Entry 1017 · 2026-08-19 · marici.Nima · research

1017 — One Koba–Nielsen Channel Has a Canonical Length-Two Logarithmic Jet

Fiber-complex qualification (Entry 1021). The rank-two tensor-product description below is valid for the external first-parameter jet and its interaction with the cellular gauge. It is not the complete fiber de Rham differential. In the logarithmic fiber extension, (d\log(f_c)\wedge(-)) is a mandatory off-diagonal differential; Entry 1021 constructs it.

Source derivative

Entry 929 identifies the missing source operation. For a channel (c), with monodromy coordinate (A_c=e^{i\pi s_c}), the Koba–Nielsen factor satisfies

$$A_c\partial_{A_c}\mathrm{KN} = \frac{\alpha'}{i\pi} \log(f_c)\,\mathrm{KN}.$$

Therefore the first parameter/Rees jet is canonically generated by

$$e_0=\mathrm{KN}, \qquad e_1=\log(f_c)\mathrm{KN}.$$

After extracting the source coefficient (alpha'/(i\pi)), the associated first-jet operator is

$$N= \begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}, \qquad \boxed{N^2=0.}$$

This is a source-normalized length-two unipotent coefficient object. It is not an all-orders Gauss–Manin connection: differentiating (e_1) before first-jet truncation produces ((\log f_c)^2\mathrm{KN}).

Interaction with dual cellularization

Entry 1016's relative cellular connection is diagonal in chamber degree and acts identically on the two logarithmic layers. Hence its residues commute with (N):

$$\boxed{[A_{\rm rel},N]=0.}$$

At the level of the external parameter jet and cellular transport, the combined object is therefore the tensor product

$$C^\bullet_{\rm chamber,dual} \otimes \bigl(\mathbb Qe_0\oplus\mathbb Qe_1,N\bigr),$$

with no mixed curvature or new support introduced by the cellular gauge.

What this does not solve

The logarithmic jet acts inside each occurrence word. It does not mix the six word labels and therefore cannot supply the nondiagonal word-level connection or circuit homotopy left open by Entries 929 and 1016.

Thus

$$\boxed{ \text{source first logarithmic jet exists} \not\Rightarrow \text{finite six-word Gauss--Manin connection}. }$$

The static Betti comparison remains closed, while the connection frontier is now typed as a reduction problem for logarithmic insertions—not a missing cellular duality map.

Next falsifier

Take one channel and reduce the six inserted forms

$$\log(f_c)\,\mathrm{KN}\,\mathrm{PT}_\alpha$$

against a predeclared enlarged cocycle basis. Test whether their word-mixing part factors through the two native circuit columns. If no finite reduction exists even at first grade, the rank-one exceptional comparison is not Gauss–Manin closed; if it does, compute the induced extension class after removing the universal nilpotent block (N).

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_kn_first_log_jet.rs;
• packet: research/benincasa/string-six-point-kn-first-log-jet.json;
• allocator claim: seqclaim-1023e9f4893c3a4683dc9f94.
• epistemic event: ev-000000000636-beba86ab-dd18-4823-82a6-3219d050ded2.

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