TFPT scientific source excerpt Original repository path: experiments/theory-contracts/e-series-source-reconstruction-20261024/SIMPLIFICATION.txt Original file SHA-256: 51bb029fdaa33e327d2d24d98b47972a19bacf6a5987452a1ea3010be7130dab Inclusive original lines: 1-169 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 / simplification amendment 000002 | 24 October 2026 000003 | 000004 | The marked generator forces the whole extension 000005 | 000006 | 1. Exact strengthening and correction 000007 | 000008 | Put U=V_D5 tensor V_A3 with its fixed unitary conformal structure. Retain 000009 | the original discriminant group A=C4 x C4 and its marked h=(1,1). 000010 | 000011 | Consider all connected local unitary VOA extensions W of U with the SAME 000012 | conformal vector. Isomorphisms restrict to the identity on U. Do not 000013 | prescribe the full U-module support, its multiplicities, holomorphicity, 000014 | finite extension index, or a separate minimality criterion. 000015 | 000016 | THEOREM. If Hom_U(M_h,W) is nonzero, then W is the unique marked extension 000017 | V_E8 over U. Its U-module is automatically 000018 | 000019 | W = M_0 + M_h + M_2h + M_3h, 000020 | 000021 | and every summand occurs exactly once. 000022 | 000023 | This strengthens the reconstruction theorem in THEOREMS.txt. The exact 000024 | support and multiplicity-one requirements there are redundant within the 000025 | connected local unitary SAME-CONFORMAL extension class once M_h occurs. 000026 | In particular the last sentence of its Section5 must be qualified: merely 000027 | containing the four sectors IS sufficient under these remaining hypotheses. 000028 | A tensor spectator changes the total conformal vector and is outside this 000029 | class. A repeated vacuum violates connectedness. 000030 | 000031 | The original theorem remains true in its narrower stated class. This 000032 | amendment makes the stronger class and the correction explicit rather than 000033 | silently rewriting the earlier source or its reviewed premise identity. 000034 | 000035 | 2. Finiteness is a consequence 000036 | 000037 | U is regular. Therefore W, regarded as a weak U-module, decomposes as a 000038 | direct sum of the sixteen irreducible lattice modules M_a, with possible 000039 | multiplicities n_a. The common conformal vector identifies their L0 action 000040 | with that on W. If Delta_a is the fixed lowest conformal weight of M_a, 000041 | each copy contributes its nonzero ground space to W_(Delta_a), giving 000042 | 000043 | n_a dim(M_a)_(Delta_a) <= dim W_(Delta_a) < infinity. 000044 | 000045 | If Delta_a is nonintegral this W weight space is zero. Thus every n_a is 000046 | finite. There are only sixteen module labels, so W is automatically a 000047 | finite U-module. No separate finite-index assumption is needed. 000048 | 000049 | By the standard VOA-extension/algebra correspondence, this finite unitary 000050 | extension defines a connected commutative separable algebra B in the 000051 | unitary pointed category C=Rep(U), with trivial twist. The normalized 000052 | Q-system supplies separability and the corresponding semisimple module 000053 | category. These are consequences of the unitary extension hypotheses, 000054 | not new free algebra data. 000055 | 000056 | 3. Short general multiplicity lemma 000057 | 000058 | Let B be a connected separable algebra in a pointed fusion category C. 000059 | The category C_B of right B-modules is semisimple. The regular right 000060 | module B is simple because 000061 | 000062 | End_B(B) = Hom_C(1,B) = C. 000063 | 000064 | For each invertible simple object M_a, left tensoring by M_a is an 000065 | autoequivalence of right B-modules, with inverse M_(-a). Thus M_a tensor B 000066 | is also simple. The free-module/forgetful adjunction now gives 000067 | 000068 | n_a = dim Hom_C(M_a,B) 000069 | = dim Hom_(C_B)(M_a tensor B,B) 000070 | in {0,1}. (3.1) 000071 | 000072 | Moreover n_a=1 exactly when M_a tensor B is isomorphic to B as a right 000073 | module. Consequently 000074 | 000075 | S={a:n_a=1} 000076 | 000077 | is the stabilizer subgroup of the regular module under the invertible 000078 | objects. This proves both subgroup closure and multiplicity one without 000079 | assuming a pairing on repeated labels or choosing multiplication constants. 