C02.2 · Derivation

The decomposition and its actual multiplication operators

From the Jordan product, \[ E_i\circ E_i=E_i,\qquad E_i\circ E_j=0\quad(i\ne j). \] For \(Z\in J_{ij}\), \[ E_i\circ Z=E_j\circ Z=\tfrac12 Z,\qquad E_k\circ Z=0\quad(k\notin\{i,j\}). \] Therefore, for \(D=aE_1+bE_2+cE_3\), \[ L_D|_{\mathbb R E_i}=a_i I,\qquad L_D|_{J_{ij}}=\frac{a_i+a_j}{2}I . \] The generic symbolic matrix equality in the script checks this on the full carrier, not by entering the desired eigenvalues into a diagonal test object.

Within a block, with \(W_{ij}(q)\) denoting the Hermitian off-diagonal insertion, \[ W_{ij}(q)\circ W_{ij}(r) =\langle q,r\rangle_s(E_i+E_j). \] The product of two distinct off-diagonal blocks lies in the remaining block. The code checks the within-block formula and the cross-block support on all declared basis pairs. Bilinearity extends these checks to arbitrary vectors in those blocks. The two counts, each \(192\), are printed under `Peirce.same_block_product_checks` and `Peirce.cross_block_support_checks`.

This explains the arithmetic means in the real-lift spectrum. It does not identify every eigenspace with an additional particle or generation.

Diagonal adjoint and the homogeneous map

For \(D=\operatorname{diag}(x,y,z)\), the cubic and adjoint are \[ N(D)=xyz,\qquad D^\#=\operatorname{diag}(yz,xz,xy). \] The double adjoint obeys \[ (D^\#)^\#=N(D)D. \] The latter is checked as a polynomial identity (`diagonal.double_adjoint_identity`). At unit determinant the adjoint is the Jordan inverse. This is the algebraic content of LIB2-047, not a theorem selecting the vacuum.

Because \(\widetilde T\) is quadratic, \[ T(-u)=T(u). \] It consequently descends to a map on \(\mathbb{RP}^2\). This is a **factorization through a quotient**, not a statement that the descended map is globally injective. Permuting the three diagonal coordinates permutes their complementary pair-products; the script checks every slot permutation as a symbolic identity.

Image equation, double lines and the limitation of elimination

Writing the output as \((X,Y,Z)\), \[ X^2Y^2+Y^2Z^2+Z^2X^2-XYZ =x^2y^2z^2(x^2+y^2+z^2-1). \] Thus the unit-sphere image lies on the quartic. This identity is verified by symbolic substitution in `Roman.quartic_pullback`.

When \(x=0\), the image is \((yz,0,0)\) with \(y^2+z^2=1\), so its coordinate satisfies \(|yz|\le\tfrac12\). The other two coordinate planes behave cyclically. Hence the image contains **segments** of the coordinate axes, not all their unbounded real points.

The code computes the endpoint bound from the eigenvalues of the real quadratic form \(yz\) on the unit circle. As a negative control, \((1,0,0)\) satisfies the quartic equation but is outside the parametrized sphere image. The outputs `Roman.quartic_axis_counterexample` and `Roman.unit_sphere_axis_endpoint_bound` make explicit why merely verifying the quartic does not prove equality with its entire real zero set.

For a non-axis output, the squared input coordinates can be reconstructed: \[ x^2=\frac{YZ}{X},\qquad y^2=\frac{XZ}{Y},\qquad z^2=\frac{XY}{Z}. \] On the non-axis real quartic, these are positive and sum to the unit-sphere constraint. Their signs recover the generic antipodal pair. On the double-line segments there can be distinct projective preimages. For example, the executed rational points \[ (0,\tfrac35,\tfrac45),\quad (0,\tfrac45,\tfrac35) \] are neither equal nor antipodal but have the same image \((12/25,0,0)\). This is a direct test that descent to projective space does not turn the Roman map into an embedding.

The pinch points require the tangent derivative

The full derivative is \[ d\widetilde T= \begin{pmatrix} 0&z&y\\ z&0&x\\ y&x&0 \end{pmatrix}. \] The relevant rank is its restriction to the tangent plane \(u^\perp\), not the determinant of this ambient matrix alone.

At \(x=0\) with \(yz\ne0\), an ambient kernel vector is \((0,y,-z)\). It lies in the tangent plane exactly when \(y^2=z^2\). Together with the sphere constraint this gives the preimages with the two nonzero coordinates of magnitude \(1/\sqrt2\). At the coordinate-axis preimages the tangent rank is instead full. If every coordinate is nonzero, the ambient derivative is already invertible. This exhausts the cases.

The exact restricted-rank check yields \(12\) sphere preimages and the \(6\) image points \[ (\pm\tfrac12,0,0),\quad(0,\pm\tfrac12,0),\quad(0,0,\pm\tfrac12). \] They are computed under `Roman.pinch_preimages` and `Roman.pinch_images`. Pairing antipodal preimages accounts for the difference between those counts. The rank test stacks the sphere normal under the derivative; its rank drops precisely when the tangent derivative has a kernel.

The golden point is not a point of the unit sphere

For \(j=(\phi,1,\phi^{-1})\), the code gives \(j\cdot j=4\). Thus \[ \widetilde T(j)=j^\#=(\phi^{-1},1,\phi), \] but the unit-sphere map acts on \(j/2\), not on \(j\): \[ T(j/2)=j^\#/4. \] Both identities are computed. The source's sentence that the Roman map sends the golden vacuum to its Weyl reflection is correct for the **homogeneous extension**. Without this domain qualification it suppresses the necessary sphere normalization. The original quotation is not rewritten; this is an explicit distinction between the two maps.