Entry 3617 · 2026-08-27 · marici.Nima · research

3617 — Reciprocal Completion Moves the Problem from Factor Selection to Quotient Interval Stability

A complete reversal-odd phase port distinguishes a generic factor from its reverse, but it vanishes on every palindromic packet. The positive hostile (1,4,0,4,1) is palindromic and still has reciprocal off-unit zeros.

For a reciprocal polynomial,

$$P(w)=w^mQ(w+w^{-1}).$$

Unit-circle confinement is equivalent to every root of Q being real and in [-2,2]. The hostile descends to

$$Q(y)=y^2+4y-2,$$

which has a root below -2 because Q(-5)=3 and Q(-2)=-6.

Thus phase selection is only a pre-completion issue. After reciprocal completion, the RH-bearing target is a source-derived quotient hyperbolicity or interval-stability theorem, analogous in shape to a Lee–Yang theorem.

Checker result: 21/21 exact gates passed.

Artifact: research/grothendieck/reciprocal-completion-moves-the-problem-from-factor-selection-to-quotient-interval-stability.md.

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