TFPT scientific source excerpt Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/AFFINE-DEFECT-INDEPENDENT-REVIEW.txt Original file SHA-256: 0870f40d5b6d65b3d62a435e6c11706bba84dbf1fb21706bd6c6aa6cc058f142 Inclusive original lines: 1-200 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 | AFFINE SOURCE WINDING ON THE FOUR D8 SECTORS 000002 | ============================================= 000003 | 000004 | 25 October 2026. Independent analytic review in the actual transported 000005 | source frame D8_F. No identification with the separately marked canonical 000006 | D8 is assumed. No Atlas, index, ledger, paper or original source is changed. 000007 | 000008 | Question and verdict 000009 | -------------------- 000010 | 000011 | Let L=D8_F and let 000012 | 000013 | A=L*/L={0,v,s,c}, c=v+s, 000014 | 000015 | with the four standard level one modules H_0,H_v,H_s,H_c. Let R be the 000016 | actual cyclic permutation of the eight source channels and e0 the original 000017 | unit winding charge. The affine label action is 000018 | 000019 | A_1(p)=R p+e0. (1) 000020 | 000021 | The proposed operator is the charge translation followed by an oscillator 000022 | and charge permutation, 000023 | 000024 | W_aff=T_e0 Gamma(R). (2) 000025 | 000026 | There is a mathematically complete positive answer, with one essential 000027 | category correction and one phase boundary. 000028 | 000029 | PROVED: 000030 | 000031 | * The direct sum 000032 | 000033 | H_parent=H_0 direct-sum H_v direct-sum H_s direct-sum H_c 000034 | 000035 | has the standard abelian intertwining algebra structure associated with 000036 | the discriminant form of D8. It is not one ordinary lattice VOA because 000037 | D8* is not integral. Its two familiar local subextensions are the 000038 | bosonic E8 VOA H_0+H_s and the odd lattice vertex superalgebra, equivalently 000039 | the complex CAR NS theory, H_0+H_v. 000040 | 000041 | * After choosing a normalized lattice cocycle and a lift of R, equation 000042 | (2) is a unitary finite energy operator on H_parent. It maps the four 000043 | sectors by fusion with v, 000044 | 000045 | 0 <-> v, s <-> c, (3) 000046 | 000047 | and restricts on H_0+H_v to the already supplied CAR winding W1. 000048 | 000049 | * The same affine action has the exact source Ward increment on every 000050 | charge p in D8*, not only on the integer NS charges. 000051 | 000052 | * Li's Delta operator for e0, composed with the R lift, gives the same 000053 | g twisted E8 module functor up to the displayed sign convention and 000054 | cocycle gauge. It produces H_v+H_c as a twisted module. It does not make 000055 | v or c ordinary local fields in the bosonic E8 vacuum VOA. 000056 | 000057 | * On a punctured simply connected chart the Delta fields retain formal 000058 | twisted locality. For an already defined smeared field, their analytic 000059 | coefficient only multiplies the test function and hence does not enlarge 000060 | its support. On the original NS CAR net, support preservation is already 000061 | an operator identity. Li's formal theorem alone neither constructs the 000062 | smeared charged intertwiners nor proves that they are bounded local net 000063 | operators. 000064 | 000065 | NOT FIXED BY THE ORIGINAL NS OPERATOR: 000066 | 000067 | The action on H_s+H_c has a residual lift choice. The standard modular 000068 | E8 defect lift and the geometric torus character are different operators 000069 | on that space. The standard lift phase is selected only after a defect 000070 | junction and spin/Arf convention are chosen. The original NS CAR winding 000071 | and its neutral holonomy word do not select that extra Ramond phase. 000072 | 000073 | Thus (2) is an exact mathematical four sector extension without inventing a 000074 | new physical source. Identifying one concrete Ramond lift, spin structure 000075 | and defect junction with the primitive TFPT source is additional source 000076 | data. This distinction is phase level, not a failure of the affine charge 000077 | or Ward construction. 000078 | 000079 | 1. The four sector parent and the affine map 000080 | -------------------------------------------- 000081 | 000082 | The dual of D8 is the union 000083 | 000084 | D8*=D8 union(D8+v) union(D8+s) union(D8+c) 000085 | =Z8 union(Z8+s). (4) 000086 | 000087 | The conformal quadratic form is 000088 | 000089 | q(0)=0, q(v)=1/2, q(s)=q(c)=1 modulo Z. (5) 000090 | 000091 | Consequently the direct sum of all irreducible V_D8 modules has an abelian 000092 | intertwining algebra whose abelian 3 cocycle is determined, up to cochain 000093 | gauge, by (5). This is the standard lattice module theorem. The nonintegral 000094 | pairings in D8* are exactly why the whole sum is not an ordinary local 000095 | lattice VOA. 000096 | 000097 | The cyclic coordinate permutation R preserves D8 and each discriminant 000098 | class. Translation by e0 represents fusion with v. Hence (1) gives (3) and 000099 | is a bijection of D8*. On the finite energy basis 000100 | 000101 | oscillator descendant tensor |p>, p in D8*, 000102 | 000103 | Gamma(R) permutes all Heisenberg oscillators and the lattice label, while 000104 | T_e0 shifts the label. Once the cocycle phases have been chosen, both maps 000105 | are unitary and preserve the algebraic finite energy core. Their product is 000106 | therefore a unitary on its Hilbert completion. 