{"id":"e3e5c668-c616-48bd-bbb4-1925bcfeb007","arxiv_id":"2507.18045","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For primes p = 4n^4 + 12n^2 + 1, four explicit pairs of unions of order-8 cyclotomic cosets in F_p are shown to be (p-1)/16-fold near-factorizations.","lead":"This paper constructs 'lambda-fold near-factorizations' for prime-sized cyclic groups: two residue sets mod p, each of size (p-1)/4, so that every nonzero residue is a sum from the two sets in exactly (p-1)/16 ways. The construction works for every prime of the form p = 4n^4 + 12n^2 + 1 and yields new instances of combinatorial objects behind cryptographic manipulation-detection codes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"","rationale":"The reader's weakest assumption was transcription of Lehmer's order-8 cyclotomic table. That is a real fragility, but the table is a published classical resource and the p=17 check plus row-sum consistency give independent support. The more immediately load-bearing and locally checkable gap is that the printed proof of Theorem 3.2 only verifies case (1a); the remaining three cases are asserted analogous without computation. Those cases are not identical substitutions: they use different combinations of cyclotomic numbers and the opposite sign condition y=-b, so the cancellation pattern must be rechecked. My own spot checks of the l=0,1 reductions suggest the analogous identities are true, but the paper does not display enough to make that certain. This does not overturn the reader's CONDITIONAL verdict; it strengthens the reason for conditioning on a full verification of all four cases. The Remark 2.4 flaw for odd u is also real but secondary, since it affects nonemptiness of U- rather than the product identities themselves.","tokens_in":10851,"tokens_out":13617,"duration_ms":132968,"concrete_test":"For each of S,T in Theorem 3.2 cases (1b),(2a),(2b), compute 64w_l for l=0,1,2,3 using Lemma 3.1(3)-(5), Table 1, the substitutions x+a=2 or x+a=-2, and y=b or y=-b; verify in both the 2 in C4_0 and 2 in C4_2 columns that 64w_l = 4p-4. A short CAS script or expanded table would settle whether the three 'analogous' cases hold. Also check Remark 2.4 by replacing the (0,1)/(0,5) comparison with the (1,3)/(1,6) entries when u is odd.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.2 proves only case (1a). Cases (1b), (2a), and (2b) are dismissed as 'analogous', yet the composition laws used for them in Lemma 3.1(3)-(5) involve different cyclotomic-number combinations and different cancellation conditions: (1b) needs y=b, while (2a) and (2b) need y=-b. For example, in case (2a), coefficient y_0 is 64((0,1)+(0,4)+(1,5)+(1,4)); in the 2 in C4_0 column the b- and y-terms cancel only when y=-b, and the analogous check must be repeated for l=1,2,3 and for the 2 in C4_2 column. Since cases (1b),(2a),(2b) constitute three quarters of the stated theorem, the central claim is not fully established in print until those 64w_l = 4p-4 identities are actually displayed or machine-checked. A secondary, but related, gap is in Remark 2.4: for odd u (2 in C4_2), the entries (0,1) and (0,5) are identical, so the claimed sign-flip of b is not derived from them; the proof of U- nonemptiness is therefore incomplete as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs λ-fold near-factorizations of the cyclic group F_p, equivalently two-block strong external difference families, for primes p = 4u^2 + 12u + 1 with u a nonzero square. The construction uses unions of two order-8 cyclotomic cosets and distinguishes two cases according to whether the primitive element α lies in U^+_p or U^-_p, where the sign of y relative to b in the two representations p = x^2 + 4y^2 and p = a^2 + 2b^2 determines the case. The proof computes the relevant group-ring coefficients in full only for case (1a), relying on Lehmer's table of order-8 cyclotomic numbers, and states that the other three cases are analogous. A conditional infinite-family statement under Bunyakovsky's conjecture is included.","tokens_in":10925,"tokens_out":6956,"duration_ms":72951,"significance":"If the missing computations are supplied, the paper would provide a genuinely new cyclotomic construction: unlike the earlier results in Result 1.1, both blocks of the near-factorization are unions of two cyclotomic cosets rather than single cosets. The explicit coefficient identity in case (1a), checked against Lehmer's table, is transparent and involves no fitted constants; the connection to strong external difference families gives the result cryptographic relevance. The paper also offers computational