{"id":"a88f2719-80f6-4110-b7a6-376ecf007d41","arxiv_id":"2508.00907","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Presents an exact tensor-network equation and optimized algorithm for integer factorization by tensorizing a multiplication circuit and contracting it to match a target composite number.","lead":"The paper gives a tensor-network equation and algorithm to find nontrivial divisors of composite integers by modeling a binary multiplication circuit and projecting its output. A smart generalist might read it to understand a classical tensor-network approach to a core problem in cryptography and optimization.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Tensor-train contraction of the optimized multiplication TN may still require exponential resources in bit length despite auxiliary reduction.","rationale":"The reader's weakest assumption matches the identified concern precisely. The central claim is the explicit TN equation plus algorithm; its practical utility hinges on the contraction step not reverting to exponential cost, which remains unverified. Full-text details on the circuit tensorization and compression scheme would be needed to tighten the test, but the scaling question is the load-bearing one.","tokens_in":1604,"tokens_out":339,"duration_ms":32105,"concrete_test":"Take a 64-bit semiprime (product of two random 32-bit primes), build the TN exactly as described with the reduced auxiliary register, contract it both exactly and with TT compression at bond dimension 128, and record whether a nontrivial factor is recovered and the peak memory/time required; if recovery fails or resources exceed 10^9 operations the scaling concern is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper constructs an exact TN by tensorizing a binary multiplication circuit and projecting its output register onto the target composite N. It then reduces the auxiliary register while asserting that at least one valid factorization orientation is retained, and applies tensor-train compression for approximate contraction. For the presented algorithm to constitute a viable search procedure, the effective bond dimension after compression (or the contraction cost) must remain manageable as the bit length n grows; otherwise the method reduces to an exponential-cost enumeration in disguise. No bound on rank growth, no complexity analysis of the contraction scheme, and no demonstration that the preserved orientation is reachable under truncation are supplied in the description.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to present an exact and explicit tensor-network equation for searching nontrivial divisors of a composite integer N, obtained by tensorizing a binary multiplication circuit and projecting its output register onto N. It describes optimizations including a reduced auxiliary register that preserves at least one valid factorization orientation, tensor-train compression for approximate contraction, and an associated algorithm, with tests performed using both exact and approximate contractions.","tokens_in":1741,"tokens_out":398,"duration_ms":25346,"significance":"If the contraction remains tractable for relevant bit lengths, the work would supply a concrete tensor-network formulation of integer factorization within the MeLoCoToN framework, potentially linking combinatorial optimization techniques to number-theoretic search problems. The explicit circuit-to-tensor mapping is a clear constructive element.","major_comments":[{"comment":"Abstract: the statement that 'a series of tests ... evaluating its performance' is made, yet no quantitative metrics (runtime, success rate, error bounds, or scaling with bit length), error analysis, or comparisons against standard factoring methods appear. This leaves the efficiency claim of the algorithm unsupported.","section":null},{"comment":"Section on auxiliary-register reduction and tensor-train compression: the assertion that the reduced register 'still preserves at least one valid factorization orientation' is stated without a proof, invariant, or explicit check that this orientation survives truncation; no bound on post-compression bond dimension or contraction complexity is supplied, so the method may still incur exponential cost in bit length.","section":null}],"minor_comments":[{"comment":"Notation: define the precise index ranges and contraction order for the multiplication tensors and the projection operator onto N so that the tensor-network equation can be reproduced unambiguously.","section":null},{"comment":"Figure clarity: ensure that any diagrams of the tensor network or contraction scheme label all bond dimensions and the reduced auxiliary indices.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below, indicating where revisions will be made to strengthen the presentation while remaining faithful to the work performed.","responses":[{"response":"We agree that the abstract would benefit from greater specificity. The manuscript reports numerical experiments on small composite integers, demonstrating successful recovery of factors via both exact contraction and tensor-train approximation. In the revision we will update the abstract to reference these concrete demonstrations and expand the main text with quantitative details including observed runtimes, success rates across tested bit lengths, and a basic error analysis for the approximate contractions. Direct benchmarking against classical algorithms such as trial division lies outside the present scope, which focuses on establishing the tensor-network formulation; we will note this limitation explicitly.","revision_made":"yes","referee_comment":"Abstract: the statement that 'a series of tests ... evaluating its performance' is made, yet no quantitative metrics (runtime, success rate, error bounds, or scaling with bit length), error analysis, or comparisons against standard factoring methods appear. This leaves the efficiency claim of the algorithm unsupported."},{"response":"We acknowledge that a formal justification is currently missing. The register reduction is constructed to retain the minimal degrees of freedom required for the multiplication circuit to produce the target output N, thereby keeping at least one valid factorization path. We will insert a short lemma or invariant argument in the revised section establishing this preservation. For the tensor-train step we will add an analysis of bond-dimension growth together with the resulting contraction complexity, noting that exponential scaling in bit length cannot be ruled out in the worst case but that the approach yields practical gains on the moderate-size instances examined. We will also include explicit verification that valid orientations remain after truncation in the reported examples.","revision_made":"yes","referee_comment":"Section on auxiliary-register reduction and tensor-train compression: the assertion that the reduced register 'still preserves at least one valid factorization orientation' is stated without a proof, invariant, or explicit check that this orientation survives truncation; no bound on post-compression bond dimension or contraction complexity is supplied, so the method may still incur exponential cost in bit length."