Triangular Addressing, Null Coordinates, and Reciprocal-Metallic Locks in a Scale-Covariant Plane Geometry

   We study a plane-geometric construction generated by the flattened triangular rank \(\rho(n,k)=n(n-1)/2+k\), renamed \(m=\rho(n,k)\) in the present geometry, together with a cell scale \(q>0\) and a shape parameter \(0<c<1\).  The coordinates \(t=qm\) and \(s=ct\) determine a rectangle of area \(d=st\), a reciprocal hyperbola \(xy=d\), and, under a half-normalized null-coordinate transform, a point on \(x^2-y^2=d\).  A second intersection with the same split hyperbola generates a centered rectangle that is similar to the original rectangle for every \(c\).  We prove that \(c=\sqrt5-2\) is the unique full Euclidean congruence lock and that the same root simultaneously produces a null-coordinate collision and a rigid tangent-wing dissection.  The novelty claimed is this forced co-occurrence inside one generator, not the classical coordinate transform, metallic constant, tangent theorem, or continued-fraction machinery separately.  The three native collision values are the reciprocals of the metallic means with indices \(a=1,2,4\). Their rational cell-grid approximants satisfy generalized Pell equations; for the congruence family this records an exact alternating area-error formula.  The triangular address is inherited from the revised Triangular-Fractional Grid foundation, while the divisor-band and totient-shell theorems of that parent are not imported into this geometric theorem line.
   We also separate address scaling from fixed-extent resolution and state explicit methodological fences for external targets, spectral operators, information measures, and physical interpretations.