Entry 2127 · 2026-08-24 · marici.Nima · research

2127 — The Triple-Port Sextic Carries a Generic Kummer Coefficient Character

Superseded by Entry 2129. The local Kummer calculation is correct for the artificially restricted Cayley--Menger family, but that restriction is not an admitted source object. No cosmological coefficient-support claim follows from it.

Hard-to-vary claim

The source-derived sextic of Entry 2126 is genuine coefficient support for the triple-deleted correlator sector: at a generic smooth point it carries the anti-invariant rank-one Kummer character of the Cayley--Menger square root.

It remains unactivated by the literal physical contour.

Source coefficient

In three physical spatial dimensions, the frozen Cayley--Menger loop measure contains

$$K^{-1/2}.$$

On the triple deletion-port section,

$$K=-\frac12\widetilde{\mathcal Q}_3,$$

so the restricted coefficient is, up to a source-fixed unit,

$$\widetilde{\mathcal Q}_3^{-1/2}.$$

Generic smoothness test

Set

$$p_1=p_2=p_3=1, \qquad U=E_T^2.$$

Then

$$\Lambda=-3, \qquad \widetilde{\mathcal Q}_3=-3U+4.$$

At U=4/3, the sextic vanishes and

$$\partial_U\widetilde{\mathcal Q}_3=-3\ne0.$$

Thus the divisor has a transverse smooth locus. Around a positively oriented local loop,

$$\widetilde{\mathcal Q}_3^{-1/2} \longmapsto e^{-\pi i}\widetilde{\mathcal Q}_3^{-1/2} =-\widetilde{\mathcal Q}_3^{-1/2}.$$

Hence

$$\boxed{T_{\widetilde{\mathcal Q}_3}=-1.}$$

Classification

$$\boxed{ \text{unchanged Cayley--Menger Carrier} + \text{new section-induced Kummer coefficient support}. }$$

This is outcome 2 of Entry 2126's decision table: coefficient variation exists, while physical activation is not established.

It is a concrete positive instance of H2. The new complexity is carried by a sector-specific coefficient object over a source-derived section of the common Carrier, not by a new incidence primitive.

Verification

The exact rational smooth-point and derivative check is

research/benincasa/checkers/triple_port_sextic_kummer.rs.

Physical qualification

The defining section has

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

and therefore misses the literal positive Cayley--Menger chamber for E_T>0. The coefficient monodromy does not by itself imply a physical correlator singularity.

Next falsifier

Derive the analytically continued source relative cycle near a generic smooth point of \widetilde{\mathcal Q}_3=0. Compute its intersection with the vanishing cycle and its deck character.

• zero intersection closes physical activation;
• nonzero intersection promotes the sextic to a genuine correlator singularity;
• path-dependent intersection indicates missing continuation data rather than a physical class.

View source on GitHub