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: 1-155 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. 000001 | MATH.SEAM.ESERIES.CUTS.04 000002 | 23 October 2026 000003 | 000004 | Exact compatibility dictionary for the native D5+A3 compiler split 000005 | 000006 | 1. Object class, markings, and equivalence 000007 | 000008 | Let L be the existing E8 lattice in its standard R8 chart, Phi its240 000009 | norm-two roots, and 000010 | 000011 | J(x0,x1,x2,x3,x4,x5,x6,x7) 000012 | =(-x1,x0,-x3,x2,-x5,x4,-x7,x6). 000013 | 000014 | The fixed compiler subsystem is 000015 | 000016 | D5 = {roots supported on coordinates0,...,4}, 000017 | D3 = {roots supported on coordinates5,6,7} = A3. 000018 | 000019 | Its Weyl group is 000020 | 000021 | H=W(D5)xW(D3), |H|=1920*24=46080. 000022 | 000023 | All equivalence statements below use this fixed chart and the actual H 000024 | action. J is not changed or conjugated to preserve the split. An E7 cut is 000025 | 000026 | C_alpha = Phi intersect alpha-perp, 000027 | 000028 | where alpha is a root and alpha and -alpha define the same cut. An E6 cut is 000029 | 000030 | C_A = Phi intersect span(A)-perp, 000031 | 000032 | where A is a regular A2 root subsystem. The A2 subsystem, rather than a 000033 | chosen ordered pair of simple roots, is the marking. There are1120 such A2 000034 | subsystems and therefore1120 E6 cuts. 000035 | 000036 | The original 3+2 mark is retained throughout: 000037 | 000038 | Y=(-1/3,-1/3,-1/3,1/2,1/2,0,0,0), (Y,Y)=5/6. 000039 | 000040 | 2. Complete E7 dictionary relative to D5+D3 000041 | 000042 | The120 root-normal E7 cuts form four H orbits. The table records the location 000043 | of the normal root line and the root subsystem left inside each fixed factor. 000044 | 000045 | normal line number C_alpha intersect D5 C_alpha intersect D3 000046 | ------------------------------------------------------------------ 000047 | D5 20 A3+A1 A3 000048 | D3 6 D5 A1 000049 | mixed 30 D4 A1+A1 000050 | half-root 64 A4 A2 000051 | 000052 | Here mixed means an integer root with one nonzero coordinate in each fixed 000053 | block. Half-root means one of the128 spinorial E8 roots. 000054 | 000055 | For every alpha, (alpha,J alpha)=0. J pairs the120 cuts into60 unordered 000056 | pairs. Every intersection is exactly D6 with60 roots: 000057 | 000058 | C_alpha intersect C_(J alpha) = D6. 000059 | 000060 | Exactly six E7 cuts contain the whole original D5. Their normal root lines 000061 | are the six root lines of D3. W(D3)=S4 acts transitively on them. They are the 000062 | six weights of the family representation6=exterior^2(4), viewed as root 000063 | lines. Thus the fixed split canonically supplies an unordered six-cut family, 000064 | but no distinguished member. 000065 | 000066 | J does not preserve this six-cut family. Only the two normal lines supported 000067 | on coordinates6,7 form one internal J pair. The other four cuts are paired 000068 | with mixed-normal E7 cuts. Their pair intersections are still D6. 000069 | 000070 | Proof of the four rows. For a normal root in D5, the perpendicular roots in 000071 | D5 are D3+A1, while the full family D3 remains. The analogous statement with 000072 | the factors exchanged gives the D3 row. A mixed normal removes one coordinate 000073 | from each block, leaving D4 and D2=A1+A1. A half-root selects half the roots 000074 | on every coordinate pair, giving A4 inside D5 and A2 inside D3. The four 000075 | normal-root sectors have20+6+30+64=120 lines and H is transitive on each, so 000076 | the list is complete. 