TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/THREE-FAMILY-ROOT-SEED-CLOSURE.txt Original file SHA-256: 3e3177272e48de3dcf65ae03d0739b6746b6446a4b5292d7dc4b62af6f694d1c Inclusive original lines: 1-193 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 | THREE FAMILY C/X ROOT SEED CLOSES TO D8, NOT E8 000002 | Exact restricted-source addendum, 30 October 2026. 000003 | No Atlas mutation and no change to earlier reports. 000004 | 000005 | 000006 | VERDICT 000007 | 000008 | Restrict the INITIAL source seed to 000009 | 000010 | C_a and X_i,a, a in F={1,2,3}, i=1,...,5, 000011 | 000012 | and include their compact adjoint roots. This gives 18 positive plus 18 000013 | opposite roots. Its exact recursive Chevalley closure is 000014 | 000015 | 36 -> 92 -> 112 roots, 000016 | 000017 | has rank eight, and is bracket closed. It is the D8 root subsystem of E8, 000018 | not the full 240-root E8 system. 000019 | 000020 | The restricted seed never generates any of the thirty missing positive 000021 | 10 tensor F spinor roots, nor any of their thirty adjoints. Their closure 000022 | depth is therefore infinite or unreachable, not two. Depth two applied 000023 | only when the omitted fourth family was present in the initial C4 plus X20 000024 | seed. 000025 | 000026 | Of the sixty roots in the full (10,6) target, exactly thirty occur at 000027 | depth one. The complementary thirty are unreachable. Thus the three 000028 | family seed does not close the carrier factor two found in 000029 | CARRIER-HODGE-SPAN-CORRECTION.txt. 000030 | 000031 | 000032 | 1. EXACT SEED AND CLOSURE 000033 | 000034 | The positive seed consists of 000035 | 000036 | three C roots, 000037 | fifteen X roots, 000038 | 000039 | with family labels 1,2,3. Adding their compact opposites gives 36 roots. 000040 | At the first recursive bracket step 56 new roots appear. At the second 000041 | step 20 further roots appear. A third step adds none: 000042 | 000043 | depth 0: 36, 000044 | depth 1: 56 new, 92 total, 000045 | depth 2: 20 new, 112 total, 000046 | depth 3: 0 new. (1) 000047 | 000048 | The 112 roots span rank eight and are closed under every actual Chevalley 000049 | bracket. A simply laced full-rank closed subsystem with this root count is 000050 | the actual D8 subsystem. Its complement in E8 has 128 roots. 000051 | 000052 | The identification is also checked in an explicit Weyl frame. For the 000053 | regular chamber vector 000054 | 000055 | (1,2,4,8,16,32,64,128) 000056 | 000057 | the eight simple roots are 000058 | 000059 | (-2, 2, 0, 0, 0, 0, 0, 0), 000060 | ( 0,-2, 2, 0, 0, 0, 0, 0), 000061 | ( 0, 0,-2, 2, 0, 0, 0, 0), 000062 | ( 0, 0, 0,-2, 2, 0, 0, 0), 000063 | ( 0, 0, 0, 0,-2, 2, 0, 0), 000064 | ( 0, 0, 0, 0, 0,-2, 2, 0), 000065 | ( 1,-1,-1,-1,-1,-1,-1, 1), 000066 | ( 1, 1, 1, 1, 1,-1,-1, 1). (1b) 000067 | 000068 | Their Cartan determinant is four. Their Dynkin graph is a D8 chain with 000069 | one trivalent node and degree sequence 000070 | 000071 | 1,1,1,2,2,2,2,3. 000072 | 000073 | This is therefore a nonstandard E8 coordinate frame for the standard D8 000074 | root subsystem, not an E7 or E6 closure. 000075 | 000076 | The twenty depth-two roots have a precise type. They are the ten omitted 000077 | three-minus Spin(10) weights with the excluded O family label, together 000078 | with their ten compact adjoints: 000079 | 000080 | depth-two sector = 10 tensor O plus dagger. (1a) 000081 | 000082 | Thus the recursion does generate a new carrier ten, but on the excluded 000083 | family line. It does not move that ten onto any of the three retained F 000084 | labels. 000085 | 000086 | 000087 | 2. MINIMAL ESCAPE FROM D8 BACK TO E8 000088 | 000089 | There are six omitted positive currents on family label 0: 000090 | 000091 | C_0, X_0,0, X_1,0, X_2,0, X_3,0, X_4,0. 000092 | 000093 | Add any ONE of these positive root spaces to the closed D8 algebra. Exact 000094 | recursive closure gives 000095 | 000096 | 113 -> 141 -> 211 -> 239 -> 240 roots. (2) 000097 | 000098 | If the compact source convention adds that current together with its 000099 | adjoint, the shorter closure is 000100 | 000101 | 114 -> 170 -> 240 roots. (3) 000102 | 000103 | All six candidates give the same counts. Since the unextended seed stops 000104 | at 112, one omitted positive current is minimal in the complex root-space 000105 | sense, and one root-adjoint pair is minimal in the compact source sense. 