{"id":"ec42570b-e4b2-40bd-b8fd-787a421977ab","arxiv_id":"2608.12648","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors generalize the escalation method to number fields, prove finiteness of criterion sets, and compute, conjecturally and in one case exactly, the analogue of the 15-Theorem over Q(√2), Q(√3), and Q(√5).","lead":"This paper develops a method, called escalation, that finds the small checklists of numbers guaranteeing a quadratic form is universal over a number field. It yields an exact checklist for Q(√5) and strongly supported conjectures for Q(√2) and Q(√3).","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Q(√2) criterion set rests on Conjecture 5.11, whose twelve universality assertions are supported only by a norm-250,000 Magma check; Remark 5.10 shows such checks can miss exceptions, so the stated set could silently fail above that bound.","rationale":"The reader's weakest assumption matches the main vulnerability: the conjectural criterion sets for Q(√2) and Q(√3) depend on unproved universality statements that are evidenced only by finite-range computation, and the paper itself supplies a nearby counterexample to justify caution. I do not see a more load-bearing internal flaw. The proven parts—the exact diagonal criterion for Q(√5), the minimality of Lee's set, and the universality of the escalations of E^(6)_3—are supported by arguments that appear sound. The finiteness theorems in Sections 3 and 4 do defer technical details to [KKR], but they are presented with enough pointers that this is a normal reliance on prior work rather than a circular step. Because the paper is explicitly conjectural about the headline sets and the reader already marked the verdict CONDITIONAL, the appropriate stress-test outcome is to keep that verdict: the concern is real and load-bearing but does not by itself overturn the paper's honest claims. The proposed concrete test is designed to either find a counterexample or substantially raise confidence; it does not pretend to convert a conjecture into a theorem, since no finite computation can do that without additional theory.","tokens_in":39847,"tokens_out":12611,"duration_ms":129969,"concrete_test":"Ask the authors to release the Magma code and independently re-run, with a second implementation, the representation check for the thirteen lattices in Conjecture 5.11 to norm 10^7, recording for each lattice the first non-represented totally positive element, if any. If a new exception appears above 250,000, Conjecture 5.11 is false and the Q(√2) criterion set must be enlarged; if the computation is clean to 10^7, the specific risk of a just-beyond-bound counterexample is substantially reduced, though not logically eliminated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.2(4) reduces Conjecture 5.1 to Conjecture 5.11, and Conjecture 5.11 is the only unproved input: it asserts that twelve quaternary lattices are universal and that E^(2)_4,(5) represents every element except 3(3−√2). Lemma 5.12 uses these assertions to conclude that all other escalations of E^(5)_3 are universal, and Proposition 5.7 uses the same input to dispose of the 1656 rank-five escalations. The evidence behind Conjecture 5.11 is the norm-250,000 Magma computation in Proposition 5.6, with the programs only available upon request (p. 32). Remark 5.10 explicitly documents why this kind of evidence is fragile: the analogous local-representation check for E^(5)_3 was complete through norm 200,000, yet new locally-represented-but-not-represented exceptions appear at norms 203,391 and 210,681. If any of the twelve lattices has a first non-represented element α with N(α)>250,000, then α is a truant of that lattice and hence, by Proposition 2.8, a critical element; the minimal classical criterion set would then contain an element outside the six listed ones, falsifying Conjecture 5.1 rather than merely leaving it unproved. The same structure applies to Conjectures 6.5 and 6.9 for Q(√3), where Proposition 6.3 and Theorem 6.2(4) are conditional on the same kind of norm-limited verification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops the method of escalation over totally real number fields, proving finiteness of S-criterion sets and giving algorithms to compute them. It also introduces a separate pseudoescalation theory for diagonal forms. The main arithmetic results are conjectural criterion sets for the classical universal forms over Q(√2) (Conjecture 