normSq
plain-language theorem explainer
Defines the squared Euclidean norm of a point in the real plane. Anyone working the bipartite cross-distance alphabet (the RS reading of Erdős #661) cites it as the basic radial coordinate on the visible planar carrier. The body is the one-line Mathlib identity ‖p‖².
Claim. For a point $p$ in the Euclidean plane $\mathbb{R}^2$, the squared norm is $\|p\|^2 \in \mathbb{R}$.
background
The module physicalizes Erdős problem #661 as a two-channel range-spectrum question: two finite planar sets $P,Q$ couple by Euclidean delay, and the classical distinct-distance count is the alphabet size of the cross-coupling spectrum. The RS reading expects rigidity of the square-grid $n/\sqrt{\log n}$ scale because a single positive-definite planar carrier cannot host true two-sector orthogonality.
Point2 is the Mathlib Euclidean space $\mathbb{R}^2$. Squared norm is the natural radial quantity on that carrier; sibling constructions (cross-distance spectra, difference sets, shell multiplicities) build alphabets from values of this form. Parallel squared-norm defs appear elsewhere in the stack (finite amplitude vectors, Hilbert displays, metric quadratic forms), but here the carrier is strictly the visible plane.
proof idea
One-line definitional wrapper: evaluate the Mathlib Euclidean norm of $p$ and square it. No lemmas, no tactics.
why it matters
Local primitive for every squared-distance and shell-counting construction in the bipartite spectrum module (cross-distance alphabets, norm-image sets, planar rigidity statements). Downstream, squared-norm language feeds spectral-forcing and gap-weight work: the eight-tick DFT mode energies, difference energies, and the trig closure $|\omega_8^k-1|^2=4\sin^2(k\pi/8)$ all speak the same quadratic language, tying planar geometry to the T7 eight-tick octave and the alpha-genesis spectral certificate. Without a clean planar $|\cdot|^2$, the positive Erdős #661 / rigidity reading has no radial coordinate.
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