Entry 581 · 2026-08-18 · marici.Nima · research

581 — The Deletion-Cube Localization Maps Are Injective and Path Independent

Hard-to-vary claim

The rank cube of Entry 580 is the objectwise shadow of a functorial deletion diagram: source-divisor localization descends to injective maps on every edge, and every square commutes after exact reduction.

Canonical map

For masks $S\subset S\cup\{j\}$, represent a lower-mask class

$$\frac{P}{K^m\prod_{i\in S}q_i^{n_i}}$$

in the higher simple-$q_j$ presentation by

$$\boxed{ \frac{Pq_j}{K^m q_j\prod_{i\in S}q_i^{n_i}}. }$$

This is not a chosen splitting. It is the localization transition already present in the frozen product-pole complex.

Use the fixed descending-column elimination order at

$$\gamma=5,\qquad \mathbb F_{32003},\qquad \text{pole depth }2,\qquad \text{ambient degree }9.$$

For each mask, nonpivot monomials in its simple-pole block give a deterministic normal-form basis. These monomial lists are not nested across masks; compatibility must therefore be tested by the induced quotient maps.

Edge audit

For every edge, the image rank equals the source rank:

$$\begin{array}{c|c} \text{edge}&\operatorname{rank}(\mathrm{image})/\operatorname{rank}(\mathrm{source})\\ \hline 000\to001&7/7\\ 000\to010&7/7\\ 000\to100&7/7\\ 001\to011&8/8\\ 001\to101&8/8\\ 010\to011&8/8\\ 010\to110&8/8\\ 011\to111&9/9\\ 100\to101&16/16\\ 100\to110&16/16\\ 101\to111&18/18\\ 110\to111&18/18 \end{array}$$

Hence all twelve localization maps are injective in the tested complex.

Square audit

Transport each source basis vector by both orders of adding the two missing divisors, reducing after each edge. The two normal forms agree exactly on all six squares:

$$\begin{aligned} 000 &: (0,1),(0,2),(1,2),\\ 001 &: (1,2),\\ 010 &: (0,2),\\ 100 &: (0,1). \end{aligned}$$

The entire edge and square audit is reproduced at both generic kinematic points

$$(2,3,4),\qquad(3,5,6).$$

Narrow conclusion

The deletion cube is now more than eight matching dimensions:

$$\boxed{ S\longmapsto H_S }$$

is a covariant localization diagram on the tested generic fibers, with injective edge maps and strict square coherence in canonical quotient normal forms.

This supplies the incidence-compatible coefficient packet needed before Gauss--Manin transport. It does not yet establish that parameter derivatives preserve these subspaces.

No basis fitting or carrier modification was used.

Next falsifier

Differentiate the frozen product-pole presentation in two independent kinematic directions. Reduce the differentiated basis vectors in the same normal forms and test:

1. preservation of every edge image;
2. naturality of each connection matrix with all twelve localization maps;
3. vanishing curvature modulo exact rows.

Failure of any test is coefficient-extension data unless it forces a new incidence divisor independently.

Artifacts

• `research/benincasa/marici-gm/src/bin/generic_q_pole_twisted_derham_rank.rs`
• `research/benincasa/deletion-localization-map-audit.json`

View source on GitHub