 =
    (5^(1/2)+1)*(7-1)/(1/((7^0)+1/((7^1)+1/((7^1)+1/((7^2)+1/((7^3)+1/((7^5)+1/(7^8))))))))

    So now we have enough insight to differentiate that from the cluster near 22.14 (22.13 was a low outlier in the pack).

    22.1349639135372 = 42 – 19.8650360864628
    -0.000455871974 = % error

    19.8651369935598 = 42 – 22.1348630064402
    0.000507963321 = % error

    11.8626496189237 = (29.4474891061275)*(19.8651369935598) / (29.4474891061275 + 19.8651369935598)
    0.000290528811 = % error

    Compare and contast with estimate based on 2(J-S)(J+S) above (clarified here) :
    11.8626174616523 ————————————————————————————
    11.8626151546089 = 1 / J
    0.000019448017 = % error

    Ramanujan fans will like this 2*5*24+58=298 tie-in.

    Imagine what else he would have taught US had he lived in good health and sufficient wealth for longer.

    The next 2 comments will challenge readers in a new way. (We’ve turned a corner.)

  108. Paul Vaughan says:

    Recipes

    Sudden change of plans: a pair of easy recipes.

    Start with 5.
    5^2 = 25

    152.000658869743 = ⌊(e^√25π)^(1/2)⌉^2 – e^√25π
    0.000658869743347168 = ⌊(e^√25π)^(1/1)⌉^1 – e^√25π
    difference = 152

    58 = ⌊152 * ΦΦ⌉ = ⌊58.058833710016⌉
    use as next input

    104.000034332275 = ⌊(e^√58π)^(1/2)⌉^2 – e^√58π
    0.000034332275390625 = ⌊(e^√58π)^(1/1)⌉^1 – e^√58π
    difference = 104

    loop back:
    152 = ⌊58 * φφ⌉ = ⌊151.845971347494⌉

    Another recipe.
    Note the first level to exceed the top level of M (which is 71) :

    70.6869176962472 = 27 * φφ
    73.3049516849971 = 28 * φφ

    71 = ⌊27 * φφ⌉ = ⌊70.6869176962472⌉
    73 = ⌊28 * φφ⌉ = ⌊73.3049516849971⌉
    73 mod 24 = 1 ————- 73 is the smallest integer with this property
    28 = ⌊73 * ΦΦ⌉ = ⌊27.8835188212577⌉

    So let’s take a look at level 28.

    a = 196585.011874415 = ⌊(e^√28π)^(1/4)⌉^4 – e^√28π
    b = 0.0118744149804115 = ⌊(e^√28π)^(1/1)⌉^1 – e^√28π
    c = 196585 = a – b
    d = 196883 = (59-12)*(59)*(59+12) —————— minimal faithful representation of M
    f = 298 = d – c


    as an aside, note a few other 2,5-ladders at level 28:
    744.01187441498 = ⌊(e^√28π)^(1/3)⌉^3 – e^√28π
    372, 18.6
    maybe less precise, but still loosely suggestive:
    553.01187441498 = ⌊(e^√28π)^(1/2)⌉^2 – e^√28π
    11.06, 22.12, 44.24

    leaving behind the aside, just note:

    42 = 298 – 152 – 104 = (1^2+2^2+3^2+…+22^2+23^2+24^2)^(1/2) – 28

    =
    Many mathematicians including Conway regard the monster as a beautiful and still mysterious object […] Conway said of the monster group: “There’s never been any kind of explanation of why it’s there, and it’s obviously not there just by coincidence. It’s got too many intriguing properties for it all to be just an accident.” […] Simon P. Norton, an expert on the properties of the monster group, is quoted as saying, “I can explain what Monstrous Moonshine is in one sentence, it is the voice of God.” […]
    =

  109. Paul Vaughan says:

    Scatter 4 radars learn D-electable Conway and Norton quotes right above.

    298 review: used 1 trick move (temporary mnemonic placeholders) there that got cleaned up.

    We reviewed 77ΦΦ ~= 1/S.
    Recall in concert 31ΦΦ ~= 1/J.

    Note: 7, 11, & 31 are factors shared by 8-groups M & B and pariahs J4 & Ly.

