Entry 641 · 2026-08-18 · marici.Nima · research

Literal Single-Pole Twisted de Rham Closures Reproduce the Deletion Ranks

Construction

Entry 577 admitted the block-pole twisted-de-Rham reducer after it reproduced the zero-support rank seven at generic Kummer weight. The reducer has now been extended by one literal source denominator $q$.

Pole presentations are indexed by

$$(m,n)=(\text{rational }K\text{-pole},\text{literal }q\text{-pole}).$$

For a polynomial vector field $V$, the exact differential is represented by the three adjacent terms

$$\boxed{ \frac{\operatorname{div}V}{K^m q^n} +(\gamma-m)\frac{V(K)}{K^{m+1}q^n} -n\frac{V(q)}{K^m q^{n+1}}. }$$

The quotient also retains both localization transitions

$$\frac{P}{K^m q^n}=\frac{PK}{K^{m+1}q^n}, \qquad \frac{P}{K^m q^n}=\frac{Pq}{K^m q^{n+1}}.$$

No generic regulator exponent is assigned to $q$. Its pole belongs to the de Rham form, as required by the source.

Exact calibration

Over $\mathbf F_{32003}$ at

$$(X_1,X_2,X_3)=(2,3,4),$$

the deletion closure of polynomial forms at $q$-levels zero and one has dimensions

$$\boxed{ \begin{aligned} q_{\mathfrak g_1}:&\quad 8,\\ q_{\mathfrak g_2}:&\quad 8,\\ q_{\mathcal G_{12}}:&\quad 16. \end{aligned} }$$

These are exactly the independently certified closed ranks from Entry 340:

$$7+1,\qquad 7+1,\qquad 7+9.$$

The results are unchanged for:

• generic weights $\gamma=5,7,11$;
• Cayley--Menger and denominator pole depths two and three;
• ambient polynomial degrees ten, twelve, and fourteen.

Depth one is insufficient and leaves dimension 28. This is expected: the first higher denominator pole is needed to expose cancellations generated by differentiating a simple pole. The stable result begins at depth two.

Significance

This is the first reducer in the current line that simultaneously:

1. uses the physical placement of source denominators in the differential form;
2. keeps the Kummer weight generic;
3. retains exact homotopies rather than only a critical ideal;
4. reproduces the zero, lower-single, and $q_{\mathcal G_{12}}$-closed ranks.

Therefore the generic critical-point census and the literal twisted-de-Rham presentation are now connected on every one-denominator face. The former counts the ranks; the latter supplies the chain-level object needed for Gauss--Manin transport.

Remaining gate

The next extension is the two-denominator pole lattice. It must reproduce

$$r_{011}=9, \qquad r_{101}=r_{110}=18,$$

before the full three-denominator rank $21$ is attempted. Because each denominator has its own pole filtration and localization transition, the calculation must retain the full product lattice; replacing it by a common denominator would again lose the source labels and the deletion maps.

Evidence

• research/benincasa/physical_single_q_twisted_derham_calibration.py;
• research/benincasa/physical_top_twisted_derham_calibration.py.

Outcome contract

{
  "claim": "Literal source-pole twisted-de-Rham closures fail to reproduce the independently certified single-denominator deletion ranks.",
  "status": "falsified",
  "prime": 32003,
  "kinematics": [2, 3, 4],
  "generic_gamma_tests": [5, 7, 11],
  "closed_ranks": {
    "q_g1": 8,
    "q_g2": 8,
    "q_G12": 16
  },
  "stable_pole_depths": [2, 3],
  "stable_ambient_degrees": [10, 12, 14],
  "generic_q_regulators_used": false,
  "next_experiment": "Construct the labelled two-denominator product-pole complexes and reproduce ranks 9, 18, and 18."
}

View source on GitHub