TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/D4-CONNECTION-LIFT.txt Original file SHA-256: d3647a78636a3d9d465c52ff17ed578b4961a097a7256a04638c6e4b2c2ba6a4 Inclusive original lines: 1-180 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | EXACT INVERSION TEST FOR THE v117 FUCHSIAN CONNECTION 000002 | ===================================================== 000003 | 000004 | 26 October 2026. Exact rational-gauge obstruction. No Atlas, paper, ledger 000005 | or verification source was changed. 000006 | 000007 | POSITIVE CONTINUATION AND SCOPE CORRECTION 000008 | ========================================== 000009 | 000010 | The negative D4 theorem in this file remains valid. A subsequently 000011 | recovered original source proves that the same connection already has a 000012 | finite C4 triangle descent. Therefore the absence of an inversion lift does 000013 | not leave the analytic v117 monodromy open: the C4 quotient proves that its 000014 | extended orbifold image is S4 and identifies the marked representation with 000015 | (U^(-1),M0) up to constant conjugacy. See 000016 | C4-DESCENT-FINITE-MONODROMY.txt and its exact checker. Statements below 000017 | that formerly called the analytic identification open are superseded by 000018 | that positive continuation; the rational inversion calculation itself is 000019 | unchanged. 000020 | 000021 | Question and answer 000022 | =================== 000023 | 000024 | Consider the actual connection 000025 | 000026 | Y'(z)=A(z)Y(z), 000027 | A(z)=sum_(k=0)^3 A_k/(z-i^k), 000028 | A_k=U^k A0* U^(-k), 000029 | 000030 | with 000031 | 000032 | U=diag(1,i,-i), 000033 | 000034 | A0*=[[1/2,sqrt(2)/6,0], 000035 | [sqrt(2)/6,1/4,sqrt(5)/12], 000036 | [0,sqrt(5)/12,1/4]]. 000037 | 000038 | Does it admit a rational lift of the original inversion s:z->1/z, 000039 | 000040 | Y(z)=G(z)Y(1/z), 000041 | 000042 | so that 000043 | 000044 | G'=A(z)G+z^(-2)G A(1/z), (1) 000045 | G(z)G(1/z)=I (2) 000046 | 000047 | up to an allowed constant scalar cocycle? 000048 | 000049 | No. Equation (1) has no nonzero rational matrix solution at all. Therefore 000050 | the concrete A0* differential equation is exactly C4 equivariant but is not 000051 | a genuinely D4 linearized connection for the marked inversion. The cocycle 000052 | condition (2) is never reached. 000053 | 000054 | This closes the proposed D4 route. It does not obstruct the actual C4 000055 | route. The existing exact matrix M0 and the finite group =S4 are now 000056 | identified with the analytic monodromy by the recovered C4 triangle proof. 000057 | The D4 orbifold rigidity theorem still cannot be used, because its required 000058 | connection-level D4 premise fails for the displayed ODE. 000059 | 000060 | 1. Existing result used and changed premise 000061 | ============================================ 000062 | 000063 | The Atlas at revision 1511caef8fbd449e2b24c859 records the correct split: 000064 | 000065 | * v116 makes the residue normal form and the a13=0 infinity resonance 000066 | cancellation exact inside the C4 equivariant ansatz; 000067 | 000068 | * v117 makes the candidate matrix M0, its order-three class and the finite 000069 | group =S4 exact; 000070 | 000071 | * v117 itself still labels equality with the unitarized ODE monodromy 000072 | numerical, but the older September source closes the marked local system 000073 | analytically through its C4 triangle descent. 