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Appendir The Mathematics of Timewave Zeroby PeterMeyer

l. Generel mathematical considerationsAs usual, let the set of non-negtive real numberc be denoted by [0,-).Let v(x) be any finction v:(0,-) r (0,-) such that there exist positivereal numbers c and d suchthat:

G) for allx, v(x) I therc exisa an integer n such thatx/a^n < d, so for all i > n, x/a^i < d, so by condicion (z) for all i > n,

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2I2 ,{?PENDIX

Thus:

I t u1v"^11i) = t ( (x/a^i)+a^i)which clearly erisa. Since

f (x) exists.As the first theorem ofTimewave Zero mathematicswe have:Proposition : For all x >= 0, f(a*r) = a'rf(x).PRooF:Lctx>=0 then

r. v(rix*r^i)t tdzxt = L ___;i_i=--

_ \. .xJ ycaa^(i+l))Ir_. - L r^(ir ) IIF= ,- l t t - ' , in) |t -L'=-- l

= a. fO which complaes the proof

2, The mathematical de6aition of the timewar.eThe tunction rhac epresents he timewave s c.cscnrially 6acal ransformofa saw+ooth tuncrion. Fint we shall defne this latter tunction.C-onsider he followiq set of 384natural numbers,traditionally knownx t\e daa poixx fot tIrc ttmewave:

a#l= -t{P + > (v(x/10''|1i)

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7, 4' ), 2, 6, 8,rj , t, 26, zt, 24, rt , n, 16, 14, rg,17, 24, zo, 2t, 61, 60, t6, tt, 47, t?,16, J8, 19, B, t9, 1t, zz, 1.4, a2, t,1.9, 1o, 27, 26, .6, '1, 21, 19, t7, 62,6r ' tt, t7, t7, ir, to, 4c, 29, ].8, 26,So, ,1, 11., 61, 60, 60. 42, 42. 41, $,42, 11, 4t, 4r, 46, 4, tt, )4, 2r, 21,19, tr , 40, 49, 29, 29, )1, 40, 16, ',9, 26, 1o, r5, 18, 14. 66, 64. 64, ,6,ti' ,7, 49' 'r, 47' 44' 46, 47, t6, tl,jr, zt, 37, to, 3r, 24, ao, J6, l't, 22,28, lr., 27, )2, )4, )r, tz, 49, 4A, I,,r, tt, 40, 43, 42, 26, 10' 18, tS, 4r,,t, tz, tr, 47, 6\ , 64, 6t, ]9 , 4r, 4\ ,22, zr, 2J, B, 4\, la, z4 22. z+ 14,17, 19, tz, to, 47, 42, 40, 42. 26, 27,)'7' t4, ,8, )t, 44, 44, 42, 4r, 40, 17,t3, ar, 16, 4$ 34, 3A, 46, 44, 44, 16,)7, )4, j6, )6, )6, )8, $, 18, 27, 26,)o, )2, 17, 29, ,o, 19, 48, 29, 17, )6,to, 19, 17, 24, 20, zt, u, ,2, to, t),t7 , tt, i4, 44, 4r, r) , 9, t, )4 26,12, )1, 41, 42, 1\ 3a, 10, 21, 19, 2t,4t, )6, tr, 47, 4t, $, 47, 62, t2, 4t ,t6, )8, 46, 47, p, 4:), 41., 42, 16, )8,41, ti. ,2, t), 47, 49, 48, 47, 4r, 44,rt, n! 19, tr, 40, 49, 21, zt, 7t , 11,1o, 27, 7, 4, 4, )2, 22, J2, 68, 7a,66, 68, 79, 7r, $, 4t, 41, )8, 40, 4r,.4, 2t. z?, 3, 1t. )8, 41, to, 4-8, 18,17, 25, J4, J8, tt, 18, 4c, 4\, J4, )j,10, 33, a1, )t , 28, 4, t2, 26, )o, 26,7t, 77, 7r, 61, 61, q, 37, 40, 4r, 49,47, '1 , '2 , J7, )1, 49. 47. 4+ tz. 18,28, 18, ,9, J7, ..2, zo, 17, 44, to, 4c.,)2, )), 33, 40, 44, t9, 12, )2, 4a, )9,14, 4r, )1, J1, j2, 11, g, t6, 72, zo,20, tz, \, ro

These velues are derivcd from certain transformations performed upon aset of64 numbcrs, the numbers oflines that change from cach hexagram

APPENDIX 213

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- '1-^'Fa,- --**--r*--"'1,,^r1",r- '-\.r'1 - --,-/--'in the King Ven sequence o the next, as etplained previously in thisbook. They provide the basic numerical valuesused n this marhemadcaldefinition of the timwave.

Definew(i) as he th valueofthis set,using ero-basedndexing.Thus:i o - i - . . .w(,) 4 . . .

