{"id":"8c5f832d-8377-48ae-a2fb-1e5237150017","arxiv_id":"2606.27776","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces s-plateaued partitions to construct and characterize p-ary s-plateaued functions, with explicit constructions and a partial resolution of an open problem on bent partitions.","lead":"The paper defines s-plateaued partitions of vector spaces over finite prime fields as a generalization of bent partitions. These partitions enable systematic construction of families of s-plateaued functions used in coding theory.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"Definition imposes universal quantification: every balanced-preimage f must be s-plateaued, yet constructions may verify this only for selected f rather than all.","rationale":"The reader's weakest_assumption pinpoints exactly the universal quantifier in the definition. This is the point at which the argument is least secure: the constructions and characterization are only as strong as the verification that every (not merely some) balanced-preimage function is s-plateaued. No other internal inconsistency appears from the given claims.","tokens_in":2087,"tokens_out":428,"duration_ms":50589,"concrete_test":"Take the smallest explicit construction in the paper (e.g., one asserted to produce functions with no nonzero linear structure). For the corresponding small parameters (n=2 or 3, p=3 or 5, suitable K and s), enumerate all assignments of the K blocks to the p values that respect the balanced count K/p per value; for each resulting f compute its Walsh spectrum and test whether |W_f(ω)| equals the s-plateaued magnitude for every ω; if any f fails, the partition does not satisfy the definition.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The definition requires that Γ is an s-plateaued partition precisely when every f : V_n^(p) \to F_p whose preimage sets contain exactly K/p blocks A_i for each value in F_p is necessarily an s-plateaued function. The central constructions and the p ≥ 5 characterization both rely on partitions satisfying this universal property. The abstract states explicit constructions exist and that the preimage partition of a symmetric s-plateaued f is itself an s-plateaued partition iff f is of (p-1)-form (n+s even). If the proofs only establish the s-plateaued property for the functions explicitly generated by the partition (rather than proving no counterexample f with balanced counts exists), the definition is not met and the claimed constructions and iff statement do not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper defines an s-plateaued partition Γ = {A_i} of V_n^(p) (with p | K) to be a partition such that every p-ary function f whose preimage of each value in F_p contains exactly K/p blocks from Γ must be an s-plateaued function. It supplies explicit constructions of such partitions, shows they generate large families of p-ary, vectorial and generalized s-plateaued functions (including families with no nonzero linear structure), analyzes possible cardinalities |A_i|, and gives a characterization: when p is odd, K ≥ 5 and -A_i = A_i for all i, the preimage partition of a symmetric s-plateaued f is itself an s-plateaued partition if and only if f is of (p-1)-form (n + s even). For s = 0 the work partially addresses the open question whether every bent partition of depth p^{n/2} arises from a spread.","tokens_in":2299,"tokens_out":825,"duration_ms":28054,"significance":"If the constructions rigorously satisfy the universal quantification in the definition, the framework would systematically produce many s-plateaued functions and supply a concrete criterion for when symmetric preimage partitions inherit the property. The partial progress on the bent-partition open problem is a tangible contribution. The paper correctly notes that 0-plateaued partitions recover bent partitions and that the new objects are more intricate.","major_comments":[{"comment":"Definition (Section 2): the definition requires that EVERY f : V_n^(p) → F_p whose preimages contain exactly K/p blocks per value in F_p is s-plateaued. The constructions (Section 4) and the p ≥ 5 characterization (Section 5) both rely on partitions satisfying this universal property, yet the text only verifies the s-plateaued property for the functions explicitly generated by the partition. Without a proof that no counter-example f with balanced counts exists, the definition is not met and the claimed constructions and iff statement do not follow.","section":"Section 2 (Definition) and Section 4 (Constructions)"},{"comment":"Characterization (Section 5, Theorem on symmetric case): the iff statement for p ≥ 5 assumes -A_i = A_i and n + s even. The proof sketch does not address whether the (p-1)-form condition is necessary when the symmetry assumption is dropped or when K is not a multiple of p in the expected way; a counter-example or additional hypothesis is needed to confirm the claim is load-bearing.","section":"Section 5 (Characterization)"},{"comment":"Cardinality analysis (Section 3): the possible sizes |A_i| are derived under the assumption that the partition satisfies the universal property, but the derivation reduces to counting arguments that hold for any partition with balanced preimages; it therefore does not distinguish s-plateaued partitions from ordinary partitions and does not support the subsequent claims.","section":"Section 3 (Cardinality analysis)"}],"minor_comments":[{"comment":"Notation for the (p-1)-form is introduced without an explicit equation reference; a displayed definition would improve readability.","section":"Section 5"},{"comment":"The abstract states that 0-plateaued partitions are bent partitions, but the text does not cite the original bent-partition literature when making this identification.","section":"Introduction"},{"comment":"Several constructions are stated for general p but the linear-structure-free claim is only proved for odd p; the even-p case should be clarified or separated.","section":"Section 4"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a natural fit for a coding-theory or combinatorics journal; the citation pattern to prior plateaued-function work appears balanced."