TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/seam-joint-ward-state-20261014/PROOF.txt
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001003 | 21. COMMON GLOBAL OPERATOR FACTORIZATION WITH THE ACTUAL CAR KLEIN FACTORS
001004 | 
001005 | The previously open common-lift existence question can be closed on the
001006 | even D8 sector. This uses the stipulated E8 extension; it does NOT derive
001007 | that extension or its lift selection from P1/P2. R_c below is an adjusted
001008 | lift of rho and is not silently identified with the fixed Tits R of20.
001009 | 
001010 | Fix the actual ordered source CAR cocycle on Z8:
001011 | 
001012 |   eps_Z(n,m)=(-1)^(sum_(i>j) n_i m_j).
001013 | 
001014 | Its commutator is(-1)^(Q(n)Q(m)-n.m), which reduces to the canonical
001015 | even-lattice commutator(-1)^(n.m) on D8. Write A(n)=sum(first4 n),
001016 | B(n)=sum(last4 n). The actual channel permutation lift has
001017 | 
001018 |   P_CAR|n>=eta(n)|Pn>,  eta(n)=(-1)^(A(n)B(n)).
001019 | 
001020 | On D8, A=B modulo2, so eta(n)=(-1)^A(n); also eps_Z(Pn,Pm)=eps_Z(n,m).
001021 | These are identities for all lattice charges, not just112 tested roots.
001022 | For E_t the group-algebra charge shift, conjugation by E_a on an odd
001023 | source fermion of charge e_j gives the Klein sign(-1)^a_j and its channel
001024 | mode carry. Define the explicit torus operator Z_a=exp(i pi a.H_0).
001025 | It contributes the same Klein sign, so the signs cancel in Z_a E_a.
001026 | Consequently the actual sign-free source mode shift satisfies
001027 | 
001028 |   W4=Z_a E_a P_CAR
001029 | 
001030 | up to its global phase, fixed by W4|0>=|a>. Omitting Z_a would fail on
001031 | cross-block fermion signs. This is the extra information missing from
001032 | the charge/energy comparison.
001033 | 
001034 | Here is an explicit common cocycle extension, not an unspecified matching
001035 | assumption. Adjoin X=E_s to the exact D8 source group algebra with relations
001036 | 
001037 |   X E_d X^-1=(-1)^(s.d) E_d,   X^2=-E_q,   d in D8.
001038 | 
001039 | They are consistent: s.d is integral, q=2s lies in D8, E_q commutes with
001040 | D8, and X commutes with E_q. Every extended charge has unique form
001041 | p=d+c s, d in D8, c in{0,1}. In the section E_p=E_d X^c its cocycle is
001042 | 
001043 |   eps_L(d+c s,e+f s)=(-1)^(c s.e+c f)
001044 |                     eps_Z(d,e) eps_Z(d+e,c f q).
001045 | 
001046 | Group multiplication proves associativity for arbitrary charges. Its
001047 | commutator is(-1)^(p.p'), and the restriction to D8 is exactly eps_Z.
001048 | This is the even E8 lattice cocycle, with no simultaneous ordinary-local
001049 | inclusion of the odd Z8 vector fields.
001050 | 
001051 | Extend P_CAR by P_hat X P_hat^-1=kappa X. Both kappa=+1 and-1 are allowed:
001052 | Ps=s and eta(q)=1, so all displayed relations are preserved and P_hat^2=I.
001053 | The original even source does not select between these two extensions.
001054 | Choose kappa=-1 for an explicit convention, and define
001055 | 
001056 |   R_c=Z_a E_r0 P_hat,   U_s=-X.
001057 | 
001058 | Since r0=s-Pa, X E_r0=-E_a in the displayed cocycle. Since a.s=2,
001059 | Z_a commutes with X. Therefore exactly
001060 | 
001061 |   U_s R_c=Z_a E_a P_hat=W4  on the original even D8 Fock sector.
001062 | 
001063 | This includes every oscillator and the actual ordered Klein factors.
001064 | On all D8 roots b it gives chi_Rc(b)=(-1)^(r0.b), while the X factor gives
001065 | (-1)^(s.Pb); their product is the actual source sign(-1)^(a.b).
001066 | The root-mode shifts are respectively m+r0.b and m-s.Pb, so their sum
001067 | is m-Pa.b, precisely the source current shift. Thus agreement includes
001068 | the complete current algebra and its central term, not only states.
001069 | 
001070 | The action of R_c on charges is rho; its oscillators transform by P.
001071 | Hence [Q,R_c]=[K,R_c]=0 and the unchanged quarter transition is retained.
001072 | In the explicit convention eta(r0)=-1, eps_L(r0,-r0)=-1, and
001073 | Z_a Z_Pa=exp(i pi Q)=I on Ls; it follows that R_c^2=I. Other kappa/phase
001074 | conventions can be normalized with opposite global phases in R_c,U_s
001075 | without changing their product. The independent unitary-current proof
001076 | in20 and this calibrated construction realize the same affine action,
001077 | but their lifts can differ by a torus character.
001078 | 
001079 | Precise scope: W4 initially acts on the entire raw CAR space, but the
001080 | displayed common equality is on its even D8 subspace and its chosen E8
001081 | completion. Odd vector fields require the spin/defect-module transport
001082 | or the full complement construction of18. There exp(i pi Q)=-I; an
001083 | ordinary common local CAR/E8 field algebra has not been constructed.
001084 | U_s and R_c are global charge transports, not local vertex insertions.
001085 | Since R_c commutes with the retained K, this still supplies no autonomous
001086 | memory transition/event law. The E8 extension and its two P-lift choices
001087 | are explicit premises, not a source selection from P1/P2.