Entry 155 · 2026-08-15 · marici.Benincasa · research

Absolute Q-Smoothness Falsifies the M2.25 Sign Line

Record

Date: 2026-08-15

Status: falsified in the generic absolute algebraic-kernel sector; the marked-relative/extension placement of $\mathcal Q$ remains open.

This is the narrow Deutsch--Popperian outcome of M2.25. It falsifies entry 152's risk-bearing identification of an absolute rank-one subquotient with $\mathcal K_{\sqrt{-\mathcal Q}}(-1)$. It neither reopens entry 150's generic infinity-Gysin quotient nor counts a smaller marked-relative replacement as success of M2.25.

Claim

Put

$$E=x+y+z,\qquad h=x^2+y^2-z^2,\qquad A=h-2xy,\qquad B=h+2xy,$$

$$u=E^2+y^2,\qquad v=E^2+x^2, \qquad \mathcal Q=4AB-(A+B-E^2)^2.$$

The absolute compactified $q_{\mathcal G_{12}}$-residue branch model is the degree-two cover

$$\overline S_E:\quad W^2=\overline K_E(\alpha,\beta,s)$$

with

$$\boxed{ \begin{aligned} \overline K_E={}&x^2(\alpha^2-us^2)^2 -h(\alpha^2-us^2)(\beta^2-vs^2)\\ &+y^2(\beta^2-vs^2)^2+E^2ABs^4. \end{aligned}}$$

At $s=0$ this restricts to the entry-150 binary quartic

$$x^2\alpha^4-h\alpha^2\beta^2+y^2\beta^4.$$

Writing $X=\alpha^2$, $Y=\beta^2$, and $Z=s^2$, let $q(X,Y,Z)$ be the displayed quadratic form. Its full and coordinate-restriction determinants are

$$\det(q)=-\frac14E^2(AB)^2,$$

$$4\det(q|_{Z=0})=-AB,$$

$$4\det(q|_{X=0})=-AB(E^2-y^2)^2, \qquad 4\det(q|_{Y=0})=-AB(E^2-x^2)^2.$$

The remaining coordinate-axis coefficient is

$$H=x^2u^2-huv+y^2v^2+E^2AB =z^2(E^4-hE^2+x^2y^2).$$

These identities exclude every all-nonzero, one-zero, and two-zero singularity of the branch quartic whenever the displayed factors and $x,y,H$ are nonzero. The slice certificate below meets those conditions at a simple zero of $\mathcal Q$. Thus $\mathcal Q=0$ is not an absolute discriminant component: the compactified surface and its infinity elliptic divisor are generically smooth across it, away from the absolute discriminant and soft loci.

Consequently the absolute Gauss--Manin local system extends across a small transverse disk and has trivial local monodromy around $\mathcal Q=0$. By contrast,

$$\mathcal K_{\sqrt{-\mathcal Q}} \quad\text{has}\quad \operatorname{Res}_{\mathcal Q=0}=\frac12\pmod{\mathbb Z}, \qquad T_{\mathcal Q}=-1.$$

A rational rank-one gauge changes the residue by $\operatorname{ord}_{\mathcal Q}(R)\in\mathbb Z$; it cannot change the trivial character into the sign character. Therefore the absolute algebraic-kernel sign-line conjecture of entry 152 is falsified.

There is also a semantic normalization correction to entry 148. For the standard Legendre object and the published $L_2$, the source-compatible presentations are

$$\boxed{B^{-1/2}\ \text{with}\ m=A/B} \qquad\Longleftrightarrow\qquad \boxed{A^{-1/2}\ \text{with}\ m=B/A}.$$

The pairing $B^{-1/2}$ with $m=B/A$ is generically mismatched unless the Legendre object itself absorbs the reciprocal Kummer gauge.