    31.0569193417858 = (104)*(44.28) / (104 + 44.28)
    11.8626876026982 = 31.0569193417858 * ΦΦ
    0.000610726120 = % error
    Know Not?
    Now Knot:
    58 = 4428 – 4370 —— minimum faithful representation of B

    Forgot to tie opposite direction (bidirectional relation) into last comment:
    196585 – 196560 = 25 = 5^2

    What I haven’t clarified (but hinted on XR thread) is building in steps with centered hexagonal, 12-gonal, & star numbers …but first the (11 each, let US) pair of comments I promised:

  110. Paul Vaughan says:

    171.406220601552 = UN beat
    55.6462717641972 = UN axial period
    111.292543528394 = UN harmonic mean

    UN analogy to JS “61”:
    4270.09258127429 = slip(164.791315640078,84.016845922161)
    1 / 4270.09258127428 = -1U+2N
    UN analogy to JS “836”:
    48590.8284812209 = slip(4270.09258127429,171.406220601552)
    1 / 48590.8284812705 = -26U+51N
    77 = 26+51

    UN harmonic mean split (Bollinger slip method generalized hierarchically)
    111.038221334937 = (48590.8284812209)*(111.292543528394) / (48590.8284812209 + 111.292543528394)

    nodal (half period)
    55.5191106674687 = 111.038221334937 / 2
    1 / 55.5191106674688 = -51U+103N

    1962.57776108676 = slip(171.406220601552,55.5191106674687)
    7300.99712676728 = slip(1962.57776108676,55.6462717641972)
    7301 = 49 * 149 = 7^2 * 149
    0.000039353977 = % error
    1 / 7300.99712675476 = +1891U-3709N

  111. Paul Vaughan says:

    jovian harmonic mean split
    29.5658781541637 = (4270.09258127429)*(29.3625733662892) / (4270.09258127429 – 29.3625733662892)

    quarter-cycle:
    7.39146953854093 = 29.5658781541637 / 4
    1 / 7.39146953854094 = +1J+1S+5U-7N

    903.397527534612 = slip(171.406220601552,7.39146953854093)
    1 / 903.397527534829 = +1J+1S-18U+16N

    7700.08638842647 = slip(903.397527534612,111.292543528394)

    7700 = (2+3+5+7+11+13+17+19) * (2+3+5) * (2+3) * (2)
    7700 = (2+3+5+7+11+13+17+19) * (2+3+5+7+11+13+17+19+23)
    = 100^2+200^2+300^2+…+2200^2+2300^2+2400^2 + 10^2+20^2+30^2+…+220^2+230^2+240^2
    -0.001121915029 = % error

    3850 = (2+3+5) * (3*5*7*11*13) / (3+5+7+11+13) = 7700/2 = 770*5

    1 / 3850.04319415152 = -16J-16S+289U-255N
    576 = 16+16+289+255 = 4^2+4^2+17^2+17*15 = 4*144 = 24^2

    196883=59*(59^2-576/4)

    196560

  112. Paul Vaughan says:

    Mods: The other comment from the pair got stuck in the filter (many numbers with few words).

  113. Paul Vaughan says:

    typo correction in italics:
    ” 73 mod 24 = 1 ————- 73 is the smallest prime integer with this property”

  114. Paul Vaughan says:

    Splitting up the filtered comment into 2 pieces, here is piece 1.
    Piece 1 derives the values needed for the calculation in piece 2.

    Sentences like these serve no purpose other than helping the filter accept uncensored math.

    To make this block of text longer (since the filter readily accepts rants including wild rants), we’re testing weather automatically-admissible speech on “the word press” includes the expression of numbers and weather some numbers are more likely than others to be filtered.

    UN analogy to JS “61”:
    4270.09258127429 = slip(164.791315640078,84.016845922161)
    1 / 4270.09258127428 = -1U+2N
    UN analogy to JS “836”:
    48590.8284812209 = slip(4270.09258127429,171.406220601552)
    1 / 48590.8284812705 = -26U+51N
    77 = 26+51

    UN harmonic mean split (Bollinger slip method generalized hierarchically)
    111.038221334937 = (48590.8284812209)*(111.292543528394) / (48590.8284812209 + 111.292543528394)

    nodal (half period)
    55.5191106674687 = 111.038221334937 / 2
    1 / 55.5191106674688 = -51U+103N

    These numbers will be used in piece 2, which will immediately follow if the filter is permissive.

  115. Paul Vaughan says:

    Mods: The other comment from the pair got stuck in the filter (many numbers with few words).”

    1962.57776108676 = slip(171.406220601552,55.5191106674687)
    7300.99712676728 = slip(1962.57776108676,55.6462717641972)
    7301 = 49 * 149 = 7^2 * 149
    0.000039353977 = % error
    1 / 7300.99712675476 = +1891U-3709N

    This is a segment of comment.