000080 | 000081 | The lemma uses standard semisimplicity and adjunction. Its application 000082 | eliminates the redundant assumptions in the existing TFPT reconstruction. 000083 | 000084 | 4. One marked generator saturates the allowed support 000085 | 000086 | In the native discriminant chart 000087 | 000088 | q(a,b)=(5a^2+3b^2)/8 mod1, 000089 | b((a,b),(c,d))=(5ac+3bd)/4 mod1. 000090 | 000091 | The common conformal vector and the integer VOA grading give trivial 000092 | twist on the support, hence q restricted to S is zero. Since S is a 000093 | subgroup, polarization then gives b(S,S)=0. 000094 | 000095 | The hypothesis M_h subset W gives H= subset S. Isotropy implies 000096 | S subset H-perp. But the native lattice already gives 000097 | 000098 | H-perp={(a,b):5a+3b=0 mod4} 000099 | ={(a,a):a in C4}=H. 000100 | 000101 | Therefore 000102 | 000103 | H subset S subset H-perp=H, 000104 | 000105 | so S=H. Equation(3.1) gives each summand once. The coherent-product and 000106 | fixed-U unitary uniqueness proof in THEOREMS.txt applies immediately. 000107 | Thus the existing marked generator forces the full E8 extension, its 000108 | canonical vacuum, all vacuum correlation functions and conformal dynamics. 000109 | 000110 | The local realization and the occurrence of M_h are still hypotheses. 000111 | This theorem does not derive their realization by the primitive P1 kernel. 000112 | 000113 | 5. Complete classification before choosing the generator 000114 | 000115 | The same lemma classifies all connected local unitary same-conformal 000116 | extensions of this U. The only isotropic support subgroups are 000117 | 000118 | S0={0}, 000119 | S2={0,(2,2)}, 000120 | Splus={(a,a):a in C4}, 000121 | Sminus={(a,-a):a in C4}. (5.1) 000122 | 000123 | Completeness is elementary: q=0 has exactly the six labels 000124 | 000125 | (0,0),(2,2),(1,1),(3,3),(1,3),(3,1). 000126 | 000127 | Its only nonzero order-two label is(2,2). Every order-four label generates 000128 | either Splus or Sminus; these cannot occur together in an isotropic subgroup. 000129 | This exhausts every subgroup of the zero set. The checker enumerates all 000130 | subsets of that six-element set and confirms precisely(5.1). 000131 | 000132 | Each subgroup is cyclic. The even lattice overlattice gives existence, 000133 | and the existing abelian-coherence argument gives one multiplication class 000134 | for each support (H2(C_n,C*)=0 for these cyclic groups). Thus there are 000135 | exactly four extension classes over the fixed U: 000136 | 000137 | support lattice realization dimension of weight-one space 000138 | S0 V_(D5+A3) 60 000139 | S2 V_D8 120 000140 | Splus V_E8 248 000141 | Sminus V_E8 248. (5.2) 000142 | 000143 | The two E8 entries have different labelled embeddings over fixed U. They 000144 | become equivalent after allowing the appropriate A3 diagram reversal. 000145 | The original marked h=(1,1) selects Splus. It does not belong to Sminus. 000146 | No variational choice of a preferred extension is needed once that marked 000147 | generator is present in the stated local conformal realization. 000148 | 000149 | 6. Sources and remaining first hypothesis 000150 | 000151 | Kirillov and Ostrik, Theorem1.6 (adjunction), Theorem3.3 (semisimplicity): 000152 | https://arxiv.org/html/math/0101219 000153 | Davydov, Mueger, Nikshych and Ostrik, Sections3.3-3.4 (module categories 000154 | and free-module functor for connected etale algebras): 000155 | https://arxiv.org/html/1009.2117 000156 | Huang, Kirillov and Lepowsky, Theorem3.2 (VOA extensions): 000157 | https://arxiv.org/html/1406.3420 000158 | Carpi, Gaudio, Giorgetti and Hillier, Theorems3.9,4.7,4.10 and Corollary4.11 000159 | (unitary extensions, normalized Q-systems and local conformal nets): 000160 | https://link.springer.com/article/10.1007/s00220-023-04722-9 000161 | 000162 | The first additional realization premise is now smaller: the primitive 000163 | source must realize the given U as a connected local unitary extension 000164 | sharing its conformal vector and containing the marked charged module M_h. 000165 | Exact support, multiplicities, holomorphicity, vacuum 000166 | functional and conformal generator then follow. Original physical time, 000167 | source localization and spacetime identifications are not supplied by this 000168 | conditional chiral theorem. All earlier scoped clock/selection results 000169 | and the original character-valued nonsplitting remain unchanged.