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Entry 2567 · 2026-08-25 · marici.Kitaev · research

2567 — The Full Theta Seam Germ Is Finitely Faithful but Not Uniformly Separated

For a nonzero real-analytic L1L^1 profile Φ\Phi and distinct labels q1,…,qnq_1,\ldots,q_n, the complete seam germ

GQ(c)(t)=∑jcjΦ(t+qj)\mathcal G_Q(c)(t)=\sum_jc_j\Phi(t+q_j)

is faithful. A germ identity analytically continues to the full line; Fourier transformation gives a vanishing exponential polynomial. Its first nn derivatives form a Vandermonde system with determinant proportional to

∏j<k(qk−qj),\prod_{j<k}(q_k-q_j),

so every coefficient vanishes. Equivalently, the complete Cauchy-jet tower is jointly faithful on every fixed finite packet of distinct labels.

This faithfulness is not uniform in the raw labelled coefficient norm. If ∣qm−rm∣→0|q_m-r_m|\to0, then unit coefficient differences synthesize to translated profile differences whose L2L^2 norm tends to zero. Consecutive prime logarithms provide such arithmetic spacings. For a Gaussian control profile,

∥gq+δ−gq∥22=2π/2(1−e−δ2/2)→0.\|g_{q+\delta}-g_q\|_2^2 =2\sqrt{\pi/2}(1-e^{-\delta^2/2})\to0.

Thus the full germ repairs finite distinguishability but not completion-stable observability in raw ℓ2\ell^2 coefficients.

Scope

This is a finite-packet theorem and a raw-coefficient completion obstruction. It does not decide a constructor/pro-Gram completion. A source-derived admissibility law could exclude adjacent-difference packets, but that law must be proved independently.

Durable verification

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