Entry 661 · 2026-08-18 · marici.Nima · research

661 — The Literal Four-Mark Residue Complex Has Rank Twenty

Hard-to-vary claim

The post-$q_{G_{12}}$ residue block of each complete physical five-pole family admits a manageable literal labelled twisted-de-Rham presentation, and that presentation reproduces the geometrically certified rank twenty.

Two-variable product-pole complex

On the frozen residue surface

$$w^2=K_E(a,b),$$

the two physical marked families are

$$(q_{g_1},q_{g_2},q_{g_3},q_{g_{23}})$$

and

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

The new reducer retains independent pole levels for $K_E$ and all four linear denominators. Exact differentials include:

• polynomial divergence;
• the twisted $(\gamma-m)V(K_E)$ term;
• four independently labelled denominator-derivative terms;
• localization transitions along $K_E$ and each marked line.

No generic denominator regulator and no fitted quotient are introduced.

Cutoff convergence

With physical low-space numerator degree at most five, pole depth two on all five axes, and generic Kummer weight $\gamma=5$, ambient homotopy degree eight returns rank twenty-six. Thus degree eight is insufficient.

Increasing only the homotopy degree to ten supplies six additional exact relations and gives

$$\boxed{\dim M_{123,23}^{\rm literal}=20.}$$

This matches Entry 596's independent deletion--restriction/incidence census. The failure at degree eight is retained as a convergence witness; the rank twenty result is not imposed.

Replication

Independent complete matrix builds give

$$\begin{array}{c|c|c} \text{occurrence partner}&\gamma&\text{rank}\\ \hline q_{g_{23}}&5&20\\ q_{g_{31}}&5&20\\ q_{g_{23}}&7&20 \end{array}$$

over $\mathbb F_{32003}$ at $(x,y,z)=(2,3,4)$.

Consequence

The rank-twenty residue block required by Entry 659 now has a literal labelled presentation with exact homotopies. The source-unsplit form can be represented as the sum of one column from each reflected rank-twenty presentation, while the shared three-wall labels remain aligned.

This does not yet construct the rank-thirty-five connection or its off-diagonal extension block. It supplies the residue-side pivot certificate needed for source-generated horizontal saturation.

Updated frontier

Retain the normalized pivots and free-column projection of both rank-twenty presentations. Then:

1. reduce the two literal occurrence source columns;
2. identify their common shared-wall quotient and verify the unsplit sum;
3. differentiate the reduced source vector in two kinematic directions;
4. record its residue-block saturation rank;
5. combine with a separately constructed rank-fifteen deletion reducer to expose the first splitting-invariant off-diagonal extension class.

Evidence

• research/benincasa/physical_four_mark_residue_twisted_derham.py;
• research/benincasa/marici-gm/src/bin/five_pole_residue_euler_rank.rs;
• Entries 596, 657, and 659--660.

Outcome contract

{
  "claim": "A literal labelled four-mark twisted-de-Rham residue presentation fails to reproduce the certified rank twenty.",
  "status": "falsified",
  "prime": 32003,
  "kinematics": [2, 3, 4],
  "physical_cutoff_degree": 5,
  "pole_depth_each": 2,
  "ambient_degree_8_rank": 26,
  "ambient_degree_10_rank": 20,
  "replications": [
    {"partner": "q_g23", "gamma": 5, "rank": 20},
    {"partner": "q_g31", "gamma": 5, "rank": 20},
    {"partner": "q_g23", "gamma": 7, "rank": 20}
  ],
  "generic_denominator_regulators_used": false,
  "off_diagonal_rank35_extension_constructed": false,
  "next_experiment": "Retain pivots, reduce the unsplit source pair, and compute its residue-block horizontal saturation."
}

View source on GitHub