evidence and a conditional infinite-family theorem. The main value is therefore modular and incremental, but the construction is concrete and checkable.","major_comments":[{"comment":"The proof proves only case (1a) in full; cases (1b), (2a), and (2b) are dismissed with the sentence 'the proof of cases (1b), (2a), (2b) is analogous.' These cases are load-bearing and not immediate transcriptions: Lemma 3.1(3)-(5) give different coefficient combinations, and the cancellation conditions differ, for example (1b) uses y = b while (2a) and (2b) use y = -b. In case (2a), for instance, Lemma 3.1(4) gives y_0 = (1,0)+(4,0)+(4,3)+(7,3) = (0,1)+(0,4)+(1,5)+(1,4), and the required identity 64y_l = 4p - 4 must be verified for l = 0,1,2,3 in both columns of Table 1; none of these computations are displayed. The theorem is therefore not fully established in print as it stands; the analogous computations should be written out or supplied as a machine-checkable appendix.","section":"Section 3, Theorem 3.2 proof"},{"comment":"The proof that both U^+_p and U^-_p are nonempty is incomplete for odd u. When 2 ∈ C^4_2, Table 1 gives (0,1) and (0,5) identical expressions, p - 7 + 2x + 4a, with no dependence on y or b. Thus the displayed consequence (0,1)_α = (0,5)_β carries no information about b_β, and the conclusion y_{α^{8y+5}} = -b_{α^{8y+5}} does not follow for odd u from the relations used in the remark. Since the theorem's 'depending on the choice of primitive element' phrasing and the claimed dichotomy rely on both sets being nonempty, a separate argument for the odd-u case is needed.","section":"Remark 2.4"}],"minor_comments":[{"comment":"The abstract states p = 4n^4 + 12n^2 + 1 while Theorem 1.3 states p = 4u^2 + 12u + 1 with u a nonzero square; the substitution u = n^2 should be made explicit at first use.","section":"Introduction, Theorem 1.3 and abstract"},{"comment":"The phrase 'first108 nonzero squares' appears to be a typesetting error for 'first 10^8 nonzero squares'; please clarify, and if a computational search is reported, state the software or provide reproducible code.","section":"Section 3, before Proposition 3.3"},{"comment":"Every later coefficient identity in Theorem 3.2 is fetched from the transcribed Table 1, which is imported from [16, Appendix] without re-derivation; the authors should state explicitly whether the table was independently verified or should provide a derivation or verification script for reproducibility.","section":"Lemma 2.2 and Table 1"}],"recommendation":"major_revision","confidential_remarks":"The core idea and the verified case (1a) suggest the construction is correct, and the gaps are local rather than conceptual. However, the missing three cases form three quarters of the theorem and involve genuinely different coefficient identities, so the paper is not yet complete in print. I recommend major revision rather than rejection, assuming the authors can supply the omitted computations and repair the odd-u sign-flip argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The new thing is real: this is the first cyclotomic construction of λ-fold near-factorizations using order-8 cosets and unions of two cosets, giving explicit pairs for primes p=4u^2+12u+1 with u a square, contingent on Bunyakovsky. The authors are upfront about the contingency and about importing Lehmer's classical cyclotomic table. I spot-checked the one fully computed case (1a) in Theorem 3.2; the cancellations work, the row-sum identities in Table 1 are consistent, and the p=17 instance is a genuine 1-fold near-factorization. That is real progress, not a repackaging.\n\nThe soft spots are all about what is not shown. The proof of Theorem 3.2 only writes out case (1a); cases (1b), (2a), (2b) are dispatched as 'analogous.' That is not a formality: they use different cyclotomic-number combinations and different sign conditions (y=b vs y=-b), and the cancellations have to be rechecked for each parity class. In (2a), for example, the coefficient y_0 involves a different combination and the b/y terms cancel only when y=-b, so the computation is not literally analogous to (1a). Three quarters of the main theorem is unverified in print.\n\nThe second real gap is Remark 2.4. The argument that both U^+_p and U^-_p are nonempty uses the equations (0,1) and (0,5) to flip the sign of b, but in the 2∈C^4_2 column those entries are identical, so they cannot force the sign flip. The conclusion may be salvageable with a different entry pair, but as written the proof is incomplete. This matters because Theorem 3.2 splits on which set α belongs to.