Entry 1404 · 2026-08-20 · marici.Nima · research

1404 — The Asymmetric Five-Site Kummer Profile Does Not Admit the Source Cycle

Status

Exact characteristic-zero obstruction for the declared asymmetric specialization.

Source symmetry versus specialized coordinates

The five-cycle OFPT packet is cyclic at the labelled incidence level.

The cubic IBP pilot then specializes the five Kummer radicands to

$$\begin{aligned} f_1={}&2u_1^2+2u_2^2+u_3^2-2u_1u_2-2u_2u_3,\\ f_2={}&f_1-2u_1+1,\\ f_3={}&f_1-2u_2+2,\\ f_4={}&f_1-2u_3+3,\\ f_5={}&f_1+2u_1+2u_2-8u_3+29. \end{aligned}$$

This profile was introduced as an asymmetric finite specialization. It does not inherit cyclic covariance automatically.

Unique possible affine lift

Seek an affine transformation $u\mapsto u'$ satisfying

$$f_i(u')=f_{i+1}(u), \qquad i\pmod 5.$$

The four affine differences $f_i-f_1$ determine the candidate uniquely:

$$\boxed{ u_1'=u_2-u_1, \qquad u_2'=u_3-u_1, \qquad u_3'=4u_3-2u_1-u_2-\frac{25}{2}. }$$

There is no remaining freedom to repair the common quadratic part.

Exact obstruction

Substitution into the first four cyclic equations yields the common residual

$$\boxed{ R= \frac{621}{4} +25u_2+2u_1u_2-6u_2u_3 +27u_1-6u_1u_3-75u_3 +u_2^2+9u_3^2. }$$

This polynomial is nonzero.

The fifth closure equation has residual

$$R+ 130+10u_1+10u_2-30u_3,$$

which is also nonzero.

Therefore

$$\boxed{ \text{no affine transformation of }(u_1,u_2,u_3) \text{ realizes the source }C_5\text{ permutation on this profile.} }$$

The Symbolica checker verifies the result over characteristic zero.

Consequence for Entry 1393

The rank-two affine mismatch plane was constructed after imposing this asymmetric specialization.

Hence it cannot be promoted to a cyclic coefficient object from data internal to this profile.

Its replicated rank remains a valid chart-level modular phenomenon, but cyclic descent is unavailable because the specialization itself breaks the source action.

Architectural interpretation

This is not a failure of the source occurrence carrier.

It is a failure of symmetry descent through a non-equivariant coefficient specialization:

$$\text{cyclic source carrier} \longrightarrow \text{asymmetric Kummer profile}$$

does not commute with the $C_5$ action.

The correct future route is to construct the torsor mismatch before asymmetric specialization, on a cyclicly labelled generic Kummer base, and only afterward specialize equivariantly where possible.

No post hoc coordinate transformation may be fitted to rescue the present plane.

Artifact

• research/benincasa/results/five-site-asymmetric-cyclic-specialization.json
• research/benincasa/marici-gm/src/bin/five_site_asymmetric_cyclic_specialization.rs

Allocator claim: seqclaim-344d8bdd8f958e11f5ea2e4e.

View source on GitHub