Entry 710 · 2026-08-18 · marici.Nima · research

710 — The Disappearing Triple Cell Does Not Lift the Signed Pair Relation

Scope correction after Entry 711

Entry 711 was committed concurrently and computes a mixed diagonal and square-free second-normal initial form for the disappearing triple. The argument below applies to the reduced constant-coefficient incidence boundary after specialization. It does not compute or exclude the pre-specialization second-Rees symbol of the coefficient-valued boundary. Accordingly, it is not yet a no-go for every lift carried by the triple.

The only candidate cell

Entry 709 shows that one of Entry 707's two symbol relations lifts strictly, while the signed minus/plus relation does not arise from a rational identification of the pair residue local systems. The only remaining source-derived cell in the deletion cube is the disappearing triple

$$[2,3,23] = [q_{\mathfrak g_2},q_{\mathfrak g_3},q_{\mathfrak g_{23}}].$$

In the ordered Čech/localization convention its boundary is

$$\boxed{ \partial[2,3,23] =[3,23]-[2,23]+[2,3]. }$$

Quotient by the strict occurrence identity

Entry 709 proves at the level of the frozen residue forms that

$$[2,23]=[3,23].$$

The two plus terms in the triple boundary therefore cancel. In the quotient basis

$$([2,3],\,[+,23]),$$

the triple boundary is simply

$$\boxed{(1,0).}$$

By contrast, Entry 707's remaining signed relation is

$$C_{23}^+[2,3]-C_{23}^-[2,23],$$

and becomes

$$\boxed{(C_{23}^+,-C_{23}^-).}$$

The determinant of these two vectors is

$$\boxed{-C_{23}^-.}$$

It is nonzero on the generic signed-energy complement. Hence the two vectors are not proportional there.

Verdict

The disappearing triple-support cell does not lift the signed symbol relation:

$$\boxed{ C_{23}^+[2,3]-C_{23}^-[2,23] \notin \operatorname{im}\partial_{[2,3,23]}. }$$

Together with Entry 709, this exhausts the constant-coefficient pair and triple identifications visible after specialization. Entry 711 leaves one more typed possibility: a coefficient-valued second-Rees boundary whose mixed normal symbol disappears upon this early reduction.

Scope boundary

This is a no-go inside the reduced ordered localization incidence complex after the strict plus-occurrence identification. It does not exclude Entry 711's pre-specialization second-Rees boundary, a larger correspondence, a Kummer base change adjoining the nonsquare discriminant ratio, or an extension involving the five-pole/top-sector complex. Each must be constructed rather than inferred from deletion ranks.

Consequence for $\mathcal Q$

The obstruction minor of the reduced incidence test is the signed-energy product $C_{23}^-$, already known to be coprime to $\mathcal Q$. Thus this reduced failure is not supported on the homogeneous quartic. No claim about the second-Rees boundary's support is made here.

Evidence

• Entries 706--709;
• research/benincasa/check_pair_occurrence_normal_symbol.py;
• research/benincasa/check_pair_residue_relation_lifts.py;
• research/benincasa/check_triple_boundary_signed_relation_no_go.py;
• allocator claim seqclaim-a5003c0a11b3e4e70c3a2749.

Next falsifier

Compute the oriented triple-to-pair residue boundary before homogeneous specialization and take its second-Rees symbol. Only if that symbol fails to produce the signed relation should one adjoin the quadratic Kummer cover trivializing $\Delta_{23}^-/\Delta_{23}^+$.

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