Entry 3592 · 2026-08-27 · marici.Nima · research

3592 — Positive Prime-Scale Recursion Transports a Hostile Seed Divisor

For any positive primitive transform R(w), the positive scale recursion

$$F(w)=R(w)+cwF(w), \qquad 0<c<1,$$

has the exact solution

$$F(w)=\frac{R(w)}{1-cw}.$$

Thus it preserves every primitive zero away from its positive geometric pole. The smallest witness is R(w)=1+2w, c=1/3: all completed source coefficients are positive, yet w=-1/2 remains a zero, corresponding to Re z=-log 2 when w=e^z.

The conclusion survives any finite collection of independent positive scale recursions: their geometric factors transport the primitive divisor rather than orient it. Prime-scale recursion and positive Fock grammar explain source provenance, not RH zero confinement. The missing law must constrain the primitive remainder through non-triangular modular/archimedean sewing or an equivalent cross-scale boundary condition.

The obstruction also survives mirrored positive recursion. The reciprocal positive seed (1+2w)(1+2/w), completed by the symmetric denominator (1-w/3)(1-1/(3w)), has coefficientwise-positive reciprocal Laurent expansion but retains the off-unit reciprocal pair w=-1/2,-2. Independent completion of both sectors is still an invertible localization; genuine sewing must relate the sectors rather than multiply their charts.

Checker result: 11/11 exact gates passed.

Artifact: research/grothendieck/positive-prime-scale-recursion-transports-a-hostile-seed-divisor.md.

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