    [mod] 2 comments restored from spam

  116. Paul Vaughan says:

    This is a test-comment (related to this) to verify a suspicion about what does not pass as automatically-accepted freedom of expression.

    Uranus-Neptune analogy to Jupiter-Saturn “61”: 1 / 4270.09258127428 = -1U+2N
    Uranus-Neptune analogy to Jupiter-Saturn “836”: 1 / 48590.8284812705 = -26U+51N

    77 = 26+51

  117. Paul Vaughan says:

    Reflection

    Late November 2007 was when I began searching online for information after finding nonrandom patterns (not simple cycles) on these ladders:

    11, 5.5
    52, 26
    48, 24

    I now understand their origin as a natural consequence of discrete-continuous relations.

    Consistently for years I maintained that climate discussion participants were ignorant of aggregation criteria.

    I now also understand the discursive methods applied at one of the climate hubs as technocratic governance, either ignorantly based on false assumptions about aggregation criteria or deceptively based on “expert” decision to leverage public ignorance of aggregation criteria.

    One conclusion I draw is that technocratic governance misapplied to discourse as observed is a dangerous enterprise.

  118. Paul Vaughan says:

    Solar system structure is based on discrete-continuous relations.
    Let’s review from a number of perspectives.

    Readers will note that these recipes are simple and accurate.

    4370 — minimal faithful representation of B (baby monster)
    4428 = 4370 + 58

    104.000034332275 = ⌊(e^√58π)^(1/2)⌉^2 – e^√58π ——— Ramanujan
    0.000034332275390625 = ⌊(e^√58π)^(1/1)⌉^1 – e^√58π
    difference = 104

    44.28 = 4428 / 100

    77.1118553248493 = (104)*(44.28) / (104 – 44.28)
    31.0569193417858 = (104)*(44.28) / (104 + 44.28)

    Note that with golden-angle scaling these are approximately S & J periods:

    29.4541077985279 = 77.1118553248493 * ΦΦ
    29.4474984673838 = 1/S
    0.022444457044 = % error

    11.8626876026982 = 31.0569193417858 * ΦΦ
    11.8626151546089 = 1/J
    0.000610726120 = % error

    Also note:

    0.0679022435064206 = 2 / 29.4541077985279
    0.0679047747389632 = ⌊(e^√7π)^(1/1)⌉^1 – e^√7π
    -0.003727620852 = % error

  119. Paul Vaughan says:

    Now work the other way.
    Start at the end of the last comment. Work backwards. Develop discrete insight.

    0.0679047747389632 = ⌊(e^√7π)^(1/1)⌉^1 – e^√7π

    29.4530098610638 = 2 / 0.0679047747389632
    29.4474984673838 = 1/S
    0.018715999548 = % error

    77.1089808872509 = 29.4530098610638 * ΦΦ

    104.005228959558 = (77.1089808872509)*(44.28) / (77.1089808872509 – 44.28)
    298.216069039649 = (104)*(77.1089808872509) / (104 – 77.1089808872509)

    104 = ⌊104.005228959558⌉
    298 = ⌊298.216069039649⌉

    77.0945273631841 = (298)*(104) / (298 + 104)
    29.4474891061275 = 77.0945273631841 * ΦΦ
    29.4474984673838 = 1/S
    -0.000031789649 = % error

  120. Paul Vaughan says:

    This is a review of something that was covered above to help readers see a little more clearly.

    19.8650360864628 = (29.4474984673838)*(11.8626151546089) / (29.4474984673838 – 11.8626151546089)
    8.4561457463176 = (29.4474984673838)*(11.8626151546089) / (29.4474984673838 + 11.8626151546089)

    9.93251804323141 = 19.8650360864628 / 2
    83.9908202014941 = 9.93251804323141 * 8.4561457463176

    Equivalently since 2(J-S)(J+S) = 2(J^2-S^2) in frequencies (i.e. 1/periods):
    83.9908202014941
    = 1 / ( 1 / 11.8626151546089 ^ 2 – 1 / 29.4474984673838 ^ 2 ) / 2

    83.9908695437061 = 84 – ( ⌊(e^√6π)^(1/1)⌉^1 – e^√6π )
    -0.000058747114 = % error

    0.00913045629386033 = ⌊(e^√6π)^(1/1)⌉^1 – e^√6π
    0.99086954370614 = e^√6π – ⌊(e^√6π)^(1/3)⌉^3 = 1 – ⌊(e^√6π)^(1/1)⌉^1 – e^√6π

    This gives a way to accurately estimate J period using S period from the last comment.

