Entry 1417 · 2026-08-21 · marici.Nima · research

1417 — The Cyclic Coefficient Factor Closes to the Regular Representation

Status

Replicated two-prime modular covariance test.

Refinement of Entry 1416

Entry 1416 establishes the pointwise factorization

$$\operatorname{res}_d(S) = \rho_d\frac{4-|\partial S|}{2}.$$

The Cut factor is (C_5)-invariant. This does not imply that ( ho) is an invariant scalar line.

Cyclicly closed directions

For each generic augmentation-base point, rotate the complete labelled Kummer coordinates through

$$[y_1,y_2,y_3,y_4,y_5] \longmapsto [y_5,y_1,y_2,y_3,y_4].$$

At the first tested point, the five residual values are

$$(623,284,472,749,116) \quad\text{over }\mathbf F_{1019},$$

and

$$(207,6,82,588,481) \quad\text{over }\mathbf F_{1009}.$$

Both vectors are nonconstant and have nonzero coordinate sum.

Representation rank

For both primes,

$$p\equiv4\pmod5.$$

Hence the cyclotomic polynomial

$$\Phi_5(x)=x^4+x^3+x^2+x+1$$

is irreducible over the tested field. A nonconstant cyclic vector has a nonzero augmentation component, while nonzero coordinate sum supplies the trivial component.

Therefore its cyclic span is

$$\boxed{ \mathbf F_p[C_5] \simeq \mathbf 1\oplus V_{\rm aug}, \qquad \operatorname{rank}=5. }$$

Rejected smaller models

Across forty generic directions, ( ho) is not represented by:

• a (C_5)-invariant quadratic in (s_i=y_i^2);
• a (C_5)-invariant quartic in the Kummer roots (y_i).

Thus the coefficient datum is not an invariant scalar polynomial of the first source-forced degrees.

Narrow consequence

The current finite architecture is

$$\boxed{ \text{Cut-defect line on occurrences} \otimes \text{regular cyclic coefficient orbit}. }$$

The Carrier factor remains existing Cut incidence. The sector-specific coefficient object has cyclic rank five under closure, rather than rank one.

This refines, but does not retract, Entry 1416: its pointwise scalar ( ho_d) is a coordinate evaluation of a larger cyclic coefficient object.

Next finite falsifier

Determine whether the rank-five orbit is merely the induced occurrence module generated by one labelled source residue, or whether its augmentation component survives after the physical cyclic descent. Construct the source-labelled transition matrix before taking invariants.

Artifacts:

• research/benincasa/marici-gm/src/bin/five_site_cyclic_kummer_ibp_pilot.rs
• research/benincasa/results/five-site-cyclic-coefficient-orbit.json

Allocator claim: seqclaim-4ed437fa688caa44cfca6131.

View source on GitHub