{"id":"1cea8717-e75c-40af-babe-41bfc8d8d0c3","arxiv_id":"2411.18389","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors classify all weakly norming systems of linear equations of rank at most two and prove that such systems are variable-transitive and forcing.","lead":"This paper initiates a systematic study of when systems of linear equations define norms on functions over finite vector spaces, generalizing Gowers uniformity norms. It proves an isomorphism theorem, shows weakly norming systems are variable-transitive, and classifies all rank-two examples.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified to the central isomorphism theorem; the main gaps are omitted proofs of supporting results, which appear fillable.","rationale":"The central theorem's proof is internally consistent and the key identities verify. The omissions flagged by the reader are real but not correctness-threatening: Proposition 2.2 has a straightforward proof by tensor powers, and Theorem 5.8's conclusion can be completed in one line. I therefore do not change the reader's conditional verdict; the paper should still be accepted only after the omitted proofs and private-communication dependence are addressed.","tokens_in":30085,"tokens_out":53237,"duration_ms":441314,"concrete_test":"Supply the full proof of Proposition 2.2 by adapting the tensor-power argument: for nonnegative f_i with ||f_i||_{r(L)}=1, assume t_L(f_1,...,f_k)=c>1, set F = ∑ f_i^{⊗m}, and use t_L(F) ≥ c^m versus ||F||_{r(L)} ≤ k to contradict for large even m; also handle any zero-norm f_i by a small perturbation. This verifies the weak Hölder inequality used in Corollary 3.2, Lemma 4.5, and Theorem 5.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the proof of Theorem 4.1 in detail, I find no load-bearing flaw in the central claim. The random-sign identity in Lemma 4.2 is correct (the k arguments are all the same summed function, so only permutations survive), and the complex-unit averaging similarly selects permutations. Proposition 4.4's dimension counting is sound: the double sums count intersections of m-dimensional subspaces, and non-isomorphism forces intersection dimension at most m-1. The reader's flagged Proposition 2.2 is genuinely omitted, but the tensor-power argument from Proposition 2.1 adapts directly to nonnegative functions, so it is a presentation gap rather than a correctness threat. The terse final step of Theorem 5.8 ('implying that L does not have the Hölder property') needs one extra sentence combining t_L(f) ≤ T - 2α^k with t_L(g) ≤ T, but it is repairable. Other omitted proofs (Theorem 4.8, Corollary 8.3) are claimed to follow from cited analogues and do not affect Theorem 4.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper initiates a systematic study of norming and weakly norming systems of linear equations over F_q^n. For an m x k system L it defines the functional t_L(f)=E_{x in Sol(L)} prod_i f(x_i) and asks when |t_L(f)|^{1/k} or t_L(|f|)^{1/k} defines a norm. The main results are an isomorphism theorem (Theorem 4.1) saying that equality of t_L and t_M on all nonnegative functions implies L and M are isomorphic up to row operations and column permutations; the corollary that weakly norming systems are variable-transitive (Corollary 4.6); necessary conditions including translation invariance, the Sidorenko property, even girth, and the existence of Schatten vectors (Theorem 5.1); a complete classification of weakly norming systems of rank at most two (Theorem 5.8); a proof that weakly norming systems are forcing (Theorem 6.3); a construction of norming systems from norming hypergraphs (Section 7); and a proof that every real-norming system admits a conjugation assignment making it complex-norming (Theorem 8.1). The arguments combine Fourier analysis, tensor-power manipulations, and an adaptation of Sidorenko's graph technique.","tokens_in":30236,"tokens_out":14840,"duration_ms":128418,"significance":"If the central theorems are correct, this is a substantial contribution to additive combinatorics. Theorem 4.1 is a powerful uniqueness criterion with no direct arithmetic predecessor and underpins the variable-transitivity result; the rank-two classification is the first complete classification in this setting; and the forcing and complex-norming results transfer a substantial body of graph-norm theory to linear systems. The paper is clearly written and gives detailed proofs for the main line, with the Fourier and tensor-product arguments checked carefully. The principal weaknesses are the