3670 — The Reciprocal Sector Pair Is a Lagrangian Graph Before Seam Extension
Result
Before adding an independent seam state, Fourier-related reciprocal sectors form the graph of a symplectomorphism in the oppositely signed doubled phase space. That graph is Lagrangian.
Restricting the ten ambient quadratic sp4 symbols to the graph has exact
rank three. The second local triple and four mixed quadratics are therefore
not independent bulk controls; they collapse to the original one-mode sl2.
Consequence
The two half-plane sectors are distinct typed presentations of one canonical
state, not two independent oscillator modes. The endpoint sp4 comparison
can become relevant only after a source-derived seam extension supplies
transverse normal or conormal data.
Scope
This proves the finite symplectic graph statement. It does not yet construct the completed theta seam as a normal bundle or determine its rank.
Durable verification
research/grothendieck/the-reciprocal-sector-pair-is-a-lagrangian-graph-before-seam-extension.md;research/grothendieck/checkers/check_reciprocal_lagrangian_graph_quadratic_collapse.py.
Allocator authority: grothendieck-reciprocal-sector-graph-lagrangian-20260827.