The inverse of the colour tetraquaternion, M 12 (C)
Abstract
The colour sector of the unified theory built on the tetraquaternion is M 12 (C) = T ⊗ M 3 (C), the tetraquaternion T = H ⊗ H = M 4 (R) tensored with Connes' colour factor, of real dimension 288; its field level is M 6 (C) = M 2 (C) ⊗ M 3 (C) and its Clifford container Cl(9, 1) = T ⊗ Cl(0, 6) = M 32 (R). This third paper on inverses gives the inverse of an element of the colour sector in the same closed structural forms as its two companions and shows what changes when the third factor is complex rather than real or quaternionic. Read as a 3 × 3 matrix over the complex tetraquaternion M 4 (C), an element is inverted by nine-term quasideterminants of inversion height three, every entry inverse being the sixteendimensional closed form with complex components, and its determinant is the product of three sixteen-dimensional quartics, det R = r(d 1 ) r(d 2 ) r(d 3 ); read as a 4 × 4 matrix over M 3 (C), by the 82-term quasideterminants of the first paper with Cramer inverses; and as a polynomial of degree eleven in R with coefficients in the twelve scalar parts ⟨R k ⟩ 0 , 12! det R having 77 terms, written out. The determinant is complex, of degree twelve, its zero set has real codimension two, the tetraquaternion sits inside with det = r 3 , which keeps the sign that the hyperquaternion lost, and the expansion into components, 12! = 479,001,600 monomials, is out of reach as before. The gluon field element a (F a •σ) ⊗ T a , F a = E a + iB a , is a zero divisor exactly when one colour sees no field: for one colour direction det = (-F•F) 3 (det N ) 2 with det N = d abc n a n b n c /12, so that every generator but λ 8 gives a zero divisor; for the Cartan family det =i (w i • w i ) over the three weight fields seen by the three quark colours; every field in an isospin su(2) subalgebra (λ 1 , λ 2 , λ 3 or a conjugate), the hedgehog included, is singular, the third colour being blind. For real fields the sector-level determinant is | det 6 | 2 ≥ 0, the phase of the field-level determinant is six times the electric-magnetic duality angle, three times the phase of the quadratic invariant whose imaginary part is the θ-term, and the phase of the algebraic determinant is turned by the U (1) of U (3) that the unimodularity condition removes. Whether the complex determinant is related to confinement is answered: it is not; it is the Frobenius type of M 3 (C), the same fact that makes 3 ̸ = 3, and the singular set of the colour element is colour blindness, the opposite of confinement, which is a dynamical statement about the compact group where the inverse is the conjugate. In the container the gluon element is always singular, because the two colour singlets of C ⊗ O, the lepton slots of the fourth colour, see no gluon.
Related
BibTeX
@misc{reynaud2026inverse,
title = {{The inverse of the colour tetraquaternion, M 12 (C)}},
author = {Patrick .B Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05749770v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The inverse of the colour tetraquaternion, M 12 (C)
AU - Patrick .B Reynaud
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05749770v2
AB - The colour sector of the unified theory built on the tetraquaternion is M 12 (C) = T ⊗ M 3 (C), the tetraquaternion T = H ⊗ H = M 4 (R) tensored with Connes' colour factor, of real dimension 288; its field level is M 6 (C) = M 2 (C) ⊗ M 3 (C) and its Clifford container Cl(9, 1) = T ⊗ Cl(0, 6) = M 32 (R). This third paper on inverses gives the inverse of an element of the colour sector in the same closed structural forms as its two companions and shows what changes when the third factor is complex rather than real or quaternionic. Read as a 3 × 3 matrix over the complex tetraquaternion M 4 (C), an element is inverted by nine-term quasideterminants of inversion height three, every entry inverse being the sixteendimensional closed form with complex components, and its determinant is the product of three sixteen-dimensional quartics, det R = r(d 1 ) r(d 2 ) r(d 3 ); read as a 4 × 4 matrix over M 3 (C), by the 82-term quasideterminants of the first paper with Cramer inverses; and as a polynomial of degree eleven in R with coefficients in the twelve scalar parts ⟨R k ⟩ 0 , 12! det R having 77 terms, written out. The determinant is complex, of degree twelve, its zero set has real codimension two, the tetraquaternion sits inside with det = r 3 , which keeps the sign that the hyperquaternion lost, and the expansion into components, 12! = 479,001,600 monomials, is out of reach as before. The gluon field element a (F a •σ) ⊗ T a , F a = E a + iB a , is a zero divisor exactly when one colour sees no field: for one colour direction det = (-F•F) 3 (det N ) 2 with det N = d abc n a n b n c /12, so that every generator but λ 8 gives a zero divisor; for the Cartan family det =i (w i • w i ) over the three weight fields seen by the three quark colours; every field in an isospin su(2) subalgebra (λ 1 , λ 2 , λ 3 or a conjugate), the hedgehog included, is singular, the third colour being blind. For real fields the sector-level determinant is | det 6 | 2 ≥ 0, the phase of the field-level determinant is six times the electric-magnetic duality angle, three times the phase of the quadratic invariant whose imaginary part is the θ-term, and the phase of the algebraic determinant is turned by the U (1) of U (3) that the unimodularity condition removes. Whether the complex determinant is related to confinement is answered: it is not; it is the Frobenius type of M 3 (C), the same fact that makes 3 ̸ = 3, and the singular set of the colour element is colour blindness, the opposite of confinement, which is a dynamical statement about the compact group where the inverse is the conjugate. In the container the gluon element is always singular, because the two colour singlets of C ⊗ O, the lepton slots of the fourth colour, see no gluon.
ER -