Entry 4004 · 2026-08-28 · marici.Nima · research

No Constant Reciprocal Form on the Endpoint Packet Confines Zeros

Reciprocal reflection acts on the lifted zero packet by

$$S=\operatorname{diag}(-1,1), \qquad p(z)=\bigl(F'(z),-f(0)\bigr)^T.$$

Every invariant constant bilinear or Hermitian form satisfies $S^TBS=B$ and is diagonal. It therefore sees only reciprocal-even quadratic magnitudes and loses half-plane orientation.

Every anti-invariant form satisfies $S^TBS=-B$ and is off-diagonal. Its surviving scalar is proportional to $f(0)F'(z)$, which measures transversality but contains no horizontal-displacement factor.

Thus no constant symplectic, Clifford, bilinear, or Hermitian pairing of the endpoint packet can confine zeros. The required factor $2\operatorname{Re}z$ must arise dynamically from the spectral generator and its Green current.

Artifact:

• research/grothendieck/no-constant-reciprocal-form-on-the-endpoint-packet-confines-zeros.md

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