TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/C4-DESCENT-FINITE-MONODROMY.txt Original file SHA-256: b707e2d720a6dad3a74115b62f16d76077dfda3fe0d3eb8f0f8f76edaf5d7bd5 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 | RECOVERED C4 TRIANGLE SOURCE FOR THE ACTUAL v117 CONNECTION 000002 | =========================================================== 000003 | 000004 | 26 October 2026. Exact source recovery and independent replay. No Atlas, 000005 | paper, ledger, or canonical verification file was changed. 000006 | 000007 | Verdict 000008 | ======= 000009 | 000010 | PROVED, BUT NOT NEW: the exact analytic closure of the v117 monodromy was 000011 | already present in the 28 September Gesamtsynthese. The actual four point 000012 | connection descends by t=z^4 to a three point Fuchsian system with local 000013 | orders (4,3,2). Its orbifold monodromy is therefore the faithful 000014 | three dimensional S4 representation. With the original orientation the 000015 | order four generator is U^(-1), and the marked representation is conjugate 000016 | to the exact v117 pair (U^(-1),M0). The numerical ODE comparison in v117 is 000017 | only a basis calibration, not the proof of finiteness. 000018 | 000019 | The shorter gauge used below makes the residue at infinity semisimple with 000020 | no integral resonance. It is meromorphically equivalent to the older 000021 | printed gauge, whose one possible resonance was explicitly cancelled. The 000022 | two arguments prove the same result. 000023 | 000024 | This positive C4 descent does not contradict the separately proved absence 000025 | of a rational inversion lift. C4 is sufficient. The connection is not a 000026 | D4 linearized connection. 000027 | 000028 | 1. Recovered primary source 000029 | =========================== 000030 | 000031 | The result is printed in: 000032 | 000033 | [OMITTED: non-scientific conversation or execution metadata] 000034 | [OMITTED: non-scientific conversation or execution metadata] 000035 | [OMITTED: non-scientific conversation or execution metadata] 000036 | [OMITTED: non-scientific conversation or execution metadata] 000037 | [OMITTED: non-scientific conversation or execution metadata] 000038 | [OMITTED: non-scientific conversation or execution metadata] 000039 | 000040 | Those lines contain the substitution t=z^4, the old gauge 000041 | diag(1,z,z^3), the three residue spectra, the infinity resonance check, 000042 | the triangle group argument, the marking U^(-1), the exact diagonal of M, 000043 | and the distinction between the A4 four puncture image and the extended S4 000044 | image. They also continue to the signed permutation basis and the genus 000045 | two curve. The present report only restores the part needed for the v117 000046 | analytic monodromy claim and checks it independently. 000047 | 000048 | The active source modules against which it is reconciled are: 000049 | 000050 | source:80af15193155b4dcf60c 000051 | verification/v116_resonance_uniqueness.py:1-170 000052 | SHA256 e9cbeb665cf8c1a2bed3f34281be06aee4c661d784e7f13a5bd8e9be37f577ca; 000053 | 000054 | source:830238c9699fad6ae325 000055 | verification/v117_monodromy_weyl_a3.py:1-206 000056 | SHA256 55eca6a64144f6e49789f3f1bbe84e256764e7ce92b207044be693f8684d5df5. 000057 | 000058 | The [N] status of the analytic identification still printed in v117 is 000059 | therefore stale relative to the older source. The exact matrix properties 000060 | in v117 remain correct. 000061 | 000062 | 2. Exact C4 descent in the short gauge 000063 | ====================================== 000064 | 000065 | Start with the actual connection 000066 | 000067 | Y'(z)=A(z)Y(z), 000068 | A(z)=sum_(k=0)^3 U^k A0 U^(-k)/(z-i^k), 000069 | 000070 | U=diag(1,i,-i), 000071 | 000072 | [ 1/2 sqrt(2)/6 0 ] 000073 | A0=[ sqrt(2)/6 1/4 sqrt(5)/12 ], 000074 | [ 0 sqrt(5)/12 1/4 ] 000075 | 000076 | whose exact rational form is 000077 | 000078 | 1 [ 2z^3 2sqrt(2)z^2/3 0 ] 000079 | A(z)=------------- [ 2sqrt(2)/3 z^3 sqrt(5)z/3 ]. 