  121. Paul Vaughan says:

    Typo correction:
    0.99086954370614 = e^√6π – ⌊(e^√6π)^(1/3)⌉^3 = 1 – ( ⌊(e^√6π)^(1/1)⌉^1 – e^√6π )

  122. Paul Vaughan says:

    Another way to do the last calculation:

    140.721638306358 = 11.8626151546089 ^ 2
    867.155165986572 = 29.4474984673838 ^ 2

    167.981640402988 = (867.155165986572)*(140.721638306358) / (867.155165986572 – 140.721638306358)

    83.9908202014941 = 167.981640402988 / 2

    So if we turn that around we can estimate 1/J using the 1/S estimate from above:

    167.981739087412 = 2 * 83.9908695437061
    867.154614655499 = 29.4474891061275 ^ 2

    140.721693041498 = (867.154614655499)*(167.981739087412) / (867.154614655499 + 167.981739087412)

    11.8626174616523 = 140.721693041498 ^ (1/2)
    11.8626151546089 = 1/J
    0.000019448017 = % error

    Another way to write it:
    11.8626174616523 = 1 / ( 1 / 29.4474891061275 ^ 2 + 1 / 83.9908695437061 / 2 ) ^ (1/2)

    where:
    83.9908695437061 = 84 – ( ⌊(e^√6π)^(1/1)⌉^1 – e^√6π )
    29.4474891061275 = ΦΦ * (298) * (104) / (298 + 104)

  123. Paul Vaughan says:

    Mods: final stage of clarified calculation review — ending with J period estimate — is stuck in filter.

  124. Paul Vaughan says:

    Piecing together from clues given above:

    72 = (59^2-196883/59)/2
    77 = (298-59^2+196883/59)/2

    72 = (59^2-71*47)/2
    77 = (298-59^2+71*47)/2

    19.8649298181927 = 3/(2/(2*59^(1/2)-MOD(59^(1/2),1))-2/(298-59^2+71*47)+4/(59^2-71*47))
    19.8650360864628 = 1 / (J-S)
    -0.000534951306 = % error

  125. Paul Vaughan says:

    Review

    √28 = 2√7

    28 = (84)*(42) / (84 + 42) = (1^2+2^2+3^2+…+22^2+23^2+24^2)^(1/2) – 42 = 70 – 42

    Pick powers that shoot between Leech (196560) & Monster (196883).
    196585.011874415 = ⌊(e^√28π)^(1/p)⌉^p – e^√28π for p = 4,6,8
    subtract power 1:
    0.0118744149804115 = ⌊(e^√28π)^(1/1)⌉^1 – e^√28π

    196585 = ⌊(e^√28π)^(1/p)⌉^p – ⌊(e^√28π)⌉ for p = 4,6,8
    298 = 196883 – 196585 ——— minimal faithful representation of M (monster)
    25 = 196585 – 196560 ——- Leech Lattice

    152.000658869743 = ⌊(e^√25π)^(1/2)⌉^2 – e^√25π
    0.000658869743347168 = ⌊(e^√25π)^(1/1)⌉^1 – e^√25π
    difference = 152

    298 = 104 + 152 + 42 = ((104+152) mod 24)^2 + 42 = ⌊42 * ΦΦ⌉ ^ 2 + 42
    42 = ⌊φφ*(298-42)^(1/2)⌉ = (1^2+2^2+3^2+…+22^2+23^2+24^2)^(1/2) – 28

    58 = ⌊152 * ΦΦ⌉ = ⌊58.058833710016⌉

    104.000034332275 = ⌊(e^√58π)^(1/2)⌉^2 – e^√58π
    0.000034332275390625 = ⌊(e^√58π)^(1/1)⌉^1 – e^√58π
    difference = 104

    152 = ⌊58 * φφ⌉ = ⌊151.845971347494⌉

  126. Paul Vaughan says:

    Recognizing and remembering symmetry structure:

    __9
    _49
    149

    1 / 49 / 149 ~= ((28-9)(149-49)-9)U+((-28-9)(149-49)-9)N

    7^2 = 49 = 42 + 7
    28 = (84)*(42) / (84 + 42) = 70 – 42 = 7^2 – 42/2 = 77-7*7
    70^2 = 49*(149-49) = 1^2+2^2+3^2+…+22^2+23^2+24^2

  127. Paul Vaughan says:

    The spectacular symmetry last noted is sharp.

    Note that 168 = 2*84 to build a simple test of how well precision transfers.

