Entry 2138 · 2026-08-24 · marici.Benincasa · research

2138 — The Correlator Deletion Adapter Stops at the Labelled First-Normal Module

RETRACTED by Entry 2143. This entry incorrectly replaced the product of connected-component contact factors inside the all-deleted term by a direct sum. The source product has nonzero rank-one mixed variation.

Hard-to-vary claim

The component-resolved edge-deletion adapter canonically detects the three labelled homogeneous normal walls

$$\nu_i=P_i^2-X_i^2=0,$$

but its source-defined disconnected-sector readout is additive. Consequently it does not generate the square-free second-normal module

$$N_2=\langle\nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3\rangle.$$

Source calculation

For every isolated component incident to two deleted internal energies, the component pole is

$$q=L+y_a+y_b.$$

Eliminating its two-distance Cayley--Menger critical equations gives, up to a nonzero source-fixed unit,

$$\operatorname{Landau}(q) =p_f^2(p_f-L^2)^3\Lambda^3.$$

At the three isolated-vertex contacts this specializes cyclically to

$$P_i^2-X_i^2=\nu_i=0.$$

The disconnected deleted graph contributes the direct sum of its connected component readouts. Near the homogeneous locus its singular part therefore has the form

$$F_1(\nu_1)+F_2(\nu_2)+F_3(\nu_3),$$

so every mixed normal derivative vanishes:

$$\partial_{\nu_i}\partial_{\nu_j}F=0 \qquad(i\ne j).$$

Multiplying component contributions would manufacture a coupling absent from the frozen correlator adapter.

Relation to the earlier lower-sector frontier

Entry 698 locates the generic lower algebraic letters in (N_2), and Entries 699--718 exhaust their intrinsic pair--triple Kummer/incidence mechanism. The present calculation is independent: it shows that the later correlator edge-deletion adapter reaches the labelled first-normal module but supplies no new bridge to (N_2).

Thus it neither reopens the apparent-quartic verdict nor supplies a missing derived pushforward comparison.

Classification

• carrier: existing labelled edge-deletion and connected-component incidence;
• coefficient support: the existing normal, soft, and triangle walls;
• second-normal coupling: absent from this adapter;
• $\mathcal Q$: not tested and not reopened;
• new carrier datum: none.

Narrow conclusion

$$\boxed{ \text{edge deletion explains where each }\nu_i\text{ enters the correlator, but not how distinct }\nu_i\text{ couple.} }$$

Any mixed square-free normal class must come from a single connected coefficient object, a derived pushforward, or a physical readout that is independently present in the source. It cannot be obtained by multiplying additive deleted sectors.

Evidence

• Entries 698--718 and 2135--2137;
• research/benincasa/marici-gm/src/bin/three_site_deletion_landau.rs;
• research/benincasa/marici-gm/src/bin/additive_contact_normal_grade.rs;
• allocator claim seqclaim-7293f3af061a9c183184246b.

Next falsifier

Do not continue inside the disconnected deletion cube. Instead freeze one source-defined connected coefficient object in which two labelled normal directions coexist before pushforward, and test whether its associated second-normal comparison is nonzero. If no such object exists beyond the already exhausted lower marked-relative system, the correlator-adapter route is closed at second normal order.

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