C05.2 · Definitions/conventions

The active projector and the word

Let \(P^2=P\), \(Q=I-P\), \(A=PAP\), and \[ A^2-3A+P=0,\qquad W_h=Q+A. \] These are the algebraic premises extracted from the registered word, not a proof that a chosen matrix has them. In the programme the active and complementary dimensions are those of LIB2-014. The script's padded example uses those dimensions as **declared inputs**, not as independently recovered multiplicities.

Define the candidate positive-spectrum branch \[ H=Q+\frac{A+P}{\sqrt5}. \] “Positive-spectrum” does not assert Euclidean symmetry or positive definiteness of the matrix in an arbitrary frame.

Symplectic adjoint versus Euclidean transpose

For a nondegenerate symplectic matrix \(\Omega\), define \(X^\dagger=\Omega^{-1}X^T\Omega\). Symplecticity is \(X^\dagger X=I\). This adjoint differs from the Euclidean transpose used by the Frobenius form.

For the Gram statement, set \[ (G_T)_{ij}=\operatorname{tr}((W_iT)^T W_jT),\qquad i,j\in\{0,1\}. \] A sufficient hypothesis for the advertised formula is \[ P^T=P,\quad W_i^T=W_i,\quad W_i^2=P/24,\quad W_0W_1+W_1W_0=0. \] The symmetric-generator convention is a necessary part of this **proof's premises**. It must be checked against the frozen representation; it is not supplied merely by writing down abstract Clifford relations. The countercontrol below demonstrates that distinction.

The tick

A \(\mathbb Z/4\mathbb Z\)-grading assigns a degree \(d\) to homogeneous products with \(d(xy)=d(x)+d(y)\pmod4\). A degree-reversing mirror sends \(d\) to \(-d\). These are finite algebraic conditions. They do not define a coordinate, differential operator, Hilbert-space momentum, or compactification radius.