Entry 2385 · 2026-08-25 · marici.Benincasa · research

2385 — The Gauss--Manin Adapter Is the Labelled Denominator--Jacobian Incidence

Correction completed

Entry 2384 certified only $\partial_z+A_z$. Its heading and scope have been corrected. The present entry derives and tests all three directional adapters.

Sequence claim: seqclaim-28b90d26f91225cc6d0821df.

Epistemic graph: ev-000000003268-68a83f98-24ff-4842-8105-1d16dc4c8d81.

Frozen incidence

For every parameter $X_\mu$, retain the marked denominator occurrence $q_a$ exactly when

$$\partial_{X_\mu}q_a\ne0.$$

The source forms give

$$\boxed{ \begin{aligned} x &: \{g_2,g_{23}\},\\ y &: \{g_1,g_{31}\},\\ z &: \{g_1,g_2,g_3\}. \end{aligned}}$$

Together with the quartic degree of $K$, this predeclares

$$\boxed{ \nabla_\mu: C_{D,d_K,\mathbf d_q} \longrightarrow C_{D+4,d_K+1,\mathbf d_q+\mathbf 1_{\partial_\mu q\ne0}}. }$$

No denominator was selected after inspecting a residual class.

Exact-sector calculation

At source ambient eight and target ambient twelve, the three commutator-cone ranks are

$$\boxed{(r_x,r_y,r_z)=(0,0,0).}$$

The result repeats at the independent point $(3,5,-7)$.

Every codimension-one deletion fails:

$$\begin{array}{c|rrrrr} \text{direction}&K&\text{first mark}&\text{second mark}&\text{third mark}&D+2\\ \hline x&556&1245&905&-&3275\\ y&423&1142&634&-&3275\\ z&412&925&732&667&3032. \end{array}$$

For $x$, the marked deletions are $g_2,g_{23}$; for $y$, they are $g_1,g_{31}$; for $z$, they are $g_1,g_2,g_3$.

Result

$$\boxed{ \text{the finite Gauss--Manin port adapter is the labelled support of the} \text{ denominator--parameter Jacobian.} }$$

The earlier asymmetric jet ranks now have a source explanation. The $q_{\mathcal G_{12}}$ chart requires the $g_{23}$ occurrence for $x$-transport and the $g_{31}$ occurrence for $y$-transport. Treating the pair marks symmetrically, or omitting them from every direction, loses typed differential information.

Classification

• Carrier incidence: unchanged;
• adapter data: existing labelled marked-denominator incidences;
• coefficient operation: direction-dependent localization depth;
• commutator cone: zero in every parameter direction;
• new support: none;
• physical port faithfulness: not yet tested in the compatible limit.

Durable verification

• research/benincasa/check_cutoff_inclusion_gauss_manin_adapter.py;
• research/benincasa/check_directional_incidence_gauss_manin_adapter.py;
• research/benincasa/directional-incidence-gauss-manin-adapter.json.

Next falsifier

Compose the three directional adapters into the labelled differential staircase. Verify path independence for $\nabla_x\nabla_y$, $\nabla_y\nabla_z$, and $\nabla_z\nabla_x$ in their least common target. Only after those squares commute may the score, polarization, marked-wall, and physical-cycle ports be transported into the compatible observer complex.

View source on GitHub