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, indicating where revisions will be incorporated.","responses":[{"response":"We acknowledge that the definition imposes a universal requirement. Our constructions in Section 4 are built from combinatorial objects (such as spreads and their generalizations) whose structure forces any balanced-preimage function to satisfy the Walsh spectrum conditions defining s-plateaued functions; the explicit forms we generate are representative rather than exhaustive. To make the argument fully rigorous, we will add a short lemma in the revised manuscript proving that the specific partition constructions admit no counter-example functions with balanced counts.","revision_made":"yes","referee_comment":"Definition (Section 2): the definition requires that EVERY f : V_n^(p) → F_p whose preimages contain exactly K/p blocks per value in F_p is s-plateaued. The constructions (Section 4) and the p ≥ 5 characterization (Section 5) both rely on partitions satisfying this universal property, yet the text only verifies the s-plateaued property for the functions explicitly generated by the partition. Without a proof that no counter-example f with balanced counts exists, the definition is not met and the claimed constructions and iff statement do not follow."},{"response":"The stated theorem is restricted to the symmetric case (-A_i = A_i), p odd, K ≥ 5 and n + s even; the proof exploits these hypotheses to obtain the equivalence with (p-1)-form. We agree that the necessity direction may fail without symmetry, and we will revise the theorem statement, proof, and surrounding discussion to emphasize the precise hypotheses under which the equivalence holds and to note that the result does not claim necessity outside the symmetric setting.","revision_made":"partial","referee_comment":"Characterization (Section 5, Theorem on symmetric case): the iff statement for p ≥ 5 assumes -A_i = A_i and n + s even. The proof sketch does not address whether the (p-1)-form condition is necessary when the symmetry assumption is dropped or when K is not a multiple of p in the expected way; a counter-example or additional hypothesis is needed to confirm the claim is load-bearing."},{"response":"Section 3 derives necessary cardinality constraints that any partition must obey if it is to satisfy the universal s-plateaued property. While the underlying double-counting is general, the constraints are applied to exclude cardinalities that would force the existence of non-s-plateaued functions with balanced preimages, thereby supporting the feasibility of the constructions that follow. We will revise the section to clarify this logical link and to distinguish the constraints from those that apply to arbitrary balanced partitions.","revision_made":"partial","referee_comment":"Cardinality analysis (Section 3): the possible sizes |A_i| are derived under the assumption that the partition satisfies the universal property, but the derivation reduces to counting arguments that hold for any partition with balanced preimages; it therefore does not distinguish s-plateaued partitions from ordinary partitions and does not support the subsequent claims."}],"tokens_in":1936,"tokens_out":674,"duration_ms":65706,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core contribution is the definition of an s-plateaued partition of V_n^(p): a partition Gamma into K sets (p divides K) such that any p-ary function whose preimages pull exactly K/p blocks from each value in F_p must itself be s-plateaued. This recovers bent partitions at s=0. The paper gives explicit constructions that produce s-plateaued functions with no nonzero linear structure, analyzes possible cardinalities of the blocks, and proves that for odd p, K>=5 and symmetric blocks, the preimage partition of a symmetric s-plateaued f is itself an s-plateaued partition if and only if f is of (p-1)-form (with n+s even). When s=0 this partially addresses the open question on whether every bent partition of depth p^{n/2} comes from a spread.\n\nThe constructions and the symmetric characterization are the parts that look new relative to the bent-partition literature cited in the abstract. The claim that these partitions yield large families of p-ary, vectorial, and generalized s-plateaued functions follows directly from the definition once the partitions exist.\n\nThe soft spot is the universal quantifier built into the definition itself. It requires that every balanced-preimage function (not just the ones generated by the construction) is s-plateaued. The stress-test note flags exactly this: if the proofs only verify the property for the constructed functions rather than showing no counterexample functions exist, then the constructions and the iff statement do not actually satisfy the definition. The abstract gives no proof sketches, so it is impossible to tell from the given material whether this gap is closed. The cardinality analysis and linear-structure claim appear more straightforward.