    84.0168 = 84*(1+1/5000)
    84.016845922161 = 1/U
    -0.000054658278 = % error

    164.79122556704=((-28-9)*(149-49)-9)/(1/49/149-((28-9)*(149-49)-9)/84/(1+1/5000))
    164.791315640078 = 1/N
    -0.000054658850 = % error

    The change in % error is negligible.

    171.406126914883 = (164.79122556704)*(84.0168) / (164.79122556704 – 84.0168)
    171.406220601552 = (164.791315640078)*(84.016845922161) / (164.791315640078 – 84.016845922161)
    -0.000054657683 = % error

    55.6462413487959 = (164.79122556704)*(84.0168) / (164.79122556704 + 84.0168)
    55.6462717641972 = (164.791315640078)*(84.016845922161) / (164.791315640078 + 84.016845922161)
    -0.000054658471 = % error

    It’s remarkable that this symmetry wasn’t pointed out to us by so-called “experts” “guiding” us a dozen years ago. Was it naive ignorance or willful deception? Dark either way, but we can find the light for ourselves.

    “There’s no one here — and people everywhere: you’re on your own.” — Queens of the Stone Age

  128. Paul Vaughan says:

    Mods: Symmetry test is caught in the filter.

    Soon I’ll post a “numerical mystery” comment that for sure will not pass the filter. I may pause — or focus on clarifying other things — to allow readers time to ponder it’s utility before elaborating.

  129. Paul Vaughan says:

    Previously noted:

    1 / 49 / 149 ~= ((28-9)(149-49)-9)U – ((28+9)(149-49)+9)N
    2 / 7700 + 16(J+S) ~= (16+1)(16+1)U – (16-1)(16+1)N

    Thus:

    U ~= ( 1/49/149/((28-9)*(149-49)-9) – ((28+9)*(149-49)+9)/((28-9)*(149-49)-9)*16/((16-1)*(16+1))*(J+S) – ((28+9)*(149-49)+9)/((28-9)*(149-49)-9)/(25*(298-59^2+47*71))/((16-1)*(16+1)) ) / (1-((28+9)*(149-49)+9)/((28-9)*(149-49)-9)*((16+1)*(16+1))/((16-1)*(16+1)))

    N ~= (-1/49/149/((28+9)*(149-49)+9) + ((28-9)*(149-49)-9)/((28+9)*(149-49)+9)*16/((16+1)*(16+1))*(J+S) + ((28-9)*(149-49)-9)/((28+9)*(149-49)+9)/(25*(298-59^2+47*71))/((16+1)*(16+1)) ) / (1-((28-9)*(149-49)-9)/((28+9)*(149-49)+9)*((16-1)*(16+1))/((16+1)*(16+1)))

  130. Paul Vaughan says:

    3850 = 7700 / 2
    3850 = 7920/2 – 5*22
    3850 = 25 * ( 298 – 59^2 + 47*71 )
    3850 = 2*5*(3*5*7*11*13)/(3+5+7+11+13)
    3850 = (2+3+5+7+11+13+17+19) * (2+3+5) * (2+3)
    3850 = (2+3+5+7+11+13+17+19) * (2+3+5+7+11+13+17+19+23) / 2
    3850 = ( √(100^2+200^2+300^2+…+2200^2+2300^2+2400^2) + √(10^2+20^2+30^2+…+220^2+230^2+240^2) ) / 2

  131. Paul Vaughan says:

    using

    a: 8.4561457463176 = 1 / (J+S)
    b: 8.45614614668225 = 1/((1/(AVERAGE(1,5^(1/2))^(-2)*104*298/(298+104))^2+1/(84-(ROUND(EXP(6^(1/2)*PI()),0)-EXP(6^(1/2)*PI())))/2)^(1/2)+AVERAGE(1,5^(1/2))^(2)/104/298*(298+104))
    0.000004734599 = % error

    gives

    a: 84.0168457926186 ;-0.000000154186 = % error
    b: 84.0168497699537 ; 0.000004579787 = % error
    __ 84.016845922161 = 1/U

    a: 164.791315385596 ;-0.000000154427 = % error
    b: 164.791323186821 ; 0.000004579576 = % error
    __ 164.791315640078 = 1/N

  132. Paul Vaughan says:

    149 = ⌊148.990844193858⌉ = ⌊φφ/(1/29.4474984673838-2^(-11.8626151546089/2))⌉
    52 = ⌊52.0011375365569⌉ = ⌊φφ/(1/29.4474984673838+2^(-11.8626151546089/2))⌉

    0.999978124775518 = 52*ΦΦ*(1/29.4474984673838+2^(-11.8626151546089/2))
    0.999978614647502 = 1 / T where T = terrestrial tropical year frequency
    -0.000048988246 = % error

