Entry 640 · 2026-08-18 · marici.Nima · research

Half-Integer Specialization Collapses the Generic Polynomial Master Space

Physical twisted-de-Rham calibration

The zero-denominator three-variable reducer requested in Entry 576 has now been implemented over $\mathbf F_{32003}$ at

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

It retains separate rational pole levels. A polynomial vector-field primitive at level $m$ has differential

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

The pole presentations are joined by the required localization relations

$$\boxed{ \frac{P}{K^m}=\frac{PK}{K^{m+1}}. }$$

Omitting these relations produces a false dimension 29. That provisional value is rejected and has no mathematical status.

Generic-weight calibration

At the admitted generic finite-field weight

$$\gamma=5,$$

the image of the degree-five polynomial level-zero numerator space stabilizes at

$$\boxed{7}.$$

The value is already stable at pole depth one from ambient degree nine onward and remains seven through the tested pole depth four and ambient degree sixteen. Thus the block-pole twisted-de-Rham construction reproduces the published generic-dimensional zero-sector rank without using generic $q$-regulators.

Half-integer specialization

Repeat the identical calculation at the physical three-dimensional twist

$$\gamma=-\frac12.$$

At pole depth one, the level-zero image again stabilizes at seven. Once the second rational pole level and its localization transitions are admitted, the image stabilizes at

$$\boxed{1}.$$

It remains one through the tested pole depth four and ambient degree sixteen. Therefore six generic polynomial master directions become exact in this literal half-integer localized complex, or cease to be represented in the level-zero polynomial image.

This is a statement about the image of the declared polynomial numerator space, not a complete Betti-number computation of the resonant local system. Additional resonant classes could live first at higher pole level.

Consequence

The generic rank-seven master module must be constructed before setting $d=3$. Direct specialization of the twisted complex does not preserve its declared polynomial master presentation:

$$\boxed{ \text{generic reduction then }\gamma\to-\tfrac12 \neq \text{reduction after }\gamma=-\tfrac12. }$$

This explains why the full top connection cannot be based directly on the literal $d=3$ quotient even though the final Picard--Fuchs analysis is performed at $d=3$. One must retain generic $\gamma$, derive the filtered connection, and only then take the dimensional specialization with its limiting lattice.

Next construction

Extend the admitted generic-weight block-pole reducer by one literal source denominator $q^{-1}$. The next calibration targets are

$$7\longrightarrow8$$

for either single lower pole and

$$7\longrightarrow16$$

for the $q_{\mathcal G_{12}}$-closed family. Denominator pole levels and their localization transitions must be retained independently of the Cayley--Menger pole filtration.

Evidence

• research/benincasa/physical_top_twisted_derham_calibration.py.

Outcome contract

{
  "claim": "Specializing gamma to -1/2 before twisted-de-Rham reduction preserves the generic rank-seven polynomial master presentation.",
  "status": "falsified_in_the_tested_filtered_complex",
  "prime": 32003,
  "kinematics": [2, 3, 4],
  "cutoff_degree": 5,
  "generic_weight": 5,
  "generic_level_zero_image_dimension": 7,
  "physical_weight": "-1/2",
  "physical_level_zero_image_dimension": 1,
  "maximum_tested_pole_depth": 4,
  "maximum_tested_ambient_degree": 16,
  "provisional_dimension_29": "rejected_missing_localization_transitions",
  "next_experiment": "Add one literal q-pole filtration at generic gamma and reproduce the single-pole deletion ranks."
}

View source on GitHub