\n\nMinor issues: the computational claim about 7,565,563 primes is unreproducible as printed—no code, no data, and the typesetting is garbled. And Lemma 2.2 depends entirely on a 1955 table; that is acceptable, but a transcription error would hit whole parity classes, so a referee should verify at least the entries used.\n\nOverall: the central construction is sound where actually shown, and the gaps look fillable. The paper deserves serious refereeing. I would send it out and ask for the missing computations, a corrected Remark 2.4, and a reproducible statement of the computational check. It is not ready as is.","headline":"New order-8 cyclotomic construction with honest gaps: the one fully proved case checks out, but three quarters of the main theorem is unverified in print.","tokens_in":11720,"tokens_out":3924,"would_cite":false,"duration_ms":36593,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05E16","05B10","11T22","94A13"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every prime $p = 4u^2 + 12u + 1$ with $u$ a nonzero square, the paper constructs four explicit $\\lambda$-fold near-factorizations of the cyclic group $\\mathbb{F}_p$ with $\\lambda = (p-1)/16$.","keywords":["λ-fold near-factorization","cyclotomic numbers","cyclotomic cosets","strong external difference family","cyclic group","group ring","near-factorization","prime values of polynomials"],"falsifier":"For $p = 113$ ($u = 4$, even) and $p = 17$ ($u = 1$, odd), enumerate the order-8 cyclotomic numbers directly over the primitive elements and compare all fifteen critical entries with Table 1; a mismatch in any entry, or a failure of $64v_\\ell = 4p - 4$ for either parity class, would refute the construction.","tokens_in":10427,"feed_emoji":"🧮","tokens_out":6784,"duration_ms":62603,"temperature":0.7,"pith_summary":"The paper extends cyclotomic near-factorization constructions to unions of order-8 cyclotomic cosets. For every prime $p = 4u^2 + 12u + 1$ with $u$ a nonzero square, it proves that four explicit pairs $(S,T)$ of size $(p-1)/4$ each cover every nonzero element of $\\mathbb{F}_p$ exactly $(p-1)/16$ times. Each pair is equivalent to a strong external difference family, a structure used in cryptographic manipulation-detection codes. The construction works for both parity classes of $u$, selecting the pair according to the primitive element; assuming a standard conjecture on prime values of integer polynomials, the prime family is infinite.","feed_headline":"Order-8 cosets yield four near-factorizations per prime","feed_subtitle":"For each prime p=4u²+12u+1 with u a square, four explicit pairs cover every nonzero element equally often.","key_machinery":"The machinery is the table of order-8 cyclotomic numbers $(i,j)_{8,\\alpha}$ over $\\mathbb{F}_p$ for $p \\equiv 1 \\pmod{16}$, which counts solutions to $1 + \\alpha^{8u+i} = \\alpha^{8v+j}$. The paper combines this table with the group-ring expansion $C_i^8 C_j^8 = \\sum_{\\ell} (j-i, \\ell-i) C_\\ell^8$ to express products of two-coset unions as $\\sum_{\\ell} v_\\ell C_\\ell^8$, then verifies $64v_\\ell = 4p - 4$ for $\\ell = 0,1,2,3$, so that each nonzero coset receives exactly $(p-1)/16$ copies of each element.","core_discovery":"For $p = 4u^2 + 12u + 1$ prime with $u$ a nonzero square, $p \\equiv 1 \\pmod{16}$, and the paper gives explicit integer representations $p = x^2 + 4y^2$ and $p = a^2 + 2b^2$ with $x,a$ determined by $u$ and $y,b$ determined up to sign by $u$ and the primitive element $\\alpha$. Depending on whether $2$ lies in the fourth-power coset $C_0^4$ or $C_2^4$ (equivalently, whether $u$ is even or odd), and on whether $y = b$ or $y = -b$, one of two described pairs of unions of two order-8 cyclotomic cosets satisfies $ST = ((p-1)/16)(\\mathbb{F}_p - \\{0\\})$ in the group ring. The proof expands $S \\cdot T$ into cosets using the order-8 cyclotomic numbers and shows every coefficient is $(p-1)/16$.","pith_inferences":["The same coefficient-checking recipe could be run at order 16 or 24 using published cyclotomic number tables; any prime family with explicit $x,a,y,b$ would give analogous two-coset-union pairs, so the paper's method is a template, not a one-off.","Because two-set strong external difference families are exactly $\\lambda$-fold near-factorizations, the new pairs translate directly into algebraic manipulation detection codes with the stated parameters — a cryptographic consequence the paper notes but does not develop.","A quick independent check of the fifteen critical cyclotomic entries for one small prime in each parity class (e.g. $p = 17$ and $p = 113$) would validate the transcribed table and thus