    148.990844193858
    148.990869543706 = 149 – 0.00913045629386033 = 148 – (-0.99086954370614)
    = 149 – ( ⌊(e^√6π)^(1/1)⌉^1 – e^√6π )
    = 148 – ( ⌊(e^√6π)^(1/3)⌉^3 – e^√6π )
    0.000017014366 = % error

  133. Paul Vaughan says:

    Jupiter, Saturn, and Ramanujan

    First, a quick review:

    83.9908202014941 = 1/(J^2-S^2)/2
    83.9908695437061 = 84 – ROUND(EXP(6^(1/2)*PI()),0) + EXP(6^(1/2)*PI())

    Going next-level, 65 + that matches new insight (see last comment).

    Reorganizing — in frequencies:

    a: 0.00595303151940295 = J^2-S^2
    b: 0.00595302802216872 = 1 / ( 84 – ROUND(EXP(6^(1/2)*PI()),0) + EXP(6^(1/2)*PI()) ) / 2

    a: 0.0175717776680523 = S-2^(-1/J/2)
    b: 0.0175717746783262 = φφ / ( 149 – ROUND(EXP(6^(1/2)*PI()),0) + EXP(6^(1/2)*PI()) )

    Prompted equation-solver to solve system of a-equations for periods s=1/S & j=1/J using b-constants as estimates for a-constants — results:

    11.8626197095694
    11.8626151546089 = 1/J
    0.000038397608 = % error

    29.4475234923263
    29.4474984673838 = 1/S
    0.000084981556 = % error

    It’s a home run — as in baseball (the sport).

  134. Paul Vaughan says:

    Mods: key calculation’s in the filter.

  135. Paul Vaughan says:

    Mods: Key calculation’s in the filter.

  136. Paul Vaughan says:

    This may be more intuitive.
    Here’s what you get from Seidelmann (1992) :

    83.9908202014941 = 1/(J^2-S^2)/2
    148.990844193858 = φφ/(S-2^(-1/J/2))
    difference ~= 65 = 149 – 84 and thus 298 = 2*84 + 130 is the Mayan tie-in.

    Notation: I’m using lower case for period (j & s) and upper case for frequency (J & S).

    Use the right-hand side of the above equations with estimates of left-hand side from the following:

    83.9908695437061 = 84 – ROUND(EXP(6^(1/2)*PI()),0) + EXP(6^(1/2)*PI())
    148.990869543706 = 149 – ROUND(EXP(6^(1/2)*PI()),0) + EXP(6^(1/2)*PI())

    The solver accepts and handles these expressions:

    83.9908695437061*((1/j)^2-(1/s)^2) = 1/2
    148.990869543706*((1/s)-2^(-j/2)) = 2.61803398874989

    A solution it gives:

    11.8626197095695
    11.8626151546089 = 1/J
    0.000038397608 = % error

    29.4475234923265
    29.4474984673838 = 1/S
    0.000084981557 = % error

  137. Paul Vaughan says:

    Ch.in.e58LANturns Dub11 Heil: Ten Four’s at Turn

    Put.in’ the Baby Monster on Ramanujan’s scale, 2-weigh climb ITerror:

    44.2769539553646 = harmonic mean of φφj & φφs
    where s & j are saturn & jupiter periods

    ⌊4427.69539553646⌉ = 4428
    58 = 4428 – 4370

    104.000034332275 = ⌊(e^√58π)^(1/4)⌉^4 – e^√58π

    29.4474963123504 = ΦΦ*(298)*(104.000034332275) / (298 + 104.000034332275)
    29.4474984673838 = 1/S
    -0.000007318222 = % error

    “In God We Trust”

    JEV, SEV, UEV, & NEV will never look the same on the course of history threw monstrous moonshine.

  138. Paul Vaughan says:

    This is an expression for j=1/J that equalizes magnitude and direction of very small bias for J, S, J-S, J+S, U, N, U-N, & U+N (-0.000008025115 = %error average across all periods with very small variance) :

    11.8626141905972 = -2*LOG(1/29.4474963123504-2.61803398874989/(149 – ROUND(EXP(6^(1/2)*PI()),0) + EXP(6^(1/2)*PI())),2)

    Deeply appreciated number: 4428.
    A salute to:

    General Ramanujan

    Next: deepening perspective on what makes levels unique, including level 28, 104 levels, & 152’s level (i.e. level 25). Also: intro to fifth roots (note that up until now we’ve been focused on square-roots). If there’s time: concise notes to help fib & luc fans see select tie-ins to a more general framework of discrete-continuous relations. There may also be a Mayan 260 “numerical mystery” left as a hint …and quite rightly: a “1/(U-N) surprise”.

