feat: GKR Add Instance

[Tabaie]

• Change the GKR API so that defining the circuit and assigning instances are separated. It removes the necessity of defining hints for the output and deferred "finalize" functions.

• Remove the Println function in favor of GetValue.

• gkrgates.Register now checks if a gate of the same name has already been registered. If so, it compares the gates and returns an error if they are not equal. If they are equal, it returns false, nil.

• Remove all logic pertaining to reordering of wires and instances.

• Unless the user explicitly provides their own initial challenge, automatically use all input and output of the circuit as initial Fiat-Shamir challenge.

• Rename GkrCompressions to GkrPermutations to correctly reflect its function. (Note: a future PR should implement compressions instead and revert the name change)

• A benchmark for gkr-poseidon2

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Note

Introduce a compile-then-add-instances GKR API with GetValue, enforce gate registry validation (incl. curve support and equality checks), refactor hints/solvers, and update Poseidon2 compressor, generators, and tests.

• GKR API:
  • Add compile-then-add pattern: API.NewInput, API.Compile, Circuit.AddInstance, and Circuit.GetValue; remove old Solve/Verify flow and Println.
  • Default Fiat–Shamir seed uses commitment to all inputs/outputs unless overridden.
• Gate Registry (gkrgates):
  • New RegisterOptions with validation; prevent conflicting duplicate registrations by function equality checks; support per-gate allowed curves; add EqualGateFunction.
• Solvers/Hints:
  • Override hints with new gkrhints engine (GetAssignment, Solve, Prove) across curves; generate-new solver_hints and fix MulAcc to multiply.
• GKR internals:
  • Gates carry allowed curves; simplify gkrinfo (remove dependency/permutation logic); export ComputeLogNbInstances; add utils SliceOfRefs, ExtendRepeatLast.
• Poseidon2:
  • Replace GkrCompressions with GkrCompressor; add RegisterGkrGates; adapt circuits/tests.
• Generator:
  • Include gate_testing.go generation; adjust outputs when not generating test vectors.
• Examples/Tests:
  • Update examples and unit tests to new API and default challenges; add coverage for single/no-instance cases and gate registration behaviors.

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[cursor]

Bug: Intermediate wires not padded in finalize

The padding logic only extends input and output wire assignments to the next power of 2, but intermediate wires are not padded. This creates an inconsistency because the ProveHint calls repeatUntilEnd which pads all wires including intermediate ones. The mismatch between frontend and hint padding could cause the GKR proof verification to fail or produce incorrect results when intermediate wire values are accessed after padding.