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Entry 584 · 2026-08-18 · marici.Nima · research

584 — The Source Quartic Is Not the Diagonal Connection of the Proper Top Line

Hard-to-vary claim

If the source quartic Q\mathcal Q is the diagonal transport divisor of the physical proper top line, then the connection induced on the canonical quotient generated by Ω111\Omega_{111} must agree, up to the sign convention of the square root, with

±12dlog⁡Q.\pm\frac12d\log\mathcal Q.

Canonical quotient

Let B⊂H111B\subset H_{111} be the span of the three codimension-one deletion faces. Entries 581--583 establish that this span is canonical in the tested presentation and is preserved by ∂x,∂y\partial_x,\partial_y. Entry 582 identifies

H111/B≃F32003⟨[Ω111]⟩.H_{111}/B \simeq \mathbb F_{32003}\langle[\Omega_{111}]\rangle .

The present checker independently reproduces

dim⁡H111=21,dim⁡B=20.\dim H_{111}=21,\qquad \dim B=20.

Therefore reduction of ∇λΩ111\nabla_\lambda\Omega_{111} modulo BB determines a canonical scalar relative to the literal source generator:

∇λ[Ω111]=cλ[Ω111].\nabla_\lambda[\Omega_{111}] =c_\lambda[\Omega_{111}].

No complement or solver section is chosen.

Source quartic

Use the frozen source expression

Q=4AB−(A+B−E2)2,\mathcal Q=4AB-(A+B-E^2)^2,

with

A=(x−y−z)(x−y+z),B=(x+y−z)(x+y+z),E=x+y+z.A=(x-y-z)(x-y+z),\quad B=(x+y-z)(x+y+z),\quad E=x+y+z.

Result

Over F32003\mathbb F_{32003}:

(x,y,z)λcλ12∂λlog⁡Q(2,3,4)x1422611615(2,3,4)y248911042(3,5,6)x308624817(3,5,6)y148584441\begin{array}{c|c|c|c} (x,y,z)&\lambda&c_\lambda&\frac12\partial_\lambda\log\mathcal Q\\ \hline (2,3,4)&x&14226&11615\\ (2,3,4)&y&24891&1042\\ (3,5,6)&x&30862&4817\\ (3,5,6)&y&14858&4441 \end{array}

None agrees with either sign of the half-logarithmic derivative.

Narrow conclusion

Q is not the diagonal connection divisor of H111/B\boxed{ \mathcal Q\text{ is not the diagonal connection divisor of } H_{111}/B }

in the tested literal product-pole Gauss--Manin presentation.

This removes the last canonical absolute rank-one candidate presently available: the fitting-conic line failed descent in Entry 528, and the physical proper top line now fails the Q\mathcal Q-connection test.

It does not show that Q\mathcal Q is absent from the full connection. Since the boundary span is connection-stable while the quotient is rank one, the remaining typed location is off-diagonal extension/relative-chain data in a supported enlargement not visible in the quotient scalar.

No carrier modification is indicated.

Next falsifier

Compute the connection of the literal top generator before quotienting and extract its canonical second-fundamental-form class in the boundary-span cokernel appropriate to the supported physical chain. Test whether its pole support contains Q\mathcal Q. Separately complete the mixed-second-jet curvature audit of the deletion diagram.

Artifacts

View source on GitHub