TFPT scientific source excerpt Original repository path: experiments/theory-contracts/atlas-defect-synthesis-20261024/CLOCK-TWISTED.txt Original file SHA-256: aa357c45619309104d147013ff56e99e5e1723e9a026149745cc0a627522ed75 Inclusive original lines: 1-124 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 | MATH.SEAM.ATLAS-DEFECT-SYNTHESIS / native clock twisted sectors 000002 | 24 October 2026 000003 | 000004 | Question and category 000005 | 000006 | Work in the already fixed holomorphic lattice VOA V_E8 with its original 000007 | conformal vector and cocycle. Use as an INPUT the proved full-field inner 000008 | conjugacy between the native clock Jhat and 000009 | 000010 | kappa=exp(2 pi i v(0)), v=(1/2,1/2,1/2,1/2,1/2,0,0,0). 000011 | 000012 | The conjugator transports the Cartan and the D5+A3 subnet. Nothing below 000013 | asserts that it preserves the originally named Gaussian marks. In particular, 000014 | this is a statement about VOA twisted sectors, not a C12/C24 base cover or a 000015 | physical time variable. 000016 | 000017 | 1. The four exact shifted-lattice ground spaces 000018 | 000019 | In the Li Delta-operator convention sigma_h=exp(-2 pi i h(0)), kappa uses 000020 | h=-v. Hence the kappa^a-twisted conformal Hamiltonian on a charge beta is 000021 | 000022 | L^(a)(0)=L(0)-a v(0)+a^2 (v,v)/2, 000023 | h_a(beta)=|beta-a v|^2/2. 000024 | 000025 | The checker enumerates the actual doubled-coordinate E8 condition: all eight 000026 | coordinates have one parity and their sum is divisible by4. It obtains 000027 | 000028 | a min |beta-a v|^2 ground h multiplicity D5 x A3 000029 | 0 0 0 1 (1,1) 000030 | 1 3/4 3/8 4 (1,4) 000031 | 2 1 1/2 16 (10,1)+(1,6) 000032 | 3 3/4 3/8 4 (1,bar4). 000033 | 000034 | For a=1 the four shifted vectors have zero first five coordinates and last 000035 | three coordinates 000036 | 000037 | (1/2,1/2,1/2), (1/2,-1/2,-1/2), 000038 | (-1/2,1/2,-1/2), (-1/2,-1/2,1/2). 000039 | 000040 | The existing D3-to-A3 isometry T maps them exactly to the four permutations 000041 | of (3/4,-1/4,-1/4,-1/4), the weights of the A3 fundamental. For a=3 the 000042 | opposite parity gives the antifundamental. For a=2 the ten doubled vectors 000043 | supported in the first five coordinates are the D5 vector weights and the six 000044 | supported in the last three are the D3=A3 vector weights. Thus the middle 000045 | ground space is exactly (10,1)+(1,6), not a dimension-only match. 000046 | 000047 | This also identifies the 5/8+3/8 glue arithmetic. The unshifted glue state has 000048 | the D5 half-spinor contribution 5/8 and A3 fundamental contribution3/8. 000049 | Kappa spectral flow subtracts the D5 charge v and leaves the A3 fundamental 000050 | as the four-dimensional twisted ground space of weight3/8. 000051 | 000052 | 2. Native oscillator and inner-shift characters 000053 | 000054 | For J with four +i and four -i Heisenberg eigenvectors, the normalized twisted 000055 | oscillator character is 000056 | 000057 | 4 q^(1/24) product_(n>=0) 000058 | (1-q^(n+1/4))^-4 (1-q^(n+3/4))^-4. (2.1) 000059 | 000060 | For the inner-conjugate kappa description it is 000061 | 000062 | Theta_(E8-v)(q) / eta(q)^8. (2.2) 000063 | 000064 | With x=q^(1/4), after removing their common factor x^(1/6), the checker 000065 | computes both sides independently through x^24. Every integer coefficient is 000066 | equal. This finite coefficient certificate is a check of the exact character 000067 | identity supplied by the already proved conformal-vector-preserving inner 000068 | conjugacy; it is not used to infer that conjugacy from finitely many terms. 000069 | 000070 | 3. Three different phase statements 000071 | 000072 | The twisted ground rotation without the Casimir shift is 000073 | 000074 | exp(2 pi i rho), rho=3/8, 000075 | 000076 | and has order8. With c=8, the normalized top T phase is 000077 | 000078 | exp(2 pi i (3/8-8/24))=exp(2 pi i/24)=zeta_24. 000079 | 000080 | All homogeneous weights are3/8+k/4. Their normalized phases have exponents 000081 | 000082 | 1/24, 7/24, 13/24, 19/24, 000083 | 000084 | and all have exact order24. T^4 is the common scalar zeta_6. The uninserted 000085 | twisted trace is not by itself a zeta_24 eigenfunction: modular T changes the 000086 | insertion (kappa,1) to (kappa,kappa). Diagonal phase statements apply to 000087 | homogeneous states or normalized fixed-point-subalgebra summands. 000088 | 000089 | The cyclic orbifold type is 000090 | 000091 | 4{16 rho mod4}=4{2}. 000092 | 000093 | For n=4 and type t=2, the discriminant-form level is 000094 | N=n^2/gcd(t,n)=8. Since c=8 is not divisible by24, the cyclic-orbifold trace 000095 | theorem gives congruence level lcm(3,8)=24. This is the precise mathematical 000096 | level24 statement. It is not a new cyclic group of order24. 000097 | 000098 | 4. Sources and limits 000099 | 000100 | Bakalov--Kac, Twisted Modules over Lattice Vertex Algebras, equations4.53, 000101 | Theorem4.2 and Remark4.4: classification, finite Heisenberg defect and twisted 000102 | module structure, under an integral lattice and a finite-order lattice lift: 000103 | https://arxiv.org/abs/math/0402315 000104 | 000105 | Li Delta-operator as stated in Lam--Shimakura, equation3.2: for a finite-order 000106 | inner automorphism of a CFT-type VOA with semisimple rational h(0), 000107 | L^(h)(0)=L(0)+h(0)+/2: 000108 | https://arxiv.org/abs/1501.05094 000109 | 000110 | van Ekeren--Moller--Scheithauer, Theorems5.10--5.14: for a simple, rational, 000111 | C2-cofinite holomorphic VOA of CFT type and a finite cyclic automorphism, the 000112 | orbifold type, discriminant-form level and trace-function congruence level: 000113 | https://arxiv.org/abs/1507.08142 000114 | 000115 | The finite E8 shell arithmetic, T chart, representations, phases and displayed 000116 | character coefficients are checked by clock_twisted_sectors.py with exact 000117 | integer/Fraction arithmetic. The native twisted-module data already existed in 000118 | tfpt_explorer/twisted_source.py. The new result is the exact spectral-flow 000119 | identification of that native four-dimensional top with the transported A3 000120 | fundamental glue sector. 000121 | 000122 | Verdict: exact within the existing V_E8 and inner-conjugacy theorem. No 000123 | microscopic source map, physical state selection, base-space cover or physical 000124 | time is supplied.