TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/GEOMETRY.txt Original file SHA-256: be6dca12fe366f6aeba7953dd7ba1c100ddfe40d5faca43d8ea1a883fe85c4ce 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 | DIVIDING REAL FORMS OF THE ORIGINAL ELLIPTIC C2 COVER 000002 | 26 October 2026 000003 | 000004 | Question 000005 | ======== 000006 | 000007 | The previous kernel audit tested only 000008 | 000009 | kappa(x,y)=(bar x,bar y) 000010 | 000011 | on the existing curve 000012 | 000013 | E: y^2=x^4-1. 000014 | 000015 | That real form has one nondividing oval. Does the already given group 000016 | generated by 000017 | 000018 | h(x,y)=(x,-y), Q(x,y)=(i x,y), kappa 000019 | 000020 | contain a dividing real form, a genuine bordered half and the same bond 000021 | reflection that passed the original v519 free NS reflection-positivity test? 000022 | 000023 | Verdict 000024 | ======= 000025 | 000026 | Yes. The omission of the deck factor h in the previous real-form test was 000027 | decisive. 000028 | 000029 | Among all eight existing antiholomorphic involutions 000030 | 000031 | rho_(a,b)=h^a Q^b kappa, a in {0,1}, b in Z/4, 000032 | 000033 | exactly 000034 | 000035 | rho_+ = h Q kappa, 000036 | rho_- = h Q^3 kappa (1) 000037 | 000038 | have two ovals. They are the two dividing real forms. Their quotients are 000039 | conformally identical annuli of modulus 1/2. Q exchanges the two annulus 000040 | charts. 000041 | 000042 | On the marked base they induce 000043 | 000044 | x -> +i bar x, x -> -i bar x, (2) 000045 | 000046 | and hence exchange the mu4 marks in adjacent pairs with no fixed mark. This 000047 | is exactly the mu=+i and mu=-i midpoint-axis orbit in v519, not a resemblance 000048 | by eigenvalues. The normal v519 checker confirms that this clock-inverting 000049 | family has square +1 spin/Fock lifts and that its bond placement is 000050 | reflection positive, while the site placement is indefinite. 000051 | 000052 | There is a second exact structural closure. The existing word 000053 | 000054 | delta = h Q^2 (3) 000055 | 000056 | is the unique fixed-point-free holomorphic involution among h, Q^2 and 000057 | hQ^2. It commutes with rho_+ and rho_-. On either annulus it descends to 000058 | the free orientation-reversing map 000059 | 000060 | (t,s) -> (1/2-t,s+1/2), (4) 000061 | 000062 | whose quotient is a Moebius band. Thus the same old curve contains the 000063 | full topological tower 000064 | 000065 | closed elliptic Schottky double E 000066 | -- quotient by rho_+ or rho_- --> bordered annulus A 000067 | -- quotient by delta --> Moebius band M. (5) 000068 | 000069 | The three roles stay separate: 000070 | 000071 | rho_+/- is the OS/Schottky reflection; 000072 | delta is the free Moebius deck on the annulus; 000073 | Q is the quarter clock and exchanges the two cut charts. 000074 | 000075 | This repairs the too-narrow statement in ELLIPTIC-KERNEL-TEST.txt that the 000076 | existing elliptic geometry supplies no positive half. Kappa alone does not; 000077 | the full already given h,Q,kappa group does. 000078 | 000079 | The repair does not identify the finite-annulus Calderon projector with the 000080 | old one-circle csc covariance. The actual annulus Cauchy data form a 000081 | two-boundary graph with fixed finite-width terms. That distinction is 000082 | computed below. 