Entry 582 · 2026-08-18 · marici.Nima · research

The Literal Top Form Generates the Proper Three-Pole Quotient

Question

Entry 580 established that the full labelled three-pole twisted-de-Rham presentation has rank (21), while Möbius inversion predicted a one-dimensional proper top grade. Entry 581 then supplied coherent, injective localization maps on the entire deletion cube. The remaining question was whether the literal constant-numerator source form

$$\Omega_{111}=\frac{dc\wedge da\wedge db}{K^\gamma q_1q_2q_3}$$

actually represents that grade.

Certificate calculation

The finite-field reducer now retains its complete pivot dictionary. Every binary-pole low column is reduced to quotient normal form. Inside the full (111) presentation, let (B) be the span of the columns belonging to the seven proper pole faces, equivalently those with at least one pole label equal to zero.

At

$$(X_1,X_2,X_3)=(2,3,4),\qquad \gamma=5,$$

over $\mathbf F_{32003}$, at pole depth two, ambient degree nine, and measured numerator degree five, the exact ranks are

$$\dim H_{111}=21, \qquad \dim B=20.$$

The normal form of the literal constant-numerator (111) column is nonzero, and adjoining it to (B) gives rank (21):

$$\boxed{ H_{111}/B\simeq\mathbf F_{32003}\langle[\Omega_{111}]\rangle . }$$

The quotient and the class are canonical relative to the declared deletion diagram. The displayed normal-form coordinates themselves depend on the monomial order and are not promoted to geometric data.

Meaning

This closes the representative-identification gap left by Entry 580. The unique proper top direction is not merely a rank remainder and does not require a fitted complement: it is generated by the literal physical three-pole form.

It does not yet identify the Gauss--Manin extension of this line by the twenty-dimensional boundary span. That requires differentiating the full presentation in at least two independent kinematic directions and reducing the derivatives with compatible certificates, as required by Entry 575.

Evidence

• research/benincasa/physical_top_proper_quotient_certificate.py;
• research/benincasa/physical_two_q_twisted_derham_calibration.py;
• Entries 580--581.

Outcome contract

{
  "claim": "The literal constant-numerator three-pole form vanishes in, or fails to generate, the proper top quotient.",
  "status": "falsified",
  "prime": 32003,
  "kinematics": [2, 3, 4],
  "gamma": 5,
  "pole_depth_each": 2,
  "ambient_degree": 9,
  "cutoff_degree": 5,
  "top_rank": 21,
  "proper_face_span_rank": 20,
  "proper_top_quotient_rank": 1,
  "literal_omega_111_generates": true,
  "next_experiment": "Differentiate the retained presentation in two independent kinematic directions and compute the induced filtered Gauss-Manin extension modulo the coherent deletion-face span."
}

View source on GitHub