TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/ORIGINAL-FUCHS-ALGEBRAIC-FRAME.txt
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000040 |  w^3=(z^2-1)/(z^2+1), eta=z(1-w^3), eta^2=1-w^6,
000041 |  H_a=(w+omega^a)/(w+omega^(a+1)), h_a^2=H_a,
000042 |  h0 h1 h2=1.
000043 | 
000044 | All indices below are modulo3. Set h=h0 and
000045 | 
000046 |  p3(w)=((1+omega)w-omega)/(w+1),
000047 |  p2(w)=((1+omega)w+omega)/(w+1),
000048 |  p1(w)=4[-(1+omega)w^3+(1-omega)w^2+(omega+2)w+omega]
000049 |                               /[9(w^3-1)].
000050 | 
000051 | Then the explicit column is
000052 | 
000053 |  g(w,h)=( h p1/(a b), h z p2/b, h z p3 )^t.
000054 | 
000055 | The two-coefficient RR ansatz in this character was row3=h z(c+d/H0).
000056 | The exact answer is d=omega c. Normalizing c=1 gives the p_j above.
000057 | 
000058 | IMPORTANT SOLVER CORRECTION: an initial intermediate test split both w
000059 | and omega coefficients while treating c,d as independent rational constants.
000060 | That incorrectly excluded d=omega c and returned zero. Constants must lie
000061 | in Q(omega), as the question specified. The corrected solver works modulo
000062 | omega^2+omega+1, finds the solution above, and checks its residual exactly.
000063 | That temporary solver error is not counterevidence against any old source.
000064 | 
000065 | 3. EXACT DIFFERENTIAL CHECK AND THREE INDEPENDENT COLUMNS
000066 | 
000067 | Write s=z^2=(1+w^3)/(1-w^3), D=s^2-1,
000068 | l=(dz/dw)/z=3w^2/(1-w^6), k=h'/h=H0'/(2H0),
000069 | and g_e(f)=e f+(f'+k f)/l, where primes here mean d/dw.
000070 | The three original row equations, after the explicit nonzero factors are
000071 | removed, are exactly
000072 | 
000073 |  D g_0(p1)-s^2(2p1+a^2 p2)=0,
000074 |  D g_1(p2)-p1-s^2 p2-b^2 s p3=0,
000075 |  D g_1(p3)-s p2-s^2 p3=0.
000076 | 
000077 | Each rational remainder is zero in Q(omega)(w), with a^2=8/9,b^2=5/9.
000078 | These equations retain the irrational a,b in the actual solution column;
000079 | they have only been scaled out of the polynomial remainder calculation.
000080 | 
000081 | The cubic deck action is c:(w,eta,h_a)->(omega w,eta,h_(a-1)).
000082 | Since it fixes z, form the three solution columns
000083 | 
000084 |  G_j=g(omega^j w,h_(-j)), j=0,1,2,   G=[G0column,G1column,G2column].
000085 | 
000086 | Their determinant is
000087 | 
000088 |  det G=-27sqrt(2)(2omega+1)(z^4-1)/10.
000089 | 
000090 | It is nonzero at every ordinary z. Thus G is a full differential gauge
000091 | trivialization on the punctured cover, not merely a monodromy count.
000092 | The Wronskian also agrees with tr A=4z^3/(z^4-1).
000093 | 
000094 | 4. BASEPOINT AND Psi(0)=I
000095 | 
000096 | Choose the lift over z=0 with w=-1, eta~2z, and write
000097 | 
000098 |  h0~rho z, rho^2=2/[3(omega-1)],
000099 |  h1(0)=tau, tau^2=-omega, h2=(h0h1)^-1.
000100 | 
000101 | Put t=2omega+1. Direct limits of the displayed meromorphic expressions give
000102 | 
000103 |  Gbase=G(0)=
000104 |  [[0,0,-3tau t/sqrt(10)],
000105 |   [-3rho/(2b),t/(3b rho tau),0],
000106 |   [-3rho t/2,1/(rho tau),0]].
000107 | 
000108 | Its determinant is +27sqrt(2)t/10. Therefore
000109 | 
000110 |  Psi=G Gbase^-1, Psi(0)=I, det Psi=1-z^4.
000111 | 
000112 | The apparent branchpoint poles in individual expressions cancel in these
000113 | limits; z=0 is an ordinary point of the original ODE. Other consistent
000114 | base-sheet/root choices merely change the associated constant frame.
000115 | 
000116 | 5. ACTUAL DECK AND QUARTER ACTIONS, INCLUDING THEIR SIGNS
000117 | 
000118 | For deck V4, h_a->epsilon_a h_a, product epsilon_a=1:
000119 | 
000120 |  G -> G diag(epsilon0,epsilon2,epsilon1).
000121 | 
000122 | For c, G -> G Pc with Pc=[[0,0,1],[1,0,0],[0,1,0]]. Thus the normalized
000123 | monodromy matrices are respectively Gbase Depsilon Gbase^-1 and
000124 | Gbase Pc Gbase^-1. They realize the faithful signed-permutation A4 action.
000125 | In particular Yq is not only a cover on which an abstract matching group
000126 | exists: these actual differential solutions transform by that deck action.
000127 | 
000128 | For the source quarter map use the calibrated actual Kummer lift
000129 | 
000130 |  Q:z->i z, w->1/w, eta->-i eta/w^3,
000131 |  Qh0=+omega/h2, Qh1=-omega/h1, Qh2=-omega/h0.
000132 | 
000133 | Choose tau=i omega^2, so the selected point over z=0 is fixed by this Q.
000134 | The missing minus signs cannot be dropped. Substitution into each entry
000135 | of G gives the EXACT identity
000136 | 
000137 |  G(Qy)=U G(y) KQ,
000138 |  KQ=[[0,1,0],[-1,0,0],[0,0,1]]=Gbase^-1 U^-1 Gbase.
000139 | 
000140 | All nine reduced Q-entry residuals are zero. Hence
000141 | 
000142 |  Psi(Qy)=U Psi(y)U^-1.
000143 | 
000144 | The same normalized identity follows independently from ODE uniqueness
000145 | and the printed square covariance; no new M0 numerical comparison enters.
000146 | KQ has eigenvalues1,i,-i and the required exchanged-slot square signs.