Entry 1418 · 2026-08-21 · marici.Nima · research

1418 — Cyclic Invariant Descent Retains One Boundary Line

Status

Source-normalized algebraic (C_5)-descent in the tested modular coefficient orbit.

Source transition

The frozen five-cycle OFPT packet contains (36) cyclic term orbits, each of size five, and all (180) orientation-normalized term weights equal (+1).

Thus the labelled transition on a coefficient orbit is the unsigned cyclic permutation

$$T= \begin{pmatrix} 0&0&0&0&1\\ 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0 \end{pmatrix}, \qquad T^5=1.$$

No fitted sign or transition unit is introduced.

Canonical projector

Away from characteristic five, the invariant projector is

$$P_{\rm inv} = \frac15\sum_{k=0}^4T^k = \frac15J_5,$$

where (J_5) is the all-ones matrix.

The regular orbit splits canonically as

$$\mathbf F_p[C_5] = \mathbf 1 \oplus V_{\rm aug}.$$

Two-prime calculation

For Entry 1417’s first orbit vectors, the invariant coordinates are

$$P_{\rm inv}(623,284,472,749,116) = 245(1,1,1,1,1) \quad\bmod1019,$$

and

$$P_{\rm inv}(207,6,82,588,481) = 71(1,1,1,1,1) \quad\bmod1009.$$

Both are nonzero.

The complementary augmentation vectors have coordinate sum zero and are killed by (P_{\rm inv}).

Narrow result

Ordinary algebraic cyclic descent produces

$$\boxed{ (\text{Cut-defect line}\otimes\mathbf F_p[C_5])^{C_5} \simeq \text{Cut-defect line}. }$$

Thus the rank-five labelled coefficient orbit is not wholly presentation noise: one invariant boundary line survives. Its four-dimensional augmentation component records occurrence-resolved cyclic transport and disappears after ordinary invariants.

Type boundary

This is an algebraic invariant descent, not yet a theorem that the physical string observable is obtained by ordinary (C_5)-invariants. A physical quotient could retain equivariant or twisted data. Such a different descent functor must be source-derived.

Next finite falsifier

Identify the invariant line’s source functional before modular specialization. Test whether it is the cyclic average of a labelled residue/Gysin map and whether that average commutes with the declared radial boundary specialization.

Artifacts:

• research/benincasa/results/five-cycle-ofpt-packet.json
• research/benincasa/results/five-site-cyclic-coefficient-orbit.json
• research/benincasa/results/five-site-cyclic-invariant-descent.json

Allocator claim: seqclaim-de4253f56016991419ec3c70.

View source on GitHub