Spacetime 3, 4, 6, 10 from the division-algebra determinant

Correction (2026-09-16). This note previously described a Buckingham-π / dimension-matrix result. That did not match the source (log L147/L195), and the dimension-matrix reading was already carried as a coordinate in v5 §1 in any case. The claim is re-filed here to what the source actually says: a determinant, but the algebraic determinant of a Hermitian matrix over a division algebra — not a dimension matrix. The earlier framing was a reconstruction from memory that invented the wrong mechanism, an instance of the compression-loss the CHARTER names.

Take a 2×2 Hermitian matrix over a normed division algebra 𝔸 (𝔸 = ℝ, ℂ, ℍ, 𝕆), with diagonal entries t+z, t−z and off-diagonal x, x̄. Its determinant is t² − z² − |x|², a Minkowski quadratic form of signature (dim 𝔸 + 1, 1). Möbius transformations act on these matrices preserving the determinant, which is exactly why sl(2, 𝔸) = so(dim 𝔸 + 1, 1). The spacetime dimensions 3, 4, 6, 10 fall out as dim 𝔸 + 2 for 𝔸 = ℝ, ℂ, ℍ, 𝕆 (that is, 1+2, 2+2, 4+2, 8+2) — and this is why superstrings live in those dimensions.

The single fact doing all the work is norm-multiplicativity, |xy| = |x||y|: it is what makes the off-diagonal term |x|² well-defined and the determinant Möbius-invariant, and it is Hurwitz's condition — satisfiable only in dimensions 1, 2, 4, 8. So this note is a resident of the norm-multiplicativity / Hurwitz through-line (the same condition behind the four-floor ladder and the exceptional-list selection effect, [[CASE-02]]; the ladder statement itself lives in synthesist-init/dictionary.md and docs/algebra-geometry-dictionary.md), read one determinant deeper: the algebra's norm becomes a Lorentzian metric, and its four permitted dimensions become the four permitted spacetimes. Nothing about unit systems, exponent matrices, or dimensionless groups is involved — that is a different determinant (v5 §1).