Fold path and field geometry

Akita uses one fold engine for every field tier. The schedule selects the opening method for each nonterminal fold.

See Coefficients and evaluation points have different roles. For the packing path, read the three obligations in order: the direct scalar opening, the source/fold consistency identity, and the physical quotient and ring switch.

The extension degree fixes the number of field coordinates in one claim. It does not select the opening method. In particular, an fp128 schedule may use coefficient packing even though its extension degree is one. That choice can reduce the D input width even though there is no EOR proof to remove.

Single-field path

When EXT_DEGREE == 1:

1. prove_root prepares coefficient packing for every root group.
2. Recursive folds use packing at level 1 and every multi-chunk fold; later single-chunk folds may dispatch to packing or evaluation trace.
3. Shared prove_fold runs the ring relation, ring switch, and sumchecks.
4. No EOR payload is produced because the extension degree is one.

The verifier mirrors this path. A scalar root is the one-group case of the grouped root verifier.

Extension claim path

When EXT_DEGREE > 1, the schedule still selects the method. Coefficient packing opens the original extension valued claim directly and emits no EOR. Evaluation trace uses EOR to bridge base field coefficients to the extension valued opening. The verifier accepts an EOR payload if and only if the scheduled method is evaluation trace and the extension degree is greater than one. See Extension-opening reduction.

Every group consumed by one fold uses the same method family. Packing groups may use different challenge subring dimensions. A setup prefix inherits the method selected by the fold that consumes it. The commitment itself does not record that method.

For coefficient packing, ring-switch finalization prepares one compact batch of checked group semantics. The batch retains shared sparse challenges, alpha powers, row weights, gadgets, and witness addresses. It supplies both the Stage 2 opening terms and the CPU relation events. The CPU calls coefficient_packing_relation_events on those groups when it needs expanded weights; shared preparation does not evaluate the challenges or allocate events. This keeps both consumers tied to the same fold inputs while allowing other backends to consume the compact batch directly. The verifier evaluates its challenges once per group when building compact factors.

Implementation map

• crates/akita-prover/src/protocol/prove/fold/.
• crates/akita-verifier/src/fold/ and crates/akita-verifier/src/stages/.
• crates/akita-params/src/layout/params/precommitted.rs.
• crates/akita-params/src/layout/subring_packing_geometry.rs.
• crates/akita-types/src/subring_coefficient_packing.rs.