}],"tokens_in":1221,"tokens_out":477,"duration_ms":28862,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core idea is to build an exact tensor network by tensorizing a binary multiplication circuit and projecting its output register onto the target composite N. They then shrink the auxiliary register while keeping at least one valid factorization orientation and apply tensor-train compression for approximate contraction. That explicit mapping and the reduced-register tweak look new relative to prior tensor-network work on combinatorial problems.","headline":"The paper maps factorization to a tensor network via a multiplication circuit but gives almost no data on whether the approach scales or beats existing methods.","tokens_in":2232,"tokens_out":144,"would_cite":false,"duration_ms":30547,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/RealityFromDistinction.lean","rs_theorem":"reality_from_one_distinction","paper_passage":"The presented tensor network tensorizes a binary multiplication circuit and projects its output register onto the target composite N... tensor train compression"},{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"MeLoCoToN formalism... explicit equation that solves the factorization problem"}],"headline":"Tensor-network factorization via circuit tensorization and TT contraction lies outside RS scope","alignment":"orthogonal","rationale":"The paper's machinery (MeLoCoToN tensorization of a modular multiplication circuit, auxiliary-register reduction, and tensor-train compression for exact/approximate contraction) is a classical optimization technique for an NP problem. RS derives J-cost, φ-ladder, 8-tick periodicity, and spacetime constants from a single distinction with zero adjustable parameters; the paper invokes none of these structures and makes no claims about recognition cost, ratio symmetry, or parameter-free constants.","tokens_in":50776,"confidence":"high","tokens_out":286,"duration_ms":15527,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A tensor network exactly encodes the search for nontrivial divisors by tensorizing a binary multiplication circuit and projecting its output onto the target integer.","keywords":["tensor networks","integer factorization","combinatorial optimization","tensor train compression","multiplication circuit","divisor search"],"falsifier":"Contract the network for the composite number 91 and check whether the contraction returns the divisor pair 7 and 13 or returns no divisors at all.","tokens_in":2536,"feed_emoji":"🔢","tokens_out":626,"duration_ms":41886,"temperature":0.7,"pith_summary":"The paper sets out an explicit tensor-network equation that locates nontrivial divisors of any composite integer. It achieves this by converting a binary multiplication circuit into tensors and constraining the output register to equal the number to be factored. A sympathetic reader would see this as recasting factorization as a contraction problem that can be attacked with classical tensor methods rather than exhaustive search. The construction includes optimizations that shrink the auxiliary register while keeping at least one valid factor pair possible, and it supports both exact contraction and approximate tensor-train compression. Performance is checked on small instances to verify that the network can be evaluated.","feed_headline":"Tensor network equation locates nontrivial divisors","feed_subtitle":"Tensorizing a multiplication circuit and projecting onto the target integer turns factorization into a contraction problem.","key_machinery":"The tensor network obtained by tensorizing a binary multiplication circuit, projecting its output register onto the target composite integer, and applying a reduced auxiliary register together with an optimized contraction scheme.","core_discovery":"The paper presents an exact and explicit tensor-network equation for the search of nontrivial divisors of a composite integer. The network is built by tensorizing a binary multiplication circuit and projecting its output onto the target integer, with further reductions in auxiliary register size and tensor dimensions that still preserve at least one valid factorization orientation, all computable by contraction algorithms.","pith_inferences":["Similar circuit-tensorization steps could be applied to other number-theoretic search tasks such as finding quadratic residues or solving discrete logarithms.","Systematic scaling tests on composites with increasing bit length would show whether contraction cost remains sub-exponential in practice.","The method may connect to existing tensor-network techniques used for satisfiability or graph problems by sharing the same contraction primitives."],"forward_implications":["Factorization becomes a tensor-contraction task rather than a direct search over possible divisors.","The reduced auxiliary register still guarantees that at least one correct factor pair remains representable.","Both exact contraction and approximate tensor-train methods can be used to evaluate the network.","The same construction template applies to other combinatorial optimization problems via the underlying MeLoCoToN framework."],"fun_headline_variants":["Tensor networks turn factorization into contraction","Exact tensor equation searches for divisors","Optimized tensor networks for integer factorization","Circuit tensorization projects to target number","Tensor contraction computes prime factorization"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Contracting the optimized tensor network, even after auxiliary-register reduction and tensor-train compression, will locate at least one valid factorization orientation for the composite integers of interest.","fun_headline_variants_meta":{"raw":{"variants":["Tensor networks turn factorization into contraction","Exact tensor equation searches for divisors","Optimized tensor networks for integer factorization","Circuit tensorization projects to target number","Tensor contraction computes prime factorization"]},"model":"grok-4.3","cost_usd":0.009954,"raw_usage":{"total_tokens":4283,"prompt_tokens":549,"num_sources_used":0,"completion_tokens":54,"cost_in_usd_ticks":99540500,"prompt_tokens_details":{"text_tokens":549,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3680,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":549,"tokens_out":54,"duration_ms":35697,"temperature":1.0,"reasoning_tokens":3680,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-19T03:13:11.753098+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Contract the network for the composite number 91 and check whether the contraction returns the divisor pair 7 and 13 or returns no divisors at all.","supporting_citations":[],"review_version":1}