000077 | 000078 | 3. Complete E6 dictionary relative to D5+D3 000079 | 000080 | Each row below is one complete H orbit. The normal column gives the sector 000081 | composition of the six A2 roots. The cut columns give C_A intersect D5 and 000082 | C_A intersect D3. The last two columns refine the orbit by the actual 000083 | intersection C_A intersect C_(JA). Those counts are counts of ordered cuts. 000084 | 000085 | A2 normal roots orbit stabilizer D5 cut D3 cut A2+A2 D4 000086 | ----------------------------------------------------------------- 000087 | 6 D5 40 1152 A1+A1 A3 16 24 000088 | 2 D5+4 mixed 120 384 A3 A1+A1 80 40 000089 | 2 D5+4 half 320 144 A2+A1 A2 192 128 000090 | 2 D3+4 mixed 60 768 D4 - 32 28 000091 | 4 half+2 mixed 480 96 A3 A1 256 224 000092 | 6 D3 4 11520 D5 - 0 4 000093 | 2 D3+4 half 96 480 A4 A1 64 32 000094 | ----------------------------------------------------------------- 000095 | total 1120 640 480 000096 | 000097 | Consequently J has no fixed regular E6 cut. It gives560 unordered pairs. 000098 | Their intersections are320 pairs of type A2+A2 and240 pairs of type D4. 000099 | This identifies the intersection root systems that were left unnamed in the 000100 | preceding contract's12-root and24-root census. 000101 | 000102 | Completeness proof. Every A2 is the six-root closure 000103 | 000104 | {+/-alpha,+/-beta,+/-(alpha+beta)}, (alpha,beta)=-1. 000105 | 000106 | Exact enumeration in the native shell gives1120 distinct sets. Reflections in 000107 | the five standard D5 simple roots and the three standard D3 simple roots 000108 | generate H. Breadth-first orbit closure gives exactly the seven rows above. 000109 | Within each row the normal-sector composition and both fixed-factor root types 000110 | are constant. Orbit-stabilizer gives the displayed stabilizer orders. Their 000111 | sum is1120. Hence this is a classification under the stated equivalence, not 000112 | only a list of root-count coincidences. 000113 | 000114 | 4. The E6 cuts that preserve the original D5 carrier 000115 | 000116 | C_A contains the original D5 exactly when A is an A2 subsystem of the family 000117 | D3. The A3 root system contains exactly four A2 subsystems. Therefore exactly 000118 | four of the1120 E6 cuts contain D5. 000119 | 000120 | Their A2 planes have the four projective normal axes 000121 | 000122 | (1,1,1), (1,1,-1), (1,-1,1), (1,-1,-1) 000123 | 000124 | in the last three coordinates. They are the vertices of the family 000125 | tetrahedron. W(D3)=S4 acts transitively, with stabilizer S3. Thus the fixed 000126 | D5+D3 split canonically determines the unordered set of four E6 cuts. It does 000127 | not determine one cut without an additional root, A2 plane, or tetrahedral 000128 | vertex marking. 000129 | 000130 | J sends every one of these four cuts outside the D5-containing family because 000131 | it rotates coordinate4 into coordinate5. Each cut is paired with a distinct 000132 | member of the 2D3+4mixed orbit. For all four pairs, 000133 | 000134 | C_A intersect C_(JA) = D4 on coordinates0,1,2,3. 000135 | 000136 | Thus the four J pairs have a common carrier intersection, but neither member 000137 | selection nor a J-invariant E6 reduction follows from the fixed split. 000138 | 000139 | 5. Exact character dictionary 000140 | 000141 | The native root partition is the actual D5xA3 restriction of the E8 adjoint: 000142 | 000143 | 248 = (45,1) + (1,15) + (10,6) + (16,4) + (16bar,4bar). (5.1) 000144 | 000145 | This is a weight identity in the fixed coordinates: 000146 | 000147 | * the40 D5 roots plus its five Cartans give(45,1); 000148 | * the12 D3 roots plus its three Cartans give(1,15); 000149 | * the60 mixed roots are the full Cartesian weight product10x6; 000150 | * the128 half-roots split into the two parity-correlated products 000151 | 16x4 and16barx4bar, each with64 weights. 000152 | 000153 | For any one of the four D5-compatible E6 cuts, choose its A2 orientation so 000154 | that the standard family branching is 000155 |