Entry 528 · 2026-08-18 · marici.Benincasa · research

Primitive Transport Does Not Canonically Descend to the Fitting-Conic Line

Question

For the canonical special-fiber line

$$\mathcal L_{\rm fit}=G_{C_{\rm fit}}/G_{\rm lim},$$

test whether the complete primitive equation

$$A(u)r'(u)=-A'(u)r(u)$$

canonically induces line transport. Two choices must be removed independently:

1. the primitive exact lift of a fixed twelve-master representative;
2. the affine solver section for $r'$.

Frozen computation

Along $C_{\rm fit}$, the entries of $A(u)$ have degree at most $30$: $K$ has degree at most $12$, and the largest common-denominator column has degree at most $30$. Compute $A'$ by exact interpolation and require agreement between derivative stencils of orders $32$ and $40$.

Use the normalized $e_6$-pivot representative of Entry 525, solve all primitive exact coordinates, and retain the complete relation. Test exact-form degrees $8,10$ at $u=3,5,7$ and at both nonsoft roots of

$$8u^2-29u+8.$$

At every point the derivative stencils agree and the primitive transport equation is solvable.

Invariance audit

For four independent exact-only null vectors, vary the primitive lift while holding all twelve master coordinates fixed. The difference between the two transport solutions satisfies

$$\operatorname{rank}\langle G_{\rm lim},\Delta r'_{\rm master}\rangle=2.$$

Thus primitive-lift ambiguity dies in the persistent plane.

For eight homogeneous null vectors with nonzero master projection, vary the affine solver section. Uniformly,

$$\operatorname{rank}\langle G_{\rm lim},\Delta r'_{\rm master}\rangle=3,$$

while

$$\operatorname{rank}\langle G_{C_{\rm fit}},\Delta r'_{\rm master}\rangle=3.$$

Hence solver ambiguity vanishes modulo the full special gauge plane, but not modulo the persistent two-plane.

The raw transported master vector itself is transverse to the full gauge plane:

$$operatorname{rank}\langle G_{C_{\rm fit}},r'_{\rm master}\rangle=4.$$

Verdict

$$\boxed{ A r'=-A'r \text{ defines ambient Grassmannian transport, but does not by itself define a canonical connection on }\mathcal L_{\rm fit}. }$$

A choice of solver section changes the would-be line component. Selecting a splitting after seeing this defect would be post hoc and is prohibited.

Therefore no diagonal pole divisor—and in particular no $8u^2-29u+8$ factor—may yet be assigned to $\mathcal L_{\rm fit}$.

Classification

Epistemic boundary

This calculation rejects only the claim that differentiating the homogeneous primitive relation canonically supplies a line connection. Independently, Entry 527 uses ambient connection regularity to exclude $\mathcal L_{\rm fit}$ as the intrinsic $\mathcal Q$-sector of the generic nine-master module. The remaining admissible possibility is source-derived physical relative-chain data not present in that absolute module.

Next falsifier

Construct the source-defined inhomogeneous Gauss--Manin target along $C_{\rm fit}$, including the elliptic quotient, and test whether its block decomposition canonically fixes the solver section. Then distinguish:

• a diagonal $8u^2-29u+8$ pole;
• only an off-diagonal algebraic--elliptic factor;
• no quartic support.

No fitted splitting or carrier modification is admissible.

Evidence

• research/benincasa/marici-gm/src/bin/marked_tangency_support.rs;
• research/benincasa/gauge-fitting-conic-primitive-transport.json;
• Entries 525--526.

View source on GitHub