Circulant And Block-Cyclic Mixers

A circulant mixer is a structured linear map whose rows are cyclic shifts of a kernel. The useful claim is structural: it commutes with cyclic shifts and has a fixed parameter budget before any learning happens.

Goal

Read circulant and block-cyclic mixer certificates as structure and accounting contracts, not as performance claims.

Concept

The circular object is the channel or token index modulo a period. A circulant kernel reuses the same parameters around the circle, so cyclic shifts in the input correspond to cyclic shifts in the output. A block-cyclic adapter applies the same idea at a block level and exposes a declared parameter budget.

period -> kernel -> circulant output
       -> dense reference parity
       -> parameter accounting

Example

The public fixture uses period 8. The certifier compares the structured circulant output with its dense reference and reports:

max_abs_dense_delta = 0
circulant_parameters = 8
dense_parameters = 64
block_cyclic_parameters = 128
block_to_dense_ratio = 0.0625
python scripts/circulant_block_cyclic_mixer_certify.py --format json
python scripts/circle_ai_contract_ready.py --kind circulant_block_cyclic_mixer --digest --field max_abs_dense_delta --field block_to_dense_ratio --include-recommendations

Theorem Trail

Certificate Reading

The useful fields are:

  • max_abs_dense_delta: parity between the structured and dense reference on the fixture.
  • circulant_parameters: parameter count for the circulant kernel.
  • dense_parameters: parameter count for the dense reference matrix.
  • block_cyclic_parameters: parameter count for the declared block-cyclic adapter.
  • block_to_dense_ratio: the finite accounting ratio for the fixture.

Planner recommendations:

  • MIXER-AUDIT-CIRCULANT-DENSE-PARITY says downstream code should keep a dense-reference parity check for the structured mixer fixture.
  • MIXER-AUDIT-BLOCK-CYCLIC-PARAMETER-BUDGET says a downstream planner should read the explicit parameter budget rather than infer savings from the word “cyclic.”

This contract proves finite structural parity and accounting facts. It does not prove training speed, inference speed, accuracy, memory savings at deployment, or that cyclic mixing is the right inductive bias for a task. Python fixtures are executable references, not proof artifacts and not performance claims.

Dictionary

Source Trail

Architecture paper: Circle AI Architectures

Quickstart: Circulant And Block-Cyclic Mixer Certifier

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