n = 7 verification (Lemma 2.7): Z7(α,t)=-1

ω=(Sqrt[((4-11 t+8 (t^2))/t)]);

αα=((-13+81 (t^1)-169 (t^2)+64 (t^3)+260 (t^4)-416 (t^5)+256 (t^6)-72 (t^7)+8 (t^8))/(8 (((-1+t))^3) (t^2)   (-1+5 (t^1)-6 (t^2)+(t^3))))+(((-1-7 (t^1)+65 (t^2)-164 (t^3)+180 (t^4)-88 (t^5)+16 (t^6)) ωω)/(8 (((-1+t))^3) (t^2)  (-1+5 (t^1)-6 (t^2)+(t^3))));

q=(2t-1)^10;

p61=16t^5(-10+27 t+288 (t^2)-2022 (t^3)+5825 (t^4)-9477 (t^5)+9336 (t^6)-5494 (t^7)+1712 (t^8)-122 (t^9)-80 (t^10)+16 (t^11));
p62=(-16 t^6)(-25+206 t-726 (t^2)+1459 (t^3)-1927 (t^4)+1870 (t^5)-1394 (t^6)+724 (t^7)-210 (t^8)+24 (t^9));
p41=(-4t^3)(50-687 t+4203 (t^2)-15695 (t^3)+41282 (t^4)-82204 (t^5)+125852 (t^6)-144150 (t^7)+117216 (t^8)-62520 (t^9)+18496 (t^10)-1080 (t^11)-960 (t^12)+192 (t^13));
p42=4t^3(10-121 t+677 (t^2)-2521 (t^3)+7132 (t^4)-15412 (t^5)+24800 (t^6)-29742 (t^7)+27184 (t^8)-18840 (t^9)+9072 (t^10)-2520 (t^11)+288 (t^12));
p21=(-2+40 t-360 (t^2)+2245 (t^3)-11098 (t^4)+42924 (t^5)-126314 (t^6)+282304 (t^7)-483968 (t^8)+638896 (t^9)-640864 (t^10)+470400 (t^11)-234144 (t^12)+65792 (t^13)-2784 (t^14)-3840 (t^15)+768 (t^16));
p22=-t^3(65-774 t+4344 (t^2)-15694 (t^3)+41224 (t^4)-82096 (t^5)+124784 (t^6)-144768 (t^7)+127840 (t^8)-83808 (t^9)+37824 (t^10)-10080 (t^11)+1152 (t^12));
p01=(1-t) (1-19 t+161 (t^2)-924 (t^3)+4066 (t^4)-13822 (t^5)+35840 (t^6)-70584 (t^7)+105400 (t^8)-118168 (t^9)+96704 (t^10)-54208 (t^11)+17760 (t^12)-1440 (t^13)-1024 (t^14)+256 (t^15));
p02=(t^3) (25-290 t+1636 (t^2)-6010 (t^3)+15992 (t^4)-32064 (t^5)+48928 (t^6)-56632 (t^7)+49024 (t^8)-30752 (t^9)+13120 (t^10)-3360 (t^11)+384 (t^12));
p11=8t^3(t-1)(-125+4005 t-60169 (t^2)+570158 (t^3)-3862166 (t^4)+20045945 (t^5)-83223823 (t^6)+284217439 (t^7)-813613881 (t^8)+1978202656 (t^9)-4124892528 (t^10)+7432364016 (t^11)-11645707296 (t^12)+15960414880 (t^13)-19242695952 (t^14)+20530254512 (t^15)-19487643408 (t^16)+16504985216 (t^17)-12441863808 (t^18)+8263236224 (t^19)-4749495424 (t^20)+2309017088 (t^21)-926839040 (t^22)+300215040 (t^23)-76503296 (t^24)+14585856 (t^25)-1794048 (t^26)+73728 (t^27)+8192 (t^28));
p12=8t^3(1-t)(-25+765 t-11171 (t^2)+105634 (t^3)-736638 (t^4)+4055851 (t^5)-18297419 (t^6)+69029409 (t^7)-220386161 (t^8)+599953820 (t^9)-1399432680 (t^10)+2804818368 (t^11)-4835265904 (t^12)+7165985152 (t^13)-9113381872 (t^14)+9916520688 (t^15)-9197207856 (t^16)+7238388416 (t^17)-4811313152 (t^18)+2688147584 (t^19)-1256283520 (t^20)+488377856 (t^21)-156990208 (t^22)+41689856 (t^23)-9309952 (t^24)+1799168 (t^25)-280576 (t^26)+24576 (t^27));
