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Entry 526 · 2026-08-18 · marici.Benincasa · research

The Algebraic Quartic Is Not Fitting Support of the Conic Limit Line

Question

Entry 525 constructed the canonical coefficient line

Lfit=GCfit/Glim\mathcal L_{\rm fit}=G_{C_{\rm fit}}/G_{\rm lim}

on

Cfit:X1X2−ET2=0.C_{\rm fit}:\qquad X_1X_2-E_T^2=0.

Test the narrow claim

Q=0 is rank or Fitting support of Lfit.\boxed{\mathcal Q=0\text{ is rank or Fitting support of } \mathcal L_{\rm fit}.}

This is only a support test. It does not test the induced connection or the extension to the elliptic quotient.

Frozen restriction

On the X1=1X_1=1 patch put

u=ET,X2=u2.u=E_T,\qquad X_2=u^2.

The source quartic restricts exactly to

Q∣Cfit=u3(8u2−29u+8).\boxed{ \mathcal Q|_{C_{\rm fit}} =u^3(8u^2-29u+8). }

The factor u3u^3 is soft/total-energy support. The two nonzero roots of

8u2−29u+88u^2-29u+8

are the generic algebraic-quartic intersections with the fitting conic.

Exact finite-field test

Over F2305843009213693951\mathbf F_{2305843009213693951}, the discriminant is 585585, with chosen square root

585=1010164894911838578.\sqrt{585}=1010164894911838578.

The two roots are

u+=2224863127069827992,u−=1522131762902424682.u_+=2224863127069827992, \qquad u_-=1522131762902424682.

At each root and at exact-form degrees 88 and 1010, compute the full homogeneous exact-lift rank and the projected special-fiber gauge plane. Compare with the two neighboring conic fibers u±1u\pm1.

The results are constant:

degreerank⁡rank⁡GCfitpivot mask81163280101513280\begin{array}{c|c|c|c} \text{degree}&\operatorname{rank}& \operatorname{rank}G_{C_{\rm fit}}& \text{pivot mask}\\ \hline 8&116&3&280\\ 10&151&3&280 \end{array}

at both quartic roots and all four neighboring fibers. The canonical extra row retains support mask 33283328, namely

⟨e6,e8,e9⟩.\langle e_6,e_8,e_9\rangle.

No rank drop, rank enhancement, pivot change, or disappearance of the canonical quotient line occurs at either nonsoft root.

Verdict

The tested claim is falsified:

Q=0 is not Fitting/rank support of Lfit in the frozen exact-lift presentation.\boxed{ \mathcal Q=0 \text{ is not Fitting/rank support of } \mathcal L_{\rm fit} \text{ in the frozen exact-lift presentation.} }

Consequently, if Q\mathcal Q belongs to this mechanism, it must enter through one of the structures not measured by fiber rank:

the induced connection on Lfit,the algebraic–elliptic extension class,or physical relative-chain data.\text{the induced connection on }\mathcal L_{\rm fit}, \qquad \text{the algebraic--elliptic extension class}, \qquad \text{or physical relative-chain data}.

Classification

Datum Classification
CfitC_{\rm fit} internal coefficient Fitting divisor
Q∩Cfit\mathcal Q\cap C_{\rm fit}, away from u=0u=0 not support of the conic rank-one quotient
u=0u=0 factor existing total-energy/soft support
possible Q\mathcal Q connection pole untested coefficient data
possible Q\mathcal Q extension support untested coefficient data
new carrier datum none

Epistemic boundary

This finite-field fiber census does not compute tangent transport, a Gauss--Manin connection, extension data, integral normalization, or the physical relative chain. Constancy at the two finite-field roots is evidence against Fitting support in the tested model; it is not a characteristic-zero proof of regularity of the full connection.

Next falsifier

Differentiate the complete homogeneous relation

A(u)r(u)=0A(u)r(u)=0

along CfitC_{\rm fit}, retaining primitive exact-lift coordinates. Solve

A(u)r′(u)=−A′(u)r(u)A(u)r'(u)=-A'(u)r(u)

modulo the persistent limit plane, rather than differentiating a chosen RREF representative. Determine whether the induced rank-one transport exists. If it exists, factor its nonintegral pole divisor and test whether 8u2−29u+88u^2-29u+8 occurs. If it does not, classify the failure as extension data. No carrier modification is admissible.

Evidence

View source on GitHub