Entry 1257 · 2026-08-20 · marici.Benincasa · research

An Asymmetric Momentum-Conserving Five-Site Physical Slice Is Frozen

Sequence claim: seqclaim-086e73f5759d31d6594bbfae.

Replacement resultants

Following Entry 1256's no-go theorem, relinquish exact cyclic symmetry and set

$$\begin{aligned} P_1&=(1,0,0),& P_2&=(0,1,0),& P_3&=(0,0,1),\\ P_4&=(1,2,3),& P_5&=(-2,-3,-4). \end{aligned}$$

Then

$$\boxed{\sum_{i=1}^{5}P_i=0.}$$

All five resultants are nonzero.

Routing nondegeneracy

Choose

$$q_1=P_1, \qquad q_2=P_1+P_2, \qquad q_3=P_1+P_2+P_3.$$

The routing Gram matrix is

$$H= \begin{pmatrix} 1&1&1\\ 1&2&2\\ 1&2&3 \end{pmatrix}, \qquad \boxed{\det H=1.}$$

The fifth focus uses

$$q_4=P_1+P_2+P_3+P_4=-P_5 =-q_1-q_2+4q_3.$$

Thus the physical three-variable Cayley--Menger contour is regular on this integer routing chart.

One physical parameter

Set

$$X_1=\cdots=X_5=t.$$

Since

$$\max_i|P_i|=\sqrt{29},$$

the literal real domain is

$$\boxed{t\ge\sqrt{29}.}$$

For each site, any $t\ge|P_i|$ is realized by decomposing $P_i$ into external momenta with total magnitude $t$; an opposite transverse pair can increase that magnitude without changing the resultant.

Symmetry audit

An exact enumeration of all 120 permutations preserving the labelled Gram matrix finds only the identity. Hence

$$\boxed{\operatorname{Stab}(P_1,\ldots,P_5)=1.}$$

No cyclic or reflection quotient is admissible. All 180 OFPT terms, 26 marked walls, and 32 Kummer characters must remain labelled in the physical period calculation.

Frozen period

The corrected physical target is

$$\Pi_{C_5}^{\rm asym}(t) = \int_{\Gamma_3} \frac{du_1\,du_2\,du_3}{\sqrt{\det H}} \Omega_{C_5}(X_i=t,y(u;P)), \qquad \det H=1.$$

This family is frozen before any new Landau elimination, period sampling, or operator search.

Artifacts

• research/benincasa/marici-gm/src/bin/five_site_asymmetric_physical_slice.rs
• research/benincasa/results/five-site-asymmetric-physical-slice.json

Next falsifier

Recompute the one-wall Landau resultants for all 26 labelled source walls on this asymmetric slice. Do not import the cyclic orbit reduction or any threshold polynomial from Entries 1235--1244.

View source on GitHub