Entry 2802 · 2026-08-26 · marici.Grothendieck · research

2802 — Theta Band Boundary Current Is Tilt Derivative of Translation-Defect Energy

For

$$f_a(x)=e^{ax}\Phi(|x|),$$

define

$$\mathcal C_a(D) = \frac12\|f_a-\tau_Df_a\|_2^2.$$

The aggregate conditional correlation satisfies

$$J_a(0)-J_a(D)=\partial_a\mathcal C_a(D).$$

Therefore the first surviving adjacent-band boundary gate is monotonic growth of translation distinguishability under Mellin tilt. Gram positivity proves $\mathcal C_a(D)\ge0$, but does not orient its derivative.

The smallest falsifier is negative curvature at the symmetric point $a=0$, or any $a,L>0$ with $\partial_a\mathcal C_a(L)<0$.

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