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Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/SAME-SOURCE-AFFINE-DEFECT.txt
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000001 | SAME SOURCE AFFINE DEFECT ON ALL FOUR D8 MODULES
000002 | ================================================
000003 | 
000004 | Date: 2026-10-25
000005 | 
000006 | Verdict
000007 | -------
000008 | 
000009 | PROVED, for the exact mathematical object specified below:
000010 | 
000011 | The original cyclic CAR operator W1 extends from the integer charge Fock
000012 | space to a unitary finite-energy operator W_ext on the standard abelian
000013 | intertwining algebra formed by all four D8 Heisenberg modules.  It maps the
000014 | E8 vacuum representation 0+s to the
000015 | corresponding g-twisted representation v+c, preserves the full fermionic NS
000016 | sum 0+v, preserves the Ramond sum s+c, and satisfies the original charge and
000017 | quarter-time Ward identities on integer and half-integer charges.  Its
000018 | restriction to the NS space is the old ordered-CAR W1, including its Klein
000019 | phases.  No global odd Phi map and no identification with the separately
000020 | marked compiler D8 are used.
000021 | 
000022 | This closes the mathematical affine W1 defect construction in the actual
000023 | fold frame D8_F=M(D8_CAR), uniquely up to intertwiner basis.  It does not
000024 | select that four-module extension from primitive P1 data, does not identify
000025 | the geometric Cartan operator with the distinct standard defect lift on the
000026 | twisted module, and does not identify the resulting defect with a local 3+1
000027 | dimensional source.
000028 | 
000029 | 
000030 | 1. Fixed inputs and exact object
000031 | --------------------------------
000032 | 
000033 | Let
000034 | 
000035 |   D8 = { n in Z^8 : Q(n)=sum_j n_j is even },
000036 |   s  = (1/2,...,1/2),
000037 |   v  = e0 + D8,
000038 |   c  = e0+s+D8.
000039 | 
000040 | Every charge in D8* has a unique expression
000041 | 
000042 |   p = n + c0 s,       n in Z^8, c0 in {0,1}.
000043 | 
000044 | The four D8 sectors are therefore
000045 | 
000046 |   0 : c0=0 and Q(n) even,
000047 |   v : c0=0 and Q(n) odd,
000048 |   s : c0=1 and Q(n) even,
000049 |   c : c0=1 and Q(n) odd.
000050 | 
000051 | The finite-energy core and its Hilbert completion are
000052 | 
000053 |   H_fin = direct_sum_(mu=0,v,s,c)
000054 |           M(1)_fin tensor C[D8+mu],
000055 | 
000056 |   H_*   = completion(H_fin).
000057 | 
000058 | Here M(1) is the rank-eight Heisenberg Fock representation with the standard
000059 | positive Hermitian form.  This is one source Fock system with four charge
000060 | sectors.  It is not an eight-copy construction.
000061 | 
000062 | The full sum H_* is the standard abelian intertwining algebra attached to
000063 | the discriminant form of D8.  It is not an ordinary lattice VOA because D8*
000064 | is not integral.  Its discriminant quadratic form is
000065 | 
000066 |   q(0)=0, q(v)=1/2, q(s)=q(c)=1 modulo Z.
000067 | 
000068 | This nontrivial quadratic form supplies the abelian 3-cocycle and the
000069 | fractional monodromy.  The two local subextensions used here are
000070 | 
000071 |   H_0+H_s, the bosonic E8 lattice VOA,
000072 |   H_0+H_v, the odd-lattice vertex superalgebra or complex CAR NS theory.
000073 | 
000074 | The E8 vacuum representation is the even extension
000075 | 
000076 |   V_E8 = H_0 direct_sum H_s.
000077 | 
000078 | The integer fermionic NS space and the Ramond space are
000079 | 
000080 |   H_NS = H_0 direct_sum H_v,
000081 |   H_R  = H_s direct_sum H_c.
000082 | 
000083 | The inputs R, e0, M and delta are the already fixed source fold, charge carry,
000084 | source-to-target dictionary and quarter-time vector.  No new Weyl rotation is
000085 | chosen here.
000086 | 
000087 | 
000088 | 2. Exact common charge cocycle
000089 | ------------------------------
000090 | 
000091 | On Z^8 retain the original ordered-CAR cocycle
000092 | 
000093 |   eps_Z(n,m) = (-1)^(sum_(i>j) n_i m_j).
000094 | 
000095 | Adjoin X=E_s to its charge algebra with relations
000096 | 
000097 |   X E_n X^(-1) = i^Q(n) E_n,
000098 |   X^2            = -E_1,
000099 | 
000100 | where 1=(1,...,1)=2s.  These relations are consistent because conjugation by
000101 | X^2 gives (-1)^Q(n), and conjugation by E_1 in the ordered-CAR charge algebra
000102 | gives the same sign.
