TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/SPIN-RIGIDIFIED-THETA-MODULE-INDEPENDENT-REVIEW.txt Original file SHA-256: fadd56e04d9e10af0d0f7e9cd94010a35563ee3dc092659499502e410dbb23f8 Inclusive original lines: 1-130 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 OF THE SPIN-RIGIDIFIED THETA MODULE 000002 | 2026-10-26 000003 | 000004 | Reviewed source 000005 | 000006 | SPIN-RIGIDIFIED-THETA-MODULE.txt 000007 | SHA-256 4bd17ffbb2be4b31c4eb7a4efe9cd8a1191c9065655889ce189f3b1f6c2499d2 000008 | 000009 | VERDICT 000010 | 000011 | ACCEPTED WITH THE SCOPE STATED IN THE REPORT. Conditional on the already 000012 | isolated one-copy insertion rule, the source theta module is the unique 000013 | four-dimensional weight-one representation of the same finite Heisenberg 000014 | group N64. The cubic and quarter lifts then land in the single native joint 000015 | class proved by the exact census. This is a complex finite-module theorem; 000016 | it does not select an integral E8 lattice, a physical Hermitian structure, a 000017 | state, or a readout. 000018 | 000019 | 1. THE FOURTH-ROOT SET IS A GROUP OF ORDER 64 000020 | 000021 | The theta group G(O(2Theta)) is a class-two central extension of A[2] by C*. 000022 | Its commutator on two lifts is the Weil pairing and therefore lies in 000023 | {+1,-1}. For g^4=h^4=1, 000024 | 000025 | (gh)^4=g^4 h^4 [h,g]^6=1. 000026 | 000027 | The convention for the order of the commutator can invert the last factor, 000028 | but its sixth power is one in either convention. Inverses retain fourth 000029 | power one, so 000030 | 000031 | N={g:g^4=1} 000032 | 000033 | is a subgroup. Above each of the sixteen points of A[2], a theta lift can be 000034 | rescaled by exactly four scalars to have fourth power one. Thus |N|=64 and 000035 | its kernel is mu4. Nondegeneracy of the Weil pairing forces every central 000036 | element to have zero image in A[2], hence Z(N)=mu4. This is the standard 000037 | two-qubit finite Heisenberg group, defined intrinsically inside the theta 000038 | group. 000039 | 000040 | 2. SYMPLECTIC LIFTS AND STONE-VON NEUMANN 000041 | 000042 | For a symplectic basis of A[2], multiplying a chosen lift by i reverses its 000043 | square. The four basis lifts can therefore be normalized to square +1, with 000044 | the two standard commutators equal to -1 and all other basis commutators 000045 | trivial. The resulting presentation is the Pauli presentation with center 000046 | mu4. Every specified symplectic Phi consequently lifts to a 000047 | center-preserving isomorphism N->N64. 000048 | 000049 | The lift is not canonical. Two lifts differ by a character A[2]->mu2, and 000050 | nondegeneracy of the commutator identifies every such character with 000051 | conjugation by a Pauli element. This is exactly the Heisenberg-frame freedom 000052 | retained in the report. 000053 | 000054 | Riemann-Roch gives dim H^0(O(2Theta))=4. Its theta-group representation has 000055 | central weight one. Finite Stone-von Neumann gives one irreducible 000056 | representation with this faithful central character, of dimension four. 000057 | The constant line N_q(S_0)^(-1) changes the alpha and beta characters but not 000058 | the N action. Therefore the resulting intertwiner with the native C^4 is 000059 | unique up to a nonzero scalar once Phi and its Heisenberg lift are fixed. 000060 | 000061 | 3. THE CUBIC LIFT HAS NO RESIDUAL TRANSLATION OR PHASE 000062 | 000063 | The preliminary rescaling gamma=lambda gamma_0 is correctly typed as the 000064 | normalization of an initially projective theta linearization. The relations 000065 | kappa^3=1 and kappa lambda^2=1 have one solution lambda in mu3, and this 000066 | rescaling preserves gamma^3=1. It defines the source operator used in the 000067 | subsequent comparison; it is not a later alteration of an already fixed 000068 | operator. Its action on the three-point norm line is unchanged because the 000069 | rescaling enters there through lambda^3=1. 000070 | 000071 | Before changing Heisenberg frame, the transported cubic operator has form 000072 | 000073 | lambda p sigma. 000074 | 000075 | Conjugation by a Pauli q changes p by (1-Sigma)q, up to a central scalar that 000076 | is absorbed in lambda. The cubic action is fixed-free on the Pauli quotient, 000077 | so 1-Sigma is invertible over F2 and one unique q removes p. 000078 | 000079 | The remaining scalar is also fixed. Gamma has order three and is conjugate 000080 | to gamma^(-1) by alpha, so its trace is real. On End(W_q), Ad gamma fixes 000081 | only the identity Pauli label. Every other label belongs to a three-cycle; 000082 | possible phase weights do not change the zero trace of such a cycle. Hence 000083 | 000084 | tr(Ad gamma)=1=|tr gamma|^2. 000085 | 000086 | For an order-three operator in dimension four, reality and this norm force 000087 | eigenvalue multiplicities (2,1,1), so tr gamma=1. The frozen sigma also has 000088 | trace one. Thus the scalar in lambda sigma is lambda=1. The argument uses 000089 | the full operator relation and does not infer equality merely from quotient 000090 | actions. 000091 | 000092 | The unitarity used here introduces no physical metric. The group generated 000093 | by the finite Heisenberg group and the finite-order normalized alpha and gamma 000094 | is finite: its quotient is contained in the finite symplectic group and its 000095 | remaining central phases have finite order. Averaging any positive Hermitian 000096 | form over this group therefore unitarizes the representation. This is enough 000097 | for tr(gamma^(-1))=conjugate(tr gamma) in the trace argument, without selecting 000098 | a source Hermitian structure. 000099 | 000100 | 4. THE CONTINUOUS NORMALIZER PHASE ADDS NO SOLUTIONS 000101 | 000102 | For a fixed quotient action, every unitary normalizer lift differs from one 000103 | discrete Clifford lift by a Pauli factor and a scalar z in U(1). If the 000104 | discrete square is nonscalar, no scalar can repair it. If it is scalar, the 000105 | condition alpha^2=iI is one equation for z^2 and leaves exactly the two signs 000106 | already represented in the G31 and zeta8 G31 fiber census. The additional 000107 | relation 000108 | 000109 | alpha sigma alpha^(-1)=sigma^(-1) 000110 | 000111 | then leaves alpha and -alpha in each of the nine compatible quotient fibers. 000112 | Thus the continuous U(1) kernel does not create missed solutions outside the 000113 | enumerated cosets. The centralizer orbit proves one mathematical joint class 000114 | for the fixed native pair (J,sigma); it does not prove that every conjugating 000115 | element preserves further physical markings. 000116 | 000117 | 5. EXACT SCOPE 000118 | 000119 | The proved identification is 000120 | 000121 | (A[2],N,W_q,gamma,alpha) -> (N64,C^4,sigma,alpha_native) 000122 | 000123 | up to Heisenberg frame, the unavoidable sign of alpha, and the stated native 000124 | centralizer orbit. Pulling back a native lattice or Hermitian form along the 000125 | complex intertwiner constructs such extra data, but does not prove that the 000126 | curve, boundary source, or P1 and P2 markings selected them. The result also 000127 | remains conditional on the physical one-copy insertion of N_q(S_0)^(-1), 000128 | which supplies alpha^2=iI. The report preserves both limits correctly. 000129 | 000130 | No material correction is required.