TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SEAM-GRAPH-REVIEW.txt Original file SHA-256: d5701f6475333f77ac7870a7917518bbe69f637fe6727e2eeb90e59edf3345c8 Inclusive original lines: 1-190 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 | INDEPENDENT REVIEW: EQUATOR LIFT, REAL FORMS, AND THE SEAM GRAPH 000002 | ================================================================ 000003 | 000004 | 26 October 2026. Exact analytic review plus independent affine checker. No 000005 | Atlas, paper, ledger, or earlier contract was changed. This review tests one 000006 | new geometric identification only. It introduces no field model and makes no 000007 | phenomenological claim. 000008 | 000009 | Question 000010 | ======== 000011 | 000012 | On 000013 | 000014 | E: y^2=x^4-1 000015 | 000016 | let 000017 | 000018 | Q(x,y)=(i x,y), 000019 | h(x,y)=(x,-y), 000020 | kappa(x,y)=(bar x,bar y), 000021 | r(x,y)=(1/x,i y/x^2). 000022 | 000023 | The base equator is fixed by x -> 1/bar x. Its lift is S=r kappa. Does the 000024 | full inverse image of that marked equator coincide with the previously used 000025 | smooth P1/NS seam circle? What group and real-form census actually results 000026 | after adjoining r? 000027 | 000028 | Verdict 000029 | ======= 000030 | 000031 | The proposed affine formulas and the enlarged census are exact: 000032 | 000033 | r^2=h, 000034 | r(z)=i z+1/2, 000035 | S(z)=bar z+i/2, 000036 | hS(z)=-bar z+1/2 mod Z+iZ. 000037 | 000038 | The marked equator does not lift to one smooth seam circle. Its complete 000039 | inverse image is 000040 | 000041 | pi^(-1)(|x|=1)=Fix(S) union Fix(hS), 000042 | 000043 | namely two horizontal and two vertical ovals. They meet at the four branch 000044 | points of pi:E->P1. The resulting embedded torus graph has 000045 | 000046 | V=4, E=8, F=4, b1=5. 000047 | 000048 | Therefore the old smooth one-circle NS source and the full marked equator 000049 | pullback are different geometric objects. Choosing one oval, normalizing the 000050 | nodal graph, or smoothing its four crossings adds a choice that is absent 000051 | from the full pullback. 000052 | 000053 | The group correction is equally precise. The old holomorphic group 000054 | has order eight and is C4 x C2, hence is abelian and not D4. Adjoining 000055 | the actual lift r gives a holomorphic group of order sixteen. It is the 000056 | central deck extension of the base D4 action. After adjoining kappa the full 000057 | group has 32 elements, split into 16 holomorphic and 16 antiholomorphic maps. 000058 | 000059 | Among the 16 antiholomorphic maps, 12 are real involutions: 000060 | 000061 | dividing 4, 000062 | nondividing 4, 000063 | empty 4. 000064 | 000065 | The remaining four antiholomorphic maps have order four and are not real 000066 | involutions. The original eight antiholomorphic maps have census 000067 | 000068 | dividing 2, nondividing 4, empty 2. 000069 | 000070 | The eight new maps contribute 000071 | 000072 | dividing involutions 2, empty involutions 2, order-four maps 4. 000073 | 000074 | The two new dividing involutions are exactly S and hS. They fix all four 000075 | branch marks. The two old dividing OS forms hQkappa and hQ^3kappa fix no 000076 | branch mark. Thus the original OS circle real forms and the marked-equator 000077 | real forms must not be identified. 000078 | 000079 | 1. Algebraic lift r 000080 | =================== 000081 | 000082 | The map r preserves the curve. Put 000083 | 000084 | x'=1/x, 000085 | y'=i y/x^2. 