{"id":"4d7e4f7a-2083-4c6e-a386-43ae8fb45625","arxiv_id":"2606.13367","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves p-time ≈ on circuits implies equivalence with ≤7-gate rewrites and that EF/CF proof size s yields ≈-chain of length s^O(1).","lead":"The paper proves there exists a polynomial-time relation on circuits implying logical equivalence, allowing one circuit to be rewritten into the other by deleting gates and adding at most seven new ones, and that Extended Frege or Circuit Frege proof size s bounds the length of chains under this relation by s to a polynomial power. A smart generalist might read it to see how proof sizes in logic connect to practical circuit transformations.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's identification of dependence on background definitions and the Jain-Jin citation is the only potential grounding issue visible; the abstract claims do not introduce additional unsupported assumptions or circularities that would alter the UNVERDICTED status.","tokens_in":1652,"tokens_out":225,"duration_ms":18414,"concrete_test":"Extract the explicit construction mapping each EF/CF inference rule to a bounded number of ≈ applications and verify that the total number of ≈ steps is at most s^O(1) with no hidden exponential dependence on circuit size or depth.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The three statements form a coherent chain from a p-time sound relation ≈ (implying equivalence) through bounded local rewrites (at most 7 new gates) to a polynomial-length chain simulating EF/CF proofs of size s. No internal inconsistency appears in the claimed implications or in the translation from proof steps to ≈-steps; the argument structure is consistent with standard definitions of the cited proof systems.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"Inspired by Jain and Jin (FOCS 2022), the paper proves three statements: there is a p-time binary relation ≈ between circuits that implies their logical equivalence; the relation ≈ implies that each of the two circuits can be rewritten into the other one by possibly deleting some gates and adding at most seven new gates; if the equivalence C ≡ D has a size s proof in an Extended Frege or a Circuit Frege proof system then there is a chain of circuits E_i with C = E_0 ≈ … ≈ E_t = D with t ≤ s^{O(1)}.","tokens_in":1723,"tokens_out":260,"duration_ms":30556,"significance":"If these results hold, they establish a link between proof size in EF and CF systems and the length of chains under a local rewriting relation ≈, which could be significant for proof complexity theory. The p-time computability of ≈ and the constant bound of 7 on new gates are particular features that, if proved, would be noteworthy.","major_comments":[{"comment":"The abstract states the three theorems cleanly but supplies no derivation steps; without the full manuscript the support for the central claims cannot be checked.","section":"Abstract"}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their comments on our manuscript. We address the single major comment below.","responses":[{"response":"Abstracts are intended to state main results concisely without full derivations; the complete proofs and supporting arguments appear in the body of the manuscript (Sections 3--5). The arXiv version supplies the full text, so the claims can be verified directly from the provided derivations.","revision_made":"no","referee_comment":"[Abstract] The abstract states the three theorems cleanly but supplies no derivation steps; without the full manuscript the support for the central claims cannot be checked."}],"tokens_in":1175,"tokens_out":149,"duration_ms":16321,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that an Extended Frege or Circuit Frege proof of size s for circuit equivalence C ≡ D yields a chain C = E0 ≈ … ≈ Et = D with t polynomial in s, where ≈ is a p-time relation that implies logical equivalence and allows each circuit to be rewritten into the other by deleting gates and adding at most seven new ones.\n\nThe three statements are new. They do not reduce to earlier published results and they make explicit a technical connection between proof size and bounded local rewriting that was only suggested before. The paper stays inside the standard definitions of EF and CF systems and uses the Jain-Jin observation as a starting point without adding extra machinery.\n\nThe argument chain is consistent: the p-time soundness of ≈ gives equivalence, the local rewrite bound follows from the definition of ≈, and the polynomial chain length follows from simulating each proof step by a bounded number of ≈ steps. No circularity or invented entities appear.\n\nThe only real limitation is that the abstract states the theorems without derivation steps, so the actual proofs need to be checked in the full text. The dependence on the cited FOCS 2022 statement is ordinary and not a flaw here. Nothing in the claimed implications looks shaky.