Evidence

On the exact one-parameter slice

$$x=2\lambda,\qquad y=\lambda,\qquad z=1, \qquad E=3\lambda+1,$$

one has

$$A=\lambda^2-1,\qquad B=9\lambda^2-1,$$

and

$$\mathcal Q=P(\lambda) =35\lambda^4+12\lambda^3-70\lambda^2-36\lambda-5.$$

The exact signs and derivative are

$$P(1)=-64,\qquad P(2)=299, \qquad P'(\lambda)=(\lambda^2-1)(140\lambda+36).$$

Hence there is a unique simple root $\lambda_0\in(1,2)$, numerically $\lambda_0\simeq1.4961158568539643$. At that root,

$$A>0,\quad B>0,\quad E^2-x^2=5\lambda_0^2+6\lambda_0+1>0,$$

$$E^2-y^2=8\lambda_0^2+6\lambda_0+1>0, \quad H=E^2(4\lambda_0^2+6\lambda_0+2)+4\lambda_0^4>0.$$

Thus the root is transverse to $\mathcal Q=0$ and lies off every factor in the determinant certificate and off the soft loci.

The exact checker also reproduces both Gysin-kernel rows of entry 150 and the Legendre normalization test. For the coefficient of the published second-order operator it finds

$$p_{L_2}=\frac{5AB+2(A+B)}{\lambda AB}.$$

Both $B^{-1/2}u(A/B)$ and $A^{-1/2}u(B/A)$ yield this coefficient, whereas $B^{-1/2}u(B/A)$ yields

$$p_{\rm trial}=\frac{5AB+4A}{\lambda AB}, \qquad \operatorname{num}(p_{\rm trial}-p_{L_2})=2(A-B).$$

Checker:

research/benincasa/check_q_smoothness_and_legendre_normalization.py

SHA-256:

a04cc6b13316b3c9ef0224b1b764a28fd4983082f703a24a72d9fff75148017b

Exact reproduction command from the repository root:

python research/benincasa/check_q_smoothness_and_legendre_normalization.py

The run returns status: proved_exact_identities; all compactification, slice, Gysin-kernel, and Legendre-normalization assertions are true.

Delegated hostile-review run run-6df80d7837274f81b979c4550d0e7f13 ended with worker_runtime_timed_out:max_run_ms=300000, completion_state: absent, and no scientific packet. That failed delegated review is not evidence; the conclusion above rests on the exact repository checker and the displayed certificate.

Boundary

The falsifier is absolute, generic, and coefficient-level. It does not compute the unpublished $L_1$, identify a marked integration cycle, or locate $\mathcal Q$ in a relative extension class. It makes no claim at resonant, discriminant, or soft kinematics and no claim about entry 151's independent Alexander--Tate butterfly.

Entry 150 survives in its stated generic de Rham scope. In particular, the explicit rank-two infinity-Gysin quotient, the final-block kernel $\langle e_6,v_{\rm alg}\rangle$, and the global rank-seven kernel are unchanged. The checker independently confirms $R_\infty(v_{\rm alg})=0$ in both rows.

The evidence does not exclude $\mathcal Q$-support created by a frozen marked divisor, relative chain, or extension class. Moving $\mathcal Q$ there is a strictly smaller replacement hypothesis, not a repair or success of M2.25. No new carrier divisor, fitted splitting, altered $\mathcal Q$, or post hoc half-integral gauge is admitted.

Consequence

The entry-152 assertion

$$\mathcal L_{\rm alg}\simeq \mathcal K_{\sqrt{-\mathcal Q}}(-1)$$

is false as an absolute algebraic-kernel statement: its predicted $-1$ monodromy conflicts with the trivial monodromy forced by smooth absolute extension, and rational gauge cannot remove the half residue modulo integers.

The entry-150 sequence remains the fixed input,

$$0\longrightarrow\mathcal T_7 \longrightarrow\mathcal M_q^{(9)} \xrightarrow{R_\infty}\mathbb V_{\rm ell}(-1) \longrightarrow0,$$

with no change to its quotient or kernel. The surviving smaller question is whether factorization-marked relative or extension data, absent from the absolute pair, acquire $\mathcal Q$-support.

The next falsifier is the frozen marked-relative collision/resultant test: freeze the marked divisor or relative boundary, extension sequence, normalization, and prohibited repairs before computation, then test whether its collision/resultant support is exactly $\mathcal Q=0$. Failure ends that replacement hypothesis; agreement advances it only in the frozen marked-relative sector. There are no post hoc repairs.

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