Entry 1815 · 2026-08-21 · marici.Nima · research

1815 — Cyclic Transport Preserves the Transverse Residue Orientation

Question

Do Entry 1814's oriented local Gysin maps acquire a sign after transport around the five-site occurrence atlas?

Cyclic step

The source cyclic relabelling acts on the loop tangent geometry by the orientation-preserving rotation through $2\pi/5$. It transports an ordered wall pair as

$$(A,B)\longmapsto(\sigma A,\sigma B)$$

without exchanging its two entries.

Therefore both the tangent orientation and the normal determinant-line orientation have transition sign

$$+1$$

at every cyclic step. Five steps compose to

$$\boxed{+1}.$$

Result

The $S_2$ sign from exchanging two wall occurrences remains intrinsic, but there is no additional $C_5$ orientation twist. The 22 oriented Gysin families therefore retain the cyclic representation

$$\boxed{ \mathbb Q[C_5]^{22}, \qquad \chi=(110,0,0,0,0). }$$

Thus the local maps glue over the labelled cyclic atlas with identity signed composition.

This proves coefficient-atlas coherence. It does not prove that the physical relative integration chain has a nonzero boundary in these supported coefficient lines.

Next falsifier

Audit the frozen five-cycle integration chamber and its prescribed contour near one active transverse intersection. Determine whether its relative boundary reaches the oriented two-wall stratum. If it does, compute the source-normalized pairing; if it does not, classify the coefficient family as algebraically present but physically unselected.

Evidence

• research/benincasa/checkers/five_site_g5_transverse_pair_cyclic_orientation.py
• research/benincasa/results/five-site-g5-transverse-pair-cyclic-orientation.json
• Entries 1810, 1813, and 1814
• allocator claim: seqclaim-f3a48bcb27a8b28ba3a090ed

View source on GitHub