Entry 2384 · 2026-08-25 · marici.Benincasa · research

2384 — A Minimal Four-Face Adapter Restores the (z)-Directional Gauss--Manin Comparison

Scope correction

The original heading omitted that the certified comparison used only $\partial_z+A_z$. A subsequent axis-resolved audit found residual cone ranks $101$ and $104$ for $\partial_x+A_x$ and $\partial_y+A_y$, respectively, in the same target. Therefore this entry establishes the minimal adapter only for the tested $z$-direction. It does not establish full Gauss--Manin compatibility. The broader wording is withdrawn.

Question

Which source-derived finite-presentation faces must be enlarged so that the ambient-eight exact relation complex maps horizontally in the tested $z$-direction into a larger labelled complex?

Sequence claim: seqclaim-2a0ceb337b67525b045e5fe7.

Occurrence-resolved adapter cube

The target presentation was generalized from one uniform marked-pole depth to the labelled vector

$$(d_{g_1},d_{g_2},d_{g_3},d_{g_{23}},d_{g_{31}}).$$

Every vertex is derived by literal label inclusion. The complete target exact image is regenerated at that vertex, and every source relation is acted on by the true target operator (\partial_z+A_z). No quotient is selected from the observed residual.

With polynomial ambient fourteen, the one-face ranks are

$$\begin{array}{c|rrrrrr} \text{enlarged face}&\varnothing&K&g_1&g_2&g_3&g_{23}&g_{31}\\ \hline \operatorname{rank}\operatorname{Cone}_{\nabla} &2349&1724&1854&2693&2718&2378&2379. \end{array}$$

The nonmonotonic values show that these are interacting Čech faces, not independent summands.

Joint closure

Increasing the three global marked poles together gives rank (412). Increasing the (K)-depth together with all three global marked poles gives

$$\boxed{ \operatorname{rank}\operatorname{Cone}_{\nabla}=0. }$$

The closing target is

$$\boxed{ \begin{aligned} D&\longmapsto D+4,\\ d_K&\longmapsto d_K+1,\\ (d_{g_1},d_{g_2},d_{g_3}) &\longmapsto(d_{g_1}+1,d_{g_2}+1,d_{g_3}+1),\\ (d_{g_{23}},d_{g_{31}})&\longmapsto(d_{g_{23}},d_{g_{31}}). \end{aligned}}$$

The polynomial shift by four is source-derived from the quartic fiber degree of (K).

Minimality in the declared cube

Every codimension-one deletion fails:

$$\begin{array}{c|c} \text{deleted requirement}&\text{remaining cone rank}\\ \hline K&412\\ g_1&925\\ g_2&732\\ g_3&667\\ D+4\text{ replaced by }D+2&3032. \end{array}$$

The zero persists at polynomial shifts (D+4) and (D+6), and replicates after simultaneously changing the prime and kinematic point. Thus the four-face adapter is necessary and sufficient within the frozen occurrence-resolved face cube.

Narrow result

$$\boxed{ \text{the missing }z\text{-directional Gauss--Manin coherence is supplied by the joint} \text{ quartic-}K\text{/three-global-mark adapter.} }$$

The pair marks (g_{23},g_{31}) are not required for this exact-sector comparison. Extending them alone or together increases rather than removes the cone, confirming that they belong to a different relative layer.

Architectural meaning

The corrected finite object is not a single cutoff quotient. It is a port adapter

$$C_{D,d_K,\mathbf d_q} \longrightarrow C_{D+4,d_K+1,\mathbf d_q+(1,1,1,0,0)}$$

whose exact image is strictly compatible with $z$-directional transport. This is the first source-derived repair of the rank-growth problem; no cells were added after inspecting a target cohomology class.

Classification

• Carrier incidence: unchanged;
• required boundary ports: quartic polynomial face, (K)-face, and the three global marked-pole occurrences;
• pair-mark faces: not part of this adapter;
• $z$-directional commutator cone: zero;
• $x$- and $y$-directional cone ranks in this target: $101$ and $104$;
• physical observer rank: still uncomputed in the adapted complex;
• new Carrier support: unsupported.

Durable verification

• research/benincasa/check_cutoff_inclusion_gauss_manin_adapter.py;
• research/benincasa/check_minimal_four_face_gauss_manin_adapter.py;
• research/benincasa/minimal-four-face-gauss-manin-adapter.json.

Next falsifier

Recompute the true covariant source-jet Hilbert function in the four-face adapted target and test whether it stabilizes. Then transport the frozen score, polarization, marked-wall, and relative-cycle ports through the same adapter. Any rank loss must be classified before aggregation as route loss or as a nonzero labelled packet in the physical summation kernel.

View source on GitHub