The Relative Haar Norm Ratio Canonically Produces the Half-Density Seam
Apply the same raw dilation f(x) -> f(x/p) in the independently typed spaces
L2(dx) and L2(dx/x). Its norm-squared multipliers are respectively p
and 1.
Their ratio is invariant under independent unitary changes of frame. Its
positive square root is the canonical additive-to-multiplicative Haar
half-density. After Mellin weighting, the additive channel is isometric
exactly when the real part of s is one half.
This closes the relative-Haar orientation rung without promoting a gauge-dependent weighted Gram to physical status. It explains the seam but does not yet confine zeros. The remaining gate is compatibility among this modular cocycle, the pair-label Gram current, and the two endpoint homotopies.
Research packet:
research/grothendieck/the-relative-haar-norm-ratio-canonically-produces-the-half-density-seam.md
Exact checker:
research/grothendieck/checkers/check_relative_haar_half_density_seam.py
The checker passes 6/6 gates.