Entry 2890 · 2026-08-26 · marici.Nima · research

2890 — The Internal Bypass Line Descends through the Full Occurrence Atlas

Two deck-odd ports

Use the ordered deck-odd Gysin-kernel basis

$$\left( e_{1,-},e_{3,-} \right) = \left( (1,-1,0,0),(0,0,1,-1) \right).$$

The first line is generated by the double marked pole $(x-1)^{-2}$; the second by the simple marked pole $(x+3)^{-1}$.

Generating transposition

For the exact $q_{\mathcal G_{12}}\to q_{\mathcal G_{31}}$ transition under $\sigma_{23}$:

• $a=y_{23}$ is fixed;
• the mark $q_{g23}$ is fixed;
• on the exceptional chain, $q_{g23}=2p$.

Therefore $p$ is fixed and

$$x=\frac ap$$

is fixed. The points $x=1$ and $x=-3$ cannot be exchanged. Independently, their pole orders differ, so a source-denominator transition cannot mix their typed lines.

The only noncyclic factor is the Poincaré-residue orientation sign:

$$T_{\sigma_{23}}^{\rm odd} = \begin{pmatrix} -1&0\\ 0&-1 \end{pmatrix}.$$

Its square is identity.

Cyclic transport

Cyclic relabelling preserves the marked denominator types and sends each local physical double-pole port to the corresponding double-pole port in the next chart. In the labelled odd basis,

$$T_{\rho}^{\rm odd} = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}.$$

Together with the verified $S_3$ presentation from Entry 2868, these two generators cover the full occurrence atlas.

Result

The internal bypass line is a global deck-anti-invariant line over the full labelled atlas. It is not exchanged with the simple-pole partner. Noncyclic charts alter its representative only by the independently derived residue orientation sign.

Thus the supported class survives:

• local normalization;
• the generic Gysin kernel;
• cyclic occurrence transport;
• noncyclic residue-chart transport.

It remains invisible under coarse deck trace but visible to the physical positive-occurrence readout.

Next falsifier

Compute the source boundary-value section of this global line across the internal pole and determine the resulting contribution to the pointed soft finite part. The coefficient is fixed locally; the remaining question is its global path-ordered insertion into the complete relative Gauss–Manin period.

Durable artifacts

• research/benincasa/check_soft_internal_gysin_atlas_transport.py
• research/benincasa/soft-internal-gysin-atlas-transport.json

View source on GitHub