number of supporting statements whose proofs are deferred or sketched (Proposition 2.2, Theorem 4.8, Corollary 8.3), a density-scaling issue in the proof of Proposition 6.1, and one terse step in Theorem 5.8; all appear repairable within the manuscript's scope.","major_comments":[{"comment":"Proposition 2.2 states that L is weakly semi-norming if and only if the Holder-type inequality t_L(f_1,...,f_k) <= prod_i ||f_i||_{r(L)} holds for all nonnegative functions, but its proof is omitted. This equivalence is used repeatedly in the rest of the paper (Corollary 3.2, Theorem 5.1, Theorem 6.3), so the omission is load-bearing. Since the reader is only referred to 'a similar argument' to Proposition 2.1, please include the proof or a precise reference; the nonnegative setting requires care because the usual absolute-value renormalisation must be replaced by positivity and monotonicity of t_L on nonnegative inputs.","section":"Section 2, Proposition 2.2"},{"comment":"The first sentence of the proof says 'We may assume that n>1 by extending f to a larger F_q^n through adding an extra zero to each point in the support.' If 'adding an extra zero' means defining f' by f'(x,0)=f(x) and f'=0 outside the embedded copy of F_q^n, then the hypothesis is not preserved: t_L(f') = q^{m-k} t_L(f) while E[f'] = q^{-1} E[f], so the sign of t_L(f') - E[f']^k can change. The reduction should instead use the constant extension f'(x,z)=f(x), which preserves both t_L and the expectation, or the proof should be modified accordingly.","section":"Section 6, Proposition 6.1"},{"comment":"Corollary 8.3, which asserts that s_L^{1/k} is a norm on F, is essential to the proof of Theorem 8.1, but its proof is omitted as 'verbatim the same' as [15, Lemma 5.5]. Given that Theorem 8.1 is a main result and the omitted proof concerns a weak decoration functional, the authors should include the argument rather than rely on an unstated matching.","section":"Section 8, Corollary 8.3"},{"comment":"At the end of the proof of Theorem 5.8, the sentence 'implying that L does not have the Holder property' omits the comparison that makes the contradiction explicit. Starting from t_L(f) <= t_L(f_1,...,f_k) - 2 alpha^k, one still has to compare t_L(f_1,...,f_k) with the upper bound supplied by Proposition 2.2 for the same tuple; the authors should insert this final step so that the displayed strict gap is shown to contradict the Holder inequality.","section":"Section 5, proof of Theorem 5.8"},{"comment":"The proof of Theorem 4.8 (components of a weakly norming system are isomorphic and norming) is omitted, with the text saying it follows from Corollary 3.3 'but is otherwise exactly the same as' the simplified proof of [10, Theorem 1.2] in [7, Lemma 2.4]. Since this theorem is stated as an application of the isomorphism theorem and as a structural statement about norming systems, a proof sketch or an explicit reduction to the cited lemma should be included.","section":"Section 4, Theorem 4.8"},{"comment":"In the proof of Corollary 5.3, the claim that if m >= 2 then 'by using row operations, one can clearly make a vector with smaller support' is not immediate and is needed for the rank-two classification. The claim is true, but it requires the argument that a subspace of dimension at least 2 over F_q contains a nonzero vector with a zero coordinate, obtained by taking a suitable linear combination of two independent row vectors; please supply this justification.","section":"Section 5, Corollary 5.3"}],"minor_comments":[{"comment":"The displayed matrix for L_H in the case a=b=3 is not a valid 4 x 9 matrix as printed: it appears to have too few entries and two identical rows. Please correct the typesetting or the matrix itself.","section":"Example 7.6"},{"comment":"The proof invokes Theorem 5.1 for norming systems, although Theorem 5.1 is stated for weakly norming systems; please add a one-sentence justification that norming systems are weakly norming, for instance via Proposition 3.5 and Proposition 2.1.","section":"Section 7, Theorem 7.7"},{"comment":"The reliance on [13], a private communication, for the statement that every norming system is equivalent to some U_k norm makes that remark unverifiable; the authors should mark it as conditional or provide a public reference.","section":"Example 7.8 and references"},{"comment":"There are several typographical errors, including 't o' in the abstract, a stray ')' in the displayed equation in Proposition 4.4, and the 'bracehtipupleft' artifacts in Section 2; a thorough proofreading pass is