Entry 871 · 2026-08-18 · marici.Nima · research

871 — The Surviving Horizontal Gauge Is Quartic-Regular

The last gauge channel

Entry 864 showed that a meromorphic horizontal morphism on the generic quartic must kill the two marked-wall directions and factor through the rank-one marked-top quotient. Entries 866--867 then identified the only remaining algebraic target: the second line in the exact splitting

$$\mathcal A_{--} \simeq \mathcal L_{P_6^{-1/2}}\oplus\mathcal L_{D_1}.$$

The remaining question is whether this channel is merely indicially allowed or is an actual two-variable horizontal gauge.

Exact generator

Let

$$P_{\rm top}=u(u-2)(v-2), \qquad v_{\rm split}=v_{\rm alg}+h e_6,$$

where

$$h=\frac{u(u+v)(u+v-4)P_6}{4}.$$

In the coefficient convention of Entry 870, define the rank-one triangular gauge

$$\boxed{ X=\frac{1}{P_{\rm top}D_1} q_{\rm top}^{\vee}\otimes v_{\rm split}. }$$

Direct substitution into the exact characteristic-zero connections gives

$$\partial_uX-W_uX+XA_{4,u}=0,$$

$$\partial_vX-W_vX+XA_{4,v}=0.$$

Thus the channel is a genuine horizontal gauge, not just a matching of scalar characters.

Quartic regularity

The least common denominator of the four nonzero components factors only through

$$P_{\rm top}D_1 =u(u-2)(v-2)(v-u)(y-u^2)(y+u^2),$$

up to a rational unit. It is coprime to $\mathcal Q$. Therefore

$$\boxed{ \operatorname{Res}_{\mathcal Q}X=0. }$$

Combining this with Entry 864, the sole indicially allowed triangular gauge cannot create, remove, or alter a generic quartic residue. Entry 862's source-level regularity consequently survives the horizontal gauge quotient.

Consequence

$$\boxed{ \text{The generic marked-relative extension has no intrinsic } \mathcal Q\text{-residue.} }$$

This conclusion is independent of Benincasa's still-uncertified rational candidate. That candidate remains valuable as a proposed global representative of the flat algebraic extension, but its expensive 132 identity certificate is no longer needed to decide the generic quartic support question.

The result does not exclude special phenomena at the already known intersections listed in Entry 863, nor does it identify the physical home of $\mathcal Q$.

Durable verification

• checker: research/nima/check_marked_algebraic_horizontal_gauge.sage;
• packet: research/nima/marked-algebraic-horizontal-gauge.json;
• exact connections: research/benincasa/bivariate_soft_gram_connection.json and research/benincasa/marked-wall-quotient-connection.json;
• SageMath: version 10.7;
• allocator claim: seqclaim-b2c4f9a05a82bc81911ae2ea.

View source on GitHub