TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/D4-RIGIDITY-INDEPENDENT-REVIEW.txt Original file SHA-256: a4c9114146fdeb4bf325a12f1a48b68d5d64814c57e789906a07d5ed9ccc750d 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 | CURRENT MARKED SOURCE CORRECTION, 26 OCTOBER 2026 000002 | ================================================ 000003 | The constructions and scoped theorems below remain valid in their stated 000004 | D4 or finite-target class. They are NOT the marked descent of the actual 000005 | A0* connection. The recovered28September C4 triangle proof is authoritative 000006 | for that connection: orders(4,3,2), positive quotient generator U inverse, 000007 | and J_actual(z)=1-z^-4. The D4/Phi pullback has orders(4,3,4) and a different 000008 | marking. A rational z->1/z lift of the actual connection does not exist. 000009 | The analytic A0* to v117 group identification is already proved, not open. 000010 | See C4-DESCENT-FINITE-MONODROMY.txt and its independent review. Reusing an 000011 | abstract S4 generator class alone does not identify these base paths. 000012 | 000013 | INDEPENDENT REVIEW OF THE D4 ORBIFOLD RIGIDITY ARGUMENT 000014 | ====================================================== 000015 | 000016 | 26 October 2026. Exact algebraic review with a bounded original source 000017 | comparison. No Atlas, original paper, or verification source was changed. 000018 | 000019 | Verdict 000020 | ======= 000021 | 000022 | The rigidity theorem is correct under its explicitly stated equivariant 000023 | local system premise. It proves more than the preceding modular existence 000024 | construction: a genuine unitary determinant one descent through the actual 000025 | D4 quotient, with the marked order four fiber operator c and a regular cubic 000026 | source peripheral a^2, has exactly one stable irreducible conjugacy class. 000027 | That class is the v117 pair. 000028 | 000029 | The theorem does not follow from the D4 symmetry of the puncture set alone. 000030 | The bounded original sources checked here provide the base D4 action, its 000031 | natural action on H1, and a separate flat Hermitian SU(3) family connection. 000032 | They do not yet provide one D4 equivariant linearization of that chosen flat 000033 | connection whose order four stabilizer is the marked c below. This is the 000034 | single missing source premise. 000035 | 000036 | There is no conflict with the earlier Q distinction. Here c is the marked 000037 | fiber linearization of the order four stabilizer in the source quotient. It 000038 | is not identified with the modular S path. If the equivariant descent exists, 000039 | the quotient theorem derives the finite S4 branch from the source geometry 000040 | instead of importing the modular Level 4 representation. 000041 | 000042 | 1. The actual quotient and its generators 000043 | =========================================== 000044 | 000045 | For 000046 | 000047 | X=P1 minus mu4 000048 | 000049 | the D4 invariant quotient function is 000050 | 000051 | J(x)=1/2-(x^4+x^(-4))/4 000052 | =-(x^4-1)^2/(4x^4). 000053 | 000054 | The points x in mu4 are removed and lie over J=0 with ramification index two. 000055 | The four points x^4=-1 lie over J=1 with index two, while zero and infinity 000056 | lie over J=infinity with index four. Thus the quotient orbifold has type 000057 | 000058 | (infinity,2,4). 000059 | 000060 | Choose its positively oriented generators with 000061 | 000062 | a b c=I, b^2=I, c^4=I. 000063 | 000064 | The pullback peripheral at each original puncture is 000065 | 000066 | M=a^2. (1) 000067 | 000068 | The marked order four fiber operator is fixed as 000069 | 000070 | c=diag(1,i,-i), (2) 000071 | 000072 | the determinant one normalization of the original homological Coxeter 000073 | rotation. Equations (1) and (2) concern a representation of the quotient 000074 | orbifold fundamental group. Its image need not be the finite geometric D4 000075 | group. This prevents the common quotient versus fiber action confusion. 