TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-compiler-closure-20261023/CUTS.txt
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Inclusive original lines: 219-324
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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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 | 
000253 |   B = {(s,t): s is one of the16 D5 half-spin weights,
000254 |               t is one of the four even-parity family sign triples}.
000255 | 
000256 | Thus B is the existing (16,4) root-current block and dim B=64. Let
000257 | 
000258 |   T={(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)}.
000259 | 
000260 | The four D5-compatible E6 cuts are naturally indexed by t in T. On the root
000261 | current basis define
000262 | 
000263 |   P_t E_(s,u) = delta_(t,u) E_(s,u),       Q_t=I_B-P_t.          (8.1)
000264 | 
000265 | These are explicit coordinate projectors, not fitted embeddings. They satisfy
000266 | 
000267 |   rank P_t=16,       rank Q_t=48,
000268 |   P_t P_u=delta_(t,u)P_t,       sum_(t in T) P_t=I_B.           (8.2)
000269 | 
000270 | For the cut indexed by t, the E6 adjoint contains exactly P_t B. The
000271 | complement is
000272 | 
000273 |   Q_t B = 16 tensor 3,                                           (8.3)
000274 | 
000275 | where the A2 is the normal root subsystem of that same cut. W(D3)=S4
000276 | conjugates these projectors by
000277 | 
000278 |   w P_t w^-1 = P_(wt).
000279 | 
000280 | Hence the whole four-projector resolution is canonical, while no individual
000281 | projector is canonical.
000282 | 
000283 | The module action in (8.3) is literal in the native root chart. D5 root
000284 | addition changes only s and gives480 directed nonzero root-step transitions on
000285 | Q_t B. A2 root addition changes only u and gives96 directed transitions. The
000286 | three family weights form the complete A2 triplet graph, with six directed
000287 | root steps for every fixed D5 spinor weight. A2 has no transition out of the
000288 | family singlet P_t B within B. This proves the tensor-product action rather
000289 | than inferring it from dimension48.
000290 | 
000291 | The positive off-block character is the actual term
000292 | 
000293 |   (27,3) = (16,3)_1 + (10,3)_-2 + (1,3)_4,                    (8.4)
000294 | 
000295 | so its81 root spaces split exactly as
000296 | 
000297 |   81 = 48 + 30 + 3.                                            (8.5)
000298 | 
000299 | For every one of the four cuts, the script constructs the three explicit root
000300 | subsets in (8.5). Their family weight sets are three isomorphic A2 triplets,
000301 | and the native A2 roots act on each. The conjugate (27bar,3bar) block is the
000302 | oppositely oriented statement.
000303 | 
000304 | The existing seed ratio now has an exact representation dictionary. Taking
000305 | the ordinary trace on the same root-current basis gives, for every cut,
000306 | 
000307 |   Tr(I_B)/Tr(Q_t) = 64/48 = 4/3
000308 |                    = (1/(6 pi))/(1/(8 pi)) = phi_base/c3.       (8.6)
000309 | 
000310 | This equality is cut independent because S4 conjugates all four Q_t. It is an
000311 | identity between the existing coefficient and the actual ranks of the native
000312 | projectors. It does not require a preferred tetrahedral vertex.
000313 | 
000314 | The same weight basis also retains the original glue norms pointwise. Every
000315 | half-root r in B, and hence every r in Q_t B, obeys
000316 | 
000317 |   ||projection_D5 r||^2 = 5*(1/2)^2 = 5/4,
000318 |   ||projection_A3 r||^2 = 3*(1/2)^2 = 3/4,
000319 |   ||r||^2 = 2.                                                  (8.7)
000320 | 
000321 | Equations (8.3), (8.6), and (8.7) give the requested single dictionary for
000322 | the actual48 module, the coefficient4/3, and the D5 discriminant norm5/4.
000323 | The actual48 here is the off-block lattice-current module (16,3). The E7
000324 | closure sector of Section7 remains the different module (12,2,2).