The Secp256k1 Boundary Operator: E8 Lattice Reduced Form Transformation Map
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Abstract
We give an explicit algebraic and lattice-theoretic construction of the (4 + 1p) endogenous rack vector R4+1p(x, y) = (x4, x3y, x2y, xy2, y3)
generated by a (3, 2) rank–lane split of a two-dimensional public point. Its squared Euclidean
norm is the non-homogeneous polynomial r2m(x, y) = x8 + x6y2 + x4y2 + x2y4 + y6, which admits a manifest sum-of-squares decomposition. We define the divisor operation directly on the endogenous lane pair and prove the exact scaling law r2m,div = (2n − 7)2r2m, rank.
Under the curve relation y2 = x3 +7, the rack coordinates recover 7x and 7y, while the metric collapses to a degree-nine polynomial in x. The central lattice theorem is completely explicit. Index the eight coordinates of R8 by u ∈ F3,2 and assign to every a ∈ F3,2 the character root r,a=(1/2(−1)a·u) u∈F3,2.
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These are eight mutually orthogonal roots of the E8 lattice. The seven nonzero indices are the
seven Fano points, and the Fano line law a + b + c = 0 is represented exactly by the Hadamard
identity 2(ra ⊙ rb) = rc. The remaining root gives the closure stride τ = 2r0 = (1, . . . , 1) ∈ ΛE8 ,
with ∥τ ∥1 = 8 and ∥τ ∥2,2 = 8. An explicit integral height functional and lattice path then realize
2n − 7 = (2n + 1) − 8 as an actual E8 lattice translation rather than a dimensional analogy.
Dr. Charles Tibedo
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Topological_Stride_E8_Explicit_Theorem_4_2_Revised (2).pdf
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Dates
- Copyrighted
-
2026-09-19