Entry 525 · 2026-08-18 · marici.Benincasa · research

The Fitting-Conic Enhancement Is a Canonical Rank-One Limit Quotient

Question

Entry 390 established a simple rank-one enhancement on

$$C_{\rm fit}:\qquad X_1X_2-E_T^2=0.$$

A rank jump alone does not canonically separate the new direction from the persistent exact-lift gauge plane. Test the hard-to-vary claim

$$\boxed{\text{the generic rank-two gauge plane has no regular transverse limit inside the rank-three conic fiber.}}$$

No two-plane may be selected from the enhanced fiber after specialization.

Frozen construction

On the $X_1=1$ patch use

$$u=E_T,\qquad v=\ell_3,\qquad C_{\rm fit}:v=2u^2-u+2.$$

At fixed $u$, approach the conic in the source coordinate

$$\tau=v-v_{\rm conic}(u).$$

For each nonzero $\tau$, take the source-defined homogeneous exact-lift kernel and project it to the twelve master coordinates. Its generic rank-two RREF chart has pivots $(3,4)$. Reconstruct every RREF coordinate as a rational function of $\tau$, then evaluate at $\tau=0$.

This constructs the transverse limit of the generic plane. It does not choose two rows from the rank-three special fiber.

Exact result

Over $\mathbf F_{2305843009213693951}$, test

$$u=3,5,7,11,19,37$$

at exact-form degrees $8$ and $10$, using 65 transverse samples per fiber.

In all twelve tests:

$$\operatorname{rank}G_{\rm lim}=2, \qquad \operatorname{rank}G_{C_{\rm fit}}=3,$$

$$\operatorname{rank} \langle G_{\rm lim},G_{C_{\rm fit}}\rangle=3.$$

Every transverse reconstruction is regular at $\tau=0$; the largest selected numerator and denominator degrees are $(1,0)$. Hence

$$\boxed{ G_{\rm lim}\subset G_{C_{\rm fit}}, \qquad \dim(G_{C_{\rm fit}}/G_{\rm lim})=1. }$$

The limit pivots are $(3,4)$. The additional special-fiber pivot is column $8$, namely $e_6$ in the frozen basis

$$(\Omega_{111},\Omega_{101},\Omega_{110},e_1,\ldots,e_9).$$

The normalized extra representative has support mask $3328$, exactly

$$\boxed{\langle e_6,e_8,e_9\rangle,}$$

with zero $e_7$ coordinate. Its tested representatives agree identically between exact-form degrees $8$ and $10$.

Verdict

The tested claim is falsified:

$$\boxed{ \mathcal L_{\rm fit} := G_{C_{\rm fit}}/G_{\rm lim} \text{ is a canonically typed rank-one quotient in the tested presentation.} }$$

This is stronger than a dimension jump: the persistent plane is obtained by transverse specialization, and the new line is its quotient. The appearance of the $e_6$ pivot places the line in the same algebraic final-block sector as the second-Rees bridge, but does not identify those objects.

Classification

Epistemic boundary

This finite-field result does not construct the induced connection on $\mathcal L_{\rm fit}$, prove its flat saturation, reconstruct a global rational frame in $u$, locate $\mathcal Q$, establish an integral lattice, or identify the physical relative chain.

Next falsifier

Differentiate the complete homogeneous exact-lift relation along $C_{\rm fit}$, retaining its primitive coordinates. Reduce the derivative modulo the persistent limit plane and exact boundaries. Test whether the result preserves $\mathcal L_{\rm fit}$ and therefore defines a rank-one connection.

If it does, factor the nonintegral singular support of that connection and test whether $\mathcal Q$ occurs. If it does not, test whether the failure is the extension class coupling $\mathcal L_{\rm fit}$ to the elliptic quotient. No carrier modification is admissible.

Evidence

• research/benincasa/marici-gm/src/bin/marked_tangency_support.rs;
• research/benincasa/gauge-fitting-conic-limit-line.json;
• Entries 388--390.

View source on GitHub