p31=8 t^3(t-1)(-125+4005 t-59369 (t^2)+553998 (t^3)-3730470 (t^4)+19565545 (t^5)-83537119 (t^6)+297885767 (t^7)-901975729 (t^8)+2350136384 (t^9)-5333296784 (t^10)+10652922784 (t^11)-18880321184 (t^12)+29852879392 (t^13)-42255474560 (t^14)+53630116864 (t^15)-60971031168 (t^16)+61759492352 (t^17)-55128623872 (t^18)+42657332224 (t^19)-28037451008 (t^20)+15314942464 (t^21)-6803466496 (t^22)+2405327616 (t^23)-657957120 (t^24)+131469312 (t^25)-16441344 (t^26)+663552 (t^27)+73728 (t^28));
p32=8 t^3(1-t)(-25+765 t-11171 (t^2)+107634 (t^3)-787518 (t^4)+4665067 (t^5)-22973819 (t^6)+95297577 (t^7)-336259225 (t^8)+1016942748 (t^9)-2647457192 (t^10)+5940364352 (t^11)-11479425312 (t^12)+19070848736 (t^13)-27171006848 (t^14)+33097766976 (t^15)-34341234112 (t^16)+30212895232 (t^17)-22415990528 (t^18)+13930544128 (t^19)-7190272256 (t^20)+3053491200 (t^21)-1060255488 (t^22)+303176448 (t^23)-73578240 (t^24)+15455232 (t^25)-2525184 (t^26)+221184 (t^27));
p51=128t^8(t-1)(-1500+28960 t-257939 (t^2)+1404604 (t^3)-5211955 (t^4)+13795890 (t^5)-25855137 (t^6)+29614248 (t^7)+3063112 (t^8)-113012293 (t^9)+333031055 (t^10)-652968101 (t^11)+993241608 (t^12)-1221269400 (t^13)+1224164696 (t^14)-994694744 (t^15)+647101024 (t^16)-331790224 (t^17)+131533872 (t^18)-39147024 (t^19)+8229120 (t^20)-1046016 (t^21)+41472 (t^22)+4608 (t^23));
p52=128 t^8(1-t)(-100+80 t+20039 (t^2)-260728 (t^3)+1786957 (t^4)-8269540 (t^5)+28531279 (t^6)-77414974 (t^7)+170435054 (t^8)-309998835 (t^9)+470441223 (t^10)-598385907 (t^11)+638090092 (t^12)-567729488 (t^13)+417123064 (t^14)-249429768 (t^15)+119783328 (t^16)-46124976 (t^17)+14587824 (t^18)-3960432 (t^19)+919872 (t^20)-157824 (t^21)+13824 (t^22));
p71=2048 t^13(t-1)(-25+56 t+2520 (t^2)-25565 (t^3)+120032 (t^4)-330197 (t^5)+533239 (t^6)-339915 (t^7)-575358 (t^8)+1899107 (t^9)-2748394 (t^10)+2566194 (t^11)-1675233 (t^12)+781067 (t^13)-257689 (t^14)+57232 (t^15)-7392 (t^16)+288 (t^17)+32 (t^18));
p72=2048 t^13(1-t)(-1-50 t+626 (t^2)-2319 (t^3)-1948 (t^4)+47711 (t^5)-201331 (t^6)+485563 (t^7)-779658 (t^8)+878711 (t^9)-712736 (t^10)+425072 (t^11)-193639 (t^12)+72003 (t^13)-23071 (t^14)+6068 (t^15)-1096 (t^16)+96 (t^17));

