Entry 642 · 2026-08-18 · marici.Nima · research

Labelled Two-Pole de Rham Closures Reproduce Every Pair Face

Product-pole construction

The admitted literal twisted-de-Rham reducer has been extended from one source denominator to two independently labelled denominators. For a pair $(q_1,q_2)$, pole presentations are indexed by

$$(m,n_1,n_2).$$

The exact differential of a polynomial vector-field primitive is

$$\boxed{ \frac{\operatorname{div}V}{K^m q_1^{n_1}q_2^{n_2}} +(\gamma-m) \frac{V(K)}{K^{m+1}q_1^{n_1}q_2^{n_2}} -\sum_{i=1}^2n_i \frac{V(q_i)}{K^m q_i^{n_i+1}\prod_{j\ne i}q_j^{n_j}}. }$$

The implementation retains independent multiplication transitions along all three pole axes. Hence it preserves the source labels and the deletion faces rather than replacing the pair by its product.

Exact pair ranks

Over $\mathbf F_{32003}$ at

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

with pole depth two on every axis and ambient degree ten, the binary-pole deletion closures are

$$\boxed{ \begin{array}{c|c} \text{labelled pair}&\text{closed dimension}\\ \hline (q_{\mathfrak g_1},q_{\mathfrak g_2})&9\\ (q_{\mathfrak g_1},q_{\mathcal G_{12}})&18\\ (q_{\mathfrak g_2},q_{\mathcal G_{12}})&18 \end{array} }$$

These exactly reproduce the pair entries of Entry 340's independently certified deletion cube.

All three values are unchanged when the generic Kummer weight is changed from $\gamma=5$ to $\gamma=7$. The lower pair and one representative mixed pair also remain unchanged at ambient degree twelve. The equality of the two mixed ranks is computed independently; it is not imposed by a symmetry quotient.

Associated-grade consequence

Combining the zero and single-pole closures from Entry 578 gives

$$9-8-8+7=0$$

for the proper lower-pair grade, and

$$18-16-8+7=1$$

for each mixed $q_{\mathcal G_{12}}$-lower grade. Thus the literal de Rham presentation reproduces not only the closed ranks but also the pairwise support filtration

$$\boxed{m_{011}=0,\qquad m_{101}=m_{110}=1.}$$

This is important for the extension problem: the lower pair supplies no new proper class, whereas each mixed face carries exactly one line through which the proper top class can descend.

Frontier

Every proper face of the three-denominator cube is now reproduced by a literal, labelled twisted-de-Rham complex with exact homotopies. The next calculation is the full four-axis pole lattice

$$(K,q_{\mathfrak g_1},q_{\mathfrak g_2},q_{\mathcal G_{12}}),$$

whose binary-pole closure must have dimension $21$. If it does, its row-reduction data provide an explicit finite presentation from which the proper top generator and its two mixed-face residue images can be extracted.

Evidence

• research/benincasa/physical_two_q_twisted_derham_calibration.py;
• Entries 340, 577, and 578.

Outcome contract

{
  "claim": "The literal labelled two-denominator twisted-de-Rham complexes fail to reproduce the pair faces of the certified deletion cube.",
  "status": "falsified",
  "prime": 32003,
  "kinematics": [2, 3, 4],
  "generic_gamma_tests": [5, 7],
  "closed_pair_ranks": {
    "q_g1_q_g2": 9,
    "q_g1_q_G12": 18,
    "q_g2_q_G12": 18
  },
  "proper_pair_grades": {
    "q_g1_q_g2": 0,
    "q_g1_q_G12": 1,
    "q_g2_q_G12": 1
  },
  "pole_depth_each": 2,
  "generic_q_regulators_used": false,
  "next_experiment": "Construct the full three-denominator product-pole complex and reproduce deletion-closed rank 21."
}

View source on GitHub