TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/EQUATORIAL-LIFTS.txt Original file SHA-256: 434eaab2fa67fbf157967a78f4b3f742cb7731d61c9632cdb06f12773c0ba1dd 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 | ORIGINAL EQUATORIAL REAL LIFTS ON THE EXISTING C2 CURVE 000002 | ======================================================= 000003 | 000004 | 26 October 2026. Exact source and curve audit. No Atlas, paper, ledger, or 000005 | previous report was changed. Atlas revision used for navigation: 000006 | 4a17a404b58c05f460121df8. 000007 | 000008 | Question 000009 | ======== 000010 | 000011 | The previous elliptic calculation used the eight anti-holomorphic maps 000012 | 000013 | h^a Q^b kappa, 000014 | 000015 | on 000016 | 000017 | E: y^2=x^4-1, 000018 | h(x,y)=(x,-y), Q(x,y)=(i x,y), 000019 | kappa(x,y)=(bar x,bar y). 000020 | 000021 | It found the dividing pair hQkappa and hQ^3kappa. The original P1 source, 000022 | however, also contains the mark-fixing base mirror x -> 1/x and the pointwise 000023 | fixed equatorial Real form x -> 1/bar x. The decisive question is whether 000024 | their lifts add a missing original seam geometry, and whether the same 000025 | Q-invariant spin line is compatible with it. 000026 | 000027 | Verdict 000028 | ======= 000029 | 000030 | [E] There are exactly two deck-related lifts of the equatorial base Real 000031 | form: 000032 | 000033 | j_+(x,y)=(1/bar x, +i bar y/bar x^2), 000034 | j_-(x,y)=(1/bar x, -i bar y/bar x^2)=h j_+(x,y). (1) 000035 | 000036 | Both square to one and both have two ovals. Their quotients are annuli. The 000037 | quarter Q exchanges them. They arise without a new cover or a chosen Real 000038 | form: if 000039 | 000040 | S(x,y)=(1/x, i y/x^2), (2) 000041 | 000042 | then S^2=h and 000043 | 000044 | j_+=S kappa, j_-=hS kappa. (3) 000045 | 000046 | [E] Adding S completes the old base D4 action on the concrete curve. The 000047 | sixteen anti-holomorphic lifts split exactly into the two old v510/v519 base 000048 | families: 000049 | 000050 | h^a Q^b kappa -> x |-> i^b bar x family D, 000051 | h^a Q^b S kappa -> x |-> i^b/bar x family A. (4) 000052 | 000053 | Among these sixteen maps, twelve are strict Real involutions and four have 000054 | order four. The previous eight-map census was correct inside , 000055 | but it was not the full marked D4 lift census. In the full census there are 000056 | two different dividing orbits: the old OS midpoint pair 000057 | {hQkappa,hQ^3kappa}, and the new equatorial pair {Skappa,hSkappa}. 000058 | 000059 | [E] This does not identify the two slots. v519 says explicitly that family A 000060 | defines the Euclidean seam section and that its equatorial member fixes the 000061 | seam pointwise, with no OS half-space decomposition. Family D supplies the 000062 | OS reflection and inverts the clock. Thus the old pair remains the correct 000063 | lift of the two mark-swapping family-D midpoint axes. The new pair supplies 000064 | the missing lift of the pointwise fixed family-A equator. 000065 | 000066 | [E] The smallest source-identification test produces a real obstruction to 000067 | calling one new oval the original full seam. If Gamma is the unit circle, 000068 | which contains all four mu4 branch marks, then 000069 | 000070 | p^{-1}(Gamma)=Fix(j_+) union Fix(j_-), p(x,y)=x. (5) 000071 | 000072 | In the square uniformization this is two horizontal plus two vertical 000073 | circles. They meet at the four ramification points. The union is a connected 000074 | graph with V=4, E=8 and b1=5; its complement consists of four disks. Each 000075 | individual oval maps two-to-one only onto one alternating quarter arc of 000076 | Gamma. Therefore the original full pointwise fixed seam does not pull back 000077 | to one ordinary boundary circle or to the boundary of one of the two 000078 | annuli. 000079 | 000080 | [E] As an unlinearized complex flat line class, the existing Q-fixed 000081 | nontrivial elliptic spin class alpha is invariant under the full marked lift 000082 | group. On each of the four equatorial ovals its restriction is NS. At the 000083 | four-valent graph vertices, however, a one 000084 | dimensional boundary Dirac operator needs additional vertex matching data. 000085 | The sources checked here provide no such matching rule and no unitary taking 000086 | the original circle kernel to this graph. 