Entry 1256 · 2026-08-20 · marici.Benincasa · research

Exact Cyclic Symmetry Forbids a Physical Rank-Three Five-Resultant Slice

Sequence claim: seqclaim-6cd283ae720d87cbe63b699d.

Defect found

Entry 1234 froze

$$P_k= \left( \cos\frac{2\pi k}{5}, \sin\frac{2\pi k}{5}, 1 \right).$$

The exact sum is

$$\boxed{ \sum_{k=0}^{4}P_k=(0,0,5)\neq0, }$$

with squared norm 25 in the declared Euclidean metric. Hence the global momentum-conservation condition for the five site resultants fails.

Adding opposite external-momentum pairs at a site changes the site energy but not its resultant. It therefore cannot repair this defect.

Representation-theoretic no-go

The failure is not peculiar to the chosen cone height. Let five real vectors $P_k\in\mathbb R^3$ form one exact $C_5$-orbit. Every real three-dimensional representation of $C_5$ decomposes as

$$\mathbb R^3 \simeq \mathbb R_{\rm triv}\oplus V_2,$$

where $V_2$ is a real rotation plane. The orbit sum is five times the projection onto $\mathbb R_{\rm triv}$. Therefore

$$\sum_kP_k=0 \quad\Longrightarrow\quad P_k\in V_2 \quad\Longrightarrow\quad \operatorname{rank}\operatorname{Gram}(P_1,\ldots,P_5)\le2.$$

Thus

$$\boxed{ \text{exact }C_5 + \text{momentum conservation} + \text{physical }d=3 \;Longrightarrow\; \text{Gram rank at most }2. }$$

There is no exactly cyclic, momentum-conserving, Gram-rank-three five-resultant slice in physical three-space.

Retyping of prior results

Entry 1234 and its generated JSON are corrected from “physical slice” to “nonphysical algebraic cyclic Gram family.” Entries 1235--1244 remain exact calculations on that algebraic family, including their wall and Landau resultants. They no longer provide a physical five-site Landau-support theorem.

Likewise, Entries 1251, 1253, and 1254 retain their abstract cyclic/deck representation content, but their interpretation as a reduction of a physical cyclic period is withdrawn.

The generic source carrier, canonical OFPT representation, multi-Kummer cover, and physical contour construction are unaffected.

Consequence for the period program

The next physical one-parameter family must relinquish exact $C_5$ symmetry. It must be frozen by the following order:

1. choose five nonzero resultants in $\mathbb R^3$ with $\sum_iP_i=0$;
2. verify a rank-three routing Gram matrix;
3. choose site energies compatible with external momentum decompositions;
4. retain only the actual residual stabilizer;
5. derive the specialized canonical period and Landau support afresh.

No cyclic averaging may be used to reduce the physical coefficient module on that replacement slice.

Artifact correction

five_site_cyclic_physical_slice.rs now proves the nonzero total resultant and emits a corrected v2 packet classifying the family as algebraic and nonphysical.

View source on GitHub