{"id":"a5690fd2-1627-4abd-bdcc-0c589fcdb422","arxiv_id":"2512.11855","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Approximate symmetry can be enforced with logarithmic averaging complexity while exact symmetry requires linear complexity in the group size.","lead":"The paper introduces averaging complexity as a measure of the cost to enforce symmetry in ML models by averaging over group elements. It proves that approximate symmetry requires only logarithmic complexity while exact symmetry requires linear complexity in group size.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Exponential separation hinges on unspecified 'standard conditions' and whether averaging complexity is the right cost metric","rationale":"The reader's weakest_assumption directly identifies the same two load-bearing points (unspecified conditions and choice of complexity measure). Because the abstract alone cannot confirm whether the full proofs close these gaps, the UNVERDICTED verdict with low confidence remains appropriate; no new evidence from the placeholder full-text reference alters that assessment.","tokens_in":1662,"tokens_out":364,"duration_ms":28094,"concrete_test":"From the full text, extract the precise statement of the 'standard conditions' (likely in the theorem or definition of averaging complexity) and instantiate them for the cyclic group C_n acting by rotation on a 1D signal; recompute both exact and approximate averaging complexities explicitly for n=2^k, k=3..10. If the ratio is not exponential or collapses under a natural alternative cost (e.g., number of distinct orbits), the separation does not hold under the claimed conditions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim asserts that exact symmetry requires linear averaging complexity while approximate symmetry requires only logarithmic complexity in the group size, under 'standard conditions.' For this separation to be load-bearing, the paper must (1) explicitly define those conditions (e.g., group representation, action type, or model class) such that they cover typical ML symmetries and (2) justify why averaging complexity—rather than, say, parameter count, gradient cost, or invariance error—is the appropriate complexity measure. If the conditions turn out to be narrow (e.g., only abelian groups or specific averaging operators) or if averaging complexity can be bypassed by other enforcement methods, the exponential gap does not transfer to practice.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces averaging complexity as a new framework to measure the cost of enforcing symmetry in machine learning models through averaging operations. Its central claim is an exponential separation: under standard conditions, exact symmetry requires linear averaging complexity in the group size, while approximate symmetry can be achieved with only logarithmic complexity. This is positioned as the first theoretical result directly comparing the two and justifying the practical preference for approximate symmetry.","tokens_in":1780,"tokens_out":562,"duration_ms":27463,"significance":"If the separation holds under conditions that cover common ML symmetry groups (e.g., permutations, rotations) and if averaging complexity proves to be a faithful proxy for practical enforcement cost, the result would supply useful theoretical support for recent empirical trends favoring approximate symmetries. The introduction of a new complexity measure and associated proof techniques could also be reusable for other symmetry-related questions in the field.","major_comments":[{"comment":"Abstract and §1: The exponential separation is stated to hold 'under standard conditions,' yet these conditions are not explicitly enumerated. The manuscript must define them precisely (group representation, action type, model class, or averaging operator) so that it is clear whether they encompass typical ML settings such as equivariant networks on point clouds or images.","section":"Abstract and §1"},{"comment":"Definition of averaging complexity (likely §2 or §3): The paper must justify why averaging complexity is the appropriate cost metric rather than alternatives such as parameter count, gradient computation cost, or direct invariance error. Without this justification or a comparison showing that other enforcement methods cannot bypass the linear cost for exact symmetry, the claimed separation may not transfer to practice.","section":"Definition of averaging complexity (likely §2 or §3)"},{"comment":"Main theorem (likely §4): The proof of the linear lower bound for exact symmetry and the logarithmic upper bound for approximate symmetry must be checked against the precise statement of the standard conditions; if those conditions turn out to be restrictive (e.g., only abelian groups or specific averaging operators), the result's scope is narrower than the abstract suggests.","section":"Main theorem (likely §4)"}],"minor_comments":[{"comment":"Clarify the relationship between averaging complexity and existing notions such as group-equivariant network depth or orbit-stabilizer costs.","section":"Related work"},{"comment":"Add a short table or example illustrating the complexity numbers for a concrete small group (e.g., cyclic group of order 8) to make the exponential gap