000087 | 000088 | [O] The exact remaining source step is consequently smaller and sharper 000089 | than before: identify whether the original operational seed uses the base 000090 | circle Gamma itself, a chosen family-D OS cut acting on Gamma, or a specified 000091 | smooth push-off of p^{-1}(Gamma), and supply its boundary trace and state 000092 | transport. The curve and its spin line now carry both required Real-form 000093 | slots. They do not by themselves choose the smooth carrier on which the 000094 | original quarter Hardy or NS kernel acts. 000095 | 000096 | 1. What the old sources actually contain 000097 | ========================================= 000098 | 000099 | 1.1 v507: torus origin and the holomorphic D4 000100 | ------------------------------------------------ 000101 | 000102 | v507 proves that the four marks are the fixed points of the hyperelliptic 000103 | involution, hence the four half-periods of the elliptic double. It proves 000104 | that mark-free collar decks are half-period translations, and that at 000105 | tau=i the stabilizer is D4 with the CM-fixed half-period central. Its 000106 | explicit descent and curve-lift calculations are at 000107 | 000108 | verification/v507_seam_bit_origin.py:407-477,607-701 000109 | SHA256 926ea236729dffb3ac531ecd34c9b8efa02afea1632abf9798786e65b40ada1f. 000110 | 000111 | The source-leading paper record is 000112 | 000113 | paper-source:acdfeae924036b5d7fbf9676 000114 | documents/changelog/sections/006-source.tex:1-65 000115 | SHA256 c5024dbed35b13842c60e4cdb0911568f8b94d4bdfa0c438e5aadf12a29c2db5. 000116 | 000117 | It constructs the holomorphic torus and deck source. It does not enumerate 000118 | the anti-holomorphic lifts of x -> 1/bar x on y^2=x^4-1. 000119 | 000120 | 1.2 v510: the two base Real-form families 000121 | ------------------------------------------- 000122 | 000123 | v510 classifies all deck-normalizing anti-holomorphic maps on the base as 000124 | 000125 | x -> mu bar x, x -> mu/bar x. (6) 000126 | 000127 | For mu4 it selects x -> 1/bar x as the unique mark-fixing equatorial 000128 | structure. Its fixed circle avoids the base deck poles 0 and infinity, so 000129 | the base deck is free along that circle. See 000130 | 000131 | verification/v510_seam_bit_freedom.py:747-798,801-858 000132 | SHA256 5b174a8914b14308621e0be98fc9d44e23b89713400ea5c09e1cae17d3546193, 000133 | 000134 | and the source record 000135 | 000136 | paper-source:3ff6da26622ba2984cb0e64c 000137 | documents/changelog/sections/005-source.tex:186-242 000138 | SHA256 f0f5a3c3add4d0a0ec714e3ef03fd838e4a47708ccc34244c25398da738e6987. 000139 | 000140 | This is a base-sphere statement. It does not say that p^{-1}(Gamma) is a 000141 | smooth circle, nor does it lift both families to the concrete C2 curve. 000142 | 000143 | 1.3 v512: the missing base mirror was already present 000144 | ------------------------------------------------------ 000145 | 000146 | v512 proves that the complete holomorphic base symmetry census consists of 000147 | 000148 | x -> lambda x, x -> lambda/x, 000149 | 000150 | and that at mu4 it closes to D4. In particular x -> 1/x is the mark-fixing 000151 | mirror. See 000152 | 000153 | verification/v512_seam_tau_flag.py:301-388 000154 | SHA256 78f7b1087e030226ab095ee8b391254f03ec23425e32a8a20d66438414c08782, 000155 | 000156 | and 000157 | 000158 | paper-source:bea881269e785d645bba31c6 000159 | documents/changelog/sections/005-source.tex:69-104 000160 | SHA256 f0f5a3c3add4d0a0ec714e3ef03fd838e4a47708ccc34244c25398da738e6987. 000161 | 000162 | Thus S in (2) is not a newly fitted symmetry. The new step is its exact lift 000163 | to the already existing C2 curve and its composition with the already 000164 | existing kappa. 000165 | 000166 | 1.4 v519: decisive role separation 000167 | ----------------------------------- 000168 | 000169 | v519 calls x -> mu bar x family D and x -> mu/bar x family A. It proves: 000170 | 000171 | * family D inverts the clock, acts on Gamma as a reflection with two cut 000172 | points, and is the OS family; 000173 | * family A centralizes the clock projectively; 000174 | * x -> 1/bar x fixes Gamma pointwise and therefore has no OS cut; 000175 | * family A defines the Euclidean section while family D reflects it. 000176 | 000177 | The decisive executable lines are 000178 | 000179 | verification/v519_woit_theta_rp_free.py:356-439,442-485,698-724 000180 | SHA256 e86e3a055b58051c7978d8f10ee527948bac758fd4665ac216e18a71cc8c1ee9. 000181 | 000182 | The same distinction is written into the active source chain at 000183 | 000184 | paper-source:d4a5d7ca6dd1a9006b1a448f 000185 | documents/changelog/sections/004-source.tex:75-123 000186 | SHA256 f1b117a014f2b1397182f975c03881140094d6d9e566fc331696c8b1b6a450e5, 000187 | 000188 | and into the explicit research contract at 000189 | 000190 | paper-source:fbdc575f08b751529cc80783