needed.","section":"Throughout"},{"comment":"In the parametrisation of Sol(L), the notation is somewhat compressed because the variables (x_1,...,x_6) appear in the first line and again in the description of the incidence structure; a concise statement of which variables correspond to which edges of the singleton-pair incidence graph would improve readability.","section":"Example 5.9"}],"recommendation":"major_revision","confidential_remarks":"The paper is a good fit for the journal and the central isomorphism theorem appears correct. The main concerns are the number of deferred proofs and the density-scaling issue in Proposition 6.1; both are fixable. I would also ask the editor to require a public reference or a proof for the result attributed to private communication [13], since it is used in a nonnegligible remark."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe two things you should know about arXiv:2411.18389: the main theorems are new and they hold up on inspection, but the paper leans on a few omitted proofs and an unpublished note from Hatami. None of that looks load-bearing, but it should be cleaned up before the authors call it final.\n\nWhat is actually new: the isomorphism theorem (Theorem 4.1), which says t_L and t_M agree on nonnegative functions iff L and M are isomorphic up to column permutation. That is a real arithmetic analogue of the graph result, and the proof via the symmetrised functional and dimension counting is sound. It also yields the variable-transitivity result (Corollary 4.6), which is the arithmetic version of Sidorenko's edge-transitivity theorem. The rank-two classification (Theorem 5.8) is a nice concrete payoff, and the forcing result (Theorem 6.3) and the complex-norming theorem (Theorem 8.1) round out the framework. The paper does a good job connecting to the graph-norm literature without hiding the extra difficulty of the Fourier setting.\n\nThe weak spots are exactly where the reader flagged them. Proposition 2.2 (the Hölder-type equivalence for weakly semi-norming systems) is stated without proof, and it is used repeatedly, so the omission is uncomfortable even though the tensor-power argument from Proposition 2.1 plausibly adapts. Theorem 4.8 and Corollary 8.3 are also deferred to cited analogues—less concerning, but still worth writing out. And [13] is a private communication from Hatami that supports substantive input; that should be replaced by a public reference or an appendix. The 'clearly' step in Corollary 5.3 is minor; the row-operation argument is easy to fill in. The last line of Theorem 5.8's proof needs one extra sentence combining the two bounds, but it is repairable.\n\nThe citation pattern is fine; the authors cite the relevant graph-norm work and their own prior results where appropriate. I don't see circularity or fitting.\n\nThis paper is for people working in additive combinatorics and graph limits. It is a serious contribution that a good referee could improve with requests for the missing proofs. I would accept it for peer review without hesitation.","headline":"A substantial and convincing framework for norming linear systems; the central theorems hold up, but the paper should supply a few omitted proofs and reduce reliance on an unpublished reference before final acceptance.","tokens_in":30805,"tokens_out":2835,"would_cite":true,"duration_ms":21512,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C65","11B30","05C35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Weighted solution counts identify a linear system up to isomorphism.","keywords":["norming linear systems","weakly norming systems","isomorphism theorem","variable-transitivity","forcing systems","uniformity norms","linear systems over finite fields","rank-two classification"],"falsifier":"A computer search over $3\\times k$ systems over $\\mathbb{F}_2$ could try to find a weakly norming system with two non-isomorphic variable-deleted subsystems; by Corollary 4.6 no such system exists, so one explicit example would refute the paper's variable-transitivity theorem.","tokens_in":29863,"feed_emoji":"🧮","tokens_out":10336,"duration_ms":88241,"temperature":0.7,"pith_summary":"This paper asks which systems of linear equations over the vector spaces $\\mathbb{F}_q^n$ are norming: the average $t_L(f)$ of $f(x_1)\\cdots f(x_k)$ over all solutions to the system should define a norm on functions. The paper's central result is an isomorphism theorem: if two systems $L$ and $M$ give the same value $t_L(f)=t_M(f)$ for every non-negative function $f$ on every $\\mathbb{F}_q^n$, then $L$ and $M$ are the same system up to row operations and column permutations. Using this, the paper proves that every weakly norming system is variable-transitive, in the sense that deleting any variable leaves an isomorphic subsystem, and it classifies all weakly norming systems of rank at most two. It also proves that every weakly norming system is forcing, meaning that the constant function is the unique minimizer of $t_L$ at a fixed mean. The motivating examples are the uniformity norms of additive combinatorics, and the paper shows how norming hypergraphs generate further norming systems.","feed_headline":"One functional tells linear systems apart","feed_subtitle":"Equality of solution-counting averages forces isomorphism, and the rank-two norming systems are fully classified.","key_machinery":"The load-bearing object is the solution-count functional $t_L(f_1,\\dots,f_k)=\\mathbb{E}_{(x_1,\\dots,x_k)\\in \\operatorname{Sol}(L)} f_1(x_1)\\cdots f_k(x_k)$, with $t_L(f)=t_L(f,\\dots,f)$. Its Fourier-inversion form, $t_L(f_1,\\dots,f_k)=\\sum_{\\xi\\in \\widehat{G}^m}\\prod_{j=1}^k \\widehat{f_j}(\\sum_i L_{ij}\\xi_i)$, turns norming conditions into spectral inequalities and is what lets the paper tell systems apart. The isomorphism proof also relies on the symmetrised functional $\\tau_L(f_1,\\dots,f_k)=\\sum_{\\pi\\in S_k} t_L(f_{\\pi(1)},\\dots,f_{\\pi(k)})$, the sum over all permutations of the $k$ inputs, together with tensor-power constructions that amplify any difference between non-isomorphic systems until it is visible in $\\tau_L$. The subdivision operation, which replaces each column $v$ by two columns $v$ and $-v$, preserves weak norming and norming and generates the rank-two classification.","core_discovery":"The discovery is that the functional $t_L(f)$, the average of $f(x_1)\\cdots f(x_k)$ over the solution set of $L$, is a complete fingerprint of the system on non-negative inputs: equality of $t_L$ and $t_M$ on all such functions forces $L$ and $M$ to be isomorphic. The proof passes through a symmetrised version $\\tau_L$ acting on $k$-tuples of complex functions, then uses Fourier inversion on $\\mathbb{F}_q^n$ and tensor powers to make non-isomorphic systems produce different values. From this fingerprint property the paper derives structural constraints on weakly norming systems: they must be translation invariant, in the sense that shifting all coordinates by the same group element preserves solutions; their girth, the smallest support of a non-zero row-space vector, must be even; and their minimum-support row vectors must be balanced sign vectors with equally many $a$ and $-a$ entries. The rank-two classification says that every weakly norming $2\\times k$ system is one of three explicit families, obtained from equality systems by subdivision. Finally, weak norming is shown to imply forcing, and every norming system is shown to admit a conjugation assignment under which it defines a norm for complex-valued functions.","pith_inferences":["The isomorphism theorem likely gives a finite certificate for isomorphism: it may suffice to check $t_L$ on indicator functions of small subsets of a fixed $\\mathbb{F}_q^n$, although the paper's proof uses many more functions and does not state such a bound.","The rank-two classification suggests a generative grammar for weakly norming systems through subdivision and disjoint unions; if this extends, higher-rank systems would be built from degenerate equality systems, which would connect the arithmetic theory to the reflection-group picture for graph norms.","Because weakly norming systems are forcing, they are natural candidates for quasi-randomness tests: equality of $t_L$ with its random value could certify that a function is structured, a consequence the paper states only implicitly.","The hypergraph construction gives a pipeline from norming hypergraphs to norms on functions; combining it with the forcing property may yield new lower bounds for arithmetic removal or density increment arguments, though the paper does not pursue these applications."],"forward_implications":["If $L$ is weakly norming, then deleting any one of its variables always leaves the same subsystem up to isomorphism; in particular, no column of a non-zero weakly norming system is zero.","Every weakly norming system of rank one is a single balanced equation with no zero coefficients, and every weakly norming system of rank two is isomorphic to one of three explicit families.","Every weakly norming system is forcing: whenever $t_L(f)=\\mathbb{E}[f]^k$, the function $f$ must be constant.","Every real-norming system can be given a conjugation assignment so that the same functional defines a norm on complex-valued functions.","Every weakly norming hypergraph gives rise, by parametrising its solution set by sums of vertex variables, to a weakly norming linear system; for graphs the resulting system has rank $|E(H)|-|V(H)|+\\kappa(H)$ and its minimal dependencies are exactly the even cycles."],"supporting_citations":[{"why":"Introduces the uniformity norms that motivate the definition of a norming linear system and supplies the Cauchy-Schwarz proof pattern.","marker":"[11]"},{"why":"Develops the graph-norm analogue and the equivalence between triangle inequality and a Holder-type inequality that Proposition 2.1 adapts.","marker":"[12]"},{"why":"Proves the graph left-isomorphism theorem whose arithmetic analogue is Theorem 4.1.","marker":"[16]"},{"why":"Proves edge-transitivity of weakly norming graphs; Corollary 4.6 follows its proof technique.","marker":"[19]"},{"why":"Establishes that real-norming graphs are norming for complex functions, providing the decoration-functional machinery used in Theorem 8.1.","marker":"[15]"},{"why":"Supplies the reflection-group families of weakly norming graphs used to generate norming hypergraph systems.","marker":"[5]"},{"why":"Proves that constant-minimization systems have even girth and gives prior structural results for the systems studied here.","marker":"[14]"},{"why":"Gives classification information for single-equation systems with the constant-minimization property, used in the forcing classification.","marker":"[9]"},{"why":"Supplies the domination inequalities between systems used to compare a weakly norming system with its subsystems.","marker":"[7]"}],"fun_headline_variants":["One functional completely identifies linear systems","Solution counts determine systems up to isomorphism","All rank-two norming systems classified","A solution-count fingerprint for linear systems","Equality of solution counts forces isomorphism"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Proposition 2.2, stated without proof, asserts that weak norming is equivalent to a Holder-type inequality for non-negative functions, and the paper uses that equivalence repeatedly to derive structural constraints; if the equivalence failed, several main conclusions would lack foundation.","fun_headline_variants_meta":{"raw":{"variants":["One functional completely identifies linear systems","Solution counts determine systems up to isomorphism","All rank-two norming systems classified","A solution-count fingerprint for linear systems","Equality of solution counts forces isomorphism"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001888,"raw_usage":{"total_tokens":7385,"prompt_tokens":910,"completion_tokens":6475,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":6415}},"tokens_in":526,"tokens_out":6475,"duration_ms":41415,"temperature":1.0,"reasoning_tokens":6415,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:18:01.239172+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A computer search over $3\\times k$ systems over $\\mathbb{F}_2$ could try to find a weakly norming system with two non-isomorphic variable-deleted subsystems; by Corollary 4.6 no such system exists, so one explicit example would refute the paper's variable-transitivity theorem.","supporting_citations":[{"cited_title":"Weakly norming graphs are edge-t ransitive","cited_arxiv_id":null,"evidence_quote":"Proves edge-transitivity of weakly norming graphs; Corollary 4.6 follows its proof technique."},{"cited_title":"On graph norms f or complex-valued functions","cited_arxiv_id":null,"evidence_quote":"Establishes that real-norming graphs are norming for complex functions, providing the decoration-functional machinery used in Theorem 8.1."},{"cited_title":"Towards a characterization of Sidorenko systems","cited_arxiv_id":null,"evidence_quote":"Proves that constant-minimization systems have even girth and gives prior structural results for the systems studied here."},{"cited_title":"Common and Sidor enko linear equations","cited_arxiv_id":null,"evidence_quote":"Gives classification information for single-equation systems with the constant-minimization property, used in the forcing classification."},{"cited_title":"Domination inequalitie s and dominating graphs","cited_arxiv_id":null,"evidence_quote":"Supplies the domination inequalities between systems used to compare a weakly norming system with its subsystems."}],"review_version":1}