  139. Paul Vaughan says:

    More or less a new well exploration completes spec cue la tory trio:

    0.000034332275390625 = ⌊(e^√58π)^(1/1)⌉^1 – e^√58π
    / 2 =
    0.00001716613769
    + 1 =
    1.00001716613769
    1.00001743371442 = 1/E = terrestrial sidereal year …period !
    -0.000026757206 = % error

  140. Paul Vaughan says:

    General Ramanujan disCRete11y notes leach let US truck sure:

    196560 = 2^4*3^3*5*7*13 ———– Leech Lattice

    139560 = ⌊(e^√58π)^(1/3)⌉^3 – ⌊(e^√58π)⌉
    57000 = 196560 – 139560

    19 = 57 / 3
    152 = 8 * 19
    146 = 298 – 152
    104 = 146 – 42 —————— inclining towards precise JEV review anyon? (not typo)

    58 = ROUND(139560.000034332/24.067904774739/100,0) = ROUND(139560/24/100,0)

    2406.20689714366 = 139560.000034332 / 58
    2406.31881498933
    -0.004650998237 = % error

    Look shaky until they discount you …and then:

  141. Paul Vaughan says:

    …note another Sol ID sign-post….

    24.0669187360712 = (16.9122914926352)*(9.93251804323141) / (16.9122914926352 – 9.93251804323141)
    -0.004096902813 = % error

    ….and then:

  142. Paul Vaughan says:

    Tune √(Φ-φ) N8*0!

    24.067904774739 = ⌊(e^√7π)^(1/p)⌉^p – e^√7π for p=2,3,4,6,12
    104.000034332275 = ⌊(e^√58π)^(1/p)⌉^p – e^√58π for p=2,4

    11.8626151191159 = 1/ ( ( 6 / ( ROUND(EXP(7^(1/2)*PI())^(1/4),0)^4 – EXP(7^(1/2)*PI()) ) + (1/4) / ( ROUND(EXP(58^(1/2)*PI())^(1/4),0)^4 – EXP(58^(1/2)*PI()) ) ) / 5 + ( 1 / ( ROUND(EXP(58^(1/2)*PI())^(1/4),0)^4 – EXP(58^(1/2)*PI()) ) + 1 / 298 ) * ((1+5^(1/2))/2)^2 )
    11.8626151546089 = 1/J
    -0.000000299201 = % error

    29.4474963123504 = 1 / ( 1 / ( ROUND(EXP(58^(1/2)*PI())^(1/4),0)^4 – EXP(58^(1/2)*PI()) ) + 1 / 298 ) / ((1+5^(1/2))/2)^2
    29.4474984673838 = 1/S
    -0.000007318222 = % error

    19.8650369676306 = 5 / ( 6 / ( ROUND(EXP(7^(1/2)*PI())^(1/4),0)^4 – EXP(7^(1/2)*PI()) ) + (1/4) / ( ROUND(EXP(58^(1/2)*PI())^(1/4),0)^4 – EXP(58^(1/2)*PI()) ) )
    19.8650360864628 = 1/(J-S)
    0.000004435773 = % error

    8.45614555057611 = 1 / ( ( 6 / ( ROUND(EXP(7^(1/2)*PI())^(1/4),0)^4 – EXP(7^(1/2)*PI()) ) + (1/4) / ( ROUND(EXP(58^(1/2)*PI())^(1/4),0)^4 – EXP(58^(1/2)*PI()) ) ) / 5 + 2 * ( 1 / ( ROUND(EXP(58^(1/2)*PI())^(1/4),0)^4 – EXP(58^(1/2)*PI()) ) + 1 / 298 ) * ((1+5^(1/2))/2)^2 )
    8.4561457463176 = 1/(J+S)
    -0.000002314784 = % error

    61.0464998394537 = 1/ ( ( 6 / ( ROUND(EXP(7^(1/2)*PI())^(1/4),0)^4 – EXP(7^(1/2)*PI()) ) + (1/4) / ( ROUND(EXP(58^(1/2)*PI())^(1/4),0)^4 – EXP(58^(1/2)*PI()) ) ) / 5 – ( 1 / ( ROUND(EXP(58^(1/2)*PI())^(1/4),0)^4 – EXP(58^(1/2)*PI()) ) + 1 / 298 ) * ((1+5^(1/2))/2)^2 )
    61.0464822565173 = 1/(J-2S)
    0.000028802538 = % error

    1/Φ = φ = (1+5^(1/2))/2
    0! = 1

  143. Paul Vaughan says:

    Mods: key refinement’s caught — land in.the filter.

