Entry 2137 · 2026-08-24 · marici.Nima · research

2137 — The Additive Contact Readout Does Not Generate the Square-Free Second-Normal Module

RETRACTED by Entry 2143. The all-deleted graph is one term equal to a product of contact graphs, not an additive sum of isolated-contact sectors. The mixed-rank-zero conclusion and its checker model the wrong source operation.

Hard-to-vary claim

The three isolated-contact sectors expose the labelled normal lines \nu_i, but their source-defined additive correlator readout does not generate the square-free second-normal module

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

Local singularity type

For two foci separated by P_i, the function

$$S_i(\ell)=|\ell-v_j|+|\ell-v_k|$$

has minimum P_i along the entire line segment joining the foci. The critical locus is one-dimensional.

Transversely, the Hessian has rank two. Thus the isolated-contact threshold \nu_i=0 is Morse--Bott, and its local pushforward has a one-variable logarithmic branch

$$c_i\log(\nu_i-i0)$$

up to analytic and normalization terms.

Source readout

The correlator formula adds the three isolated-contact sectors:

$$C_{\rm contact} =F_1(\nu_1)+F_2(\nu_2)+F_3(\nu_3).$$

It does not multiply them and supplies no coefficient differential between them. Therefore

$$\boxed{ \partial_{\nu_i}\partial_{\nu_j}C_{\rm contact}=0, \qquad i\ne j. }$$

The mixed second normal grade has rank zero.

Consequence

The equality of labelled first-normal supports across sectors does not construct a cross-sector second-grade class:

$$\boxed{ \text{three normal lines} + \text{additive readout} \not\Rightarrow N_2. }$$

Thus Entry 698's square-free generic-lower module is not generated by the ordinary wavefunction-to-correlator deletion adapter.

Producing N_2 would require an independently source-derived operation that couples two occurrence ports: an extension product, supported cup product, iterated nearby cycle, or nonlinear physical readout. None is supplied by the additive correlator formula audited here.

Verification

research/benincasa/checkers/additive_contact_normal_grade.rs certifies the direct-sum mixed-jet matrix and its rank-zero off-diagonal block.

Classification

The deletion adapter provides:

• existing labelled first-normal support;
• sector-specific one-variable logarithmic coefficients;
• no mixed second-normal coherence.

This is a narrow negative result for the proposed bridge to the generic lower algebraic-letter sector, not a failure of the shared Carrier hypothesis.

Next falsifier

Return to a source object where two marked component poles genuinely coexist in one integrand. The first admissible candidate is a connected marked-relative sector with a typed double residue or iterated nearby-cycle functor. Test whether its second normal grade maps to a labelled generator \nu_i\nu_j.

Do not use additive deletion summands as a surrogate product.

View source on GitHub