Fxtend w to a tuncdon wrQ such that for any non-negative nteger i, w(i)= Er( i mod 38a , where mod 384 s the remainder pon divisionofi by184.Thus, for example, u(777) =,N( 777 ft'o4 38a = w(q) = 8. .r'r0 is adiscrete unction definedonly for integas, not for all real numbers.Now for any non'negative real number x, let v(x) be rhe value obtainedby linear nterpolationbetween he valueswr(in(x)) and wr(in(x)+r),where in(x) is rhe integral parr ofx. Formally v(x) is defined as*'r(in(x)) + ( x in(x) ) * ( *r(x+t - wr(x) )

or in exPanded orm:(*) =*( in( ') m"d 18+ + ( (x- in(x) ) +( w ( in(x+t mod 38a - w ( in(x) mod 3s+ ).

Now consider rhe fractal transform f(x) ofv(x) using a = 64, as ollows:

frvr _ s \( ,64' i )

or, what is the same hing,

,.,-,t ,.,,-,.-,z- - , i ) '64' i )The tunction f(x) exisa because

(r) fbr a x, v(x) < 80, ard(z) for a.llx < 3,v(x) = 0.

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The fiactal tunction (x), which represents he timewxve, and which isgraphedby the software, s a simple transformation of f(x), as ollows:

, , ( tflx/ = 64^lwherex = time in &ys prior to 6 A.M. on the zro dare.. The scaling ac-tor of 64^3 is used so as to produce conveniennalue, on the y-axisof thegraph.Thus the va.lueofr0 at 5 a.v. on rhe zero date is

do)=#=0.The value of O at 5 a.v. on the day before the zero &te is

( '=#=ooooooS5r5ottr 'The ralue ofr0 at 5 nu. on the day ten daysbefore the zero date is

t(Ss) - oo., * f(9 t) = ooooo4-t8s69t

and the ralue at 6 A.M. on the day r,ooo,ooo,ooo,ooo &ys (about2,77,888,267 ye rsl befor the zrodate is r,192,046.511416.These valuesare ndependentofthe acnralzero date. The ralue ofthetimewave at any point io time is not a function ofthat temponl locationirself but rather of the difference between hat rime and the time assignedto th zem point ofthe wave.

' Th. tineww is zm onrye onepoint,wh.n r =0. Forx ' 0 ihe Elue ofrhe wav. r positir. Tbeu n poidt r rh oidr n tine chcs to orcpond to rh. ra1u.0 orx. Thc !i1poin(ukdn6^.M.on Dmb.r 2r, 2oD(LnM d rh. m daie). Thus th. tincRvc hs 3positirc'"luc tu .1 poinbin iin. prior o rhezrc dar, n Fb only .t rhe m point,md isu.de6ne,lalicr theanopoilt.

,5

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- *-XjJ;- =-;*- l"^rt..r- "'rl.r -\-.,-----.-1Note that the "direcrion"ofthe graph s rhe oppositeofwhat is usuglwith Carresian oordinares.he graphof a tunction f(x) normallypro-ceeds rom left to righr along he x,ais for increasing . In rhis case hegraphproceedsrom right to left for incrersing , rhar s, for increasingnumberofdavsarar to the zeroooint.

3. The mathernaticalbasisofresonanceThe phenomenonof resonance, hereby egionsof the wave rt widelysepararedntewalsmay havecacdy the same hape,s a remarkableea,ture of the tinewave.+ he marhematielbaris or rhis phenomenons asfollows:

Considera point in time x daysprior to the zerodate, hen hevalueofrhe wavear rhar point is (x), ai definedabove.Now consider he ulue,t(64*x), ofrhe wive at the point in time 6,{*xdaysprior to rhezerodate.Frorn he resultprovedar rheend ofSectionI abovewe have hat

(ra*,) = f6ri:) #- - 6o-,(,).Thus the valueofrhe waveat a point B, 64 timesasdistant iom rhezeropoint asa point A, is 64 tines the valueofthe waveat A. Since his s alsorrue br the points n the neighborhoodsfA and B, it is thusclearwhy aregionaroundB has hesame hape sa regionaroundA.This resonances called he first highermajor resonance,ince he region around he poinr C, 6+*6+ timesasdistant rom rhe zeropoinr a.sthe point A, is also esonant ith the regionaroundA and constituteshescond highermajor resonrnce fpoint A. Therearean unlimitednum-bd oahishermajorresonanas.Simildy for any point x, (t64) = (x)/64, and so the valueof the waveat a poinr B, 64 times doser o the zero poinr thar a point A, is r/64 the' The phenonenor ofconinc is gathidlly ilBtmted br meanrofthe soliwr, TimewrR Zero,*hi.h denonnnres (hn thcory. Ir is dplai0ed in deftil in ch{ter ? ofthe mrnurl provtdd wnh thesotnerc. Tinemve z.ro, which rvs dsignedand oded by de rurl'or ofthn rppendix, is ayriliblehon Dolphn Softr'rre, 43 Sl*ru.t Squ2r #r 47. B.rlele', c:lif.rni. 94704i (5I o 4al 3009).