000074 | 000075 | The new premise tested here is stronger than a13=0: the same A0* connection 000076 | must carry a rational inversion linearization. This is exactly the missing 000077 | connection-level hypothesis needed to apply D4-ORBIFOLD-RIGIDITY.txt. 000078 | 000079 | 2. The connection in one exact rational formula 000080 | ================================================ 000081 | 000082 | Summing the four residues gives 000083 | 000084 | 1 [ 2z^3 2sqrt(2)z^2/3 0 ] 000085 | A(z)= ----------------- [ 2sqrt(2)/3 z^3 sqrt(5)z/3 ] . (3) 000086 | z^4-1 [ 0 sqrt(5)z/3 z^3 ] 000087 | 000088 | Equation (3) proves the rotation covariance directly: 000089 | 000090 | i A(i z)=U A(z)U^(-1). (4) 000091 | 000092 | Thus the actual ODE has the advertised C4 lift. The issue is specifically 000093 | the reflection completing C4 to D4. 000094 | 000095 | The residue polynomial is 000096 | 000097 | det(lambda I-A0*)=lambda(3lambda-1)(3lambda-2)/9, (5) 000098 | 000099 | so every finite residue has the distinct eigenvalues 0,1/3,2/3. At infinity 000100 | 000101 | A(z)=B0/z+O(z^(-2)), B0=diag(2,1,1). (6) 000102 | 000103 | The vanishing a13 entry is precisely what removes the level-one resonance 000104 | obstruction in v116. Equations (5)-(6) now also give a complete degree bound 000105 | for every possible rational inversion lift. 000106 | 000107 | 3. Any rational lift would have to be a tiny polynomial 000108 | ======================================================== 000109 | 000110 | Let G be a rational solution of (1). 000111 | 000112 | At an ordinary point, a pole of order m>=1 is impossible: G' would have a 000113 | term of order m+1 while both coefficient matrices in (1) are regular there. 000114 | 000115 | At a puncture z=i^k, the residue operator acting on a leading matrix C is 000116 | 000117 | L_k(C)=A_k C-C A_(-k). (7) 000118 | 000119 | Since the two residues in (7) are conjugate Hermitian matrices with spectrum 000120 | (5), the spectrum of L_k consists of their pairwise differences 000121 | 000122 | {0,+/-1/3,+/-2/3}. 000123 | 000124 | A pole C t^(-m) would require L_k(C)=-m C. No negative integer occurs, so G 000125 | has no pole at any of the four punctures. 000126 | 000127 | Near z=0 the transformed infinity term is 000128 | 000129 | z^(-2)A(1/z)=B0/z+O(1). (8) 000130 | 000131 | The same leading-order argument excludes a pole because B0 has only the 000132 | positive eigenvalues 2,1,1. It also gives G(0)=0. If 000133 | 000134 | G=sum_(n>=0) G_n z^n, 000135 | 000136 | then the indicial equation is 000137 | 000138 | n G_n=G_n B0 (9) 000139 | 000140 | at the first nonzero order. Consequently the first column begins at order 000141 | z^2, while the second and third columns begin at order z. 000142 | 000143 | Finally, at infinity the leading part of (1) is 000144 | 000145 | G'=(B0/z)G+O(z^(-2)G). (10) 000146 | 000147 | For a polynomial leading term z^d C, equation (10) says dC=B0 C. Hence the 000148 | first row has degree at most two and the second and third rows degree at most 000149 | one. Every rational solution of (1) must therefore have exactly the bounded 000150 | form 000151 | 000152 | [ a z^2 b z+c z^2 d z+e z^2 ] 000153 | G(z)= [ 0 f z g z ]. (11) 000154 | [ 0 h z j z ] 000155 | 000156 | This is not a guessed truncation. Regularity, (5), the 0-infinity exchange 000157 | and the Birkhoff-Grothendieck exponents (2,1,1) force (11). 000158 | 000159 | 4. Nine coefficients, rank nine 000160 | ================================= 000161 | 000162 | Insert (11) and (3) into (1), and multiply the result by (z^4-1)/z. A small 000163 | triangular subset of the resulting coefficient equations already gives 000164 | 000165 | E_11=(2sqrt(2)/3) z^2(b+c z)=0 000166 | => b=c=0, 000167 | 000168 | E_13=-d z^3-e-(2sqrt(2)/3)g z^2=0 000169 | => d=e=g=0, 000170 | 000171 | E_12=(2sqrt(2)/3)(a z-f z^2)=0 000172 | => a=f=0, 000173 | 000174 | E_31=(2sqrt(2)/3)h z^2=0 000175 | => h=0, 000176 | 000177 | E_32=(sqrt(5)/3)j z=0 000178 | => j=0. (12) 000179 | 000180 | Thus G=0. Equivalently, the complete exact linear coefficient matrix has