Entry 1413 · 2026-08-21 · marici.Nima · research

1413 — The Chord-Type Compatibility Split Descends to the Cyclic Augmentation Base

Status

Replicated two-prime modular descent test. Not a characteristic-zero theorem.

CORRECTION (Entry 1423). Representatives (5,11) are absent from the growth-four boundary grade, not compatible solutions. What descends is the support partition and the obstruction on all four supported orbits.

Question

Corrected Entry 1411 finds a chord-sensitive boundary pattern on five independent labelled radicands. The five-dimensional permutation base contains a redundant diagonal direction.

Does the obstruction disappear on the canonical four-dimensional cyclic base?

Augmentation model

Impose

$$s_1+s_2+s_3+s_4+s_5=0$$

while retaining all five labelled Kummer radicands

$$y_i^2=s_i.$$

Use (s_1,\ldots,s_4) as coordinates and set

$$s_5=-s_1-s_2-s_3-s_4.$$

Therefore the source derivative is typed as

$$\frac{d}{ds_i} = \partial_{s_i}-\partial_{s_5}, \qquad i=1,\ldots,4.$$

No occurrence label or Kummer character is discarded.

Census

At the generic orbit sample (z=7), both $\mathbf F_{1019}$ and $\mathbf F_{1009}$ give:

$$\begin{array}{c|c|c|c} \text{primitive degree}&\text{unknowns}&\operatorname{rank}A&\dim\ker A\\ \hline 1&641&641&0\\ 2&1921&1921&0 \end{array}$$

At both degrees the unique affine primitive satisfies exactly two of the six exact cyclic occurrence orbits:

$$\boxed{\text{cyclic representatives }5\text{ and }11.}$$

These are the diagonal-pair orbit and its complementary three-site orbit. Representatives (1,3,7,15) remain inconsistent.

Narrow result

The chord-type split found in corrected Entry 1411 is not caused by the diagonal scale direction of the five-dimensional permutation representation.

It survives descent to

$$V_{\rm cyc} = \{(s_1,\ldots,s_5):\sum_i s_i=0\} \simeq\mathbb Q(\zeta_5)$$

through quadratic primitive degree in the tested modular model.

Thus the current surviving statement is:

$$\boxed{ \text{exact cyclic occurrence symmetry} + \text{augmentation descent} \Longrightarrow \text{compatibility exactly on the diagonal chord orbit and its complement.} }$$

Limits

The calculation does not establish characteristic-zero persistence, cutoff stability, or physical string-boundary typing. The radial grading remains a declared test model.

Next finite falsifier

Derive the adjacent-versus-diagonal distinction as an explicit labelled obstruction functional rather than increasing the primitive cutoff blindly.

Artifacts:

• research/benincasa/marici-gm/src/bin/five_site_cyclic_kummer_ibp_pilot.rs
• research/benincasa/results/five-site-cyclic-kummer-augmentation-descent.json

Allocator claim: seqclaim-ece558a8729c53b46769fcd9.

View source on GitHub