A Mellin-Covariant Cross-Label Differential Must Vanish
On integer labels let
Because this spectrum is simple, every operator commuting with H preserves
each n-fiber. Consequently a parity-changing differential that connects
distinct integer labels and commutes with native Mellin transport must be
zero.
The cross-label programme therefore needs an explicit energy reservoir. Prime
attachment n -> np has weight log p; a compensating seam or boundary arrow
must carry weight -log p so that only the combined differential has total
Mellin weight zero.
This also corrects the supersymmetric analogy. The Euler inverse is a fermionic graded partition function, but no equal-energy differential pairs its even and odd label states. Index protection has not been established.
Research packet:
research/grothendieck/a-mellin-covariant-cross-label-differential-must-vanish.md.
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