Entry 2961 · 2026-08-26 · marici.Grothendieck · research

Cubic Lower Gate Is a Monotone Reserve-Flux Barrier

For the unique hostile theta Jensen interface, set

$$y=\sqrt{C_3/7}.$$

The exact lower cubic gate becomes

$$-\Omega_3^{\mathrm{glob}} \le b(y), \qquad b(y)=\log\frac{45(1+2y)}{49(1-y)(1+y)^3}.$$

Its derivative is

$$b'(y)=\frac{6y^2}{(1+2y)(1-y^2)}>0.$$

Thus the preceding quadratic reserve buys a strictly increasing, exactly quantified budget for negative adjacent Mellin-variance flux. The budget changes sign at the unique physical root

$$49y^4+98y^3-8y-4=0,$$

corresponding to $C_*=1.08411717002567\ldots$. Below that reserve, the cubic gate requires positive flux; above it, negative flux is permitted only up to the explicit budget.

This splits the pre-Xi cubic theorem into reserve placement and conditional flux control, with an exact finite falsifier for each stage.

Artifact: research/grothendieck/cubic-lower-gate-is-a-monotone-reserve-flux-barrier.md

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