TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/marked-seam-monodromy-20261023/THEORY.txt
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Inclusive original lines: 252-284
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000252 | 5. Actual clock/reflection subgroup: full factor table and its class
000253 | 
000254 | Let R on C4 be diag(-1,1,1,1), the native complex reflection in the E8
000255 | root line(1+i)e0. Its real8matrix is diag(-1,-1,1,1,1,1,1,1). It is NOT a
000256 | real codimension1 Weyl reflection. It preserves L, commutes with J, and
000257 | G0=<J,R> is C4 times C2 of order8. Formula(2.1) evaluated in the actual
000258 | ordered integral basis gives the complete character-valued factor table
000259 | in results.json. Its class in H2(G0,A) is ZERO, although its normalized
000260 | generator section has nonzero factors.
000261 | 
000262 | For a self-contained certificate, list the elements as
000263 | (1,R,J,JR,J^2,J^2R,J^3,J^3R). Characters use binary coefficients in the
000264 | original integral basis, least significant bit first. The table c is
000265 |   0 0 0 0 0 0 0 0
000266 |   0 1 1 0 1 0 0 1
000267 |   0 0 1 0 0 0 1 0
000268 |   0 1 1 1 1 0 0 0
000269 |   0 1 0 1 0 1 0 1
000270 |   0 0 1 1 1 1 0 0
000271 |   0 1 1 1 0 1 1 1
000272 |   0 0 1 0 1 1 0 1.
000273 | An explicit trivializing cochain in the same order is
000274 |   b=(0,86,6,80,136,223,142,217).
000275 | Direct integer-bit multiplication gives
000276 |   b(g)+g.b(h)+c(g,h)=b(gh)                             (5.1)
000277 | on all64pairs, and the twisted cocycle law on all512triples. Alternatively
000278 | correcting the J generator by character6 and R by86 gives commuting
000279 | generators of orders4and2; their powers construct b. Thus(5.1) proves
000280 | triviality of the whole G0 class, not just separate cyclic restrictions.
000281 | This does not decide splitting on the full G31 or W(E8). The genuinely
000282 | nonzero CENTRAL class in TheoremA has a different quotient and coefficient
000283 | module; zero of this G0 class does not contradict it.
000284 |