000086 | 000087 | Then 000088 | 000089 | y'^2=-y^2/x^4 000090 | =-(x^4-1)/x^4 000091 | =1/x^4-1 000092 | =x'^4-1. 000093 | 000094 | Applying r twice gives 000095 | 000096 | x''=x, 000097 | y''=i(i y/x^2)/(1/x)^2=-y, 000098 | 000099 | so r^2=h exactly. This is the decisive distinction from a base reflection: 000100 | the base inversion has order two, while its chosen lift has order four and 000101 | squares to the elliptic deck map. 000102 | 000103 | Use the square-torus uniformization E=C/(Z+iZ) and write z=t+i s. In the 000104 | normalization used by the existing elliptic calculation, 000105 | 000106 | Q(z)=i z, 000107 | r(z)=i z+1/2, 000108 | h(z)=-z+(1+i)/2, 000109 | kappa(z)=-i bar z+(1+i)/2. (1) 000110 | 000111 | The last formula is the affine torus expression of algebraic conjugation in 000112 | this uniformization. Direct composition gives 000113 | 000114 | S=r kappa: z -> bar z+i/2, 000115 | hS: z -> -bar z+1/2. (2) 000116 | 000117 | No adjustable phase or translated real structure was inserted. 000118 | 000119 | 2. Why the full equator pullback is a graph 000120 | ============================================ 000121 | 000122 | Let pi(x,y)=x. Since S lifts x -> 1/bar x, a point p lies over the unit circle 000123 | precisely when 000124 | 000125 | pi(S p)=pi(p). 000126 | 000127 | Away from a branch point the fiber of pi is {p,hp}. Therefore 000128 | 000129 | S p=p or S p=h p. 000130 | 000131 | The second alternative is equivalent to hS p=p. At branch points the two 000132 | alternatives coincide. This proves 000133 | 000134 | pi^(-1)(|x|=1)=Fix(S) union Fix(hS). (3) 000135 | 000136 | From (2), 000137 | 000138 | Fix(S): 2s=1/2 mod 1, 000139 | s=1/4 or 3/4, 000140 | 000141 | Fix(hS): 2t=1/2 mod 1, 000142 | t=1/4 or 3/4. (4) 000143 | 000144 | Thus Fix(S) is two horizontal circles and Fix(hS) is two vertical circles. 000145 | Their intersection consists of 000146 | 000147 | (t,s) in {1/4,3/4} x {1/4,3/4}. (5) 000148 | 000149 | These are exactly the fixed points of h, because 000150 | 000151 | h(z)=z iff 2z=(1+i)/2 mod Z+iZ. 000152 | 000153 | They are therefore the four ramification points upstairs, lying over 000154 | x^4=1. Each of the four circles is cut by two vertices into two edges. Hence 000155 | 000156 | V=4, 000157 | E=4 circles times 2 arcs=8. 000158 | 000159 | The graph is connected. Its incidence matrix has rank V-1=3, so 000160 | 000161 | b1=E-rank(partial)=8-3=5. (6) 000162 | 000163 | Since the ambient surface is a torus, V-E+F=0 and therefore F=4. The checker 000164 | constructs the eight edges and its incidence matrix explicitly. It does not 000165 | insert V,E,F or b1 as a census table. 000166 | 000167 | The normalization of (3) is four disjoint circles. The unnormalized inverse 000168 | image is the connected V4/E8 graph. Neither object is the single smooth circle 000169 | on which the older D_Gamma and its alpha-twisted H1 domain were defined. 000170 | 000171 | 3. Exact group closure 000172 | ====================== 000173 | 000174 | Let 000175 | 000176 | H0=. 000177 | 000178 | Because Q has order four, h has order two, and h commutes with Q, direct 000179 | closure gives 000180 | 000181 | |H0|=8, 000182 | H0 isomorphic to C4 x C2. (7) 000183 | 000184 | Its element-order distribution is 000185 | 000186 | order 1: 1, 000187 | order 2: 3, 000188 | order 4: 4. 000189 | 000190 | This group is abelian, so it is not D4. After adjoining kappa, the old group