# FRI verifier over BabyBear: config + fold/opening relation + conformance KAT (component (d), 0x0AE8)

> **Scope caveat (2026-07-19 finding).** "Plonky3/SP1" wording below means Plonky3 `0.4.3-succinct`
> (Aere's OWN BabyBear + FRI STARK stack). It does NOT mean the pinned SP1 6.1.0, which is a Hypercube
> release whose inner recursion proof uses BaseFold, NOT FRI. This FRI low-degree test does not apply
> to SP1 6.1.0 at all; it is conformance-confirmed for a BabyBear + FRI STARK verifier (Aere's own
> Plonky3 circuits), not for an SP1 6.1.0 verifier. See
> `../../docs/AERE-STARK-SP1-RECURSION-AIR-PORT-SPEC-SUMMARY.md`.

Date: 2026-07-19. Status: FRI config CONFIRMED-FROM-SOURCE; fold + opening relation (verify_query)
IMPLEMENTED; real conformance KAT PASSED (accept genuine + reject tampered).

This records the exact FRI low-degree test Plonky3/SP1 use over BabyBear, traced from the pinned
`p3-fri` source, the fold + opening relation the verifier checks, and the real known-answer test that
confirms conformance. The top-level precompile 0x0AE8 stays FAIL-CLOSED (returns EMPTY for every input)
regardless: this component being confirmed does not port the duplex-sponge Fiat-Shamir challenger
(component (f)) that binds the transcript, nor the SP1 recursion-AIR (component (e)).

## Pinned target (exact, checksum-matched)

The FRI verifier is `p3_fri::verifier` from `p3-fri`, its commit-phase MMCS is `ExtensionMmcs` from
`p3-commit` wrapping the CONFIRMED `FieldMerkleTreeMmcs` (component (c)), and the challenger used to
emit the ground truth is `DuplexChallenger` from `p3-challenger`. All crates.io version
`0.4.3-succinct` (the Succinct Plonky3 fork), the revision pinned in the repo's Cargo.lock files:
- `p3-fri`        0.4.3-succinct, sha256 `5cbc4965ee488f3247867b7ec4bb005b8afa72cb0d461a4dcb1387ecab6426d5`
- `p3-challenger` 0.4.3-succinct, sha256 `b6a908924d43e4cfb93fb41c8346cac211b70314385a9037e9241f5b7f3eaf77`
- `p3-dft`        0.4.3-succinct, sha256 `be6408b10a2c27eb13a7d5580c546c2179a8dc7dbc10a990657311891f9b41c0`
- `p3-commit`     0.4.3-succinct, sha256 `50acacc7219fce6c01db938f82c1b21b5e7133990b7fff861f91534aeb569419`

Found in `aerenew/zk-circuits/*/Cargo.lock` and `aerenew/rollup-evm-validity/*/Cargo.lock`. A fresh
cargo build of the extractor (below) resolved the SAME checksums, so the crates executed are
byte-identical to the ones the AERE prover pins.

## The confirmed FRI config (traced from p3-fri source)

`FriConfig` (p3-fri `config.rs`) has exactly four fields:
`{ log_blowup, num_queries, proof_of_work_bits, mmcs }`. Traced from `verifier.rs`:

- Folding arity is 2 (hard-coded): each layer opens the sibling pair `(index, index ^ 1)` and folds
  `index_pair = index >> 1`. The comment in `proof.rs` says a non-2 arity would need multiple siblings;
  this pinned version is arity 2.
- `log_max_height = commit_phase_commits.len() + log_blowup`. The commit phase runs for
  `log_max_height - log_blowup` layers (`for log_folded_height in (log_blowup..log_max_height).rev()` in
  `prover.rs` commit_phase), so `num_fold_rounds = log_max_height - log_blowup`.
- The FINAL polynomial is a single CONSTANT `final_poly: F` (F = the challenge field EF). Folding runs
  all the way down; after the last fold the remaining `blowup()` evaluations are all equal, and the
  verifier checks `folded_eval == final_poly` for EVERY query (`verify_challenges`). (The source
  comment notes it "could become Vec<...> if generalized to support non-constant final polynomials"; in
  this pinned version it is a constant, so `log_final_poly_len = 0`.)
- The commit-phase MMCS is `ExtensionMmcs<Val, Challenge, ValMmcs>` (the FRI codewords live in the
  degree-4 extension EF = `BinomialExtensionField<BabyBear, 4>`). Each commit-phase leaf is a PAIR of EF
  evaluations; `ExtensionMmcs::verify_batch` flattens each EF element to its 4 base coords
  (`as_base_slice`) and concatenates, so a leaf is 8 BabyBear coords, a single width-8 row of a
  height-`2^log_folded_height` tree, checked by the CONFIRMED base `FieldMerkleTreeMmcs::verify_batch`
  (component (c)).
- Proof-of-work / grinding (`proof_of_work_bits`, `check_witness`) is part of the transcript phase
  (`verify_shape_and_sample_challenges`), i.e. component (f); it is NOT part of the per-query fold +
  opening relation this component ports.
- `num_queries`, `log_blowup`, `proof_of_work_bits` are proximity/soundness parameters read from the
  frozen SP1 config; the values are a [MEASURE] against a real exported proof (see the port spec). The
  KAT here exercises several (log_blowup, log_max_height, num_queries) shapes to validate the relation
  independently of the exact frozen numbers.

## The fold + opening relation (p3-fri verify_query, ported)

For one query at `index` (from `verifier.rs verify_query`, reproduced exactly by the references and by
`Fri.verifyQuery` in the precompile):

```
folded_eval = 0 in EF
x = two_adic_generator(log_max_height)^reverse_bits_len(index, log_max_height)   # coset point, in EF
for each layer (log_folded_height = log_max_height-1 down to log_blowup):
    folded_eval += reduced_openings[log_folded_height + 1]     # the injected input opening at this height
    index_sibling = index ^ 1;  index_pair = index >> 1
    evals = [folded_eval, folded_eval];  evals[index_sibling % 2] = sibling_value   # the opened sibling
    MMCS.verify_batch(commit_layer, dims={width:2(EF)->8(base), height:2^log_folded_height},
                      index_pair, [evals flattened to 8 base coords], opening_proof)   # component (c)
    xs = [x, x];  xs[index_sibling % 2] *= two_adic_generator(1)   # the sibling point is -x
    folded_eval = evals[0] + (beta - xs[0]) * (evals[1] - evals[0]) / (xs[1] - xs[0])   # interpolate at beta
    index = index_pair;  x = x^2
return folded_eval    # verify_challenges checks this == final_poly for every query
```

- `two_adic_generator(bits)` for `bits <= 27` is the BabyBear base generator embedded in EF (confirmed
  from `p3-baby-bear extension.rs ext_two_adic_generator`), so `x` is a base-field element embedded in
  EF; `two_adic_generator(1) = -1` (the order-2 root). The arithmetic uses F_{p^4} (component (a)).
- The reduced openings come from the input polynomials in a full STARK (component (e)); for a
  standalone single-codeword low-degree test they are zero except `reduced_openings[log_max_height] =
  codeword_value_at_index`, which the extractor emits as `ro_top`.

## Montgomery / canonical note

Field elements are serialized as canonical u32 (`PrimeField32::as_canonical_u32` for base coords). The
F_{p^4} fold arithmetic is canonical (component (a)); the Montgomery `R^{-1}` factor lives entirely
inside the Poseidon2 internal layer (component (b), resolved there) and does not surface in the FRI or
MMCS layers. As a cross-check, the extractor's `permute([0;16])` (emitted as `perm_zeros`) equals the
confirmed Poseidon2 "zeros" vector, and the extractor's `two_adic_generator(bits)` for `bits 0..=27`
equals component (a)'s `twoAdicGenerator(bits)` (this is what makes the fold's coset points come out
right, closing the old `[VERIFY]` on the generator).

