TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-compiler-closure-20261023/CUTS.txt
Original file SHA-256: e4eb9b53f944d411e887deb1783ff2d12e6a949e7550f9e6ebdd0cf77cd1ef21
Inclusive original lines: 156-252
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.

000156 |   4    = 3_1 + 1_-3,
000157 |   4bar = 3bar_-1 + 1_3,
000158 |   6    = 3bar_2 + 3_-2,
000159 |   15   = 8_0 + 1_0 + 3_4 + 3bar_-4.                            (5.2)
000160 | 
000161 | Substituting (5.2) into (5.1) gives
000162 | 
000163 |   78 = 45_0 + 1_0 + 16_-3 + 16bar_3,                           (5.3)
000164 | 
000165 | for the E6 adjoint, and the complete E6xA2 branching
000166 | 
000167 |   248 = (78,1) + (1,8) + (27,3) + (27bar,3bar),                 (5.4)
000168 | 
000169 | with
000170 | 
000171 |   27    = 1_4 + 10_-2 + 16_1,
000172 |   27bar = 1_-4 + 10_2 + 16bar_-1.                              (5.5)
000173 | 
000174 | Equations (5.1) through (5.5) are compatible term by term. In the +++ axis
000175 | chart the E6 root spaces are40 D5 roots of charge0 and two sets of16 half
000176 | roots of charges+3 and-3. The remaining162 roots split as
000177 | 
000178 |   3_(+4) + 3_(-4) + 30_(+2) + 30_(-2) + 48_(+1) + 48_(-1),
000179 | 
000180 | which are exactly the weight terms in (27,3)+(27bar,3bar). Reversing the axis
000181 | simultaneously reverses charges and exchanges barred labels.
000182 | 
000183 | The corresponding character for any E7 cut containing D5 is
000184 | 
000185 |   133 = (45,1)_0 + (1,3)_0 + (1,1)_0
000186 |         + (10,1)_2 + (10,1)_-2
000187 |         + (16,2)_1 + (16bar,2)_-1,                              (5.6)
000188 | 
000189 | under D5xA1xU1, again up to simultaneous orientation reversal. This is the
000190 | restriction of the standard E7 to D6xA1 character
000191 | 
000192 |   133=(66,1)+(1,3)+(32,2)
000193 | 
000194 | followed by D6 to D5xU1. Formula (5.6), rather than the number133 alone, is
000195 | the representation content of the six D5-compatible E7 cuts.
000196 | 
000197 | 6. The 3+2 mark and the canonical four-cut family
000198 | 
000199 | Y has support only in the D5 Cartan. It is orthogonal to every family A2
000200 | normal plane, hence lies in every one of the four D5-compatible E6 cuts. The
000201 | whole W(D3)=S4 fixes Y. Therefore Y preserves the canonical unordered four-cut
000202 | family but cannot distinguish one tetrahedral vertex. Its existing identity
000203 | 
000204 |   Tr Y^2=3(1/3)^2+2(1/2)^2=5/6
000205 | 
000206 | is unchanged by the E6 restriction.
000207 | 
000208 | 7. What the carrier numbers do and do not identify
000209 | 
000210 | There is a genuine representation bridge in (5.1): the half-root sector is
000211 | 
000212 |   128 = 16*4 + 16*4.
000213 | 
000214 | The family factor here is the four-dimensional defining representation of
000215 | A3. Correspondingly, the D5-compatible E6 cuts form a four-vertex S4 orbit.
000216 | The rank of A3 is3, but rank3 is not the multiplicity character4.
000217 | 
000218 | The tempting equality for the paired E7 closure,
000219 | 
000220 |   missing48 = 3*16,
000221 | 
000222 | fails as a representation identity. For the canonical pair with normals
000223 | alpha=e0+e1 and Jalpha=e0-e1, the missing root spaces form
000224 | 
000225 |   (12,2,2)
000226 | 
000227 | under D6xA1xA1. They are not closed as three D5 spinors. The fixed Y character
000228 | gives an immediate exact discriminator. The missing48 charge multiset is
000229 | 
000230 |   -5/6:4, -2/3:2, -1/3:12, -1/6:4, 0:4,
000231 |    1/6:4,  1/3:12,  2/3:2,  5/6:4.
000232 | 
000233 | Three copies of the two D5 chiral spinors instead give respectively
000234 | 
000235 |   -1:3, -1/3:9, -1/6:18, 0:3, 1/2:6, 2/3:9,
000236 | 
000237 | and
000238 | 
000239 |   -2/3:9, -1/2:6, 0:3, 1/6:18, 1/3:9, 1:3.
000240 | 
000241 | Neither character matches. The earlier equality16*5*3=240 remains its stated
000242 | root-count identity. It does not turn the E7 missing sector into three spinor
000243 | carriers or turn the four E6 cuts into a three-element family.
000244 | 
000245 | 8. Positive four-cut consequence inside the original (16,4) block
000246 | 
000247 | There is nevertheless a different, exact 48-dimensional module in the E6
000248 | dictionary. It must be kept separate from the missing48 sector of Section7.
000249 | 
000250 | Choose one of the two chiral half-root blocks in (5.1), and write its literal
000251 | weight basis as
000252 |