Entry 1021 · 2026-08-19 · marici.Nima · research

1021 — The Physical Logarithmic Jet Is a Triangular de Rham Extension

Correction to the constant-layer picture

Entry 1017 correctly derives the external first-parameter jet generated by

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

Inside the integration-variable de Rham complex, however, the logarithm is not a constant coefficient. For the (34) channel,

$$f_{34}=z_3-z_4, \qquad \ell_{34}=\log(f_{34}), \qquad d\ell_{34}=d\log(f_{34}).$$

Thus the first logarithmic enlargement of the Koba–Nielsen twisted de Rham complex is

$$\Omega^\bullet_{\log,\leq1} = \Omega^\bullet_{\rm KN} \oplus \ell_{34}\Omega^\bullet_{\rm KN},$$

with differential

$$\boxed{ D(\alpha,\beta) = \left( D_{\rm KN}\alpha +d\log(f_{34})\wedge\beta, D_{\rm KN}\beta \right). }$$

Exact complex identity

Set ( heta=d\log(f_{34})). Since (d\theta=0), graded Leibniz gives

$$D_{\rm KN}(\theta\wedge\beta) =-\theta\wedge D_{\rm KN}\beta.$$

The two off-diagonal terms in (D^2) cancel, while (D_{\rm KN}^2=0). Therefore

$$\boxed{D^2=0.}$$

The resulting coefficient sequence is

$$0 \longrightarrow \Omega^\bullet_{\rm KN} \longrightarrow \Omega^\bullet_{\log,\leq1} \longrightarrow \Omega^\bullet_{\rm KN} \longrightarrow0.$$

Its extension class is represented by (d\log(f_{34})), whose residue on the existing channel divisor (f_{34}=0) is (1). Hence it is generically nontrivial as a logarithmic de Rham extension.

Consequences

This supplies the missing type distinction between Entries 1017 and 1019:

• Entry 1019 computes the formal tangent of rational comparison matrices;
• the physical logarithmic insertion lives in the triangular complex above;
• their comparison requires reducing the off-diagonal (d\log(f_{34})\wedge(-)) operation through the labelled source complex.

No new carrier divisor is introduced:

$$\boxed{ \text{new first-jet complexity} = \text{coefficient extension on the existing }f_{34}\text{ wall}. }$$

This is direct evidence for the sector-lens architecture: the string carrier is unchanged, while the coefficient lens acquires a unipotent logarithmic extension.

Scope

The full reduction to a finite six-word or chamber basis is not yet computed. In top degree the wedge term vanishes dimensionally, but it acts on the lower-degree primitives used in twisted de Rham reduction; discarding it would again mistake a formal top-form derivative for Gauss–Manin transport.

Next falsifier

Build the degree-two primitive module used by the six-point Parke–Taylor reduction and apply (d\log(f_{34})\wedge(-)). Test whether its image lies in the existing two circuit columns after exact reduction. The mapping cone against Entry 1019's formal tangent is the first intrinsic physical extension class.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_log_insertion_complex.rs;
• packet: research/benincasa/string-six-point-log-insertion-complex.json;
• allocator claim: seqclaim-9ee97b681065d1dcd7e4f3ba.
• epistemic event: ev-000000000640-4905d588-cb60-4803-8da1-eb2abcbd3904.

View source on GitHub