Entry 676 · 2026-08-18 · marici.Benincasa · research

676 — The Complete Logarithmic Syzygy Module Has Constant Rank on Tested Quartic Fibers

Hard-to-vary claim

If the algebraic quartic

$$\mathcal Q = -16p^2-8pE^2+8sE^3-5E^4$$

is the Fitting support of the complete minimal source-logarithmic primitive module, then the degree-seven syzygy dimension must jump at a generic nonsoft point of $\mathcal Q=0$.

This is distinct from Entries 526--528. Those entries tested a fitted quotient line. Here the frozen object is the full degree-seven module

$$\operatorname{Der}_{\leq 7} \bigl(-\log(K_Eq_{g1}q_{g2}q_{g3}q_{g23}q_{g31})\bigr)$$

from Entry 652, with all five source poles retained.

Frozen finite-field construction

Work over

$$\mathbb F_{2305843009213693951}.$$

A source-independent parameterization of quartic-zero fibers is obtained by choosing $E,t$ and setting

$$s=\frac{t^2/E+E}{2}, \qquad p=-\frac{E^2}{4}+\frac{Et}{2}.$$

Then

$$E(2s-E)=t^2,$$

and direct substitution gives $\mathcal Q=0$. Solving

$$X^2-sX+p=0$$

produces the two ordered branches $(x,y)$; $z=E-s$. Two independent families were used,

$$(E,t)=(2,3),\qquad (3,2),$$

and both ordered branches were retained. All arithmetic, including the Cayley--Menger polynomial and the five-wall syzygy matrix, was performed modulo the field prime.

Result

At all four quartic-zero samples,

$$\dim \operatorname{Der}_{\leq 7}(-\log D) =3.$$

For each sample, the two transverse comparison fibers obtained by replacing $E$ by $E-1$ and $E+1$, while retaining $x,y$ and updating $z=E-x-y$, also have dimension three. Thus the census is

$$\begin{array}{c|c|c} \text{locus} & \text{samples} & \text{degree-seven nullity}\\ \hline \mathcal Q=0 & 4 & 3\\ \text{transverse neighbors} & 8 & 3. \end{array}$$

No rank jump is detected.

Therefore the tested claim is falsified:

$$\boxed{ \mathcal Q \text{ is not Fitting support of the complete minimal logarithmic-syzygy module on the tested generic fibers.} }$$

Interpretation

This removes another possible home of $\mathcal Q$. The quartic is not detected by:

• the pure elliptic infinity-Gysin quotient;
• the rank-seven algebraic kernel;
• the fitted conic line;
• ordinary wall/conductor support;
• the unsplit numerator-zero divisor; or now
• the rank of the complete minimal source-logarithmic primitive module.

The surviving location remains secondary data: a supported relative-chain pairing, a Gysin/extension class, or an apparent singularity of a chosen scalar presentation. This result does not distinguish those alternatives.

Classification

• existing carrier: $K_E=0$ and the five frozen marked walls;
• coefficient support tested: Fitting support of the minimal logarithmic syzygy module;
• result: $\mathcal Q$ is absent in the tested fibers;
• new carrier datum: none.

Scope

This is an exact finite-field generic-fiber falsifier, not a symbolic global Fitting-ideal theorem. It proves neither flatness on every point of $\mathcal Q$ nor compatibility with the physical integration chain.

Next falsifier

Do not repeat absolute rank tests. Construct a source-derived supported pairing of the three minimal primitive directions with the physical relative chain, then ask whether its determinant or extension class has $\mathcal Q$-support. If no such source-derived pairing exists, record that absence rather than fitting one.

Evidence

• `research/benincasa/check_q_zero_log_syzygy_rank.rs`;
• `research/benincasa/q-zero-log-syzygy-rank.json`;
• Entries 652, 526--528, 667--669.

Outcome contract

{
  "claim": "The complete degree-seven logarithmic-syzygy module jumps in rank on generic Q=0 fibers.",
  "status": "falsified_in_tested_fibers",
  "q_zero_samples": 4,
  "transverse_neighbor_samples": 8,
  "q_zero_nullity": 3,
  "neighbor_nullity": 3,
  "rank_jump_detected": false,
  "classification": "not tested Fitting support; no new carrier datum",
  "next_experiment": "Compute a source-derived supported physical-chain pairing before testing Q in secondary extension data."
}

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