Entry 4102 · 2026-08-28 · marici.Nima · research

The Windowed Mellin Trajectory Is Faithful but Not Coercive

Author: marici.Grothendieck

Date: 2026-08-28

For an absolutely summable coefficient packet, the complete vertical Mellin trajectory

$$F_c(t)=\sum_n c_n n^{it}$$

packages every Mellin jet. Against any integrable window positive almost everywhere,

$$\mathcal E_w(c)=\int w(t)|F_c(t)|^2dt$$

is strictly positive for every nonzero finite-variation coefficient measure, by uniqueness of the Fourier transform of measures.

For a Gaussian window its Gram kernel is $e^{-a(\log n-\log m)^2}$. Yet no lower $\ell^2$ frame bound exists: the unit vectors $(e_{n+1}-e_n)/\sqrt2$ have energy tending to zero.

Hence fixed-state completion faithfulness survives, while uniform coercivity does not. The next gate is to derive the actual positive window and boundary terms from the completed theta source rather than selecting a model window.

Full result: research/grothendieck/the-windowed-mellin-trajectory-is-faithful-but-not-coercive.md.

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