000103 | 
000104 | Use the section
000105 | 
000106 |   E_(n+c0 s) = E_n X^c0.
000107 | 
000108 | Its normalized cocycle is
000109 | 
000110 |   eps_*(n+c0 s,m+d0 s)
000111 |     = i^(c0 Q(m)) eps_Z(n,m)                         if c0 d0=0,
000112 | 
000113 |   eps_*(n+s,m+s)
000114 |     = -i^Q(m) eps_Z(n,m) eps_Z(n+m,1).
000115 | 
000116 | Associativity of this charge multiplication follows directly from the
000117 | presented associative charge algebra.  The checker also verifies its
000118 | 2-cocycle identity on a generating set and all generator products.  This
000119 | ordinary charge 2-cocycle does not trivialize the abelian 3-cocycle of the
000120 | four-sector intertwining algebra.  The latter remains fixed by q and is the
000121 | categorical source of the fractional field monodromy.
000122 | 
000123 | Two restrictions are exact, rather than merely cohomologous:
000124 | 
000125 |   (a) On Z^8, eps_* is eps_Z.
000126 | 
000127 |   (b) If Q(n) and Q(m) are even, eps_* equals the previously fixed E8
000128 |       cocycle
000129 | 
000130 |       eps_E8(n+c0 s,m+d0 s)
000131 |         = (-1)^(c0 Q(m)/2+c0 d0)
000132 |           eps_Z(n,m) eps_Z(n+m,c0 d0 1).
000133 | 
000134 | Thus the construction simultaneously retains the original CAR Klein factors
000135 | and the already fixed even E8 lattice cocycle.  Fields from the E8 spinor
000136 | coset and the odd integer vector coset are not thereby made mutually ordinary
000137 | local.  Their joint use belongs to the sector-changing or twisted
000138 | intertwining system described below.
000139 | 
000140 | 
000141 | 3. The cyclic lift and its forced mu4 phase
000142 | -------------------------------------------
000143 | 
000144 | Let R be the cyclic permutation
000145 | 
000146 |   R(n0,n1,...,n7)=(n7,n0,...,n6).
000147 | 
000148 | The exact ordered-CAR phase is
000149 | 
000150 |   eta_R(n)=(-1)^(n7 sum_(j<7) n_j).
000151 | 
000152 | It obeys
000153 | 
000154 |   eta_R(n+m) eps_Z(n,m)
000155 |    = eta_R(n) eta_R(m) eps_Z(Rn,Rm).
000156 | 
000157 | To extend the lift to X one must set
000158 | 
000159 |   R_hat X R_hat^(-1) = kappa X.
000160 | 
000161 | Since eta_R(1)=-1 and X^2=-E_1, covariance of X^2 forces
000162 | 
000163 |   kappa^2=-1,
000164 |   kappa in {+i,-i}.
000165 | 
000166 | These are the two mu4 representatives of the cyclic lift on the Ramond
000167 | sectors.  They will be shown below to differ only by an intertwiner-basis
000168 | gauge.  They do not constitute an additional physical choice.  This basis
000169 | gauge must still be distinguished from the two genuinely different
000170 | operators obtained by choosing the geometric Cartan lift or the standard
000171 | g-defect lift, and from any finite compiler Arf word.
000172 | 
000173 | For either kappa define
000174 | 
000175 |   eta_*(n+c0 s)=eta_R(n) kappa^c0.
000176 | 
000177 | Then
000178 | 
000179 |   eta_*(p+q) eps_*(p,q)
000180 |    = eta_*(p) eta_*(q) eps_*(Rp,Rq)
000181 | 
000182 | for all p,q in D8*.  Hence R_hat is a unitary cocycle-compatible lift on the
000183 | whole four-sector charge algebra.
000184 | 
000185 | Define the diagonal basis change
000186 | 
000187 |   U_tau |n+c0 s> = (-1)^(c0 Q(n)) |n+c0 s>.
000188 | 
000189 | It is pointwise the identity on both already supplied local subtheories:
000190 | 
000191 |   * c0=0 on the whole CAR NS theory 0+v;
000192 |   * c0=1 and Q(n) even on the E8 spinor sector s, so it is also the identity
000193 |     on the whole E8 vacuum theory 0+s.
000194 | 
000195 | Since T1 changes Q(n) by one,
000196 | 
000197 |   U_tau W_kappa U_tau^* = W_(-kappa).
000198 | 
000199 | Thus kappa=+i and kappa=-i are isomorphic descriptions of the same extension
000200 | up to the normalization of the remaining v-to-c intertwiner.  In particular,