000076 | 000077 | 2. Classification of the involution b 000078 | ====================================== 000079 | 000080 | Assume the representation is unitary with determinant one and M has the 000081 | regular cubic spectrum 000082 | 000083 | spec(M)={1,omega,omega^2}. 000084 | 000085 | The choice b=I would give a=c^(-1) and 000086 | 000087 | a^2=c^(-2)=diag(1,-1,-1), 000088 | 000089 | which is not regular cubic. Hence b is a nontrivial involution in SU(3). 000090 | It is Hermitian and has eigenvalues (1,-1,-1), so uniquely 000091 | 000092 | b=2 v v^*-I (3) 000093 | 000094 | for a unit vector v. Write 000095 | 000096 | p_j=|v_j|^2, p_1+p_2+p_3=1. 000097 | 000098 | Since a=(bc)^(-1), unitarity gives 000099 | 000100 | tr(bc)=conjugate(tr(a)). (4) 000101 | 000102 | Directly from (2) and (3), 000103 | 000104 | tr(bc)=2[p_1+i(p_2-p_3)]-1. (5) 000105 | 000106 | No phase of v enters (5). 000107 | 000108 | 3. The four square root branches 000109 | ================================= 000110 | 000111 | Because a^2=M is regular cubic and det(a)=1, the possible eigenvalue sets of 000112 | a give exactly 000113 | 000114 | tr(a) in {0,2,2 omega,2 omega^2}. (6) 000115 | 000116 | This can be obtained by choosing determinant one square roots of 000117 | {1,omega,omega^2}; there are four sign branches. 000118 | 000119 | If tr(a)=2, equations (4) and (5) require 000120 | 000121 | p_1=3/2, 000122 | 000123 | which is impossible. 000124 | 000125 | If tr(a)=2 omega or 2 omega^2, they require 000126 | 000127 | p_1=0. 000128 | 000129 | Then v is orthogonal to the c eigenvalue one line Ce_1, so b e_1=-e_1. 000130 | Both b and c preserve Ce_1. Consequently a and M=a^2 preserve it as well, 000131 | and the source pair is reducible. Stability or irreducibility excludes 000132 | both branches. 000133 | 000134 | The only remaining branch has tr(a)=0. Equations (4) and (5) now force 000135 | 000136 | p_1=1/2, p_2=p_3=1/4. (7) 000137 | 000138 | In this branch the eigenvalues of a are the three cube roots of unity, so 000139 | 000140 | a^3=I. (8) 000141 | 000142 | Thus the finite triangle relation is derived from unitary descent, the 000143 | regular local class, and irreducibility. It is not inserted as a modular 000144 | Level 4 assumption. 000145 | 000146 | 4. Uniqueness and exact v117 identification 000147 | ============================================ 000148 | 000149 | The phases of a vector satisfying (7) are removed by a diagonal unitary in 000150 | the centralizer of c. A common scalar on that diagonal does not alter the 000151 | conjugation and can, if desired, make its determinant one. Therefore every 000152 | irreducible solution is centralizer conjugate to 000153 | 000154 | v_0=(1/sqrt(2),1/2,1/2)^T, 000155 | b_0=2 v_0 v_0^*-I, 000156 | a_0=(b_0 c)^(-1). (9) 000157 | 000158 | Direct multiplication gives b_0^2=I, a_0^3=I and a_0^2 regular cubic. With 000159 | 000160 | D=diag(1,-(1+i)/sqrt(2),(1-i)/sqrt(2)), 000161 | 000162 | which commutes with c, one obtains exactly 000163 | 000164 | D a_0^2 D^*=M0_v117, 000165 | D c D^*=U_v117=c. (10) 000166 | 000167 | The determinant of the displayed D is -1, but multiplying it by any scalar 000168 | whose cube is -1 leaves (10) unchanged and places the conjugator in SU(3). 000169 | Hence (10) is uniqueness in the correctly marked unitary category, not only 000170 | equality of spectra. 000171 | 000172 | 5. Exact source premise still required 000173 | ======================================= 000174 | 000175 | The archived carrier source 000176 | 000177 | documents/archive--tfpt-45--source_extracts--02_carrier_source/ 000178 | sections/006-subsection-family-monodromy-and-occupancy.tex 000179 | 000180 | does establish: