TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/e-series-source-reconstruction-20261024/SCALAR-QUARTER-REPRODUCTION.txt
Original file SHA-256: 16e123a72d0d65e48fc1777bbc0b3a10a7585114c3f82bad33c87e0794e15f94
Inclusive original lines: 28-80
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.

000028 | 1. Actual map, rather than equal norms
000029 | 
000030 | Let L0=D5+D3 in R5+R3, and put
000031 | 
000032 |     q=(1/4)^8,       s=2q=(1/2)^8.
000033 | 
000034 | Define the isometry T:R3 -> {(y1,y2,y3,y4):sum y=0} by
000035 | 
000036 |     T(x,y,z)=(x+y+z,x-y-z,-x+y-z,-x-y+z)/2.
000037 | 
000038 | Exactly T^T T=I3. It maps all12 D3 roots bijectively to all12 A3 roots
000039 | e_a-e_b, so it identifies their ENTIRE root lattices. With F=I5+T,
000040 | 
000041 |     F(s)=(1/2,1/2,1/2,1/2,1/2;3/4,-1/4,-1/4,-1/4)
000042 |         =(omega_s,omega_f)=lambda.                      (1.1)
000043 | 
000044 | This is the original v983/v1033 glue vector, not just a vector of the same
000045 | length. Its two squared projection norms are5/4 and3/4; h(s)=1, while
000046 | h(q)=1/4. The apparent D5 half-phase/A3 inverse-quarter phase description
000047 | uses NINE coordinates subject to the A3 zero-sum constraint. These cannot
000048 | be compared componentwise with eight elementary channel phases without
000049 | the map F and the corresponding dual current/charge transport.
000050 | 
000051 | 2. The two quotient orders and the corrected inclusion tower
000052 | 
000053 | s is not integral. Its double 2s=(1)^8 has odd block sums5 and3, hence is
000054 | not in L0. Its fourth multiple 4s=(2)^8 has even block sums10 and6. Thus
000055 | 
000056 |     ord(s modulo L0)=4.
000057 | 
000058 | L1=L0+Z(2s) consists of integer vectors whose two block parities agree.
000059 | That is exactly D8, the integer vectors of even total sum. Therefore
000060 | 
000061 |     ord(s modulo D8)=2,
000062 |     L=L0+Zs=D8 union(D8+s)=E8,
000063 |     U=V_D5 tensor V_A3 --index2--> V_D8 --index2--> V_E8. (2.1)
000064 | 
000065 | The total index is4 and the mu-indices are16,4,1. v125_glue_qsystem.py
000066 | explicitly constructs this intermediate subgroup {0,2}; v154 names U
000067 | as the index-four base. The charged-scaling source005 equations7.1--7.4
000068 | also uses U and is correctly typed. By contrast, the old research-contract
000069 | 017 introduction/externalization calls the base so(16)_1 while retaining
000070 | the index-four/order-four language. That mixes the two bases in(2.1).
000071 | Over D8 the missing sector is one order-two spinor with128 ground states;
000072 | over U it splits into two odd grades of64 each, plus the even intermediate.
000073 | 
000074 | Thus a doubled quarter charge is NOT excluded by the previous quarter-
000075 | weight test. It supplies the exact compiler lattice class via(1.1), with
000076 | order four or two depending on the actual base. This does NOT assert that
000077 | the square of a bounded finite flux-register operator is the local field
000078 | Y(e^s,z). Disorder fusion, coincident operator multiplication and a
000079 | renormalized smeared charged field are different operations.
000080 |