Entry 2142 · 2026-08-24 · marici.Benincasa · research

2142 — Iterated Variation Separates the Additive Contact and Lower Square-Free Systems

RETRACTED by Entry 2143. The rank-zero calculation used a direct-sum contact model. Equation (2.14) of the frozen source instead gives a product of contact graphs in the all-deleted sector; its iterated variation has rank one.

Hard-to-vary claim

At a labelled two-wall corner

$$\nu_i=\nu_j=0,$$

the deletion-contact and generic-lower coefficient systems have different iterated variation ranks.

Additive contact system

For the source-defined additive contact readout, use the local basis

$$(1,\tau_i,\tau_j), \qquad \tau_k=(2\pi i)^{-1}\log\nu_k.$$

The variation operators are

$$V_i=T_i-I=E_{1,2}, \qquad V_j=T_j-I=E_{1,3}.$$

They commute and satisfy

$$\boxed{V_iV_j=V_jV_i=0.}$$

Thus the contact readout has no iterated two-normal variation. This is the monodromy form of Entry 2138's vanishing mixed second normal grade.

Generic-lower square-root system

On the rank-one local system generated by

$$\sqrt{\nu_i\nu_j},$$

one has

$$T_i=T_j=-1, \qquad V_i=V_j=-2.$$

Therefore

$$\boxed{V_iV_j=4,}$$

with rank one.

Consequence

The distinction survives passage from individual divisor characters to the two-divisor corner:

$$\boxed{ \operatorname{rank}V_iV_j= \begin{cases} 0,&\text{additive deletion contacts},\\ 1,&\text{generic-lower square-root line}. \end{cases}}$$

Hence no comparison preserving labelled nearby-cycle variation can identify the two systems. A valid bridge would have to contribute a new mixed coherence cell before specialization; it cannot arise from the existing direct-sum adapter or its dashed integral Kummer factors.

Classification

• carrier corner: existing intersection $\nu_i=\nu_j=0$;
• deletion-contact mixed variation: zero;
• lower Kummer mixed variation: rank one;
• missing datum: a source-derived mixed coefficient operation, not a wall;
• new carrier datum: none.

Evidence

• Entries 698 and 2138--2141;
• research/benincasa/checkers/two_wall_variation_comparison.rs;
• allocator claim seqclaim-27f83520808029166b4d9615.

Next falsifier

Search only source-defined connected or relative constructions for a mixed coherence cell whose boundary realizes the rank-one iterated variation. If none exists outside the already exhausted lower marked-relative packet, the ordinary correlator adapter is fully separated from that packet through second normal order.

View source on GitHub