000083 | 000084 | 1. Uniformization fixed by the original marked points 000085 | ====================================================== 000086 | 000087 | Use O=(0,i) as the elliptic origin and T=hO=(0,-i). The original quarter 000088 | automorphism fixes O and T. On 000089 | 000090 | E = C/(Z+iZ) 000091 | 000092 | the maps therefore have the affine expressions 000093 | 000094 | Q z = i z, 000095 | h z = -z+T, 000096 | kappa z = -i bar z+T, 000097 | T = (1+i)/2. (6) 000098 | 000099 | The formula for kappa is forced by kappa(O)=T, kappa(T)=O and by its action 000100 | on dz. It satisfies kappa^2=1 and kappa Q kappa=Q^-1. 000101 | 000102 | Modulo the square lattice, all eight real forms are 000103 | 000104 | a b rho_(a,b)(z) real locus 000105 | --------------------------------------------------------------- 000106 | 0 0 -i bar z + T 1 oval, nondividing 000107 | 0 1 bar z + T empty 000108 | 0 2 i bar z + T 1 oval, nondividing 000109 | 0 3 -bar z + T empty 000110 | 1 0 i bar z 1 oval, nondividing 000111 | 1 1 -bar z 2 ovals, dividing 000112 | 1 2 -i bar z 1 oval, nondividing 000113 | 1 3 bar z 2 ovals, dividing. (7) 000114 | 000115 | This also shows why integral H1 action alone was insufficient. For example 000116 | Q kappa and h Q^3 kappa have the same linear reflection type up to the D4 000117 | frame, but the affine T shift makes the first fixed-point free and the second 000118 | dividing. 000119 | 000120 | 2. Direct algebraic fixed-point and infinity census 000121 | ==================================================== 000122 | 000123 | A fixed point of rho_(a,b) can be written 000124 | 000125 | x=e^(i b pi/4)t, y=i^a v, t,v real. 000126 | 000127 | The curve equation becomes 000128 | 000129 | v^2=(-1)^a[(-1)^b t^4-1]. (8) 000130 | 000131 | For (a,b)=(1,1) or (1,3), this is 000132 | 000133 | v^2=t^4+1. 000134 | 000135 | Its positive and negative graphs close separately at the two points at 000136 | infinity, giving two ovals. For completeness, if 000137 | 000138 | u=1/x, eta=y/x^2, 000139 | 000140 | then the two infinity points are P_+ and P_- with eta=+1 and -1. The real 000141 | form acts there by 000142 | 000143 | eta -> (-1)^(a+b) bar eta. (9) 000144 | 000145 | Thus P_+ and P_- are individually fixed exactly when a+b is even. In the 000146 | two dividing cases each graph closes at one fixed infinity point. For 000147 | (a,b)=(0,1),(0,3), equation (8) has no finite real point and (9) swaps the 000148 | infinity points, so these real forms are genuinely fixed-point free. 000149 | 000150 | In the integral cycle basis A=1, B=i, let 000151 | 000152 | J=[[0,-1],[1,0]], 000153 | K=[[0,-1],[-1,0]]. 000154 | 000155 | Then 000156 | 000157 | Q_*=J, kappa_*=K, 000158 | rho_(a,b)*=(-I)^a J^b K. (10) 000159 | 000160 | Every rho is anti-symplectic. For the dividing pair, 000161 | 000162 | rho_+* = diag(-1,+1), Fix H1 = Z B, 000163 | rho_-* = diag(+1,-1), Fix H1 = Z A. (11) 000164 | 000165 | The fixed homology classes are exactly the directions of their two real 000166 | ovals. Q sends A to B and conjugates rho_+ to rho_-. 000167 | 000168 | 3. The annulus and its exact modulus 000169 | ===================================== 000170 | 000171 | For rho_+=-conjugation, a fundamental bordered half is 000172 | 000173 | A_+={z=t+i s: 0<=t<=1/2, s in R/Z}. (12) 000174 | 000175 | Its boundary circles are t=0 and t=1/2. For rho_-=conjugation, the 000176 | corresponding half is the rotated strip 000177 | 000178 | A_-={z=s+i t: 0<=t<=1/2, s in R/Z}. (13) 000179 | 000180 | With the square flat conformal metric, both have 000181 | 000182 | circumference L=1, 000183 | width W=1/2, 000184 | conformal modulus m=W/L=1/2, 000185 | round-annulus radius ratio r=exp(-2 pi m)=exp(-pi). (14) 000186 | 000187 | There is no fitted length in (14). Overall rescaling cancels in W/L. The 000188 | Schottky double has modulus 2 i m=i, exactly the given elliptic curve. 000189 | 000190 | The quarter clock does not act inside one chosen half: