The Generalized-Flow Identity of the Alignment Matrix

Abstract

Every matrix on the alignment hyperplane sandwiches K to a determinant-scaled copy of K.

Theorem 1.1 (Generalized-flow identity on the alignment hyperplane).

$$\begin{gathered} K=\left(\begin{array}{cc}1& -2\\ 2& -1\end{array}\right),K^2=-3I, \\ \forall \beta \in V,\beta K\beta =(\det \beta )K \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/PrimeForms/AlignmentClifford.generalized_flow (✓ std3). ∎

Source. Repository-derived.

Commentary.

The alignment matrix K = [[1,-2],[2,-1]] is an integer 2x2 matrix that squares to -3 times the identity, so it behaves as a square root of -3. Its alignment hyperplane is V = { X : tr(X K) = 0 }, the integer matrices X whose trace against K vanishes.

The generalized-flow identity states that every matrix on this hyperplane rescales K by its determinant: for all beta in V, beta K beta = (det beta) K. The identity holds for every beta on the hyperplane, with no unimodularity assumption; the unimodular case det beta = +-1, where beta sends K to +-K, is a special case. Moreover the hyperplane is closed under the sandwich map: if beta and gamma lie in V then so does beta gamma beta.

The flow identity is proved by reading off the single hyperplane constraint and applying it to each of the four entries of beta K beta - (det beta) K; closure is a trace-cyclicity corollary. Only these three clauses — the square identity, the generalized flow, and closure — are recorded here. The unimodular acts-by-plus-or-minus-one reading, the flow / self-insertion / even-texture unification, the paired-and-zero census certificate, and the phase-charge parity interpretation of the wider result are not covered by this statement.

References

• Truth anchor: D5/S3/PrimeForms/AlignmentClifford.generalized_flow