TFPT scientific source excerpt Original repository path: experiments/theory-contracts/atlas-defect-synthesis-20261024/FINITE-QUADRATIC-CLOCK.txt Original file SHA-256: e1799dbd225d5571508a45f02ed65217fab78832d4a72b80ab22b70f53a8048c Inclusive original lines: 1-142 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 | COMMON GLUE AND CLOCK DISCRIMINANT 000002 | 24 October 2026 000003 | 000004 | Question 000005 | 000006 | Can the already established E8 glue extension and the native quarter-clock 000007 | twisted sectors be recovered from the same marked object, with actual 000008 | quadratic forms rather than matching counts? 000009 | 000010 | Category and inputs 000011 | 000012 | Fix the positive definite root lattice Q=D5 direct-sum A3, its unitary 000013 | same-conformal VOA U, and its marked diagonal simple-current generator 000014 | h=(1,1). The canonical diagonal extension is V_E8. The prior SOURCE06 000015 | inner-conjugacy theorem identifies its dual glue clock kappa with the native 000016 | Jhat after transporting Cartan and all other marks. This transported category 000017 | is essential; a fixed old Gaussian Cartan is not preserved by that conjugacy. 000018 | 000019 | 1. One sixteen-element quadratic module 000020 | 000021 | The discriminant group of Q is A_Q=C4 x C4, with 000022 | 000023 | q(a,b)=(5a^2+3b^2)/8 mod1, 000024 | B((a,b),(c,d))=(5ac+3bd)/4 mod1. 000025 | 000026 | This A_Q is NOT the different group A_J=E8/(1-J)E8=F2^4 in the doily theorem. 000027 | They both have sixteen elements; their additive groups and pairings differ. 000028 | Their connection is through the same V_E8 and its clock, not a group 000029 | isomorphism between C4^2 and F2^4. 000030 | 000031 | The determinant of diag(5,3) modulo4 is a unit. Thus B is nondegenerate. 000032 | All q-isotropic subgroups are 000033 | 000034 | {0}, <(2,2)>, H+={(k,k)}, H-={(k,-k)}. 000035 | 000036 | The two Lagrangians H+ and H- describe the two oriented E8 extensions over 000037 | fixed U. Their common subgroup is <(2,2)>, the D8 intermediate extension. 000038 | This reproduces the earlier extension classification rather than introducing 000039 | a new choice of lattice. 000040 | 000041 | 2. The clock sectors are the four fibers of this same module 000042 | 000043 | For x=(a,b), braiding with h has character 000044 | 000045 | B(x,h)=(a-b)/4 mod1. 000046 | 000047 | Hence the quotient/monodromy flux is f=a-b mod4. In the Li convention with 000048 | kappa=exp(+2 pi i v(0)), v=(1/2^5,0^3), the kappa^r twisted lattice is 000049 | E8-rv and has U labels (k-r,k). Thus f=-r; the two orientations must not be 000050 | silently interchanged. 000051 | 000052 | Restriction of each twisted module to U is the sum of the four simple U 000053 | modules in its flux fiber. One may see this directly by partitioning the 000054 | shifted lattice into Q cosets. This proves the branching at every conformal 000055 | grade, rather than inferring it from ground-state dimensions. 000056 | 000057 | D5 level-one lowest weights are (0,5/8,1/2,5/8), with top dimensions 000058 | (1,16,10,16). A3 weights are (0,3/8,1/2,3/8), with top dimensions 000059 | (1,4,6,4). Adding the two weights in each fiber gives 000060 | 000061 | r 0 1 2 3 000062 | ground h 0 3/8 1/2 3/8 000063 | top 1 (1,4) (10,1)+(1,6) (1,bar4) 000064 | dimension 1 4 16 4. 000065 | 000066 | The names 4 and bar4 use the T chart of CLOCK-TWISTED.txt. An A3 discriminant 000067 | label called 1 in another chart can denote the opposite fundamental; the 000068 | listed U labels and f=-r formula are the actual orientation contract. 