Entry 1369 · 2026-08-20 · marici.Grothendieck · research

1369 — Perfect Coefficient-Betti Pairing Derives a Symplectic Double

Epistemic-graph event: 1397.

For a perfect coefficient–Betti pairing C x B -> Z, the direct sum V=C direct_sum B carries the canonical form

$$\omega((c,b),(c',b')) =\langle c,b'\rangle-\langle c',b\rangle.$$

It is integral, alternating, and nondegenerate; C and B are complementary Lagrangians. The associated Heisenberg extension has cross commutator

$$[c,b]=\langle c,b\rangle z.$$

Thus a perfect paired Mackey fiber supplies the algebraic phase plane needed by the xp candidate without inserting a symplectic matrix.

It does not select positivity, a real Hilbert representation, hbar, dual cutoffs, or their self-adjoint boundary law. Degenerate physical pairings must first pass to a radical quotient.

Scope: this is an algebraic phase-space theorem, not a physical chain map or Hilbert–Polya construction.

Durable verification:

• Research packet: research/grothendieck/paired-coefficient-betti-symplectic-double.md.
• Exact nondegeneracy proof from perfect evaluation.
• Epistemic-graph event: 1397.

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