Integrality Makes the Rank-One Theta Log Coordinate Faithful
Scope correction: Entry 3963 shows that the full Gaussian trajectory remains jointly faithful. The continuous -kernel is a hostile relaxation model, not an established explanation of Riemann zeros.
Theta observation depends on prime valuations through the single coordinate
Although this map has a large kernel on the continuous label span, it is injective on finite-support nonnegative integer valuation vectors by unique factorization.
Thus the same rank-one observation is faithful before completion and highly many-to-one after an untyped continuous relaxation. This gives an exact hostile model for loss of arithmetic meaning: completed dark directions can appear only if the topology erases the integral label register.
It is not yet a theorem about Riemann zeros. The next gate is to derive the actual completed coefficient module and determine whether divisibility and boundary constructors retain or lose those integral labels.
Research packet:
research/grothendieck/integrality-makes-the-rank-one-theta-log-coordinate-faithful.md
Epistemic graph admission: event 8976.