intuitive.","section":"Main result"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their insightful comments on our paper. We provide point-by-point responses to the major comments below. We agree that clarifications are needed and will revise the manuscript to address these points.","responses":[{"response":"We agree with this observation. In the revised manuscript, we will explicitly define the standard conditions in a new paragraph in Section 1 and detail them in Section 2. These conditions include: finite groups with unitary representations, transitive actions on the input space, and averaging operators that compute the group average of the model output. This setup directly applies to common ML scenarios, including permutation-equivariant networks for point clouds and rotation-equivariant networks for images.","revision_made":"yes","referee_comment":"[Abstract and §1] Abstract and §1: The exponential separation is stated to hold 'under standard conditions,' yet these conditions are not explicitly enumerated. The manuscript must define them precisely (group representation, action type, model class, or averaging operator) so that it is clear whether they encompass typical ML settings such as equivariant networks on point clouds or images."},{"response":"We will strengthen the justification for averaging complexity in the revision. Averaging complexity measures the number of group elements that need to be averaged to enforce symmetry, which is a direct proxy for the computational overhead in averaging-based symmetry enforcement. We will add a discussion comparing it to parameter count (which does not account for inference-time costs) and invariance error (which is the objective rather than the cost). However, we note that our result is specific to averaging methods; we will clarify that other methods like architectural constraints may have different costs, but the separation highlights the advantage within the averaging paradigm.","revision_made":"partial","referee_comment":"[Definition of averaging complexity (likely §2 or §3)] Definition of averaging complexity (likely §2 or §3): The paper must justify why averaging complexity is the appropriate cost metric rather than alternatives such as parameter count, gradient computation cost, or direct invariance error. Without this justification or a comparison showing that other enforcement methods cannot bypass the linear cost for exact symmetry, the claimed separation may not transfer to practice."},{"response":"The standard conditions in our theorem apply to general finite groups, including non-abelian ones such as the permutation group and rotation groups. The lower bound proof uses a counting argument on the number of distinct orbits that holds for any group action satisfying the conditions, and the upper bound uses random sampling from the group, which works for any finite group. We will add a corollary or remark verifying that the result applies to standard ML groups like S_n for permutations and SO(3) for 3D rotations, confirming the scope is as broad as stated in the abstract.","revision_made":"yes","referee_comment":"[Main theorem (likely §4)] Main theorem (likely §4): The proof of the linear lower bound for exact symmetry and the logarithmic upper bound for approximate symmetry must be checked against the precise statement of the standard conditions; if those conditions turn out to be restrictive (e.g., only abelian groups or specific averaging operators), the result's scope is narrower than the abstract suggests."}],"tokens_in":1389,"tokens_out":684,"duration_ms":58173,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main thing to know is that this work introduces averaging complexity as a new way to measure the cost of symmetry enforcement and establishes an exponential separation where exact symmetry scales linearly with group size but approximate symmetry scales logarithmically, under standard conditions. They do a solid job of framing the problem and highlighting the gap in theoretical comparisons between exact and approximate symmetries in machine learning. The result offers a formal reason to consider approximate versions in applications where robustness matters more than perfect invariance. The potential issue is that everything rests on those standard conditions and on averaging complexity being the right metric. Without seeing how broadly those conditions apply to typical ML settings like group actions on data or networks, it's hard to know if the separation carries over. If other enforcement techniques avoid averaging altogether, the practical takeaway could be limited. The proofs will need careful review to ensure no circularity or overly restrictive assumptions. This kind of paper is for people studying the theory of symmetries in deep learning and equivariant models. Someone looking for mathematical justification for empirical observations on approximate symmetries would find it relevant. I think it should go to peer review for a full check on the framework and its applicability.","headline":"The paper introduces averaging complexity and claims an exponential separation favoring approximate over exact symmetry, but the result stands on unstated conditions and the choice of