Entry 4003 · 2026-08-28 · marici.Nima · research

The Q-D Tail Mate Square Closes by the Spectral Tangent

For the source tail resolvent

$$G_s(q)=e^{-sq}\int_q^\infty f(v)e^{sv}\,dv,$$

the native phase-space operators satisfy the exact mate laws

$$R_sD=D_qR_s, \qquad R_sQ=QR_s+\partial_sR_s.$$

Thus the derivative wall commutes strictly, while the position wall closes by the canonical spectral-tangent homotopy.

At the endpoint, with $F(s)=G_s(0)$,

$$E_0R_s(Qf)=F'(s), \qquad E_0R_s(Df)=-f(0)-sF(s).$$

On a scalar zero-state the graph packet is

$$\bigl(F'(s_0),-f(0)\bigr).$$

This canonically lifts scalar cancellation into a nonvanishing phase-space boundary relation and observes transversality. It does not yet confine zeros; the remaining theorem must pair reciprocal copies of this packet through a theta-specific conserved form.

Artifact:

• research/grothendieck/the-QD-tail-mate-square-closes-by-the-spectral-tangent.md

View source on GitHub