000080 | z^4-1 [ 0 sqrt(5)z/3 z^3 ] 000081 | 000082 | It obeys i A(iz)=U A(z)U^(-1). Put 000083 | 000084 | t=z^4, Y=diag(1,z,z^(-1)) X. 000085 | 000086 | Then 000087 | 000088 | dX/dt=B(t)X, B(t)=B0/t+B1/(t-1), 000089 | 000090 | [ 0 0 0 ] 000091 | B0 = [ -sqrt(2)/6 -1/4 -sqrt(5)/12 ], 000092 | [ 0 0 1/4 ] 000093 | 000094 | B1=A0, 000095 | 000096 | [ -1/2 -sqrt(2)/6 0 ] 000097 | Binf = [ 0 0 0 ]. 000098 | [ 0 -sqrt(5)/12 -1/2 ] 000099 | 000100 | The exact characteristic polynomials are 000101 | 000102 | det(lambda-B0) = lambda(4lambda-1)(4lambda+1)/16, 000103 | det(lambda-B1) = lambda(3lambda-1)(3lambda-2)/9, 000104 | det(lambda-Binf) = lambda(2lambda+1)^2/4. (1) 000105 | 000106 | Moreover 000107 | 000108 | Binf(Binf+I/2)=0. (2) 000109 | 000110 | Equation (2) proves that Binf is semisimple, despite the repeated 000111 | eigenvalue -1/2. At all three singularities every nonzero exponent 000112 | difference has absolute value below one. In the Frobenius recursion 000113 | n-ad(Bj), n>=1, no eigenvalue can vanish. Thus a holomorphic gauge reduces 000114 | each local connection to its residue, with no logarithmic term. The local 000115 | monodromies consequently have exact orders 4, 3, and 2. 000116 | 000117 | The older source uses 000118 | 000119 | Y=diag(1,z,z^3)v. 000120 | 000121 | Its residues are the printed B0_old, A0, Binf_old with spectra 000122 | (0,-1/4,-3/4), (0,1/3,2/3), and (-1/2,0,1/2). The two quotient frames are 000123 | related by 000124 | 000125 | X=diag(1,1,t)v. (3) 000126 | 000127 | This shifts one exponent by an integer. In the old frame the only possible 000128 | infinity resonance is cancelled by the printed coefficient 000129 | (Cinf)_(3,1)=0. In the short frame the same cancellation has already been 000130 | absorbed by (3), leaving the semisimple identity (2). Hence the new replay 000131 | is a compact alternative proof, not a different connection. 000132 | 000133 | 3. The marked monodromy is exactly S4 000134 | ====================================== 000135 | 000136 | A positive loop about t=0 lifts from z to iz. Returning the lifted fiber to 000137 | the chosen quotient fiber uses the inverse deck identification. Therefore 000138 | 000139 | c0 is conjugate to U^(-1)=diag(1,-i,i), (4) 000140 | 000141 | not to U. Let m1 be the positive local monodromy about t=1 and use the 000142 | oriented convention 000143 | 000144 | m1 c0 binf=I. 000145 | 000146 | The local calculation gives 000147 | 000148 | c0^4=m1^3=binf^2=I. 000149 | 000150 | Thus the image is a quotient of the spherical triangle group 000151 | 000152 | Delta(4,3,2)= congruent S4. (5) 000153 | 000154 | The image contains c0 of exact order four. Every proper quotient of S4 is 000155 | a quotient by V4 or A4, or is trivial, and none contains an element of order 000156 | four. The representation in (5) is therefore faithful, so its image has 000157 | 24 elements. A reducible three dimensional faithful S4 representation is 000158 | impossible because the two dimensional irreducible representation kills 000159 | V4. The determinant one and trace(c0)=1 select the sign twist of the 000160 | standard representation. 000161 | 000162 | The exact v117 matrices satisfy 000163 | 000164 | c0=U^(-1), m1=M0, binf=M0 U^(-1), 000165 | c0^4=m1^3=binf^2=I. (6) 000166 | 000167 | There is one simultaneous conjugacy orbit of ordered S4 pairs consisting 000168 | of a 4-cycle and a 3-cycle whose product is an involution. The finite 000169 | certificate enumerates its 24 representatives. Equations (4)-(6) hence 000170 | identify the analytic marked representation with the exact v117 pair up to 000171 | one constant global basis change. Choosing the printed v117 basis gives 000172 | 000173 | M0=[[0, -(1+i)/2, (1-i)/2], 000174 | [-(1+i)/2, -i/2, -1/2 ], 000175 | [(1-i)/2, -1/2, i/2 ]]. (7) 000176 | 000177 | This is the precise closure. It proves the exact local system and all 000178 | basis invariant statements. Literal equality with a numerically produced 000179 | harmonic frame still depends on that frame's constant basis convention. 000180 |