5.1) and Q(√3) (Conjecture 6.1), the exact classical criterion set for Q(√5) due to Lee is reproved with minimality (Theorem 7.1), and the exact diagonal criterion set for Q(√5) is computed (Theorem 7.2). The Q(√2) and Q(√3) criterion sets are supported by numerical verification up to norm 250,000 and by conditional proofs that reduce them to explicit conjectures about a small number of quaternary and ternary lattices (Conjectures 5.11, 6.5, 6.9).","tokens_in":40178,"tokens_out":4962,"duration_ms":50804,"significance":"If the conjectures are correct, this would be the first explicit universality criterion sets over real quadratic fields other than Q(√5), and the escalation method would be a substantial algorithmic contribution to the field. The finiteness proof for S-criterion sets, once completed, removes the reliance on the Chan–Oh theorem in [KKR] and provides a practical computational route. The pseudoescalation theory for diagonal forms is new and of independent interest. The paper is remarkably honest in labeling its main results as conjectural and in documenting the fragility of numerical evidence (Remark 5.10). The unconditional results, such as the universality of all escalations of ⟨1,2+√2,2⟩ (Proposition 5.8) and the diagonal criterion for Q(√5), are solid and valuable. However, the central claims for Q(√2) and Q(√3) remain conditional on unproved assertions that are verified only by computations to norm 250,000, and the computational programs themselves are not made available.","major_comments":[{"comment":"The proof of Lemma 3.6 omits the core local S-universality argument, stating only that it is 'virtually the same as in the proof of [KKR, Prop. 3.1]' and providing two pointers. Since Theorem 3.7, which states the finiteness of the escalation tree, depends directly on this lemma, and since the paper advertises a 'simple proof of finiteness', the omission is load-bearing. Please provide a complete proof of the local S-universality step, or at least a precise statement with the necessary modifications of the KKR argument, rather than deferring the key detail to a reference.","section":"Section 3, Lemma 3.6"},{"comment":"The claim that Conjecture 5.1 follows from Conjecture 5.11 makes the Q(√2) criterion set conditional on the universality of twelve quaternary lattices that are checked only up to norm 250,000. Remark 5.10 explicitly warns that the analogous check for E^(5)_3 was complete through norm 200,000 and yet new exceptions appear at norms 203,391 and 210,681. Therefore the numerical evidence does not strongly support Conjecture 5.11, and there is a real possibility that a first non-represented element occurs above the computed bound, which would change the minimal criterion set. The same issue applies to Conjectures 6.5 and 6.9 for Q(√3). The paper should state this limitation more prominently in the abstract and in the statements of Theorems 5.2 and 6.2, and ideally give some heuristic or further evidence that the conjectures hold beyond the checked bound.","section":"Section 5, Conjecture 5.11 and Theorem 5.2(4)"},{"comment":"The proof of Theorem 5.2(2)–(3) and Proposition 5.6 relies on a Magma computation that is described only as 'We used a program written in Magma to compute all elements up to norm 250 000 and check their representability', with the programs 'available upon request'. This is not sufficient for a reproducible proof, especially because the paper explicitly uses these computations to establish that the criterion set contains no further elements below norm 250,000, which is one of the few unconditional results. The authors should provide the program code as supplementary material, or at least give a complete, checkable algorithmic description and the full output of the computations.","section":"Section 5, Proposition 5.6 and computational verification"}],"minor_comments":[{"comment":"In the proof of Cdiag_odd = Ccl_odd, the sentence 'we just proved Cdiag_odd ⊂ Cdiag_odd' is a typo: the intended statement is that the exhibited diagonal forms prove Ccl_odd ⊂ Cdiag_odd (or the reverse inclusion, depending on the direction being discussed).","section":"Section 4, Example 4.9"},{"comment":"The proof cites '[KKR, Thm. 1.2]' for closure of criterion sets under units and conjugation, whereas the earlier text cites '[KKR, Cor. 3.5]' for the same property; please harmonize the references.","section":"Section 6, proof of Theorem 6.2"},{"comment":"The Q(√5) part of the proof states 'They can all be checked to be universal' without showing the checks; a short table listing the forms and the required representations would make the argument easier to verify.","section":"Section 7, proof of Theorem 7.