The E₈ discriminant glue theorem
The D₅ carrier lattice and the A₃ family lattice have compatible discriminant forms. Their diagonal order 4 glue gives an even unimodular lattice of rank 8. This is the lattice argument behind the TFPT E₈ construction.
Statement and assumptions
A mathematical construction with the stated positive definite D₅ and A₃ root lattices, normalized so roots have squared length 2. Identifying this lattice with a physical gauge theory requires the separate physical interfaces described in the papers.
The two discriminant forms
For an integral lattice L, its discriminant group is L*/L, where L* is the dual lattice. For an even lattice the squared norm defines a quadratic form on this group, valued modulo 2ℤ. Both lattices here have cyclic discriminant group of order 4. Their generating classes have the following norms.
In words: Both discriminant groups are cyclic of order 4. Their generating norms are five quarters and three quarters.
In the product discriminant group, the diagonal class (1,1) has norm 2, which is zero modulo 2ℤ. All its multiples also have even norm. The subgroup it generates is therefore isotropic and has order 4. Adjoining the corresponding cosets is the even lattice extension used in the source.
In words: The diagonal subgroup has four elements and its quadratic form vanishes modulo even integers.
The index 4 extension
Let L₀ = D₅ ⊕ A₃ and let L be the extension associated with H. Its index is 4. The determinant of an overlattice is the original determinant divided by the square of its index, so the extension has determinant 1.
In words: The rank 8 extension is even and unimodular because its determinant is one.
The positive definite even unimodular lattice of rank 8 is E₈. This identifies the extension. In the source, the order 4 glue is denoted μ₄, whose four elements are {1, i, −1, −i}. The norm identity also appears directly in a spinor root split into five and three coordinates.
In words: A half integer root splits into five coordinates of total norm five quarters and three of total norm three quarters.
The explicit roots and branching
The cited verification source constructs the roots in standard eight dimensional coordinates. There are 112 integer roots, obtained by choosing two coordinates and independent signs. There are 128 half integer roots, with an even number of minus signs. Every listed vector has squared norm 2.
In words: The roots are the signed two coordinate vectors and the half integer vectors with even sign parity.
In words: The construction contains 240 roots and the Lie algebra has dimension 248.
Under D₅ × A₃, the adjoint decomposes into the two adjoints, a vector product and a pair of conjugate spinor products. The root count differs from the Lie algebra dimension by the eight Cartan directions.
In words: The five representation blocks have dimensions 45, 15, 60, 64 and 64.
In words: Removing the five and three Cartan directions gives the corresponding 240 root count.
The source script also supplies a simple root basis, its integral even Gram matrix, determinant 1 and the coordinates of every root in that basis. Those are the concrete checks cited by this argument; publishing their source does not report a new execution.
Read the original sources
This reading edition follows the original arguments below. The paper and its surrounding sections retain the full publication context.
Original discriminant theorem and branching
documents/tfpt_1_architecture_e8/sections/005-source.tex · original lines 166 to 229
Source version
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Inspect the cited verification code
The following scripts support the particular calculations described here. A citation identifies a source; a new execution needs its own result.
v1_e8_glue.py
Constructs the 240 roots and checks the simple root lattice certificate, discriminant norms, ranks and glue index.