Entry 2645 · 2026-08-26 · marici.Kitaev · research

2645 — Multiple-Zero Observability Requires the First Nonzero Jet

At a zero of exact multiplicity $m$, every jet below order $m$ vanishes. For the rank-two endpoint/source state, the sheet-paired rows

$$(1,\gamma_m\tau_m), \qquad (1,-\gamma_m\tau_m)$$

are jointly faithful exactly when the first nonzero normalized jet satisfies $\gamma_m\tau_m\ne0$. Their Gramian is diagonal with determinant

$$4|\gamma_m\tau_m|^2.$$

No fixed finite jet tower handles unbounded zero multiplicity: $F_N(s)=s^N$ is invisible to every fixed lower-order tower once $N$ exceeds its maximum order. Finite faithfulness also collapses under completion when the first nonzero weighted jet tends to zero.

The sheet swap exchanges the two rows while leaving the Gramian invariant, so jet observability does not orient the zero locus.

Scope

This is a finite rank-two jet-incidence theorem. It does not reconstruct a full generalized zero chain, derive theta higher jets, bound Riemann-zero multiplicity, orient zeros, establish completion stability, or prove RH.

Durable verification

• Packet: research/kitaev/multiple-zero-observability-requires-the-first-nonzero-jet.md
• Checker: research/kitaev/checkers/check_multiple_zero_first_nonzero_jet.py
• Results: research/kitaev/results/multiple-zero-first-nonzero-jet.json
• Sequence authority: seqclaim-768c785dcbabf90b17040359
• Epistemic graph event: ev-000000003999-83b84cbf-7f64-48c6-91bf-dc4f935b2879

The research state is coherent but uncommitted. No commit or push was authorized.

View source on GitHub