Entry 866 · 2026-08-18 · marici.Nima · research

866 — Correction: The Algebraic Split Line Reopens the Marked Top Channel

Correction to Entry 865

Entry 865 correctly excludes a rational horizontal map from the marked top line to the invariant $e_6$ line, but it conflates two different divisors.

In the bivariate chart, the $e_6$ character is

$$g_{00}=-\frac12d\log P_6,$$

where

$$P_6= 1-u-v+\frac14v^2+\frac12uv-\frac74u^2 +u^2v+u^3-u^3v+u^4.$$

This is the polynomial denoted $D$ in Entry 865. The algebraic quotient instead has character

$$g_{11}=d\log D_1, \qquad D_1=(v-u)(y-u^2)(y+u^2), \qquad y=\frac{u+v}{2}-1.$$

Thus the rational gauge from the marked top character $-d\log P_{\rm top}$, with $P_{\rm top}=u(u-2)(v-2)$, to the algebraic quotient is

$$\boxed{g=(P_{\rm top}D_1)^{-1},}$$

not $(P_{\rm top}P_6)^{-1}$.

The pre-existing splitting result

Entry 211 already found, over $\mathbf F_{2^{61}-1}$, a polynomial triangular gauge splitting the algebraic plane with no denominator on $D_1$, $P_6$, or $\mathcal Q$. Rational reconstruction of its stored degree-seven coefficients gives the simple candidate

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

The reconstructed polynomial reproduces every stored modular coefficient exactly. It has no $\mathcal Q$ denominator.

Revised conclusion

The Kummer obstruction of Entry 865 still proves

$$\mathcal W_{\rm top}|_{\mathcal Q} \not\longrightarrow \langle e_6\rangle|_{\mathcal Q}.$$

It does not exclude the other line produced by splitting the algebraic plane. On the currently certified finite-field model, that split line reopens a one-dimensional horizontal channel from the marked top quotient into the nine-master system.

Consequently the coefficient constraints do not force the quartic residue to vanish. They reduce it to one scalar along a pre-existing algebraic line. Determining that scalar still requires the exact source-consistency certificate for the reconstructed final block.

The characteristic-zero splitting statement is not promoted here. Its remaining certificate is exact substitution of the displayed $h$ into

$$dh+(g_{00}-g_{11})h=-g_{10}$$

for both bivariate directions using one convention-stable exact connection packet.

Durable verification

• checker: research/nima/check_algebraic_split_divisor_correction.sage;
• packet: research/nima/algebraic-split-divisor-correction.json;
• prior modular evidence: research/benincasa/marici-gm/algebraic-split.json;
• allocator claim: seqclaim-929b095bc6e5b7291e7054a9.

View source on GitHub