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is well-written and honest about the conjectural status of its main results. The main concerns are: the incomplete proof of Lemma 3.6, which is load-bearing for the finiteness theorem; the fragility of numerical evidence in support of Conjectures 5.11, 6.5, and 6.9, as demonstrated by the paper's own Remark 5.10; and the unavailability of the Magma programs used for the claimed 250,000-norm verification. These are fixable in a revision, but they make the current version unsuitable for acceptance. I would not recommend rejection, as the theoretical framework and the unconditional results are sound and significant."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Jakub and Giuliano have written a serious paper, and the headline results are more interesting than the abstract lets on. The real news is not just the conjectured criterion sets for Q(√2) and Q(√3); it is that the escalation method finally works over number fields. Their finiteness proof via König's lemma is clean, and the pseudoescalation apparatus for diagonal forms is a genuine invention that should be useful beyond the specific fields. The proven results are also worth having: Theorem 7.1 settles the minimality of Lee's criterion set for Q(√5), and Theorem 7.2 gives the exact diagonal criterion set for the same field.\n\nThe soft spots are real but localized. Lemma 3.6 defers the local S-universality argument to 'the same as in KKR,' which makes the finiteness proof less self-contained than advertised. More importantly, the Q(√2) criterion set hangs entirely on Conjecture 5.11, which asserts universality of twelve quaternary lattices and one near-universal lattice. The evidence is a norm-250,000 Magma check, and Remark 5.10 contains the authors' own warning that such checks can miss exceptions: they found new exceptions at norms 203,391 and 210,681 for a closely related lattice. The programs are only available upon request, so an independent referee cannot verify the computational claims without asking. The same structure underpins Q(√3), via Conjectures 6.5 and 6.9.\n\nNone of this is a fatal flaw, because the paper is honest about what is conjecture and what is theorem. The conjectures are stated as conjectures, and the conditional chain from Conjecture 5.11 to Conjecture 5.1 is explicit. If the authors made the Magma code public and tightened Lemma 3.6, the paper would be much stronger. As it stands, it deserves a serious referee and a conditional path: the conjectural criteria should be clearly labeled as computational predictions, and the code should be archived.\n\nI would cite the proven parts, especially the diagonal criterion for Q(√5), and I'd bring the paper to a reading group focused on quadratic forms. For peer review: yes, send it out.","headline":"A genuine methodological advance in number-field escalation, but the headline criteria are honest conjectures resting on norm-limited computation.","tokens_in":40720,"tokens_out":2711,"would_cite":true,"duration_ms":26210,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11E12","11E20","11R04","11R11","11R80"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper develops escalation over number fields and uses it to compute universality criterion sets for Q(√2), Q(√3), and Q(√5).","keywords":["universal quadratic forms","criterion sets","escalation","real quadratic fields","totally real number fields","quadratic lattices","15-Theorem","diagonal forms"],"falsifier":"Run the paper's escalation algorithm beyond norm $250\\,000$ (or analyze the class-number-two ternary lattice $E^{(5)}_3$) and check whether any of the twelve quaternary lattices in Conjecture 5.11 misses a totally positive integer; equally decisive would be finding any critical element outside the six listed elements by producing a lattice whose truant lies outside that set. The paper records local–global failures for the parent ternary lattice at norms $203\\,391$ and $210\\,681$, so the search should be specifically target large-norm exceptions.","tokens_in":39604,"feed_emoji":"🧮","tokens_out":10295,"duration_ms":85677,"temperature":0.7,"pith_summary":"The paper aims to turn the 15-Theorem/290-Theorem question — which numbers must a quadratic form represent in order to represent every totally positive integer? — into a finite algorithmic procedure for totally real number fields. It develops escalation over number fields, and a variant called pseudoescalation for diagonal forms, proving that the process of repeatedly adjoining a vector representing the smallest missing totally positive integer always terminates. Applied to the three simplest real quadratic fields, the method yields explicit criterion sets: exact for diagonal forms over $\\mathbb{Q}(\\sqrt{5})$, and provably correct below norm $250\\,000$ with conjectural exactness for $\\mathbb{Q}(\\sqrt{2})$ and $\\mathbb{Q}(\\sqrt{3})$. The payoff is that universality criteria, previously explicit only for $\\mathbb{Q}(\\sqrt{5})$, become computable objects.","feed_headline":"Escalation computes universality criteria for real quadratic fields","feed_subtitle":"A finite 'escalation' tree yields the minimal list of integers a form must represent; exact for Q(√5), conjectural for the others.","key_machinery":"The central mechanism is the escalator tree. Starting from the zero lattice, at each step take the current lattice's truant — the least (with respect to any order refining norm) totally positive algebraic integer it fails to represent — and add a vector representing that truant; every lattice has only finitely many such escalations, and no infinite chain can exist because eventually the lattice has rank at least $5$ and the asymptotic local–global principle makes it universal. Hence the tree of escalators is finite, and the equivalence 'critical iff truant of some lattice, equivalently of some escalator' turns criterion-set computation into a finite search. For diagonal forms, pseudoescalation plays the same role, adjoining one or more diagonal coefficients just enough to represent the current truant. The remaining work for explicit fields is proving (or conjecturing) universality of the finitely many quaternary escalators.","core_discovery":"On the paper's own terms, the discovery is that the minimal criterion set — the unique finite set $C$ such that a classical quadratic lattice is universal iff it represents every element of $C$ — can be computed by escalation. A totally positive algebraic integer is critical precisely when it occurs as the truant of an escalator, i.e. the least element (in any admissible order refining norm) an escalator fails to represent; because the tree of all escalators is locally finite and has no infinite branches, the procedure terminates. The paper carries this out for $\\mathbb{Q}(\\sqrt{5})$, $\\mathbb{Q}(\\sqrt{2})$ and $\\mathbb{Q}(\\sqrt{3})$: it proves the diagonal criterion set for $\\mathbb{Q}(\\sqrt{5})$ is $\\{1,2,(5+\\sqrt{5})/2,(7\\pm\\sqrt{5})/2,2(5+\\sqrt{5})/2\\}$, proves that the previously known six-element classical criterion set for $\\mathbb{Q}(\\sqrt{5})$ is minimal, and reduces the classical criterion sets for $\\mathbb{Q}(\\sqrt{2})$ and $\\mathbb{Q}(\\sqrt{3})$ to explicit conjectures about finitely many quaternary lattices, with numerical verification up to norm $250\\,000$.","pith_inferences":["The method is, in principle, an exact algorithm: pair any universal-lattice oracle with Algorithm 3.9 and it outputs the criterion set, while a norm bound and finite verification give a certified lower bound and a heuristic upper bound; applying it to fields with class number greater than one would test how much survives non-free lattices.","The paper's own warning about exceptions at norm $203\\,391$ suggests the $250\\,000$-norm verifications for $\\mathbb{Q}(\\sqrt{2})$ and $\\mathbb{Q}(\\sqrt{3})$ should not be treated as proof; the plausible failure mode is a large-norm exception in one of the conjecturally universal quaternary lattices.","The conjectural equality $C^{\\mathrm{diag}}(\\mathbb{Q}(\\sqrt{3})) = C^{\\mathrm{cl}}(\\mathbb{Q}(\\sqrt{3}))$ raises the question of whether diagonal forms generally see the same obstructions as arbitrary classical forms; if so, diagonal criteria could serve as a cheap first test before tackling the full classical criterion set.","The paper mentions evidence from a follow-up that $\\mathbb{Q}(\\sqrt{21})$ has a small criterion set for which the largest-critical-element phenomenon fails, so the 'almost-universal largest element' pattern observed for $D=2,3,5$ is probably an accident of those fields rather than a general law."],"forward_implications":["For $\\mathbb{Q}(\\sqrt{2})$, any classical lattice representing $\\{1,2+\\sqrt{2},3,3(2+\\sqrt{2}),3(3\\pm\\sqrt{2})\\}$ represents every totally positive integer up to norm $250\\,000$; assuming Conjecture 5.11 it is universal, and these six elements are exactly the minimal criterion set.","For $\\mathbb{Q}(\\sqrt{3})$, the ten-element set $\\{1,\\varepsilon,4\\pm\\sqrt{3},\\varepsilon(4\\pm\\sqrt{3}),7\\pm2\\sqrt{3},\\varepsilon(7\\pm2\\sqrt{3})\\}$ with $\\varepsilon=2+\\sqrt{3}$ is contained in both the classical and diagonal criterion sets, with equality following from Conjectures 6.5 and 6.9.","For $\\mathbb{Q}(\\sqrt{5})$, the diagonal criterion set is exactly the five-element set $\\{1,2,(5+\\sqrt{5})/2,(7\\pm\\sqrt{5})/2,2(5+\\sqrt{5})/2\\}$, and the classical criterion set is minimal.","The finiteness theorems for escalation and pseudoescalation give an independent proof that every totally real number field has a finite criterion set, without invoking the earlier general finiteness theorem [CO].","In the three fields studied, the largest critical element has a sharp 'almost-universal' property: any lattice representing all smaller criteria represents everything except possibly the largest; the diagonal analogue holds for $\\mathbb{Q}(\\sqrt{3})$ but fails for $\\mathbb{Q}(\\sqrt{2})$ and $\\mathbb{Q}(\\sqrt{5})$."],"supporting_citations":[{"why":"Supplies the theorem that the minimal criterion set equals the set of critical elements, and the characterization that an element is critical iff it is a truant.","marker":"[KKR]"},{"why":"Asymptotic local–global principle used to prove that escalation cannot continue indefinitely, hence the escalator tree is finite.","marker":"[HKK]"},{"why":"Provides the previously known explicit classical criterion set for Q(√5) and the escalator lattices whose truants prove its minimality.","marker":"[Le]"},{"why":"Proves universality of the four ternary escalator lattices used as base cases over Q(√2) and Q(√3).","marker":"[CKR]"},{"why":"Gives the additive structure of totally positive integers in Z[√2] used in Lemma 5.9 to bound non-represented elements.","marker":"[HK]"},{"why":"Earlier finiteness theorem for S-criterion sets that escalation now reproves independently and makes effective.","marker":"[CO]"}],"fun_headline_variants":["Escalation method computes minimal criteria for Q(√5), conjectures for others","Escalation tree ends: minimal integer sets found for Q(√5), Q(√2), Q(√3)","Proven criteria set for Q(√5); conjectural lists for Q(√2), Q(√3) via escalation","Escalation method: exact criteria for Q(√5), conjectures for Q(√2), Q(√3)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that Conjecture 5.11 holds — the twelve listed quaternary lattices over $\\mathbb{Q}(\\sqrt{2})$ are universal and one further lattice represents everything except $3(3-\\sqrt{2})$ — together with the analogous Conjectures 6.5 and 6.9 for $\\mathbb{Q}(\\sqrt{3})$; if any of those lattices has an extra exceptional element, the claimed criterion sets are incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Escalation method computes minimal criteria for Q(√5), conjectures for others","Escalation tree ends: minimal integer sets found for Q(√5), Q(√2), Q(√3)","Proven criteria set for Q(√5); conjectural lists for Q(√2), Q(√3) via escalation","Escalation method: exact criteria for Q(√5), conjectures for Q(√2), Q(√3)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000667,"raw_usage":{"total_tokens":3051,"prompt_tokens":963,"completion_tokens":2088,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":579,"completion_tokens_details":{"reasoning_tokens":1972}},"tokens_in":579,"tokens_out":2088,"duration_ms":13370,"temperature":1.0,"reasoning_tokens":1972,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:02:24.974064+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the paper's escalation algorithm beyond norm $250\\,000$ (or analyze the class-number-two ternary lattice $E^{(5)}_3$) and check whether any of the twelve quaternary lattices in Conjecture 5.11 misses a totally positive integer; equally decisive would be finding any critical element outside the six listed elements by producing a lattice whose truant lies outside that set. The paper records local–global failures for the parent ternary lattice at norms $203\\,391$ and $210\\,681$, so the search should be specifically target large-norm exceptions.","supporting_citations":[],"review_version":1}