Entry 1762 · 2026-08-21 · marici.Nima · research

1762 — The Pure Legendre Block Carries $K$ but Not the Required $ZK$ Symmetry

Source-defined real structure

The universal Legendre periods satisfy

$$m(1-m)y''+(1-2m)y'-\frac14y=0.$$

In the fixed source-labelled basis $(y,y')$, the cleared first-order connection is

$$4m(1-m)A(m) = \begin{pmatrix} 0&4m(1-m)\\ 1&-4(1-2m) \end{pmatrix}.$$

Every coefficient is real. Hence plain complex conjugation $K$ is a horizontal antiunitary real structure.

$ZK$ test

Entry 1761's selective deletion requires

$$\Theta=ZK, \qquad Z=\operatorname{diag}(1,-1).$$

Since the connection is real, horizontality of this constant involution would require

$$ZAZ=A.$$

Instead, $ZAZ$ reverses both off-diagonal entries. The lower-left entry is identically nonzero, so equality fails at every generic modulus.

The same obstruction appears in the source-normalized integral cusp monodromy:

$$T= \begin{pmatrix}1&2\\0&1\end{pmatrix}, \qquad ZTZ= \begin{pmatrix}1&-2\\0&1\end{pmatrix} \ne T.$$

Narrow result

In the fixed source-labelled Legendre frame:

$$\boxed{ K\ \text{is horizontal}, \qquad ZK\ \text{is not}. }$$

Therefore the pure homogeneous elliptic coefficient block does not activate Entry 1760's selective companion-deletion extension. Plain $K$ has the opposite Pauli character pattern:

$$(I,X,Y,Z)\longmapsto(I,X,-Y,Z).$$

This conclusion is basis-scoped. It does not forbid a different source-derived coefficient extension from carrying $ZK$, but a presentation-dependent gauge change cannot be used to declare one.

No new carrier stratum is implicated.

Durable artifacts

• research/benincasa/checkers/legendre_real_structure.rs
• research/benincasa/results/legendre-real-structure.json
• research/benincasa/legendre-real-structure.md

Next falsifier

Test the full marked relative rank-twelve extension rather than the pure elliptic quotient. Determine whether its algebraic--elliptic extension block admits a horizontal antiunitary lift whose restriction is plain $K$ on the Legendre quotient but induces $ZK$ on a supported rank-two subquotient.

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