TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/theta_lattice_closure_20261027/fields/TWO-BRANCH-VERTEX-GROUPOID.txt
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000001 | TWO BRANCH LATTICE VERTEX GROUPOID
000002 | The marked alpha transport is coherent on full vertex fields without splitting G31
000003 | 2026-10-27
000004 | 
000005 | 
000006 | VERDICT
000007 | =======
000008 | 
000009 | The required full-field lift exists on the two already constructed objects
000010 | 
000011 |   Lambda_0=Lambda,              Lambda_1=zeta_8 Lambda.
000012 | 
000013 | The smallest construction does not require a same-object lift of V with
000014 | V^2=1, does not require [V,J]=1 in the lattice central extension, and does not
000015 | split the full G31 extension.
000016 | 
000017 | Start with the existing faithful C12 lattice-vertex lift on object zero,
000018 | 
000019 |   c_0^12=1,       beta_0=c_0^9,       sigma_0=c_0^4,
000020 |   beta_0^4=1,     sigma_0^3=1,        [beta_0,sigma_0]=1.      (1)
000021 | 
000022 | Let
000023 | 
000024 |   T: V_(Lambda_0) -> V_(Lambda_1)                              (2)
000025 | 
000026 | be any unitary standard-cocycle lift of the actual lattice isometry
000027 | alpha_01=zeta_8 V.  In the branch coordinate frames the underlying integral
000028 | matrix of T is V.  Define
000029 | 
000030 |   beta_1  =T beta_0 T^(-1),
000031 |   sigma_1 =T sigma_0^(-1) T^(-1),
000032 |   alpha_01=T,
000033 |   alpha_10=beta_0 T^(-1).                                     (3)
000034 | 
000035 | Then on V_(Lambda_0) direct sum V_(Lambda_1),
000036 | 
000037 |   alpha^2=diag(beta_0,beta_1),
000038 |   alpha^8=1,
000039 |   alpha sigma alpha^(-1)=sigma^(-1).                          (4)
000040 | 
000041 | Both branches carry a faithful C12 action
000042 | 
000043 |   c_j=beta_j sigma_j,
000044 |   c_j^12=1,              c_j^4=sigma_j,       c_j^9=beta_j.   (5)
000045 | 
000046 | This closes the requested coherent two-object field family.  It does not form
000047 | a single alpha-invariant lattice VOA.  Alpha remains an arrow between the two
000048 | VOAs, exactly as required by the already proved lattice obstruction.
000049 | 
000050 | An optional stronger result also holds.  For the fixed standard cocycle there
000051 | are 256 simultaneous sign-character sections for which J,V,sigma themselves
000052 | obey the full same-frame presentation of C4 x S3.  Their lifted group has
000053 | order 24 and c=J sigma has order 12.  That stronger section is unnecessary for
000054 | (2)-(5), and every strict solution changes the zero-on-basis lift of J.  It
000055 | must therefore not replace an already frozen J vertex lift without an explicit
000056 | linearization choice.
000057 | 
000058 | 
000059 | 1. FIXED STANDARD COCYCLE
000060 | =========================
000061 | 
000062 | Let B be the integral basis produced by check_theta_lattice.py and let G be its
000063 | even unimodular Gram matrix.  The checker fixes the standard bimultiplicative
000064 | cocycle
000065 | 
000066 |   epsilon(m,n)=(-1)^(m^T F n),                                (6)
000067 | 
000068 | where
000069 | 
000070 |   F_ii=G_ii/2 mod 2,
000071 |   F_ij=G_ij mod 2 for i>j,
000072 |   F_ij=0 for i<j.
000073 | 
000074 | Thus
000075 | 
000076 |   F+F^T=G mod 2,
000077 |   epsilon(m,m)=(-1)^(<m,m>/2).                                (7)
000078 | 
000079 | No cocycle is fitted to the desired group relations.  The same rule is used
000080 | on both branch coordinate frames.
000081 | 
000082 | The mod-two calculation is a complete all-charge calculation, not a finite
000083 | shell test.  Both epsilon(m,n) and every sign carry eta_A(m) depend only on the
000084 | classes of m,n in Lambda/2Lambda.  Hence the 256 charge classes and the 65536
000085 | ordered charge pairs exhaust the cocycle identities for every m,n in Lambda.
000086 | The exact integral matrix identities A^T G A=G and the geometric relations of
000087 | J,V,sigma are checked separately over Z by check_theta_lattice.py before this
000088 | parity reduction is used.