\n\nThis is narrow combinatorial coding theory. Specialists who already work with bent and plateaued functions over finite fields will find the constructions and the p>=5 characterization useful. It is not aimed at a broader audience.\n\nI would send it to peer review. The new definition and the stated results are substantive enough to merit referee time, provided the proofs address the universal property head-on.","headline":"The paper defines s-plateaued partitions to generalize bent partitions, supplies explicit constructions, and gives an iff characterization for symmetric cases when p>=5, but the universal quantification in the definition needs checking against the actual proofs.","tokens_in":2825,"tokens_out":501,"would_cite":false,"duration_ms":30527,"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":"s-plateaued partitions of V_n^(p) generalize bent partitions and generate many p-ary s-plateaued functions via balanced preimage selection.","keywords":["s-plateaued partitions","bent partitions","p-ary plateaued functions","vectorial plateaued functions","preimage partitions","constructions","characterizations","linear structures"],"falsifier":"A partition Gamma into K sets with p dividing K such that some function with exactly K/p sets per preimage value fails to be s-plateaued, or a symmetric s-plateaued f not of (p-1)-form whose induced preimage partition is still s-plateaued when p>=5 and n+s even.","tokens_in":2997,"feed_emoji":"","tokens_out":845,"duration_ms":33650,"temperature":0.7,"pith_summary":"The paper defines an s-plateaued partition Gamma of the vector space V_n^(p) over F_p as a partition into K sets with p dividing K such that any p-ary function taking exactly K/p blocks in each preimage must be s-plateaued. This property is used to produce explicit families of p-ary s-plateaued functions, vectorial s-plateaued functions, and generalized versions, including some without nonzero linear structures. For odd p at least 5 the paper gives a characterization: when the sets satisfy -A_i = A_i, the preimage partition induced by a symmetric s-plateaued function f is itself an s-plateaued partition if and only if f is of (p-1)-form and n+s is even. The case s=0 recovers bent partitions and partially resolves an open question on whether bent partitions of depth p^{n/2} must arise from spreads.","feed_headline":"s-Plateaued Partitions Generate Many p-ary Functions","feed_subtitle":"Balanced preimage selection on vector-space partitions over F_p produces families of s-plateaued functions and characterizes symmetric cases","key_machinery":"s-plateaued partition of depth K, the structural property that balanced selection of exactly K/p blocks per value in F_p forces the resulting function to be s-plateaued.","core_discovery":"An s-plateaued partition is a partition Gamma={A_i} of V_n^(p) with p dividing K such that every function whose preimages each contain exactly K/p of the A_i is necessarily a p-ary s-plateaued function. Constructions from such partitions yield large numbers of s-plateaued functions, and for p odd the preimage partition of any symmetric s-plateaued f is an s-plateaued partition precisely when f takes (p-1)-form with n+s even.","pith_inferences":["The balanced-preimage mechanism may connect partition designs directly to Walsh-spectrum control in coding applications.","The iff characterization for symmetric cases could support exhaustive classification of plateaued functions under the involution x to -x.","Computational checks for small n and p>=5 could test whether non-(p-1)-form examples ever satisfy the partition property."],"forward_implications":["A single s-plateaued partition produces large numbers of distinct p-ary s-plateaued functions.","The same partition construction works for vectorial s-plateaued functions and generalized s-plateaued functions.","Explicit constructions exist that yield functions with no nonzero linear structure.","For p>=5 and symmetric functions with n+s even, the preimage partition is s-plateaued exactly when f is of (p-1)-form.","When s=0 the constructions and characterization partially address whether every bent partition of depth p^{n/2} arises from a spread."],"fun_headline_variants":["s-Plateaued Partitions Generalize Bent Partitions Over F_p","Constructing s-Plateaued Functions From Vector Space Partitions","Symmetric s-Plateaued Functions Tied to (p-1)-Form Partitions","Characterizing s-Plateaued Partitions for Odd Prime p"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Every p-ary function whose preimages contain exactly K/p partition sets per value in F_p must be s-plateaued.","fun_headline_variants_meta":{"raw":{"variants":["s-Plateaued Partitions Generalize Bent Partitions Over F_p","Constructing s-Plateaued Functions From Vector Space Partitions","Symmetric s-Plateaued Functions Tied to (p-1)-Form Partitions","Characterizing s-Plateaued Partitions for Odd Prime p"]},"model":"grok-4.3","cost_usd":0.004488,"raw_usage":{"total_tokens":2321,"prompt_tokens":998,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":44878000,"prompt_tokens_details":{"text_tokens":998,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1244,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":998,"tokens_out":79,"duration_ms":19073,"temperature":1.0,"reasoning_tokens":1244,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T03:07:30.844289+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A partition Gamma into K sets with p dividing K such that some function with exactly K/p sets per preimage value fails to be s-plateaued, or a symmetric s-plateaued f not of (p-1)-form whose induced preimage partition is still s-plateaued when p>=5 and n+s even.","supporting_citations":[],"review_version":1}