## How the KAT vectors were obtained (primary source)

A small Rust extractor (`pq-stark/fri-extractor`, `main.rs` + `Cargo.toml` + `Cargo.lock`) depending on
the pinned crates was compiled and run (cargo 1.97.0). It builds the exact SP1 inner FRI config
(BabyBear + `BinomialExtensionField<_,4>` + the `seed_from_u64(1)` Poseidon2 permutation +
`FieldMerkleTreeMmcs` + `ExtensionMmcs` + `DuplexChallenger<_,_,16,8>`), constructs genuinely
low-degree codewords (an RS/LDE of a random low-degree polynomial via `Radix2Dit`, bit-reversed), runs
`p3_fri::prover::prove` to emit a real `FriProof`, then runs
`p3_fri::verifier::{verify_shape_and_sample_challenges, verify_challenges}` to confirm the pinned
LIBRARY itself accepts. It emits everything the reference verifier needs (commit-phase roots, betas,
query indices, reduced openings, per-query per-layer sibling values + MMCS opening proofs, final poly),
each field element as canonical u32. The output (`pq-stark/fri_ground_truth.json`) is the authoritative
source. Three FRI cases are emitted:

| case | log_blowup | log_max_height | num_queries | fold layers |
|---|---|---|---|---|
| blowup1_h6_q4 | 1 | 6 | 4 | 5 |
| blowup2_h7_q5 | 2 | 7 | 5 | 5 |
| blowup1_h8_q6 | 1 | 8 | 6 | 7 |

## Conformance KAT (real known-answer test, executed from the pinned crates)

The reference (`fri_verify_reference.py`, `.mjs`, standalone `FriVerifySelfTest.java`, and
`Fri.verifyQuery` in the precompile) re-verifies each emitted proof:
- ACCEPT: every query's MMCS openings pass AND every query's folded constant equals `final_poly`.
- REJECT (four tamper variants per case): a corrupted opened value (breaks the MMCS opening), a
  corrupted MMCS sibling digest (breaks the opening), a wrong fold challenge `beta` (breaks the
  interpolation so `folded != final_poly`, i.e. a wrong fold), and a non-matching `final_poly` (the
  final-poly check, the mechanism that catches a non-low-degree witness).

Result (ran 2026-07-19):
- `python test_fri_verify.py` -> PASS=55 FAIL=0 (sanity perm + 28 generators; 3 cases x (accept + 4
  rejects); Python + Node + Java byte-identical per-query folded values; recorded in
  `../results/kat-results-fri-verify.json`).
- `java FriVerifySelfTest fri_ground_truth.json` -> PASS=44 FAIL=0.
- The PRECOMPILE's own `Sp1StarkVerifierPrecompiledContract.Fri.verifyQuery` (driven directly)
  reproduces accept + reject for all 3 cases (15/15), so the wired code, not just the parallel
  references, is conformant.

## Status summary

- FRI config: CONFIRMED-FROM-SOURCE (arity 2; num_fold_rounds = log_max_height - log_blowup; constant
  final poly; ExtensionMmcs-over-FieldMerkleTreeMmcs commit phase; traced from p3-fri 0.4.3-succinct;
  checksums matched Cargo.lock).
- Fold + opening relation (verify_query): IMPLEMENTED (Python/Node/Java + `Fri.verifyQuery`).
- Conformance KAT: PASSED (3 real FRI proofs accepted, 4 tamper variants each rejected; three-language
  byte-identical; the precompile's own method also conformant).
- Closed by this KAT (were `[VERIFY]`): arity-2 fold, num_fold_rounds, reverse-bits index-to-coset-point
  mapping, `two_adic_generator(bits)` = Plonky3's, and the F_{p^4} non-residue `W = 11` = Plonky3's (the
  EF fold arithmetic reproduces the interpolation).
- `Fri.foldRelationConfirmed` = true (the fold + opening relation is confirmed conformant).
- `[MEASURE]` remaining: conformance against a REAL exported SP1 v6.1.0 inner proof's FRI section (needs
  the exported proof + the challenger, component (f)) and the AIR reduced openings (component (e)).
- Top-level 0x0AE8: still FAIL-CLOSED (EMPTY for every input). The duplex-sponge challenger
  (`Challenger.spongePorted = false`, component (f)) that derives the betas/indices/grinding, and the
  SP1 recursion-AIR (`StarkConstraints` = UNAVAILABLE, component (e)) that supplies the reduced
  openings, are un-ported, so `Fri.checkQuery` cannot be driven and returns UNAVAILABLE. ACCEPT is
  unreachable. DO NOT ACTIVATE.