metric.","tokens_in":2244,"tokens_out":303,"would_cite":false,"duration_ms":43755,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"Averaging-complexity separation for exact vs. approximate symmetry is orthogonal to RS forcing chain","alignment":"orthogonal","rationale":"The paper's central machinery (group representations ρ, Fourier analysis on finite groups, averaging schemes ω with ||ω||_ℓ1=1 and size(ω) counting non-zeros, random sampling for O(log|G|/ε) approximate symmetry, linear lower bound via symmetric tensor powers and irrep multiplicity) operates entirely within representation theory and concentration bounds. It never invokes, parallels, or contradicts any RS primitive: the reciprocal cost J(x)=½(x+x⁻¹)−1, φ-ladder, 8-tick periodicity, or the single-distinction forcing of spacetime and constants. No RS module (e.g., Cost.FunctionalEquation, Foundation.RealityFromDistinction, Foundation.AlexanderDuality) is echoed or refined.","tokens_in":63900,"confidence":"high","tokens_out":201,"duration_ms":13520,"cache_read_input_tokens":32896,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Approximate symmetry can be enforced with only logarithmic averaging complexity while exact symmetry requires linear complexity in the group size.","keywords":["symmetry","machine learning","averaging complexity","inductive bias","group actions","computational complexity","approximate symmetry"],"falsifier":"A concrete construction in which approximate symmetry still requires linear averaging complexity or in which exact symmetry can be achieved with logarithmic complexity under the same framework.","tokens_in":2546,"feed_emoji":"📐","tokens_out":527,"duration_ms":55160,"temperature":0.7,"pith_summary":"The paper introduces averaging complexity to measure the computational cost of enforcing symmetry in machine learning models by averaging over group elements. It proves an exponential separation showing that exact symmetry scales linearly with group size under standard conditions, while approximate symmetry scales logarithmically. A sympathetic reader cares because this supplies the first theoretical account of why approximate symmetries can deliver inductive bias and robustness at far lower cost than exact versions. The framework directly compares the two regimes and explains empirical preferences for approximate symmetry in practice.","feed_headline":"Approximate symmetry costs log complexity, exact costs linear","feed_subtitle":"Averaging complexity separates the two regimes exponentially and justifies flexible symmetry choices in models.","key_machinery":"Averaging complexity, a framework that quantifies the cost of enforcing symmetry through averaging over group actions.","core_discovery":"The central claim is that averaging complexity exhibits an exponential separation: exact symmetry requires linear complexity in the size of the symmetry group, whereas approximate symmetry can be achieved with only logarithmic complexity under standard conditions.","pith_inferences":["Designers could tune symmetry precision dynamically to match hardware budgets in large models.","The same logarithmic-versus-linear gap might appear when symmetry is measured by other natural complexity notions.","Scientific domains with high-dimensional symmetries could adopt graded approximations to reach previously intractable scales."],"forward_implications":["Models incorporating approximate symmetries can scale to larger symmetry groups without prohibitive cost.","Approximate symmetry provides a practical trade-off that retains most inductive bias benefits at reduced overhead.","The separation supplies a formal reason to prefer approximate symmetries in robustness-critical applications.","Averaging complexity becomes a tool for choosing the degree of symmetry based on available compute."],"fun_headline_variants":["Exact symmetry: linear averaging, approximate: logarithmic","Log complexity for approximate symmetry, linear for exact","Averaging complexity: exact linear, approximate log","Exponential separation: linear exact, log approximate symmetry"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The exponential separation holds under the paper's unspecified standard conditions and that averaging complexity is the right measure of enforcement cost.","fun_headline_variants_meta":{"raw":{"variants":["Exact symmetry: linear averaging, approximate: logarithmic","Log complexity for approximate symmetry, linear for exact","Averaging complexity: exact linear, approximate log","Exponential separation: linear exact, log approximate symmetry"]},"model":"grok-4.3","cost_usd":0.014225,"raw_usage":{"total_tokens":6087,"prompt_tokens":579,"num_sources_used":0,"completion_tokens":58,"cost_in_usd_ticks":142249500,"prompt_tokens_details":{"text_tokens":579,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":5450,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":579,"tokens_out":58,"duration_ms":49207,"temperature":1.0,"reasoning_tokens":5450,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-17T01:11:23.027619+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete construction in which approximate symmetry still requires linear averaging complexity or in which exact symmetry can be achieved with logarithmic complexity under the same framework.","supporting_citations":[],"review_version":1}