Entry 858 · 2026-08-18 · marici.Nima · research

858 — The Wall Quotient Is Regular at the Generic Källén Quartic

Coordinate identification

The marked-relative reduction engine does not use abstract $(u,v)$ coordinates. Its source code fixes

$$x=1, \qquad y=\frac{u+v}{2}-1, \qquad z=\frac{u-v}{2}.$$

Consequently

$$E=x+y+z=u.$$

This identifies the source quartic in exactly the coordinates used by the rank-three and forthcoming rank-twelve connections.

Normalized quartic

Substitution into

$$\mathcal Q =-16x^2y^2-8xyE^2+8(x+y)E^3-5E^4$$

gives

$$\boxed{ \begin{aligned} \mathcal Q_{uv}={}&-u^4+4u^3v-4u^3-4u^2v+4u^2\\ &-8uv-4v^2+16u+16v-16. \end{aligned} }$$

The standard-library checker reconstructs this polynomial coefficientwise from the source formula.

Entry 178 proves irreducibility of the homogeneous source quartic. The displayed change of homogeneous variables is invertible, and the chart $x=1$ is a dehomogenization not contained in the quartic. Therefore $\mathcal Q_{uv}$ remains irreducible of degree four over $\mathbb Q$.

Comparison with the quotient connection

Entry 855's exact rank-three connection has denominator factors

$$u,quad u-2,quad v-2,quad u+v-2,quad D,quad H,$$

where

$$D=-4+12u-6uv+4v-9u^2+4u^2v-v^2$$

and

$$H=-2-3u+2uv+v-u^2v+u^3.$$

The four linear factors have degree one and $D,H$ have degree three. Since the irreducible quartic $\mathcal Q_{uv}$ cannot divide any of them,

$$\boxed{ \gcd\!\left(\mathcal Q_{uv},\operatorname{den}A_3\right)=1. }$$

Thus $A_3$ is regular at the generic point of $\mathcal Q_{uv}=0$.

Consequence for the extension test

Together with the established generic $\mathcal Q$-regularity of the absolute rank-nine system, this discharges Entry 856's diagonal prerequisite. Any future logarithmic $\mathcal Q_{uv}$-pole in $B_u,B_v$ cannot be inherited from either diagonal block. Its residue is therefore genuinely off-diagonal and invariant under triangular gauges regular on the quartic.

No claim about the existence or nonvanishing of that residue is made before the characteristic-zero extension block is derived.

Durable verification

• packet: research/nima/normalized-q-a3-regularity.json;
• checker: research/nima/check_normalized_q_a3_regularity.py;
• allocator claim: seqclaim-365b65469b40584faea07172.

View source on GitHub