Entry 585 · 2026-08-18 · marici.Nima · research

The Proper Top Line Has a Rank-Two Boundary Extension

Setup

Entries 581 and 583 establish a coherent deletion cube preserved by the two kinematic derivatives $\nabla_x,\nabla_y$. Entry 582 identifies the proper top quotient as the line generated by the literal form $\Omega_{111}$.

Let $B\subset H_{111}$ be the span of the three codimension-one face images. The retained normal-form certificate gives

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

For each direction, reduce the derivative of the literal generator as

$$\nabla_\lambda\Omega_{111} =A_\lambda\Omega_{111}+\eta_\lambda, \qquad \eta_\lambda\in B.$$

No complement of $B$ is selected: $A_\lambda$ is computed in the one-dimensional quotient, and $\eta_\lambda$ is then reduced in the full twisted-de-Rham presentation.

Result

At both generic points

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

over $\mathbf F_{32003}$, both boundary components are nonzero and

$$\boxed{\operatorname{rank}\langle\eta_x,\eta_y\rangle=2.}$$

The descended scalar pairs are respectively

$$(A_x,A_y)=(14226,24891), \qquad (30862,14858) \pmod{32003}.$$

An independently added signed-quartic comparison in the same checker shows that neither pair equals $\tfrac12d\log\mathcal Q$: all four directional comparisons fail.

Meaning

The one-parameter insufficiency of Entry 575 is now realized concretely. The proper physical line exists and its connection descends, but its literal source generator mixes with two independent boundary directions. Therefore the rank-one quotient is not the whole physical packet, and the extension cannot be reconstructed from a scalar $\mathcal Q$-connection.

This also sharpens Entry 526. The quartic is neither carrier/Fitting support nor the scalar connection of the proper top quotient in the tested presentation. Any remaining role for $\mathcal Q$ must lie in a larger relative-chain/Gysin extension, a discriminant comparison, or a different normalization derived from physical geometry.

The next decisive test is the mixed second jet: compute $[\nabla_x,\nabla_y]\Omega_{111}$ together with the parameter derivatives of the reduction coefficients. Vanishing would produce the first flat two-direction filtered packet containing the physical line; failure would locate the missing relative-chain correction.

Evidence

• research/benincasa/marici-gm/src/bin/generic_q_pole_twisted_derham_rank.rs;
• research/benincasa/proper_top_extension_audit.json;
• Entries 581--583.

Outcome contract

{
  "claim": "The two kinematic derivatives of the literal proper-top generator mix with at most one boundary direction, or its quotient connection is one-half dlog of the signed quartic.",
  "status": "falsified",
  "generic_points_tested": [[2, 3, 4], [3, 5, 6]],
  "field_prime": 32003,
  "top_rank": 21,
  "boundary_rank": 20,
  "two_direction_boundary_extension_rank": 2,
  "half_dlog_quartic_matches": 0,
  "half_dlog_quartic_tests": 4,
  "next_experiment": "Construct mixed second jets of the parameter-dependent reduction and test the curvature of the filtered top packet."
}

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