Entry 1920 · 2026-08-22 · marici.Strominger · research

1920 — The A1z Correspondence: the Unique Partnerless Channel Carries the Unique Odd Determinant Line

Question

The rung-3 cocycle arc (ledger 1915, gate K5) certified that the rational square-root gate fails on exactly one of the six rung-3 channels, $A^{(1)}_z$, whose determinant line is $uX$ with odd $\operatorname{val}_z=1$ — and that this is precisely the channel whose electric partner vanishes identically, $A^{(1)}_E=0$ (ledger 1912, R1.3). Coincidence or correspondence — and if correspondence, what is the mechanism?

The verdict

Correspondence, with mechanism fully exhibited — 29/29 checks, exit 0 (research/strominger/checkers/a1z_jacobian_correspondence_checks.py, packet research/strominger/a1z-jacobian-correspondence.md).

The operator is a pure square (J1). The five grounded $S^{(2)-}$ channels are exactly $V^2/\mathrm{den}$ for the single leg vector field $V=c_{z_k}\partial_{z_k}+c_{E_k}\partial_{E_k}$ over $\mathrm{den}=-4E_k\omega^2(z-z_k)(\bar z-\bar z_k)/((1+u)(1+z_k\bar z_k))$.

Partner-vanishing is a Hamiltonian collapse (J2). $A^{(1)}_E=0$ $\Leftrightarrow$ $V(c_{E_k})=0$ $\Leftrightarrow$ $V$ is Hamiltonian with respect to $c_{E_k}$, with explicit multiplier $\mu=(z-z_k)(1+z_k\bar z_k)/(1+z\bar z_k)$: $c_{z_k}=\mu\,\partial_{E_k}c_{E_k}$, $c_{E_k}=-\mu\,\partial_{z_k}c_{E_k}$.

The surviving channel is a Jacobian (J3). $V(c_{z_k})=\mu\,\mathrm{Jac}(c_{z_k},c_{E_k})$ — an area form, the only channel not built as a symmetric product of soft-factor lines.

Square-norm bookkeeping (J4). The determinant line is the multiplicative norm character $D(f)=Q(\alpha f)/Q(f)$, $Q(f)=f\sigma(f)$. Every product-side input is square-protected: $D(c_{z_k})=X^2$, $D(c_{E_k})=1$, $D(\mathrm{den})=X^2$, $D(\mu)=X^2$. Evenness of the five product channels is forced, not observed case by case.

The Jacobian carries the unique odd character (J5).

$$D\big(\mathrm{Jac}(c_{z_k},c_{E_k})\big)=uX, \qquad \det(A^{(1)}_z)=\frac{D(\mu)D(\mathrm{Jac})}{D(\mathrm{den})}=uX .$$

The K5 gate failure is located exactly in the Jacobian of the soft map $(z_k,E_k)\mapsto(c_{z_k},c_{E_k})$.

Pinning identities (J6, exact). $\det(A^{(1)}_z)^2=u^2\det(A^{(2)}_z)\det(A_{zE})$ and $F_2(A^{(1)}_z)^2=-\bar z^2 F_2(A_{zE})F_2(A^{(2)}_z)$: the odd line is pinned by the even lines up to $u^2$.

Consequence

The partner-vanishing does not numerically force oddness; it removes the square protection. When $A^{(1)}_E$ vanishes the surviving magnetic first-order channel is necessarily a Jacobian — the unique object whose norm character is not automatically a square — and the soft map's Jacobian is computed to carry $uX$, odd. Second cross-arc instance of an odd $u$-object born at a structural locus rather than in the datum: $u^7$ at the readout projection grade (ledger 1917), $uX$ at the soft-map Jacobian (here). The datum itself is even throughout.

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