TFPT scientific source excerpt Original repository path: experiments/theory-contracts/e-series-source-reconstruction-20261024/THEOREMS.txt Original file SHA-256: 5c0d7328c583cdbc0a8c358445b57821d5b5722540ca372c1d28052b900eb584 Inclusive original lines: 1-196 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.ESERIES.SOURCE-RECONSTRUCTION.06 000002 | 24 October 2026 000003 | 000004 | Canonical reconstruction from the fixed marked C4 glue 000005 | 000006 | 0. Verdict and exact scope 000007 | 000008 | Retain as proved input, without changing or rederiving it, the native compiler 000009 | 000010 | K = D5 + A3 subset L, 000011 | H = <(s,f)> subset K*/K, 000012 | L = union_(a in H) (K+a) = E8, 000013 | 000014 | the fixed sector labels, the original epsilon convention, the compiler and 000015 | cover actions, and 000016 | 000017 | phi_base = (4/3)c3 = 1/(6 pi). 000018 | 000019 | The reconstruction class considered here is the following precise one. Put 000020 | 000021 | U = V_D5 tensor V_A3 = V_K. 000022 | 000023 | An object of Ext_H(U) is a connected local unitary VOA extension W of U, 000024 | with the same conformal vector, whose underlying labelled U-module is 000025 | 000026 | W = direct_sum_(a in H) M_a (0.1) 000027 | 000028 | with every M_a occurring exactly once. An isomorphism in Ext_H(U) restricts 000029 | to the identity on U and preserves each labelled summand M_a. It may multiply 000030 | a simple summand by a scalar. 000031 | 000032 | THEOREM. Ext_H(U) has exactly one isomorphism class. It is represented by 000033 | V_E8. The corresponding normalized connected commutative Q-system has 000034 | exactly one unitary equivalence class. Its residual automorphisms over U are 000035 | 000036 | Hom(H,U(1)) = Hom(C4,U(1)) = C4. (0.2) 000037 | 000038 | Thus the four residual choices are character gauge, not four extensions. 000039 | Within this exact class the vacuum, conformal vector, L0, unitary structure, 000040 | local products and vacuum correlation functions are fixed up to the stated 000041 | cocycle and character gauges. No spectator is left, and the extension is 000042 | generated over U by the sector M_(s,f), equivalently by its weight-one space. 000043 | 000044 | This is a mathematical reconstruction theorem from the already established 000045 | marked C4 glue. It does not prove that primitive P1 data furnish U, its 000046 | positive-energy representation, its local fields, or this reconstruction 000047 | class. No 3+1 dimensional or physical source identification is asserted. 000048 | 000049 | 000050 | 1. The pointed category and the fixed Lagrangian subgroup 000051 | 000052 | The irreducible modules of the positive definite lattice VOA V_K are indexed 000053 | by the discriminant group 000054 | 000055 | A = K*/K = (D5*/D5) x (A3*/A3) = C4 x C4. 000056 | 000057 | All sixteen modules are simple currents and have categorical dimension one. 000058 | Choose the spinor generator s of D5*/D5 and the fundamental generator f of 000059 | A3*/A3. Their discriminant quadratic forms, written as lowest conformal 000060 | weights modulo Z, are 000061 | 000062 | q_D5(0)=0, q_D5(s)=5/8, q_D5(2s)=1/2, q_D5(3s)=5/8, 000063 | q_A3(0)=0, q_A3(f)=3/8, q_A3(2f)=1/2, q_A3(3f)=3/8. (1.1) 000064 | 000065 | Let h=(s,f). The fixed diagonal subgroup is 000066 | 000067 | H = {0,h,2h,3h} = C4. 000068 | 000069 | Equation (1.1) gives actual lowest weights, not only residues, 000070 | 000071 | Delta(M_0)=0, 000072 | Delta(M_h)=5/8+3/8=1, 000073 | Delta(M_2h)=1/2+1/2=1, 000074 | Delta(M_3h)=5/8+3/8=1. (1.2) 000075 | 000076 | Consequently q restricted to H is zero modulo one. The pointed modular 000077 | category is nondegenerate, |A|=16 and |H|=4. Hence 000078 | 000079 | |H-perp|=|A|/|H|=4, 000080 | 000081 | and isotropy H subset H-perp implies H=H-perp. Thus H is Lagrangian. This is 000082 | also the lattice statement that the index-four overlattice associated with H 000083 | is even and unimodular. The fixed input identifies it with E8. 000084 | 000085 | 000086 | 2. Exact sector content and the 248 weight-one states 000087 | 000088 | Use the standard level-one labels for D5 and A3. The four H sectors and their 000089 | lowest finite-dimensional representations are 000090 | 000091 | sector U-module Delta dimension at Delta 000092 | 0 (1,1) 0 1 000093 | h (16,4) 1 64 000094 | 2h (10,6) 1 60 000095 | 3h (bar(16),bar(4)) 1 64. (2.1) 000096 | 000097 | The vacuum sector contributes at weight one 000098 | 000099 | (M_0)_1 = (45,1) + (1,15), 000100 | 000101 | of dimension 45+15=60. Combining this with (2.1) gives the exact branching 000102 | 000103 | (V_E8)_1 restricted to so(10) + sl(4) 000104 | = (45,1) + (1,15) + (16,4) + (10,6) 000105 | + (bar(16),bar(4)), 000106 | 000107 | 248 = 60 + 64 + 60 + 64. (2.2) 000108 | 000109 | This is sector by sector. It is not an inference from the number 248. 