Multicoil Phase Features

A single periodic feature is one circle. A tuple of periodic features is a finite product of circles, and the joint repeat horizon is the least common multiple of their periods.

Goal

Learn how Circle Calculus packages multi-period phase tags as explicit finite features with theorem-backed repeat and relative-shift checks.

Concept

For periods (p1, p2, ..., pk), a position becomes a phase tuple:

position -> (position mod p1, position mod p2, ..., position mod pk)

The tuple repeats after the joint horizon. For two positions, a relative phase can stay invariant under a common shift even when absolute positions change.

Example

The public fixture uses periods (5, 7) and position 37:

phase tuple = (2, 2)
joint repeat horizon = 35

Shifting by 35 returns the same tuple. The relative phase between a query and key is also preserved when both are shifted by the same amount.

python scripts/multicoil_phase_feature_certify.py --format json
python scripts/circle_ai_contract_ready.py --kind multicoil_phase_feature --digest --field joint_repeat_horizon --field relative_phase --include-recommendations

Theorem Trail

Certificate Reading

The useful fields are:

periods: the declared finite phase bank.
phase_tuple: the phase tag for the selected position.
joint_repeat_horizon: the repeat horizon for the full tuple.
relative_phase: the query-key phase difference in the declared period.
shifted_relative_phase: the same value after a common shift.

Planner recommendations:

PHASE-USE-JOINT-REPEAT-HORIZON says a downstream system can use the joint horizon as an explicit repeat boundary.
PHASE-AUDIT-RELATIVE-SHIFT-INVARIANT says query-key features should preserve relative phase under common shifts when that invariance is expected.

This contract proves finite phase-feature bookkeeping and invariance facts. It does not prove that these features improve a model, choose the right periods, or discover periodic structure in real data. Python fixtures are executable references, not proof artifacts and not performance claims.

Dictionary

Source Trail

Architecture paper: Circle AI Architectures

Quickstart: Multicoil Phase Feature Certifier