000089 | 
000090 | For a lattice isometry A define
000091 | 
000092 |   D_A=A^T F A+F mod 2,
000093 |   q_A^0(m)=sum_(i<j) (D_A)_ij m_i m_j mod 2.                  (8)
000094 | 
000095 | The checker proves exhaustively on Lambda/2Lambda that
000096 | 
000097 |   q_A^0(m+n)+q_A^0(m)+q_A^0(n)=m^T D_A n.                    (9)
000098 | 
000099 | Consequently
000100 | 
000101 |   Ahat(e^m)=(-1)^(q_A^0(m)) e^(A m)                          (10)
000102 | 
000103 | is an automorphism of the twisted lattice group algebra.  On the full lattice
000104 | VOA it acts by
000105 | 
000106 |   Ahat[h_1(-n_1)...h_r(-n_r)e^m]
000107 |     =(Ah_1)(-n_1)...(Ah_r)(-n_r)(-1)^(q_A^0(m))e^(Am).        (11)
000108 | 
000109 | Because A is orthogonal and the phase has modulus one, (11) is unitary.  It
000110 | fixes the vacuum, preserves the conformal vector and intertwines every vertex
000111 | operator.  The calculation on the group algebra together with the ordinary
000112 | isometry relations on the Heisenberg generators therefore proves the
000113 | relations on all vertex fields, not only on the root shell.
000114 | 
000115 | Multiplication by a sign character
000116 | 
000117 |   (-1)^(t^T m),              t in (F_2)^8                     (12)
000118 | 
000119 | changes the lift section but not epsilon, A, the lattice, or the vertex
000120 | algebra.  The optional strict census enumerates exactly these 256 choices per
000121 | marked generator.
000122 | 
000123 | 
000124 | 2. WHY THE TWO OBJECT CONSTRUCTION NEEDS NO SPLITTING
000125 | =====================================================
000126 | 
000127 | The identities in (3)-(5) are formal consequences of one existing C12 lift
000128 | and one invertible VOA isomorphism T.
000129 | 
000130 | First,
000131 | 
000132 |   alpha_10 alpha_01
000133 |     =beta_0 T^(-1)T
000134 |     =beta_0,
000135 | 
000136 |   alpha_01 alpha_10
000137 |     =T beta_0 T^(-1)
000138 |     =beta_1.                                                   (13)
000139 | 
000140 | Since beta_0 and beta_1 have order four, alpha has exact order eight.
000141 | 
000142 | For the cubic mark, the first arrow obeys
000143 | 
000144 |   T sigma_0
000145 |     =sigma_1^(-1) T,                                         (14)
000146 | 
000147 | because sigma_1^(-1)=T sigma_0 T^(-1).  On the return arrow,
000148 | 
000149 |   beta_0 T^(-1) sigma_1
000150 |     =beta_0 sigma_0^(-1) T^(-1)
000151 |     =sigma_0^(-1) beta_0 T^(-1),                             (15)
000152 | 
000153 | using only [beta_0,sigma_0]=1.  Equations (14),(15) are precisely the two
000154 | components of alpha sigma alpha^(-1)=sigma^(-1).
000155 | 
000156 | Conjugation transports (1) to beta_1.  The inverse in the definition of
000157 | sigma_1 does not change its order, and beta_1 still commutes with sigma_1.
000158 | Equation (5) follows.
000159 | 
000160 | The actual geometry also matches.  In branch coordinates T has matrix V.
000161 | The return arrow has matrix
000162 | 
000163 |   J V^(-1)=J V=iV,                                           (16)
000164 | 
000165 | because the geometric matrix V is an involution.  Likewise
000166 | 
000167 |   V sigma^(-1) V^(-1)=sigma,
000168 | 
000169 | so the transported branch-one cubic has the same geometric matrix sigma.
000170 | Only its cocycle section has been transported, as it must be.
000171 | 
000172 | The exact checker deliberately uses a valid standard-cocycle lift T for which
000173 | 
000174 |   T^2 != 1,                 [T,beta_0] != 1.                  (17)
000175 | 
000176 | Nevertheless every identity (13)-(15) passes on all 256 residue fields.
000177 | Thus no hidden same-frame split is being used.
000178 | 
000179 | The choice of T is unique only up to character gauge.  If T'=chi T, transport
000180 | by chi conjugates all branch-one data beta_1,sigma_1,c_1 and the return arrow.
000181 | It produces an isomorphic two-object vertex groupoid.  Thus the field family
000182 | is canonical only up to this ordinary groupoid equivalence; the proof does not
000183 | promote one character section to a physical observable.
000184 | 
000185 | 
000186 | 3. OPTIONAL STRICT ORDER 24 SECTION
000187 | ===================================
000188 | 
000189 | For completeness, the checker also asks the stronger question posed in the
000190 | task.  For every A in {J,V,sigma} it enumerates
000191 | 
000192 |   eta_A^t(m)=(-1)^(q_A^0(m)+t^T m),       t in (F_2)^8,       (18)
000193 | 
000194 | and tests the complete relations on all m mod 2.
000195 | 
000196 | Power-normalized candidate counts are
000197 | 
000198 |   J^4=1:       256,
000199 |   V^2=1:        16,
000200 |   sigma^3=1:    16.                                          (19)