000107 | 000108 | On Z8=D8 union(D8+v), choose the ordered CAR cocycle and the already fixed 000109 | normalization. Then (2) is exactly the old source W1, including its Klein 000110 | signs. This is a restriction statement, not a reconstruction from matching 000111 | dimensions. 000112 | 000113 | 2. Exact Ward identity on all four sectors 000114 | ------------------------------------------ 000115 | 000116 | Use the retained folded source energy 000117 | 000118 | K_alpha(p,N)=|p|^2/2+delta.p+N, 000119 | delta_k=(k+alpha)/8-1/2, k=0,...,7. (6) 000120 | 000121 | Take R(p_0,...,p_7)=(p_7,p_0,...,p_6). The oscillator level N is unchanged. 000122 | Direct expansion gives 000123 | 000124 | |Rp+e0|^2/2-|p|^2/2=p_7+1/2, 000125 | 000126 | and 000127 | 000128 | delta.(Rp+e0-p) 000129 | =(p_0+...+p_6)/8-7p_7/8+alpha/8-1/2. 000130 | 000131 | Therefore, for every real p and hence for every p in D8*, 000132 | 000133 | K_alpha(A_1 p,N)-K_alpha(p,N) 000134 | =(Q(p)+alpha)/8, Q(p)=sum_k p_k. (7) 000135 | 000136 | Equation (7) proves the proposed Ward increment on the Ramond cosets as 000137 | well as the NS lattice. No new energy formula is fitted to those sectors; 000138 | it is the same quadratic conformal energy restricted to the four standard 000139 | D8 modules. Absolute Ramond ground energies are fixed by (5). 000140 | 000141 | 3. Delta(e0,z) is the corresponding inner twisted functor 000142 | ---------------------------------------------------------- 000143 | 000144 | For a CFT type VOA V and semisimple h in V_1 with rational h(0) spectrum, 000145 | Li's operator is 000146 | 000147 | Delta(h,z)=z^(h(0)) exp(sum_(n>0) h(n)/(-n) (-z)^(-n)). (8) 000148 | 000149 | Lam and Shimakura, Proposition 3.1, state that 000150 | 000151 | Y_M^(h)(a,z)=Y_M(Delta(h,z)a,z) (9) 000152 | 000153 | turns a sigma twisted module M into a sigma_h sigma twisted module, where 000154 | 000155 | sigma_h=exp(-2 pi i h(0)). (10) 000156 | 000157 | The zero mode and energy become 000158 | 000159 | x_0^(h)=x_0+, 000160 | L_0^(h)=L_0+h(0)+/2. (11) 000161 | 000162 | For V=V_E8 and h=e0, the automorphism in (10) is +1 on the D8 root fields 000163 | and -1 on the spinor root fields. Its unique twisted module is the shifted 000164 | lattice module 000165 | 000166 | V_(E8+e0)=H_v direct-sum H_c. (12) 000167 | 000168 | The sign h=-e0 may be used instead if the opposite Delta convention is 000169 | chosen. Since the automorphism has order two, both signs give the same g, 000170 | although they reverse the displayed charge shift. 000171 | 000172 | For a Heisenberg current x and an E8 charge alpha, equations (8) and (9) 000173 | give 000174 | 000175 | x(z) -> x(z)+/z, 000176 | Y(e^alpha,z) -> z^()Y(e^alpha,z). (13) 000177 | 000178 | Composing with Gamma(R) replaces x and alpha by Rx and Ralpha. This is the 000179 | field version of p -> Rp+e0. The coordinate characters e_i all define the 000180 | same E8 involution because e_i-e_j lies in D8 subset E8 and E8 is self dual. 000181 | Thus R normalizes this g and the composition remains the same g twisted 000182 | functor. 000183 | 000184 | Equation (13) also gives the precise formal locality statement. On any 000185 | simply connected interval avoiding z=0, choose one branch of the fractional 000186 | power. The twisted Jacobi identity supplies the correct monodromy. If the 000187 | field has already been constructed as a smeared operator, multiplying its 000188 | test function by 1/z or the chosen analytic power does not enlarge support. 000189 | Constructing that smeared charged intertwiner, proving strong convergence 000190 | on a common core and obtaining a bounded conformal net defect still require 000191 | energy bounds and a defect localization theorem. The Delta proposition by 000192 | itself is a VOA module theorem. 000193 | 000194 | This analytic distinction is substantive. Dong and Xu construct conformal 000195 | nets and orbifold solitons for even lattices and lattice isometries, while 000196 | the present g is an inner torus twist. Carpi and Tomassini provide energy 000197 | bounds for unitary VOA extensions. Neither result, without an additional 000198 | identification, proves that the concrete cocycle calibrated W_aff is a 000199 | localized bounded defect endpoint on the original source net. 000200 |