TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-source-reconstruction-20261024/SCALAR-QUARTER-REPRODUCTION.txt
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000081 | 3. Full field action of the uniform scalar monodromy
000082 | 
000083 | On the fixed E8 lattice VOA define the toral automorphism
000084 | 
000085 |     g_q=exp(2pi i q(0)),
000086 |     g_q(oscillator e^beta)=exp(2pi i(q,beta)) oscillator e^beta.
000087 | 
000088 | It preserves the ACTUAL cocycle product because the phase is a lattice
000089 | character. All Heisenberg oscillators are fixed. Since 2q=s lies in the
000090 | self-dual lattice E8, g_q^2=1 on EVERY lattice state, not merely on roots.
000091 | It is nontrivial, so its exact order is2. On elementary odd CAR weights e_j
000092 | the original scalar character is i of order4; those weights do not belong
000093 | to the bosonic E8 vacuum lattice. Keeping these carriers distinct is the
000094 | first necessary comparison with a native clock.
000095 | 
000096 | The fixed roots satisfy (s,alpha) even. Their scalar products are
000097 | 
000098 |     D8 roots:     -1:28, 0:56, 1:28;
000099 |     half roots:   -2:1, -1:28, 0:70, 1:28, 2:1.
000100 | 
000101 | Therefore the fixed root set is exactly
000102 | 
000103 |     Phi(E7(s)) disjoint-union {s,-s},
000104 |     |Phi(E7(s))|=126,     |fixed roots|=128.              (3.1)
000105 | 
000106 | This is an ACTUAL E7 connection: E7(s) centralizes the root sl2(s).
000107 | Adding the eight Cartan currents gives the fixed Lie algebra e7+sl2,
000108 | dimension136, and the complementary112 currents, the (56,2) component.
000109 | No claim that this is E7 alone is made.
000110 | 
000111 | There is also an exact statement for the WHOLE fixed VOA. Its charge
000112 | lattice M={beta in E8:(s,beta) even} has index2 and determinant4.
000113 | The root lattice E7(s)+A1(s) is inside M and also has determinant2*2=4.
000114 | Hence they are equal, giving
000115 | 
000116 |     (V_E8)^(g_q)=V_(E7+A1).                             (3.2)
000117 | 
000118 | 4. All phase-compatible extensions over the fixed D8
000119 | 
000120 | The uniform scalar monodromy restricted to V_D8 has two, and only two,
000121 | extensions to this fixed V_E8. The second is gamma g_q, where
000122 | 
000123 |     gamma=exp(2pi i e1(0)),
000124 |     gamma=+1 on V_D8,     gamma=-1 on V_(D8+s).
000125 | 
000126 | For completeness: any two extensions differ by an automorphism fixing
000127 | V_D8 pointwise. Its action on the irreducible spinor module is scalar by
000128 | Schur's lemma. The nonzero spinor-times-spinor product into V_D8 forces
000129 | that scalar's square to1. Thus Aut_(V_D8)(V_E8)={1,gamma}. Both displayed
000130 | extensions exist, commute with gamma, and square to1. This is a complete
000131 | classification in the stated fixed-subnet category, independent of the
000132 | earlier nonsplit G31 lattice-lift problem.
000133 | 
000134 | The exact weight-one spectra, in eigenvalue order(1,i,-1,-i), are
000135 | 
000136 |     g_q:             (136,0,112,0), fixed E7+A1;
000137 |     gamma g_q:       (120,0,128,0), fixed D8';
000138 |     original Jhat:   (60,64,60,64).                     (4.1)
000139 | 
000140 | For gamma g_q, the fixed lattice has determinant4 and112 roots with actual
000141 | rank8 D8 Cartan matrix, giving the indicated full lattice VOA. The native
000142 | Jhat result uses its original cocycle and sixty four-cycles on roots, with
000143 | four+i and four-i Cartans. Thus neither extension is Jhat, even under
000144 | conjugation of the full weight-one space. No discarded field phase can
000145 | restore order four while keeping that D8 restriction fixed.
000146 | 
000147 | This excludes that particular scalar-character/native-J identification.
000148 | It does not identify toral monodromy with the flux-register shift or a
000149 | physical clock, nor exclude some different originally derived dictionary.
000150 |