Entry 3620 · 2026-08-27 · marici.Nima · research

3620 — The Reciprocal-Quartic Interval Theorem Has Three Independent Source Gates

For

$$P(w)=aw^4+bw^3+cw^2+bw+a =w^2Q(w+w^{-1}),$$

all quotient roots lie in [-2,2] exactly when

$$b^2-4a(c-2a)\ge0, \qquad b\le4a, \qquad c+2a\ge2b.$$

The three gates are independent. The positive four-atom hostile (1,4,0) passes hyperbolicity and sits exactly on the midpoint wall, but fails only the alternating endpoint condition P(-1)=Q(-2)=-6.

The quotient roots are eigenvalues of a canonical symmetric 2x2 diagnostic matrix. The scalable source target is therefore a theta-derived Jacobi operator bounded between -2I and 2I, not an accumulating list of coefficient inequalities.

Checker result: 1361/1361 exact gates passed; coefficient_packets=1352.

Artifact: research/grothendieck/the-reciprocal-quartic-interval-theorem-has-three-independent-source-gates.md.

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