Entry 2126 · 2026-08-24 · marici.Nima · research

2126 — The Triple Deletion Port Produces a Source-Derived Incidence Sextic

Superseded by Entry 2129. The sextic is the mathematically valid restriction of the Cayley--Menger determinant to three equations, but those equations come from three different additive sectors. The frozen source does not define their simultaneous fiber product. Calling the sextic source-derived coefficient incidence was a typing error.

Hard-to-vary claim

Imposing all three translated deletion normals produces a new external sextic, but it is canonically the restriction of the existing Cayley--Menger divisor. It is not the homogeneous source quartic $\mathcal Q$ and does not require a new Carrier stratum.

Frozen incidence

The three equations

$$E_T+2y_{12}=E_T+2y_{23}=E_T+2y_{31}=0$$

fix

$$A=B=C=\frac{E_T^2}{4}.$$

No loop coordinate remains. Substituting this section into the frozen Cayley--Menger determinant gives

$$K\big|_{A=B=C=E_T^2/4} =-\frac12\widetilde{\mathcal Q}_3,$$

where

$$\boxed{ \widetilde{\mathcal Q}_3 = E_T^2\Lambda(p_1,p_2,p_3) +4p_1p_2p_3, \qquad p_i=P_i^2. }$$

Type and comparison

\widetilde{\mathcal Q}_3 is homogeneous of physical degree six. The source algebraic-letter $\mathcal Q$ is a degree-four polynomial in homogeneous site energies. Their degrees and source variables differ, so they are not the same typed object.

After imposing P_i=X_i and E_T=X_1+X_2+X_3, exact factorization does not reduce the sextic to the published quartic times a source-fixed linear or quadratic carrier factor.

Interpretation

This is the first progressive result of the enlarged deletion-port microprogramme:

$$\boxed{ \text{existing Cayley--Menger carrier} \xrightarrow{\text{triple translated-port section}} \text{new source-derived external sextic}. }$$

The polynomial is new as a coefficient/readout incidence equation, not as a Carrier divisor. It predicts an independently testable analytic-continuation locus without repairing the old $\mathcal Q$ hypothesis.

Physical qualification

The section uses

$$y_{12}=y_{23}=y_{31}=-E_T/2,$$

so it does not meet the literal positive loop chamber for E_T>0. Physical activation would require an analytically continued relative cycle and must be tested independently.

Verification

The exact determinant construction and specialization are in

research/benincasa/marici-gm/src/bin/three_site_deletion_landau.rs.

Next falsifier

Test whether \widetilde{\mathcal Q}_3=0 carries any coefficient-rank loss or monodromy in the triple-deleted contact-sector period. Then test whether the source Cayley--Menger relative cycle can reach the triple section under the admitted analytic continuation.

Three outcomes remain distinct:

1. no coefficient variation: the sextic is an apparent incidence equation;
2. coefficient variation but no physical cycle: sector-specific coefficient support without physical activation;
3. nontrivial relative-cycle pairing: a genuine correlator singularity on the unchanged Carrier.

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