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valLre f the wave atA Thus the regionamund B has he sameshapeas theregjon roundA ,nd fi!a consrirurc'he6r.r lowelnajolBonance of,hcregion around A. A: wirh higher resonances,here are also second, hird,founh, and so on, lower major rcsonancso my region of rhe graph. Thelowr major resonancesrecompressed eometrically owardthe zeropoint,'o thrr onlv a fewsecondrmayseplarc rhenth and heIn+j)'h majo;towerresonancesor some,not particularly large,value ofn.

4. Further mathematiel resultsThe mathematicsof Timewave Zero srend considerablybeyond the ini-tial proposirion proved above.'Lemma r: For any natural number x, v(x) = w(x mod 3s4).PRooF:This follows rom the defnidon offinction v0, since or a nat-ural numberx = in(x) ard sox - ;n(x) = 0.Lemma z: For any natural numbers ard i, v(x/54^i)'!64^i s a narunlPRooI: By rhe definitionoffunction v0:

v(xl64^i),\64^i= 64^i * w(int(x/64^i)mod jsa)+ 64^i ''. ( t64^i - in(,i64^D )* ( (w( in(x/da^i+t modl8a)- w(in(x/6a^i) mod j8a) )= 6+^i * w(in(x/6+^i) mod t8+)+ (x - 64^i*inr(>/64^i)* ( ($( in(x/6a^i+, modr8a)- w(in(x/64^i) mod r84) )

' Tnenarh.mdi.:l rsls pqent.d in rhe n2inder ofrhir rpFndix r bsed pdtl, on md< donebr Klau Schd$of B.+cFchdhach, cerhry. App.ndixV oathenruil lor TinMve tae pe-fnB,h.{6Jb |dIor -o dkin 'n8rmdLi. r.onrnr.d..ff,ed ITUmd"r.

A?PENDIX

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Sinct the vatuesofthe firnction w0 are natural numbers,and x is a nat-ural nurnber, the value ofthis expressions a natunl number.I-emma 3: For any natural number x that is divisible by 3

(, = x v(xl6a^l*6a^i w(xx6amod3s4)/6a.i=0PRooF: Suppose is a natural number divisible by 3. By the de6nition off(x) above:

r tx l = Lv\x|64^t l+64^ri=--

=l,{*t aa"iS*5o^l X (x*64^D/64^i

=! "t*rr,arrxe,"r \ i - ,r"+64^imod 84)/64^ii=0 i=lby Lemma r, since x*64^i is integral. Thus f qx7=:) ' (x/64^i ,*64^i + w({}04 mod184)/64 t${r}6a^' modr84)/6a^i .

Since = t*y for somenaturalnumbery

i=0

Zw(,D(x+64^i)modJ8a)/5a^i= >wb*y+6a^2x6a^i mod.JB4)/64^ii=0

= Zw{jE4*y*l r*64^j mod184)/64^ii=0

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384*y*32i.64^i mod t84 is 0, and w(0) = 0, so each erm in this sum is 0.Thu!fk)=tv(x/5a^i)*54^i + w(x*64mod sa)/64.eEDPrcposition 2: For any.natural number x that is divisible by 3 rhere h anatural number k suchthat f(x) = k /5a.PRooF: Lt x b a naturel number divisible by 3then by Lemma 3

f (x) =lvlxtsat'1*5r,'i + w(x*64 mod 384)164.i-0By kmma z eachrerm in this sum is an integer,so the sum is an inceger,and so is an integral multiple of /ra. Thc second erm in chesum is alsoan intgtal multiple of 64, and so f(x) is.On thc basisofProposirion z we have:Corollary r: For any natural number x that is divisible by 3 there is a nat_ural nmbcr k suchthat (x) = k/6a^a.PRooF: Sincc(x) = f(x)/54^r.Propocition 3: For any natunl number x that is divisible by 3S4 here is ananral numbcr k suchdrat f(x) = 6a'rk.Proof kt x be a natursl number divisible by 384 then r is divisiblc by I

fG) = I v(x/61^t*54^i w(x*64mod3s4)/6a.i=0Sincc x is divisible by 384,x*54 mod 384 s 0, so at(x*64 mod .js4)t6a =w(0) = 0, so

fG) = t v(x 64^t*64^i =vG)+X vG 54^0,k64^i.

APPENDIX

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PPTNDIX

Now x is integral, so v(x) = w(x rnod r8a) = w(0) = 0, so

r DL, z v(x 64^r iro4^r.Since x is divisible by 384there is some natural numbr k such that x =

f (x) - > \ (6*6a* /(64.04^( i- l )+{64+64^{i- i l l

Now by kmma z each erm in the sum is a narural number, so the sumis, so chere s a natural number k suchrhat f(x) = 64*k. QEDThe mathematical results presentedabovar ust a beginning. There areno doubt far more profound theorems that await discovery by rnarhe-maticians.