The inverse of the algebraic element of the unified theory at eleven parameters, M 4 (C) ⊕ M 4 (H) ⊕ M 12 (C)

@misc{reynaud2026inverse,
  title = {{The inverse of the algebraic element of the unified theory at eleven parameters, M 4 (C) ⊕ M 4 (H) ⊕ M 12 (C)}},
  author = {Patrick B. Reynaud},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05751096v2},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - The inverse of the algebraic element of the unified theory at eleven parameters, M 4 (C) ⊕ M 4 (H) ⊕ M 12 (C)
AU  - Patrick B. Reynaud
PY  - 2026
DA  - 2026/09/15
PB  - HAL
UR  - https://hal.science/hal-05751096v2
AB  - The Standard-Model algebra of the unified theory built on the tetraquaternion is T ⊗ A F = M 4 (C) ⊕ M 4 (H) ⊕ M 12 (C), the tetraquaternion T = H ⊗ H = M 4 (R) tensored with Connes' finite algebra A F = C ⊕ H ⊕ M 3 (C), of real dimension 32 + 64 + 288 = 384: the hypercharge tetraquaternion, the electroweak hyperquaternion and the colour sector as one algebra. Three companion papers gave the inverse of an element of each summand; this fourth paper gives the inverse of the element of the whole, and says what the direct sum adds to the three and what it does not. The algebra is not simple: its unit is the sum of three central idempotents, 1 = e 1 +e 2 +e 3 , its centre is C⊕R⊕C, and an element is a triple (R 1 , R 2 , R 3 ) whose inverse is the triple of the three inverses, each in the closed structural forms of the companions, the cubic-over-quartic form of the tetraquaternion with complex components, the quasideterminants and the Study determinant of the hyperquaternion, the nine-term quasideterminants of the colour sector. The determinant is not one number but three, (r, Det, det) in C×R ≥0 ×C, of degrees 4, 8 and 12, whose product is the determinant of the representation on C 24 ; the inverse is a polynomial of degree 11 in R with the idempotents and of degree 23 without them, the idempotents being themselves polynomials in R with real coefficients exactly when the three conjugation-closed spectra are pairwise disjoint; the singular set is the union of the three, of real codimension two, stratified by the sector that fails; an element supported on one or two sectors is a zero divisor of the whole and has a partial inverse, the group inverse, in the ideal where it lives; and on the tetraquaternion embedded diagonally the three determinants are (r, r 2 , r 3 ), so that the sign of r, which the electroweak summand loses, is kept by the two others and lost again by the product r 6 . At the field level, M 2 (C) ⊕ M 4 (C) ⊕ M 6 (C), the twelve field strengths are one element whose determinant is the product of three: -F Y •F Y for the hypercharge, the Heron form -A 2 of the isospin Gram matrix for the weak triplet, singular when the three isospin components form a degenerate triangle, and the colour determinant of the third paper, singular when a colour sees no field. The element of a body sits in the three sectors at once, with the determinants (r, r 2 , r 3 ), and the electroweak vacuum lifts the zero of the second only: the Standard-Model element of a body is a zero divisor on the cone r = 0 of the first paper, with or without the vacuum, and only its electroweak part has its zero moved to the set of codimension three of the second. The rotation operator of the first paper, R ′ = QRQ -1 , keeps its form: the Lorentz group and the gauge group conjugate the three sectors at once by Q ⊗ u, a Lorentz element times a gauge element, while the translations and the conformal transformations carry the complex unit and act on the electroweak sector only through the choice of a complex line, on an algebra whose inner automorphisms form a group of 379 parameters against the 27 of the symmetry, the mixed ones having no counterpart in any symmetry and the container carrying a generalised Lorentz transformation of its own, the 66 rotations of signature (5, 7). The metric of the algebra, preserved by the rotation operator among all the inner automorphisms, is a five-parameter family of trace forms, one parameter per real dimension of the centre, always symmetric, the sectors orthogonal and each the tensor product of the tetraquaternion's metric with that of its finite factor; 1 its canonical member has signature (188, 196), negative exactly on the Lie algebra of the compact group, it is Minkowski's on the body with the multiplicities 1, 2, 3 of the sectors, gauged it is the metric of the gravitational theory, and at the field level its weights are the three couplings, the θ-terms belonging to the field level alone because the parity is inner at the sector level; the electroweak summand is the Clifford algebra Cl(5, 1), the two isospin directions two spacelike dimensions, and the embedded conformal algebra closes with the isospin on so(6, 2). In the container Cl(5, 7) = M 32 (H), which is simple, the determinant is one, of degree 64, the gluon element alone is always singular on the lepton slots and becomes invertible once a generic weak field dresses them, and mixed components can make a whole singular whose sector parts are invertible, which the direct sum never does. The Mandelbrot set of the direct sum is the product of the three companion sets, its scalar slice is M × [-2, 1 4 ] × M, and on a decomposable element X ⊗ a it is the intersection of the three sector slices, drawn for a hypercharge phase, an isospin quaternion and the λ 8 colour direction; iterating the rotation operator instead gives no fractal, the operator being a symmetry of the quadratic iteration rather than a dynamics.
ER  -