\n\nThis is for specialists already working on proof complexity and circuit-based systems. Someone tracking EF lower bounds or circuit equivalence algorithms will find the link useful for organizing existing ideas.\n\nSend it to peer review. The claims are precise enough and the subfield is active enough that a referee should see it.","headline":"Krajicek turns an EF/CF proof of size s into a polynomial-length chain of ≈ steps, each adding at most seven gates.","tokens_in":2189,"tokens_out":387,"would_cite":false,"duration_ms":22027,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Equivalence proofs of size s in Extended Frege or Circuit Frege systems yield chains of length at most s to the O(1) under a local circuit rewrite relation.","keywords":["extended frege","circuit frege","proof complexity","circuit equivalence","rewriting","polynomial time relation","gate addition","boolean circuits"],"falsifier":"An explicit pair of circuits C and D whose logical equivalence has an Extended Frege proof of size s but for which every possible chain under the relation ≈ has length superpolynomial in s.","tokens_in":2548,"feed_emoji":"","tokens_out":673,"duration_ms":23737,"temperature":0.7,"pith_summary":"The paper introduces a polynomial-time decidable relation ≈ on Boolean circuits such that C ≈ D implies the two circuits compute identical functions. The same relation guarantees that one circuit can be transformed into the other by deleting arbitrary gates and inserting at most seven new gates. When two circuits are shown equivalent by a proof of size s in either the Extended Frege or the Circuit Frege system, the paper constructs an explicit sequence of circuits linked by ≈ steps whose length is bounded by a polynomial in s.","feed_headline":"EF proofs reduce to short circuit rewrite chains","feed_subtitle":"A size-s proof yields an ≈-chain of length at most s^O(1) where each step deletes gates and adds at most seven new ones.","key_machinery":"The polynomial-time binary relation ≈ on circuits that implies logical equivalence and allows one circuit to be obtained from the other by deleting gates and adding at most seven new gates.","core_discovery":"If the equivalence C ≡ D has a size s proof in an Extended Frege or a Circuit Frege proof system then there is a chain of circuits E_i with C = E_0 ≈ … ≈ E_t = D where t ≤ s^{O(1)}, and the relation ≈ is polynomial-time and permits rewriting with at most seven added gates while preserving logical equivalence.","pith_inferences":["The existence of such short chains might be used to design heuristic algorithms that search for circuit equivalences by exploring bounded local changes rather than full proofs.","If circuit equivalence testing lies outside P, the construction still separates the length of ≈-chains from the existence of short proofs only if the relation itself is hard to decide.","The fixed seven-gate allowance could be tightened or relaxed in follow-up work to obtain sharper bounds on proof size versus rewrite length."],"forward_implications":["Short proofs in these systems can be converted into polynomially long sequences of local circuit modifications.","The relation ≈ supplies a certificate for circuit equivalence that is checkable in polynomial time.","Any upper bound on the length of ≈-chains immediately yields an upper bound on the size of proofs in the corresponding Frege systems.","The seven-gate bound on local rewrites is preserved uniformly across the chain."],"fun_headline_variants":["EF proofs imply ≈ rewrite chains","Size-s EF proofs bound rewrite chain length","Extended Frege implies circuit ≈ chains","CF proofs reduce to bounded gate chains"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The standard definitions of Extended Frege and Circuit Frege proof systems are taken to hold exactly as used in the cited background statements.","fun_headline_variants_meta":{"raw":{"variants":["EF proofs imply ≈ rewrite chains","Size-s EF proofs bound rewrite chain length","Extended Frege implies circuit ≈ chains","CF proofs reduce to bounded gate chains"]},"model":"grok-4.3","cost_usd":0.00939,"raw_usage":{"total_tokens":4150,"prompt_tokens":573,"num_sources_used":0,"completion_tokens":43,"cost_in_usd_ticks":93899500,"prompt_tokens_details":{"text_tokens":573,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3534,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":573,"tokens_out":43,"duration_ms":41292,"temperature":1.0,"reasoning_tokens":3534,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T04:47:32.302635+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit pair of circuits C and D whose logical equivalence has an Extended Frege proof of size s but for which every possible chain under the relation ≈ has length superpolynomial in s.","supporting_citations":[],"review_version":1}