TFPT scientific source excerpt Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/CARRIER-HODGE-SPAN-CORRECTION.txt Original file SHA-256: 9752d2b4b96b248dbb9a08e8403d85fdb07b54eacdb86d32a25114d0a759d22e Inclusive original lines: 1-200 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 | CARRIER AND HODGE SPAN CORRECTION 000002 | Independent correction to COMPACT-REAL-SIX-ADJOINT-CLOSURE-ADDENDUM, 000003 | 30 October 2026. The earlier addendum remains byte unchanged. 000004 | No Atlas mutation and no new spinor, field or source insertion is used. 000005 | 000006 | 000007 | VERDICT 000008 | 000009 | The family Hodge statement is exact, but it is not by itself a proof that 000010 | the restricted C/X input spans the full compact real (10,6). The compact 000011 | adjoint complements the family pair AND reverses the D5 vector weight. 000012 | Consequently the directly checked C/X triplet bracket has real rank 30, 000013 | whereas the full compact (10,6) has real dimension 60. 000014 | 000015 | The missing factor two is supplied algebraically if the input really is a 000016 | complete 16 tensor F. The ten Spin(10) components omitted by the C/X start 000017 | then produce the opposite five D5 vector weights. Those ten components are 000018 | indeed generated at depth two by the previously checked E8 Lie closure of 000019 | the original 24 C/X currents and their adjoints. They are not members of 000020 | the initial C/X input, however. Thus the Lie closure proves availability 000021 | inside the full current algebra, not that the geometric q/p product or a 000022 | given source insertion already excites the complete spinor input. 000023 | 000024 | Accordingly, every claim of a full 60 dimensional real (10,6) from the 000025 | restricted C/X tensor Lambda^2 F bracket must be read as conditional on 000026 | this full carrier completion. No additional O family fermion is required 000027 | once a complete 16 tensor F input is actually supplied. The verified C/X 000028 | subset alone is insufficient. 000029 | 000030 | 000031 | 1. EXACT TARGET DIMENSIONS 000032 | 000033 | For the complexified subgroup D5 times A3, 000034 | 000035 | (10,6)=10 tensor Lambda^2(4) 000036 | 000037 | has complex dimension 60. Its compact real form also has real dimension 000038 | 60. Choose family label 0 for O and labels 1,2,3 for F. Then 000039 | 000040 | dim_C Lambda^2 F=3, 000041 | dim_C O wedge F=3. (1) 000042 | 000043 | The antilinear family Hodge operator exchanges these two summands. On the 000044 | D5 vector it simultaneously exchanges the five negative and five positive 000045 | weights. Therefore the real completion of a complex subspace W in 000046 | 000047 | W5_negative tensor Lambda^2 F 000048 | 000049 | has real dimension twice dim_C W. It does not retain one D5 weight while 000050 | filling an independent six dimensional family fiber. 000051 | 000052 | 000053 | 2. THE ACTUAL C/X RESTRICTION HAS RANK 15 COMPLEX, 30 REAL 000054 | 000055 | The starting current dictionary is 000056 | 000057 | C_a in 1 tensor 4, 000058 | X_i,a in bar5 tensor 4, 000059 | [C_a,X_i,b]=Y_i,(a wedge b). (2) 000060 | 000061 | Restrict a,b to the triplet labels 1,2,3. There are five carrier labels 000062 | i and three unordered family pairs 12,23,31. The exact Chevalley root 000063 | enumeration gives 000064 | 000065 | 5 times 3 = 15 complex root directions. (3) 000066 | 000067 | All 15 have the negative D5 vector sign. Their compact adjoints have the 000068 | positive D5 sign and the complementary family pairs 03,01,02. The root 000069 | spaces and their opposites are independent, so selfadjoint combinations 000070 | have 000071 | 000072 | rank_R(C/X triplet bracket plus dagger)=2 times 15=30. (4) 000073 | 000074 | This is the exact span generated by 000075 | 000076 | h=sum_i,ab z_i,ab Y_i,ab, ab in {12,23,31}, 000077 | H=h+h^dagger. 000078 | 000079 | It is a valid compact real 30 dimensional subspace of (10,6), not the 000080 | whole 60 dimensional sector. Suppressing the carrier index made the 000081 | earlier family-only description look like a full real six for every 000082 | fixed carrier. That suppression is invalid because dagger changes the 000083 | carrier weight at the same time. 000084 | 000085 | 000086 | 3. A COMPLETE 16 TENSOR F DOES GIVE THE FULL 60 000087 | 000088 | Use the actual odd-minus Spin(10) half-spinor weights in doubled signs. 000089 | Their census is 000090 | 000091 | one weight with five minus signs, 000092 | five weights with one minus sign, 000093 | ten weights with three minus signs. (5) 000094 | 000095 | These are the U5 components 1, bar5 and 10, up to the retained naming 000096 | convention. C and X use only the first two rows of (5), namely six of the 000097 | sixteen spinor weights. The ten three-minus weights are absent from the 000098 | starting C/X input. 