the whole family; this is a finite computation, not a conjecture."],"forward_implications":["For each admissible prime $p$, the paper supplies four explicit $(p,2,(p-1)/4,(p-1)/16)$-strong external difference families, one for each combination of primitive-element case and pair.","This is the first cyclotomic construction in which both subsets are unions of two cyclotomic cosets rather than single cosets, opening a new pattern for higher-order constructions.","The construction covers both parity classes of $u$; no admissible prime of the form $4u^2 + 12u + 1$ is left out.","The families overlap with the earlier order-4 construction only at $p = 17$; elsewhere they produce new parameters.","If the relevant prime-value conjecture holds, the sequence $4n^4 + 12n^2 + 1$ produces infinitely many primes, hence infinitely many cyclic groups carrying these near-factorizations."],"supporting_citations":[{"why":"Supplies the complete table of fifteen critical order-8 cyclotomic numbers and the relation table used in Lemma 2.2; every coefficient computation in Theorem 3.2 draws on it.","marker":"[16]"},{"why":"Supplies the uniqueness of the representations $p = x^2 + 4y^2$ and $p = a^2 + 2b^2$, and the criterion $2 \\in C_0^4$ iff $8 \\mid 2y$ used in Lemma 2.3.","marker":"[3]"},{"why":"Introduces $\\lambda$-fold near-factorizations and gives the group-ring formulation and notation the paper adopts.","marker":"[13]"},{"why":"Provides the earlier order-4 and order-6 cyclotomic constructions, the baseline against which the union-of-cosets construction is new; also supplies the $p = 17$ overlap comparison.","marker":"[2]"},{"why":"Establishes the equivalence between two-subset strong external difference families and $\\lambda$-fold near-factorizations, which converts the main theorem into SEDF parameters.","marker":"[18]"},{"why":"Supplies the statement of the prime-value conjecture used in Proposition 3.3 to argue the prime family is infinite.","marker":"[5]"}],"fun_headline_variants":["Four near-factorizations from order-8 cyclotomic cosets","Cyclotomic order-8 cosets give four λ-fold near-factorizations","Four explicit pairs from order-8 cosets cover every nonzero element","Cyclotomic construction yields four near-factorizations for each prime"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof rests on the transcribed table of fifteen critical order-8 cyclotomic numbers from [16, Appendix]: if any single entry is wrong, the identities $64v_\\ell = 4p - 4$ fail for an entire parity class of primes, and the theorem collapses.","fun_headline_variants_meta":{"raw":{"variants":["Four near-factorizations from order-8 cyclotomic cosets","Cyclotomic order-8 cosets give four λ-fold near-factorizations","Four explicit pairs from order-8 cosets cover every nonzero element","Cyclotomic construction yields four near-factorizations for each prime"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000976,"raw_usage":{"total_tokens":4111,"prompt_tokens":871,"completion_tokens":3240,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":3162}},"tokens_in":487,"tokens_out":3240,"duration_ms":27728,"temperature":1.0,"reasoning_tokens":3162,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:44:40.789295+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $p = 113$ ($u = 4$, even) and $p = 17$ ($u = 1$, odd), enumerate the order-8 cyclotomic numbers directly over the primitive elements and compare all fifteen critical entries with Table 1; a mismatch in any entry, or a failure of $64v_\\ell = 4p - 4$ for either parity class, would refute the construction.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the complete table of fifteen critical order-8 cyclotomic numbers and the relation table used in Lemma 2.2; every coefficient computation in Theorem 3.2 draws on it."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the uniqueness of the representations $p = x^2 + 4y^2$ and $p = a^2 + 2b^2$, and the criterion $2 \\in C_0^4$ iff $8 \\mid 2y$ used in Lemma 2.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the earlier order-4 and order-6 cyclotomic constructions, the baseline against which the union-of-cosets construction is new; also supplies the $p = 17$ overlap comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the equivalence between two-subset strong external difference families and $\\lambda$-fold near-factorizations, which converts the main theorem into SEDF parameters."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the statement of the prime-value conjecture used in Proposition 3.3 to argue the prime family is infinite."}],"review_version":1}