The Half-Density Conjugates the Scale Differential to the Critical-Centered Operator
In logarithmic scale, the canonical relative-Haar half-density W is
multiplication by exp(q/2). It obeys
and therefore transports partial_q+s to partial_q+s-1/2.
The same operation transports logarithmic translation with half-density
factor sqrt(p), so the seam-period norm scales by p. Thus the critical
spectral offset and the relationship metric are two consequences of one
source-derived conjugation.
This closes the operator-level mate between relative Haar orientation and the endpoint homotopy system. Remaining work is the completed boundary packet: domain, primitive/square/archimedean covariance, endpoint-flux vanishing, and nonzero finite lifted period.
Research packet:
research/grothendieck/the-half-density-conjugates-the-scale-differential-to-the-critical-centered-operator.md
Exact checker:
research/grothendieck/checkers/check_half_density_centered_differential_mate.py
The checker passes 5/5 gates.