Two particular checks and a plot

RootReduce[1/q{Sqrt[p71+p72 ωω],p61+p62 ωω,-Sqrt[p51+p52 ωω],p41+p42 ωω,Sqrt[p31+p32 ωω],p21+p22 ωω,-Sqrt[p11+p12 ωω],p01+p02 ωω}.Table[Sqrt[A]^j,{j,7,0,-1}]/.A->αα/.ωω->ω/.{{t->4/5},{t->1/(11-4Sqrt[6])}}]

Plot[1/q{Sqrt[p71+p72 ωω],p61+p62 ωω,-Sqrt[p51+p52 ωω],p41+p42 ωω,Sqrt[p31+p32 ωω],p21+p22 ωω,-Sqrt[p11+p12 ωω],p01+p02 ωω}.Table[Sqrt[A]^j,{j,7,0,-1}]/.A->αα/.ωω->ω,{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),1},PlotRange->{-1.1,-.9},WorkingPrecision->30]

Note that we will need 4 lemmas:

α*b1=u1/q, α^3*b3=u3/q, α^5*b5=u5/q,α^7*b7=u7/q 

u1,u3,u5,u7 has the form (P1+P2)ω/Q ,where P1,P2,Q are suitable polynomials of t.

u1=(-10+198 t-1742 (t^2)+9029 (t^3)-30986 (t^4)+74600 (t^5)-128398 (t^6)+153928 (t^7)-113584 (t^8)+22480 (t^9)+49912 (t^10)-57728 (t^11)+28480 (t^12)-7808 (t^13)+2656 (t^14)-1280 (t^15)+256 (t^16))- ωω t(10-174 t+1409 (t^2)-7126 (t^3)+25020 (t^4)-63866 (t^5)+121016 (t^6)-171312 (t^7)+180480 (t^8)-140328 (t^9)+80640 (t^10)-35392 (t^11)+12352 (t^12)-3104 (t^13)+384 (t^14));
u3=(1/(4 (((-1+t))^3) (t^3)))(85 t-1530 (t^2)+11801 (t^3)-45890 (t^4)+41000 (t^5)+573384 (t^6)-3922012 (t^7)+14743612 (t^8)-38993060 (t^9)+77564192 (t^10)-118920088 (t^11)+141854664 (t^12)-132268192 (t^13)+97062688 (t^14)-56854208 (t^15)+27058816 (t^16)-10397056 (t^17)+2955904 (t^18)-459904 (t^19)-25728 (t^20)+24576 (t^21)-3072 (t^22)+t (5+26 t-1583 (t^2)+18062 (t^3)-114412 (t^4)+486044 (t^5)-1488148 (t^6)+3355172 (t^7)-5454140 (t^8)+5774672 (t^9)-2173480 (t^10)-4792760 (t^11)+11086592 (t^12)-12846496 (t^13)+10198080 (t^14)-6137984 (t^15)+2960896 (t^16)-1140864 (t^17)+322944 (t^18)-57216 (t^19)+4608 (t^20)) ωω);
u5=(1/(4 (((-1+t))^6) (t^3)))(-250 t+2987 (t^2)+4045 (t^3)-313967 (t^4)+2996100 (t^5)-16116136 (t^6)+58167348 (t^7)-149013636 (t^8)+272559440 (t^9)-335591112 (t^10)+197736736 (t^11)+199225392 (t^12)-723484336 (t^13)+1121333232 (t^14)-1215613648 (t^15)+1022423712 (t^16)-691412576 (t^17)+377059456 (t^18)-161992864 (t^19)+52100096 (t^20)-11373440 (t^21)+1286912 (t^22)+47232 (t^23)-33792 (t^24)+3072 (t^25)-t (10+391 t-9797 (t^2)+89447 (t^3)-426220 (t^4)+970280 (t^5)+778040 (t^6)-14382620 (t^7)+57858672 (t^8)-144875368 (t^9)+260147248 (t^10)-352440432 (t^11)+368727984 (t^12)-300424464 (t^13)+188428624 (t^14)-84962240 (t^15)+18995168 (t^16)+8535104 (t^17)-11947808 (t^18)+6943232 (t^19)-2508672 (t^20)+576000 (t^21)-77184 (t^22)+4608 (t^23)) ωω);
u7=(1/(4 (((-1+t))^6) (t^3)))(-33 t-16 (t^2)+9292 (t^3)-122404 (t^4)+793766 (t^5)-2977444 (t^6)+5622684 (t^7)+4021212 (t^8)-63726144 (t^9)+227073744 (t^10)-503694112 (t^11)+787840912 (t^12)-888389920 (t^13)+694326576 (t^14)-308427104 (t^15)-34342848 (t^16)+181724576 (t^17)-160749120 (t^18)+84638784 (t^19)-29409792 (t^20)+6561152 (t^21)-788736 (t^22)+4736 (t^23)+11264 (t^24)-1024 (t^25)+t (-1-90 t+1532 (t^2)-7096 (t^3)-28190 (t^4)+520588 (t^5)-3157956 (t^6)+11559500 (t^7)-28309248 (t^8)+46157648 (t^9)-42374448 (t^10)-6235696 (t^11)+91494176 (t^12)-165214832 (t^13)+179249632 (t^14)-132624128 (t^15)+66918944 (t^16)-20646976 (t^17)+1733440 (t^18)+1652608 (t^19)-877696 (t^20)+214528 (t^21)-27776 (t^22)+1536 (t^23)) ωω);

Together[αα^1 (p11+p12 ωω)-u1^2/.ωω->ω]

Together[αα^3 (p31+p32 ωω)-u3^2/.ωω->ω]

Together[αα^5 (p51+p52 ωω)-u5^2/.ωω->ω]

Together[αα^7 (p71+p72 ωω)-u7^2/.ωω->ω]

Together[1/q(u7+(p61+p62 ωω)αα^3+u5+(p41+p42 ωω)αα^2+u3+(p21+p22 ωω)αα+u1+p01+p02 ωω)/.ωω->ω]

Resolve[ForAll[t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ])<t<1,αα^1 (p11+p12 ωω)==u1^2/.ωω->ω],Reals]

Resolve[ForAll[t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ])<t<1,αα^3 (p31+p32 ωω)==u3^2/.ωω->ω],Reals]

Resolve[ForAll[t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ])<t<1,αα^5 (p51+p52 ωω)==u5^2/.ωω->ω],Reals]

Resolve[ForAll[t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ])<t<1,αα^7 (p71+p72 ωω)==u7^2/.ωω->ω],Reals]

Plots

Plot[-Sqrt[αα^1 (p11+p12 ωω)]-u1/.ωω->ω,{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),1},PlotRange->{-.1,.1},WorkingPrecision->30]

Plot[Sqrt[αα^3 (p31+p32 ωω)]-u3/.ωω->ω,{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),1},PlotRange->{-.1,.1},WorkingPrecision->30]

Plot[-Sqrt[αα^5 (p51+p52 ωω)]-u5/.ωω->ω,{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),1},PlotRange->{-.1,.1},WorkingPrecision->30]

Plot[Sqrt[αα^7 (p71+p72 ωω)]-u7/.ωω->ω,{t,(NumericalApproximation[Root, (RowDefault["0.777", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,2], ]),1},PlotRange->{-.1,.1},WorkingPrecision->30]