000106 | 000107 | Equations (2) and (3) do not make that omitted current a word in the 000108 | three-family seed. They identify the smallest additional current-algebra 000109 | input that leaves the D8 frame. A geometric source must still supply or 000110 | access it. 000111 | 000112 | 000113 | 3. THE MISSING SPINOR COMPONENTS REMAIN OUTSIDE 000114 | 000115 | For each retained family label, the odd Spin(10) half-spinor has 000116 | 000117 | 16=1+5+10 000118 | 000119 | weights. C and X provide the first six. The ten three-minus components 000120 | are absent. Across the three family labels this is thirty positive roots, 000121 | plus thirty compact adjoints. 000122 | 000123 | The exact recursive closure contains none of these sixty roots. Hence 000124 | 000125 | depth(missing 10 tensor F)=unreachable. (4) 000126 | 000127 | This differs materially from the full four-family seed. With C4 plus X20 000128 | and all adjoints, the same forty positive missing components over four 000129 | family labels occur at closure depth two and the closure reaches E8. The 000130 | omitted family label is therefore algebraically essential for that 000131 | generation route, while section 2 shows that one current pair from it is 000132 | already sufficient. 000133 | 000134 | 000135 | 4. ONLY HALF OF THE (10,6) TARGET IS REACHED 000136 | 000137 | The full E8 root realization contains sixty roots of type (10,6), namely 000138 | one nonzero D5 vector coordinate and one nonzero A3 vector coordinate. 000139 | The restricted seed generates thirty of them. Every generated target root 000140 | appears already at depth one through a direct mixed bracket such as 000141 | 000142 | [C_1,X_i,2], [C_2,X_i,3], [C_3,X_i,1] 000143 | 000144 | and their ordered or adjoint variants. The remaining thirty target roots 000145 | are absent from the final D8 subsystem. Thus 000146 | 000147 | target depths: 30 at depth one, 30 unreachable. (5) 000148 | 000149 | The result JSON contains exact nested bracket certificates for three 000150 | generated target roots and two additional depth-two D8 roots. Every leaf 000151 | is one of the 36 stated seed roots, and every internal coefficient is read 000152 | from the actual Chevalley bracket table. The depth-two examples explicitly 000153 | exhibit the 10 tensor O sector in (1a). 000154 | 000155 | 000156 | 5. PRECISE SCOPE 000157 | 000158 | This is a zero-mode current-algebra closure. It proves that the restricted 000159 | three-family C/X current seed does not manufacture its missing carrier 000160 | components as Lie words. 000161 | 000162 | It does not contradict the following separately proved statements: 000163 | 000164 | * the family Hodge reality is exact on the full family six; 000165 | * the full four-family C4 plus X20 source and its adjoints generate E8; 000166 | * an actually supplied complete 16 tensor F input brackets onto the full 000167 | sixty-dimensional compact real (10,6). 000168 | 000169 | Those statements use a larger input than the seed tested here. They do not 000170 | show that the geometric q/p source accesses that larger input. 000171 | 000172 | No affine descendant grade, current mode, state preparation, physical 000173 | time, finite Phi consumer or Standard Model Higgs identification follows 000174 | from (1) through (5). The remaining source question is still an explicit 000175 | arrow from the geometric three-family fields into the additional current 000176 | components. The restricted Lie closure itself does not provide it. 000177 | 000178 | 000179 | REPRODUCTION 000180 | 000181 | python3 aps_source_review/check_three_family_root_seed.py \ 000182 | aps_source_review/three-family-root-seed-results.json 000183 | 000184 | The normal and optimized Python runs produce byte-identical JSON. 000185 | 000186 | 000187 | FILES 000188 | 000189 | aps_source_review/check_three_family_root_seed.py 000190 | aps_source_review/three-family-root-seed-results.json 000191 | aps_source_review/ROOT-CARRIER-HODGE-SPAN-CHECK.json 000192 | tfpt_explorer/current_block_geometry.py 000193 |