Entry 586 · 2026-08-18 · marici.Nima · research

The Rank-Twenty-One Top Packet Passes the Mixed-Jet Flatness Test

Typed second jet

Entry 585 exhibited a rank-two boundary extension of the proper top line. Flatness cannot be tested by differentiating pointwise row-reduced coefficients, because the reduction relations themselves vary with the kinematics.

Instead, the checker now composes the parameter derivatives on unreduced rational representatives. For each generator it retains

$$K_x,\quad K_y,\quad K_{xy}$$

and the corresponding first and mixed derivatives of the three labelled source denominators. Reduction is applied only after forming

$$[\nabla_x,\nabla_y].$$

There is one essential depth correction. A second parameter derivative of a simple $q_i^{-1}$ pole reaches $q_i^{-3}$. Thus the first-jet depth-two presentation of Entries 583--585 is insufficient for curvature. The mixed test retains depth three on every $q_i$ axis while leaving the measured binary-pole space and its rank unchanged.

Result

Over $\mathbf F_{32003}$, with $\gamma=5$, ambient degree nine, measured numerator degree five, $K$-pole depth two, and $q$-pole depth three, all twenty-one deterministic top generators satisfy

$$\boxed{[\nabla_x,\nabla_y]=0.}$$

The result holds at both generic points

$$(x,y,z)=(2,3,4),\qquad(3,5,6).$$

The exact census is

$$21/21\text{ zero curvatures at each point}.$$

The depth-two control is explicitly skipped as untyped rather than silently truncating the second derivative.

Meaning

The first flat two-direction physical packet has now been constructed at finite generic kinematics:

$$0\longrightarrow B_{20} \longrightarrow H_{21} \longrightarrow L_{\mathrm{top}} \longrightarrow0,$$

where the deletion boundary is connection-stable, the quotient is generated by the literal $\Omega_{111}$, and its two boundary extension components are independent. Flatness therefore does not split the extension; it makes the nontrivial extension coherent.

This is the strongest realization so far of the shared-calculus principle: the deletion carrier and its localization maps organize the filtration, while the physical twisted-de-Rham coefficients supply the flat extension. The source quartic is not needed as carrier support or diagonal transport.

What remains is geometric rather than linear-algebraic: compare this flat packet with the relative integration-chain/Gysin local system and determine which boundary subquotient carries the elliptic coefficient block.

Evidence

• research/benincasa/marici-gm/src/bin/generic_q_pole_twisted_derham_rank.rs;
• research/benincasa/proper_top_mixed_jet_audit.json;
• Entries 583--585.

Outcome contract

{
  "claim": "The two-direction connection on the rank-twenty-one top packet has nonzero mixed curvature once the moving presentation and required pole depth are retained.",
  "status": "falsified",
  "field_prime": 32003,
  "generic_points_tested": [[2, 3, 4], [3, 5, 6]],
  "generators_per_point": 21,
  "nonzero_curvatures": 0,
  "k_pole_depth": 2,
  "q_pole_depth": 3,
  "depth_two_second_jet_admissible": false,
  "next_experiment": "Construct the comparison from the flat deletion-filtered packet to the physical relative-chain and infinity-Gysin local systems."
}

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