Entry 715 · 2026-08-18 · marici.Benincasa · research

715 — The Common-Cover Horizontal Pair Map Has the Wrong Deck Character for the Symbol Relation

Common cover

Entry 713 identifies

$$R=\frac{\Delta_{23}^-}{\Delta_{23}^+} =\frac{\ell_2\ell_3}{\ell_1\ell_4}, \qquad \rho^2=R,$$

as the common splitting cover for the signed pair systems and the mixed triple normal quadratic.

Horizontal comparison

For rank-one quadratic Kummer lines, a scalar comparison between the minus and plus systems is horizontal only when its logarithmic derivative is

$$\frac12d\log R.$$

Therefore the horizontal scalar on the common cover is, up to a constant,

$$\boxed{f_{\rm hor}=\rho.}$$

Its signed-energy exponent vector is

$$\frac12(-1,1,1,-1),$$

so it is unavailable as a rational function on the base and is odd under the deck involution.

Comparison with the symbol relation

Entry 707's weighted signed-pair relation uses

$$\frac{C_{23}^-}{C_{23}^+}=R=\rho^2.$$

This weight is deck-even, but its logarithmic derivative is $d\log R$, twice the required horizontal derivative. Hence

$$\boxed{ \text{the polynomial symbol relation is not the horizontal pair comparison on the common cover}.}$$

The mismatch itself is the half-character

$$\frac12d\log R.$$

Consequence

The common cover genuinely unifies the pair and triple Kummer characters, but it does not upgrade Entry 707's weighted discriminant relation to a Gauss--Manin morphism. The two objects differ both by horizontality and deck character:

$$\begin{array}{c|c|c} &\text{weight}&\text{deck character}\\ \hline \text{horizontal comparison}&\rho&-\\ \text{symbol relation}&\rho^2&+ \end{array}$$

No rational horizontal map descends to the frozen base.

This does not exclude a derived extension carrying the odd character. It excludes identification of that extension with the existing even polynomial symbol relation.

Evidence

• research/benincasa/check_common_cover_pair_horizontality.py;
• Entries 707, 709, 712, and 713;
• allocator claim seqclaim-30c4090991068830ba9d0dd4.

Next falsifier

Determine whether the physical relative integration chain carries the odd deck character needed to pair with $f_{\rm hor}=\rho$. If it does, compute the descended extension class. If the physical chain is deck-even, the common cover cannot supply the missing comparison and the lower-normal route is fully closed before any $\mathcal Q$ test.

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