TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/CARTAN-FAMILY-DICTIONARY.txt
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000114 | 5. The family factor cannot replace this Cartan subspace
000115 | 
000116 | [OMITTED: non-scientific conversation or execution metadata]
000117 | [OMITTED: non-scientific conversation or execution metadata]
000118 | [OMITTED: non-scientific conversation or execution metadata]
000119 | 
000120 |                   eigenvalue       1    omega    omega^2
000121 |   Gaussian H+                       2      1         1
000122 |   intrinsic family4                 0      2         2.
000123 | 
000124 | The second row holds for ALL480 toral representatives of the actual
000125 | whole native C12 class after its ninth power has been aligned with kappa.
000126 | The checker exhausts the old6561 third-torsion classes and adds this
000127 | previously unrecorded factor-level character test. It does not treat
000128 | 480 as a new number of physical sources.
000129 | 
000130 | Here are the exact family phases. Choose an integral even-sum b with
000131 | specified standard residues modulo3. Write p=sum(b_D3) mod2 and let
000132 | sigma_j=(+++),(+--),(-+-),(--+). The unique order3 SU4 lift has residues
000133 | 
000134 |   r_j=(-sigma_j dot b_D3+3p)/2 mod3.
000135 | 
000136 | The added common sign compensates the Spin/family kernel(-1,-I4).
000137 | The lift is unique because3 and the central kernel order4 are coprime.
000138 | Every full native match has a permutation of(1,1,2,2); the six ordered
000139 | patterns occur80 times each. The Spin10 vector character is(4,3,3).
000140 | 
000141 | Consequently no sigma-equivariant direct isomorphism identifies H+ with
000142 | this family4. A scalar cubic character, duality or permutation cannot
000143 | repair the different multiplicity patterns. This is compatible with
000144 | the positive inclusion c subset Delta tensor F4 and with full E8
000145 | conjugacy. It is a factor-level exclusion, not a no-go for that inclusion.
000146 | 
000147 | 6. Why some old-looking correlations nevertheless agree
000148 | 
000149 | Let G4 and F4 denote the two C3 modules in Section5. Their dimensions are
000150 | the same and some symmetric powers are isomorphic as C3 modules:
000151 | 
000152 |   module                         Gaussian          family
000153 |   Sym2                              (4,3,3)          (4,3,3)
000154 |   Sym3                              (8,6,6)          (8,6,6)
000155 |   exterior2                         (2,2,2)          (4,1,1)
000156 |   Sym4                             (13,11,11)        (9,13,13).
000157 | 
000158 | The equality pattern is completely determined in all degrees. The trace
000159 | generating functions at sigma are
000160 | 
000161 |   sum tr(sigma|Sym^n G4)t^n = 1/((1-t)(1-t^3)),
000162 |   sum tr(sigma|Sym^n F4)t^n = (1-t)^2/(1-t^3)^2.
000163 | 
000164 | Their difference is3t/(1-t^3)^2. Since total dimensions agree and the
000165 | sigma^2 traces are the conjugates, the full C3 characters agree exactly
000166 | when n is NOT1mod3. For n=3k+1 their multiplicity difference is
000167 | (2(k+1),-(k+1),-(k+1)). Thus quadratic and cubic agreement cannot supply
000168 | a tensor-natural dictionary for the original modules. Exterior square
000169 | already detects the error. Nothing here asserts equality for the full
000170 | G31 or Spin10 x SU4 action.
000171 |