  144. Paul Vaughan says:

    Review:

    1.59868955949705 = 1/(E-V) = beat period
    0.380883104686082 = 1/(E+V) = axial period
    0.761766209372164 = 2/(E+V) = harmonic mean
    where
    E = Earth sidereal frequency
    V = Venus sidereal frequency

    Generalized Bollinger (1952) method:
    0.814040387734912 = (11.8626151546089)*(0.761766209372164) / (11.8626151546089 – 0.761766209372164)
    44.2784629967674 = slip(1.59868955949705,0.814040387734912)

    At next level up hierarchy:
    146.000401793899 = slip(44.2784629967674,1.59868955949705)

    Recall now-familiar discrete-continuous relations almost-integer modular frame:
    146 = 298 – 152 = 104 + 42

    More review:

    period: 22.1392315068494 ~= (φ^22+1/11)^(e/11+1/22)
    frequency: H = 1 / 22.1392315068494 ~= (φ^22+1/11)^-(e/11+1/22)

    ( aside: limit of (φ^x+2/x)^(2e/x+1/x) as x goes to ∞ = φ(φφ)^e )

    Doing a little frequency algebra (derived from discrete-continuous slip-cycle structure):

    1/146 ~= 14H-V+E
    E ~= 1/146-14H+V
    H ~= 3V-5E+2J
    5E ~= 3V-H+2J
    E ~= 3V/5-H/5+2J/5

    E = E
    3V/5-H/5+2J/5 ~= 1/146-14H+V
    3V-H+2J ~= 5/146-70H+5V
    69H+2J-5/146 ~= 2V
    V ~= 69H/2+J-5/146/2

    Find sharp J estimate above (was stuck in filter — hopefully moderator liberated by now). Plug in H & J.
    Compare estimates with Seidelmann (1992) :

    0.615197281361715
    0.615197263396975 = 1/V
    0.000002920159 = % error

    1.00001746209129
    1.00001743371442 = 1/E
    0.000002837638 = % error

    The estimates have to be good enough to preserve the slip-cycles up the generalized Bollinger hierarchy.

    With the help of Ramanujan, Seidlemann, Conway & associates — plus sagely Mayan astronomy legacy — it’s proving doable. There’s scope for refinement & formality beyond skeletal outlines noted casually thusfar.

    We can tie the above to the smallest simple sporadic group through 5256 = 36 * 146. A few noteworthy partitions of 7920 including 5256 are noted above.

    The E & V estimates based on 146 are good to second level up the slip hierarchy, but not third.

    Note well: The J & S estimates outlined in the last comment go far enough up the hierarchy to challenge Seidelmann as key arbiter.

    It will be tragic comedy observing some conventional mainstreamers who will no doubt be illogically biased against the clear role of monstrous moonshine in solar system orbital stability only because its name “sounds crazy”. The fantastic name provides a convenient test of sound human awareness and integrity in a rapidly worsening time of extremely compromised political psy-ops.

  145. Paul Vaughan says:

    There are 3 minor typos in the last comment — will be obvious to an astute reader.
    Mods: The comment with the sharp J estimate remains stuck in the filter. Readers: Study it carefully.

  146. Paul Vaughan says:

    I’ve generalized Ramanujan’s method in several ways. There will never be time to present more than occasional crude snapshots from along the exploration trail.

    That’s to underscore that level 28 TOWERS mod 7.

    Conventional linear thinkers will be so lost in monstrous moonshine they’ll reject it out of hand. We live in a time of brutally hardening ignorance.

    Remember that level 28 power 4 splits into 25 & 298 around key dimensions (see above). As with 4370 + 58 = 4428 there are more stories, including one about 1/(U-N).

    Fib, Luc, & Binet are just a drop in this ocean — takes all of 5 seconds to note how to build them from moonshine before getting on with noticing the monstrous sea of other things.

  147. Paul Vaughan says:

    “Whatever EU DO: don’t tell any U-N.
    We’ve got somethin’: 2 reveal.
    √(Φ-φ) think EU already know:
    the art isn’t gone”
    — Queens of the Stone Age

  148. Paul Vaughan says:

    New trail on the C[ENSO]Rship weather left or right: a strong new environmental ethic (e.g. 3/4 of urban development replaced by forest) to cleanly divorce climb IT lies that AImost kill US.

    In hawk key they call this a penalty shh!OT. The referee has good reasons for awarding it.

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