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.