000110 | 000111 | 000112 | 3. The associator obstruction and why it vanishes on H 000113 | 000114 | First note that every multiplication channel in an object of Ext_H(U) is 000115 | nonzero. The invariant Frobenius pairing of a connected unitary extension is 000116 | nondegenerate, so the component M_a tensor M_(-a) -> M_0 is nonzero for every 000117 | a. If some component M_a tensor M_b -> M_(a+b) vanished, associativity after 000118 | multiplication by M_(-b) would give zero in the bracketing (M_a M_b)M_(-b), 000119 | whereas the bracketing M_a(M_b M_(-b)) contains the nonzero unit action on 000120 | M_a. This is a contradiction. Hence each one-dimensional fusion space below 000121 | is used by a nonzero intertwiner. 000122 | 000123 | For a,b in H choose a unit-norm fusion isomorphism in the unitary pointed 000124 | category (hence a unitary basis intertwiner) 000125 | 000126 | I_(a,b): M_a tensor M_b -> M_(a+b). 000127 | 000128 | Every relevant fusion space is one dimensional. Comparing the two triple 000129 | products and comparing exchanged products gives scalars 000130 | 000131 | F(a,b,c) in U(1), Omega(a,b) in U(1). 000132 | 000133 | The pentagon and the two hexagons say precisely that (F,Omega) is a normalized 000134 | abelian 3-cocycle. This datum must not be discarded. The full pointed category 000135 | on A has nontrivial quadratic trace, for example 000136 | 000137 | Q(s,0) = exp(2 pi i 5/8) != 1, 000138 | 000139 | so its associator and braiding cannot be made globally trivial. 000140 | 000141 | The Eilenberg-Mac Lane trace theorem sends an abelian 3-cocycle class to 000142 | 000143 | Q(a)=Omega(a,a). 000144 | 000145 | For a VOA module category this trace is the conformal spin 000146 | 000147 | Q(a)=exp(2 pi i q(a)) (3.1) 000148 | 000149 | in the standard convention. Using the inverse convention changes no 000150 | conclusion below. The trace map is a bijection from abelian 3-cohomology to 000151 | quadratic forms. By (1.2), Q(a)=1 for every a in H. Therefore the restriction 000152 | of the abelian class [(F,Omega)] to H is zero. This conclusion uses the full 000153 | abelian trace theorem. It is not the false statement that ordinary 000154 | H^3(C4,U(1)) vanishes; in fact 000155 | 000156 | H^3(C4,U(1)) = C4. (3.2) 000157 | 000158 | Because the restricted abelian class is zero, there is a normalized 000159 | 2-cochain mu:H x H -> U(1) such that replacing 000160 | 000161 | I_(a,b) by mu(a,b) I_(a,b) 000162 | 000163 | makes both F and Omega equal to one on H. Thus associativity and locality are 000164 | obtained simultaneously. The explicit E8 lattice vertex operators furnish 000165 | one such trivialization, so existence does not depend on an abstract 000166 | obstruction calculation alone. 000167 | 000168 | Suppose mu and mu' are two coherent trivializations. Their quotient eta is a 000169 | normalized ordinary 2-cocycle. Two quotients differing by a coboundary give 000170 | isomorphic algebra objects, by rescaling M_a with a 1-cochain. For arbitrary 000171 | nonzero VOA multiplication constants use C* coefficients; for unitary 000172 | normalized trivializations use U(1). In both cases 000173 | 000174 | H^2(C4,C*) = H^2(C4,U(1)) = 0. (3.3) 000175 | 000176 | An elementary proof of (3.3) is useful. A projective lift T of a generator 000177 | has T^4=zeta for some zeta in U(1). Choose rho with rho^4=zeta^(-1). Then 000178 | (rho T)^4=1, and the projective cyclic action linearizes. Hence every 000179 | 2-cocycle is a coboundary. The same proof works for C*, since a nonzero 000180 | complex number also has a fourth root. Thus algebraic uniqueness does not 000181 | assume that an arbitrary VOA multiplication was already normalized to unit 000182 | modulus. Section 4 supplies the unitary equivalence of normalized Q-systems. 000183 | 000184 | These scalar coefficients concern only the local C4 extension algebra. 000185 | They do not replace Hom(L,Z2), its nontrivial group action, or the fixed 000186 | character-valued G31 lift extension. No global sign-extension splitting 000187 | follows from (3.3). 000188 | 000189 | It follows that there is one algebra isomorphism class. The 1-cocycles that 000190 | stabilize its multiplication are the four characters of H. They act by 000191 | 000192 | v in M_(kh) |-> chi(kh) v, 000193 | 000194 | and give precisely (0.2). 000195 | 000196 |