000099 | 000100 | For two COMPLETE 16 tensor F inputs, the exact root bracket over the 000101 | three family pairs has 000102 | 000103 | 15 target roots with negative D5 vector sign, 000104 | 15 target roots with positive D5 vector sign. (6) 000105 | 000106 | Thus its holomorphic target has complex rank 30. Its compact adjoint has 000107 | the complementary family pairs and gives 000108 | 000109 | rank_R(full 16 tensor Lambda^2 F plus dagger) 000110 | =2 times 30=60. (7) 000111 | 000112 | Equations (6) and (7) are a direct root enumeration, not a dimension 000113 | match. The explicit Spin(10) gamma calculation independently confirms 000114 | that Sym^2(16) maps onto the vector 10. 000115 | 000116 | 000117 | 4. WHICH SPINOR BILINEARS SUPPLY THE OTHER CARRIER HALF 000118 | 000119 | The same sign enumeration distinguishes the U5 channels. Vector roots 000120 | arise as follows: 000121 | 000122 | 1 times bar5: five witnesses, all negative vector weights; 000123 | 10 times bar5: twenty witnesses, all positive vector weights; 000124 | 10 times 10: thirty witnesses, all negative vector weights. (8) 000125 | 000126 | Hence the carrier half missing from the direct C/X bracket is supplied 000127 | by the cross channel 10 times bar5. The 10 times 10 bilinears are part of 000128 | the complete Spin(10) realization but land in the same negative vector 000129 | half as 1 times bar5. They do not alone repair the missing positive half. 000130 | 000131 | This is consistent with symmetry: the vector lies in Sym^2(16), while 000132 | the family target lies in Lambda^2(4). It also shows why adding only a 000133 | family Hodge complement cannot repair the carrier deficit. 000134 | 000135 | 000136 | 5. WHAT THE ORIGINAL E8 SOURCE CLOSURE PROVES 000137 | 000138 | The original source starts with the 24 positive currents C4 plus X20 and 000139 | their 24 compact adjoints. Its exact iterative Chevalley closure has root 000140 | counts 000141 | 000142 | 48 -> 140 -> 240. (9) 000143 | 000144 | The forty positive half-spinor roots in 000145 | 000146 | 10 tensor 4 000147 | 000148 | that are missing from C/X all appear at closure depth two. Their forty 000149 | adjoints appear with them in the full closure. Consequently the already 000150 | verified full E8 current algebra contains the complete 16 tensor 4 and 000151 | can form the full (10,6) through its actual brackets. 000152 | 000153 | This is an algebraic current-generation statement. It does not show that 000154 | the two geometric q/p triplet fields are already complete 16 tensor F 000155 | inputs, nor that the source evaluation inserts the depth-two generated 000156 | currents with the required carrier coefficients. Invoking the closure is 000157 | therefore a genuine additional source step, although it introduces no new 000158 | representation or freely chosen field. 000159 | 000160 | 000161 | 6. CORRECTED SCOPE 000162 | 000163 | Unconditional from the direct C/X triplet input: 000164 | 000165 | * the family Hodge complement and all its exterior signs are exact; 000166 | * the quarter and all 60 native SU4 actions preserve that antilinear 000167 | reality; 000168 | * the C/X tensor Lambda^2 F bracket plus dagger spans 30 real directions; 000169 | * no O family input is needed for those 30 directions. 000170 | 000171 | Conditional on an ACTUAL complete 16 tensor F input: 000172 | 000173 | * the Spin(10) gamma map supplies both five-dimensional D5 vector 000174 | weight halves; 000175 | * the bracket plus compact adjoint spans all 60 real directions of 000176 | (10,6); 000177 | * no independent O family fermion is then needed for the full real 000178 | sector. 000179 | 000180 | Still required for the geometric source claim: 000181 | 000182 | * identify the q/p fields with the full sixteen carrier components, or 000183 | derive their access to the ten depth-two source currents through the 000184 | existing E8 closure; 000185 | * transport the resulting full carrier coefficient into the actual 000186 | Y current and through the original normal reader into a specified 000187 | finite Phi block. 000188 | 000189 | Thus the corrected next arrow is 000190 | 000191 | geometric q/p triplet source 000192 | -> full 16 carrier completion inside the existing E8 closure 000193 | -> full (10,6) compact current 000194 | -> original normal and finite-source evaluation. (10) 000195 | 000196 | The first arrow in (10) is not established by the family Hodge map. 000197 | 000198 | 000199 | EXACT AUDIT DATA 000200 |