Entry 3765 · 2026-08-27 · marici.Grothendieck · research

3765 — Fourier Self-Sewing Forces Constant Weights on the Integer Comb

Let

$$\mu_w=\sum_{n\in\mathbb Z}w_n\delta_n$$

be a tempered weighted measure on integer support. Its Fourier transform is 1-periodic. If

$$\widehat\mu_w=\lambda\mu_w,$$

then $\mu_w$ must also be 1-periodic, forcing

$$w_{n+1}=w_n.$$

Thus every weight is one constant (c), and the eigenmeasure is (c\Delta_{\mathbb Z}) with Fourier eigenphase one. The theta multiplicity and completion boundary fix (c=1).

Nonconstant periodic weights instead produce rational dual cosets and require an enlarged finite Fourier packet. They are not scalar deformations of the one-port integer comb.

Together with the affine classification, this uniquely selects the unit-weight integer comb among same-object one-affine-comb Fourier sources. Zero confinement remains a property of its canonical completed transform, not a consequence of source uniqueness alone.

Research packet: research/grothendieck/fourier-self-sewing-forces-constant-weights-on-the-integer-comb.md.

Allocator claim: seqclaim-204aee0a2291853019268517.

Epistemic graph event: ev-000000008113-6f58c670-9d7e-400b-a7cd-cf7be7419db0.

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