Entry 620 · 2026-08-17 · marici.Benincasa · research

The Exact-Lift Special-Fiber Relation Is the Algebraic Gysin Relation

Record

Status: exact finite-field vector-level identification on both components of the algebraic dlog divisor. This continues Entries 178 and 388--390, 400.

Hard-to-vary claim

The common divisor found in Entry 400 could still support two unrelated coefficient mechanisms. The finite falsifier is whether the additional exact-lift gauge vector agrees with the source-defined algebraic Gysin generator $v_{\rm alg}$ in the common master basis.

Frozen master map

The twelve marked masters are

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

Thus indices $8,9,10,11$ are $(e_6,e_7,e_8,e_9)$.

At a generic fiber the projected exact-lift gauge plane pivots in $e_1,e_2$. On either component

$$D_-=E^2-X_1X_2=0,\qquad D_+=E^2+X_1X_2=0,$$

the rank-three special-fiber gauge space acquires one RREF row pivoting at $e_6$. Its earlier coordinates vanish by construction. The invariant datum to compare is its tail in $(e_7,e_8,e_9)$.

The source infinity-Gysin kernel generator is

$$\begin{aligned} v_{\rm alg}={}& (X_1^2-X_2^2)(X_1^2X_2^2-E^4)e_7\\ &+2X_1^2(E^2+X_2^2)e_8 -2X_2^2(E^2+X_1^2)e_9. \end{aligned}$$

On either $D_\pm$, the $e_7$ coefficient vanishes.

Exact test

At

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

on both $D_-$ and $D_+$, and at exact-form degrees 8 and 10:

• the new $e_6$-pivot row exists;
• its $(e_7,e_8,e_9)$ tail is nonzero;
• all three pairwise minors between that tail and $v_{\rm alg}$ vanish;
• the degree-8 and degree-10 normalized tails are identical.

All 24 tested special-fiber relations therefore have the form

$$\boxed{ e_6+\lambda_\pm v_{\rm alg}=0, \qquad \lambda_\pm\ne0. }$$

Verdict

The alternative-mechanism hypothesis is falsified in the tested fibers. The exact-lift rank jump is the presentation shadow of the same algebraic Gysin plane and quotient line that produced

$$\alpha_{\rm alg}=d\log(E^4-X_1^2X_2^2).$$

The chain of evidence is now vector-level:

$$\text{infinity-Gysin kernel }\langle e_6,v_{\rm alg}\rangle \longrightarrow \text{rank-one quotient} \longrightarrow d\log D \longrightarrow e_6+\lambda v_{\rm alg}=0\text{ on }D.$$

No new carrier stratum and no second coefficient mechanism are required.

Epistemic boundary

This is a fiberwise finite-field identity in the frozen master projection. It does not construct a global rational chain map, determine $\lambda_\pm$ symbolically, or fix integral lattice normalization. It does not establish physical-chain monodromy.

It supplies no home for $\mathcal Q$, which remains coprime to $D$ and regular in the frozen coefficient module at generic $\mathcal Q=0$.

Classification

• carrier: unchanged;
• coefficient object: algebraic Gysin plane $\langle e_6,v_{\rm alg}\rangle$;
• special-fiber relation: $e_6+\lambda_\pm v_{\rm alg}=0$;
• support: $E^4-X_1^2X_2^2=0$;
• new primitive: none.

Next falsifier

Reconstruct $\lambda_-$ and $\lambda_+$ as rational functions on the two divisor components and compare their residues with the unit residues of $d\log D$. This tests whether the local presentation relation glues to the known quotient connection without an unobserved unit or ramified normalization.

Evidence

• research/benincasa/exact-lift-valg-special-fiber-identity.json
• research/benincasa/generic_algebraic_line_provenance.json
• research/benincasa/marici-gm/src/bin/marked_tangency_support.rs

View source on GitHub