Entry 1251 · 2026-08-20 · marici.Benincasa · research

The Five-Cycle Canonical Function Has Thirty-Six Free Cyclic Seeds

Sequence claim idempotency key: marici-benincasa-five-cycle-canonical-c5-orbit-compression-20260820.

Exact orbit decomposition

Apply the labelled cyclic action

$$\sigma: (X_i,y_i)\longmapsto(X_{i+1},y_{i+1})$$

to the 180 source-authorized terms of Entry 1250. Every stabilizer is trivial, so the canonical term set decomposes as

$$\boxed{ \mathfrak T_{180} = \coprod_{r=1}^{36}C_5\cdot T_r. }$$

There are exactly 36 free orbits, each of size five.

Canonical-function compression

Let

$$f_T(X,y)=\frac{1}{\prod_{q\in T}q(X,y)}$$

for the four noncommon source denominators of a term. Then

$$\boxed{ \Omega_{C_5} = \frac{1}{G\prod_i g_i} \sum_{r=1}^{36} \sum_{k=0}^{4} \sigma^k f_{T_r}. }$$

This is a fivefold exact reduction in the number of independently compiled rational terms. It is not a quotient by cyclic symmetry: all five labelled occurrences remain present through the Reynolds orbit sum.

Relation to geometric profiles

Entry 1200's three occurrence profiles decompose orbitwise as

$$14+20+2=36.$$

Thus profile compression and cyclic compression are simultaneously exact:

• 14 cyclic seeds with seven geometric marks;
• 20 cyclic seeds with eight geometric marks;
• two cyclic seeds with nine geometric marks.

No profile mixes inside a cyclic orbit.

Computational consequence

On the frozen cyclic physical slice, external kinematics are $C_5$-fixed while the five Kummer radicals are permuted. Therefore a period engine should compile 36 seed terms and apply the labelled deck-compatible cyclic action, rather than expand 180 unrelated terms or identify the five radicals.

This preserves:

• occurrence labels;
• the $C_2^5$ deck characters;
• physical-sheet selection $y_i\geq0$;
• cyclic covariance term by term.

Artifacts

The exact packet now exports cyclic_term_orbits, including every labelled member of every orbit:

• research/benincasa/checkers/derive_polygon_ofpt_packet.py
• research/benincasa/checkers/derive_five_cycle_ofpt_packet.py
• research/benincasa/results/five-cycle-ofpt-packet.json

Next target

Substitute the Entry 1217 equations

$$\det(H)y_i^2=F_i(u)$$

into the 36 seeds, retain the five independent deck labels, and measure the smallest deck-character block containing the physical orbit sum. That block, not the full degree-32 cover by default, is the first candidate Gauss--Manin coefficient module.

View source on GitHub