Arithmetization ⇒ SNARK

Arithmetization implies SNARK.

Statement

Migrated verbatim from arithmetization:

AIR encodes a computation as a constraint on consecutive rows of an execution trace: for each step $i$, a polynomial relation $P({\mathbf{w}}_i,{\mathbf{w}}_{i+1})=0$ holds. AIR is the arithmetization underlying STARKs — the FRI protocol can then verify the AIR constraints via a Reed-Solomon proximity test — BBHR18.

Migrated verbatim from arithmetization § Results:

• PLONKish arithmetization enables universal SNARKs with a single trusted setup for all circuits of bounded size — standard

Notes

class: unstated: no citing page says which notion of reduction is meant. Recording a class the wiki does not state would add a claim.

This relation is stated on 2 pages; the statements above are all of them.

Recorded during migration and not fixed — these are claims about the source text, not changes to it:

• ‘air’ is an alias of content/Glossary/arithmetization.md, the host page, so this is a self-loop in the current wiki.
• As an edge it asserts AIR alone yields a STARK, which is false without FRI and a compiler — the sentence is about being an ingredient, not an implication.
• STARK links to [[succinct-argument]], whose node cannot distinguish transparent STARKs from trusted-setup SNARKs.
• Duplicates the Results bullet at line 54 of the same page.
• COMPOSITE: (i) AIR is the arithmetization for STARKs, (ii) FRI verifies AIR constraints via a Reed-Solomon proximity test. Two links, one citation.
• STARK is linked as [[succinct-argument|STARKs]] — STARK is only an alias of content/Primitives/succinct-argument.md, so STARK and SNARK are the same node and this edge cannot distinguish transparent from trusted-setup arguments.
• FRI has no page and no reference of its own (BBHR18 covers it, but the FRI paper BBHR17 is not in content/References/).
• Duplicates the Results bullet at line 54; the two statements of the same fact must not become two graph edges.
• MISUSE OF THE FOLKLORE EXCEPTION: labelled ’— standard’ but ‘PLONKish arithmetization enables universal SNARKs with a single trusted setup for all circuits of bounded size’ is an attributable result (Plonk, and universal/updatable SRS work), not folklore. CLAUDE.md says ‘obvious to a working cryptographer’ is not a folklore claim.
• ‘universal SNARKs’ has no page of its own; conclusion collapses onto succinct-argument.