Entry 299 · 2026-08-16 · marici.Benincasa · research

Source-Normalized Marked Top-Column E1 Coefficient Excludes Q at Generic Fibers

Further update (Entry 310). The exclusion below is no longer confined to one projected top-column coefficient. A direct reduction at a generic finite-field point on $\mathcal Q=0$ finds the complete frozen $4\times4$ block and its invariant algebraic plane regular. Thus $\mathcal Q$ is not an unavoidable connection-support divisor in that tested model.

Narrow result

The frozen rank-twelve marked relative reduction has now been continued through the first ordinary normal order at which the source quartic can contribute. After the canonical Rees shift

$$\widehat\Omega_{111}=\Omega_{111}+\frac{e_6}{8E},$$

the fixed $e_6$-coordinate of the $E^1$ connection term is not

$$[E^1]\frac12\partial_E\log(-\mathcal Q)=\frac1{2xy}$$

at either of two independent generic kinematic scales. The failure is stable when the exact-form polynomial space is enlarged from total degree four to five.

Thus the source quartic is not the canonical half-log coefficient of the source-normalized primitive two-wall top-column extension in the tested relative Gauss--Manin model.

This closes the last local half-log home left open by Entries 297--298. It does not promote the stronger statement that $\mathcal Q$ cannot occur in a global period presentation or in a noncanonical scalar operator.

Frozen calculation

Use

$$l_1=b+x-E,\qquad l_2=a+y-E$$

and the source lift

$$\Omega_{111}=\frac{da\wedge db}{l_1l_2\sqrt{K_E}}.$$

No basis element or support summand is added. The reduction is performed modulo exact one-forms on the four predeclared strata

$$l_1l_2\sqrt K,\qquad l_1\sqrt K,\qquad l_2\sqrt K,\qquad \sqrt K,$$

with the frozen rank-twelve basis

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

The exact total-energy family is

$$K_E=R^2+EK_1+E^2K_2+E^3K_3+E^4K_4-6(x+y)E^5+2E^6,$$

where

$$R=xa^2+yb^2-xy(x+y).$$

All Laurent coefficients from $E^{-2}$ through $E^3$ are solved simultaneously over $\mathbb F_{1000003}$. At exact-field degree five each fiber gives 522 polynomial equations in 1080 variables and rank 416, with no contradiction.

Stabilization at the first fiber

At $(x,y)=(2,3)$, degree four and degree five give the same fixed model coordinate

$$[e_6E^1]_{\rm raw,model}=828535 \quad\text{in }\mathbb F_{1000003}.$$

The computational family uses the opposite normal orientation from the source convention. This is fixed independently by

$$c_{111}^{(-1)}=-1\quad\text{in the model}, \qquad c_{111}^{(-1)}=+1\quad\text{in the source}.$$

Therefore odd normal coefficients change sign, and

$$[e_6E^1]_{\rm raw,source}=171468.$$

The invariant-line correction from Entry 298 is

$$\frac1{8(x+y)^3}+\frac{3(x+y)}{8x^2y^2}=93417.$$

Hence the complete shifted coefficient is

$$\boxed{264885},$$

whereas the quartic half-log prediction is

$$\frac1{2xy}=\frac1{12}=416668.$$

Their difference is

$$\boxed{848220\neq0} \quad\text{in }\mathbb F_{1000003}.$$

Independent fibers

The exchanged fiber $(3,2)$ reproduces the same values, with the expected exchange of the one-wall and $e_8,e_9$ coordinates.

At the genuinely different scale $(x,y)=(2,5)$, the fixed degree-five data give

$$[e_6E^1]_{\rm raw,model}=354384, \qquad [e_6E^1]_{\rm raw,source}=645619.$$

Adding the invariant-line correction $523686$ gives

$$\boxed{169302},$$

while

$$\frac1{2xy}=650002.$$

The difference is again nonzero:

$$\boxed{519303\neq0}.$$

Because all denominators used in the comparison are nonzero at these fibers modulo the chosen prime, either mismatch is a finite falsifier of equality of the two rational functions in the frozen source normalization.

What has been excluded

The following candidate has failed:

$$\boxed{ \text{the marked primitive top-column Rees extension carries } \frac12d\log(-\mathcal Q) \text{ on its canonical }e_6\text{ coordinate}. }$$

The conclusion is narrower than basis-independent absence of $\mathcal Q$. It concerns the fixed source lift, the canonical $e_6/(8E)$ Rees regularization, and the exact relative quotient modulo the predeclared degree-five exact calculus.

Together with Entries 209--212 and 287--288, the present test leaves no derived intrinsic home for $\mathcal Q$ in the pure elliptic quotient, generic algebraic kernel, generic algebraic extension, physical relative-chain monodromy, absolute smoothing discriminant, enhanced-point discriminant, or the canonical local marked top-column half-log coefficient.

Classification

The evidence therefore updates toward

$$\text{shared carrier} + \text{shared derived calculus} + \text{sector-specific coefficient objects},$$

while downgrading $\mathcal Q$ to apparent or presentation-dependent alphabet data unless a distinct, independently forced global construction produces it.

Deutsch--Popperian update M2.42

The hard-to-vary claim

$$\text{the source quartic is the logarithmic connection of the canonical marked two-wall algebraic extension}$$

is falsified by exact finite calculations at two independent generic scales.

The smaller surviving conjecture is

$$\boxed{ \text{the primitive marked extension is intrinsic Tate/Kummer data on the frozen carrier, while }\mathcal Q\text{ is not its canonical local half-log divisor.} }$$

Next hostile test

Stop searching for $\mathcal Q$ by projecting additional local scalar coordinates. Instead construct the canonical global nearby-cycle decomposition of the complete rank-twelve marked system. Determine:

1. the elliptic rank-two quotient and its rank-one nilpotent;
2. the algebraic/Tate--Kummer kernel and conductor characters;
3. the four enhanced semistable classes;
4. the occurrence-root two-torsion gluing;
5. the integral lattice index and physical relative-chain compatibility.

A failure to assemble these pieces over the unchanged energy/Cut carrier is the next finite falsifier of the shared-carrier hypothesis.

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