000069 | 000070 | 3. Explicit match to cyclic orbifold theory 000071 | 000072 | Set phi(i,j)=(3i+j,2i+j) mod4. Its determinant is1, flux(phi(i,j))=i, and 000073 | 000074 | q(phi(i,j))=i^2/8+ij/4 mod1. 000075 | 000076 | This is exactly the discriminant form for a cyclic order-four automorphism 000077 | of type t=2. In the notation of van Ekeren, Moller and Scheithauer, 000078 | t=4^2 rho mod4=2 for rho=3/8. The match is an explicit isometry of marked 000079 | quadratic modules, not just a census of sixteen states. Twisted-eigenspace 000080 | labels allow a root-of-unity normalization, as in their Theorem5.13. 000081 | 000082 | 4. What the anomaly actually says 000083 | 000084 | There is no isotropic subgroup K of order4 with K intersect H+={0}. 000085 | Equivalently, there is no local extension selecting exactly one summand from 000086 | each of the four flux fibers. This is the obstruction to a transverse local 000087 | orbifold completion relative to this marked H+. 000088 | 000089 | There DO exist holomorphic local extensions: H+ and H-. No conclusion that 000090 | all local completions are impossible follows from the obstruction. 000091 | 000092 | 5. Full modular phases 000093 | 000094 | For the sixteen simple U modules, set 000095 | 000096 | S_xy=(1/4) exp(-2 pi i B(x,y)), 000097 | T_x=exp(2 pi i(q(x)-8/24)). 000098 | 000099 | The normalized quadratic Gauss sum equals 000100 | 000101 | (1/4) sum_(a,b) zeta8^(5a^2+3b^2) 000102 | =(2 zeta8^5)(2 zeta8^3)/4=1. 000103 | 000104 | The exact cyclotomic computation verifies S^2=(ST)^3=C, where C sends x to 000105 | -x. The quadratic form has level8. Including the c=8 Casimir term gives 000106 | T of exact order24. These are matrices on the labeled module space; distinct 000107 | ordinary characters can agree as scalar functions. 000108 | 000109 | This modular level is not a statement about the degree of an earlier C12 000110 | base cover and does not identify conformal rotations with physical time. 000111 | 000112 | 6. Reproduction and review 000113 | 000114 | python3 finite_quadratic_clock.py 000115 | python3 -OO finite_quadratic_clock.py 000116 | 000117 | All equalities are over integers, rationals or Z[zeta24], using 000118 | Phi24(X)=X^8-X^4+1. Normal and optimized results are byte-identical. An 000119 | independent review checked orientation, all isotropic supports, the explicit 000120 | orbifold isometry, Gauss sum and both modular relations. CLOCK-TWISTED.txt 000121 | independently checks actual shifted E8 weights and native oscillator data. 000122 | 000123 | Primary sources 000124 | 000125 | Bakalov and Kac, Twisted Modules over Lattice Vertex Algebras: 000126 | https://arxiv.org/abs/math/0402315 000127 | 000128 | Li spectral flow as stated by Lam and Shimakura, Proposition3.1 and eq3.2: 000129 | https://arxiv.org/abs/1501.05094 000130 | 000131 | van Ekeren, Moller and Scheithauer, Construction and classification of 000132 | holomorphic vertex operator algebras, Proposition5.10, Theorem5.13, 000133 | Proposition5.14 and the following congruence-level theorem: 000134 | https://arxiv.org/abs/1507.08142 000135 | 000136 | Verdict 000137 | 000138 | Exact common marked glue/clock structure within the previously reconstructed 000139 | unitary conformal realization. This removes an independent choice of twisted 000140 | sector data in that class. It does not select P1's microscopic source, 000141 | spacetime, a coupling normalization outside this realization, or a physical 000142 | state of the universe.