TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/canonical-seam-fermionization-20261025/LATTICE-AND-SOURCE-FRAME.txt
Original file SHA-256: 77e0687412cb79bf1d51d079946484b6e8be337adf8a72a85bf2e6079b045ae7
Inclusive original lines: 100-171
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.

000100 | 
000101 | Use the fixed standard target coordinates in which A3 is the usual D3 root
000102 | lattice on the last three coordinates.  Then
000103 | 
000104 |   K={z in Z8:
000105 |        sum_(i=0)^4 z_i is even,
000106 |        sum_(i=5)^7 z_i is even}.
000107 | 
000108 | Choose the marked Z4 glue generator
000109 | 
000110 |   g=(1/2,1/2,1/2,1/2,1/2,1/2,1/2,1/2).
000111 | 
000112 | It has order four modulo K: 2g=(1,...,1) is not in K, whereas
000113 | 4g=(2,...,2) is in K.  A cyclic group of order four has one subgroup of
000114 | order two, namely <2g>.  Its intermediate lattice is
000115 | 
000116 |   L_2=K union (K+2g).
000117 | 
000118 | For an integral vector z, membership in K means that the first-five and
000119 | last-three coordinate sums are both even.  Membership in K+2g means that
000120 | both sums are odd.  Their union is therefore
000121 | 
000122 |   L_2={z in Z8: sum_i z_i is even}=D8.                    (1)
000123 | 
000124 | Equation (1) proves the full lattice identity.  It is not inferred from the
000125 | number of roots.
000126 | 
000127 | On roots, use doubled coordinates as in the original programs.  The marked
000128 | glue degree is
000129 | 
000130 |   deg(r)=sum_(i=0)^4 r_i mod 4.
000131 | 
000132 | The exact E8 root counts in degrees 0,1,2,3 are
000133 | 
000134 |   52,64,60,64.
000135 | 
000136 | The roots in the order-two subgroup are degrees 0 and 2.  Their total is
000137 | 52+60=112, and their set is exactly
000138 | 
000139 |   R_can={+/-e_i +/-e_j: i<j},                             (2)
000140 | 
000141 | the standard D8 root set in physical target coordinates.
000142 | 
000143 | The original map in tfpt_explorer/cartan_source.py is
000144 | 
000145 |   O=T S^(-1)/sqrt(2),                                     (3)
000146 | 
000147 | where S is the original Gaussian simple-root matrix and T is the displayed
000148 | target simple-root matrix.  Exactly O^T O=I.  If r is an original norm-four
000149 | Gaussian root and n=S^(-1)r is its integral simple-root coordinate, then
000150 | 
000151 |   O(r/sqrt(2))=Tn/2.                                      (4)
000152 | 
000153 | The checker verifies that (4) maps all 240 original roots bijectively to the
000154 | fixed standard target E8 roots.  Applying the marked even glue grades gives
000155 | precisely (2).  Thus no unmarked classification is being substituted for the
000156 | actual O dictionary.
000157 | 
000158 | 2. The folded CAR D8 in the same target frame
000159 | ----------------------------------------------
000160 | 
000161 | The eight folded CAR charge components have integer charge lattice Z8.  Its
000162 | even part is
000163 | 
000164 |   D8_CAR={p in Z8: sum_i p_i is even},
000165 | 
000166 | with the same standard root set R_can in the folded component frame.
000167 | 
000168 | The existing fold dictionary does not identify that component frame with
000169 | the target axes.  It uses the fixed matrix M stored in
000170 | 
000171 |   v16-operator-pullback-20261014/fold-results.json.