{"id":"f4e3b000-6be0-4877-a0f2-c5f969e1e7ff","arxiv_id":"2411.17168","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a dihedral action on Z_{2n}, all S-invariant permutations form the affine group Aff(Z_{2n}), and the Goldbach sieve symmetry group G_N restates the Goldbach conjecture as G_N not equal to Aff(Z_N).","lead":"This paper builds a group-action language for sieves and proves that the symmetries of a dihedral sieve on numbers modulo 2n are exactly the affine maps x maps to u x plus a. It then constructs a sieve whose unselected residues are the Goldbach pairs of an even number, turning Goldbach's conjecture into a question about symmetry groups.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Corollary 3.24 is supported and the Section 4 action is well-defined; remaining proof gaps are peripheral.","rationale":"The Reader's weakest assumption is the well-definedness of the action in Section 4.1. That action is in fact well-defined: the restriction to Q_k is a transported regular action, and the complement action is the sign homomorphism D_m→Z_2 because Q_k is closed under negation. The orbit of 2q_k is Q_k by construction. The claimed non-collision of Q_k also holds for prime q not dividing N. Thus the main obstacle identified by the Reader evaporates. Corollary 3.24 is the key group-theoretic result, and its proof is a concrete normalizer computation: the derived generators are exactly translations and unit multiplications on Z_{2n}, giving the affine group. The n-even case is messier because Assumption (II) fails, but the paper directly analyzes the quotient and produces the same affine structure. I found no internal inconsistency in this central argument. The manuscript does contain genuine flaws, notably the proof of Proposition 8.5, which is opaque and appears to contain invalid inferences; however, that proposition belongs to the numerical-calculation section and is not load-bearing for the central claim that the Goldbach sieve symmetry group embeds into Aff(Z_N). The overall verdict should therefore remain CONDITIONAL as the Reader set it, not because of a specific fatal objection but because the paper's exposition and several secondary proofs need repair.","tokens_in":37657,"tokens_out":40973,"duration_ms":367267,"concrete_test":"Use GAP to compute N_Sym(Z_N)(⟨T_2, M⟩) for N=30 and N=12, where T_2(x)=x+2 mod N and M(x)=-x mod N, and compare with Agl(1,N); also brute-force verify for N=30, q=7 that the map (4.5) satisfies all (2m)^2 multiplication laws and that the orbit of 2q is Q_7. If either check fails, the central claim must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim survives scrutiny. The specific concern raised in the Reader's verdict—that (4.4)–(4.5) may not define a group action—does not land. For x∉Q_k, the prescription σ^h ρ^k·x = (-1)^h x is a homomorphism D_m→{±1} on the symmetric complement, since Q_k is symmetric; for x∈Q_k, the action is the transport of the regular D_m-action through the bijection f, hence associative on Q_k. The orbit of 2q_k is exactly Q_k. The map f is injective: if q is prime and q∤N, a collision would require N | (i+j+4)q, equivalently N | (i+j+4), but 0<i+j+4≤2⌊N/q⌋<N, impossible. Corollary 3.24 is the corresponding normalizer computation; the proofs for n odd and n even are internally consistent, and the n-even case is handled directly after Assumption (II) fails. The paper's real weaknesses are peripheral: Proposition 8.5's proof is garbled, and several cross-references and typos obscure the exposition, but these do not threaten the main isomorphism or the sieve-to-Goldbach correspondence.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a general framework of algebraic sieves, in which a finite set X is partitioned by the orbits of a list of finite groups, and studies the symmetry group of permutations that intertwine the group actions with automorphisms of the selector groups. The central structural result is Corollary 3.24, which states that for the dihedral action of D_n on Z_{2n}, the group of S-invariant functions is isomorphic to the affine group Aff(Z_{2n}) = Z_{2n} ⋊ GL(Z_{2n}). This is applied to a 'Goldbach sieve' on Z_N, where the selected set is intended to correspond to Goldbach pairs, and the paper analyzes the symmetry group G_N of that sieve and proposes two conjectures relating the size of G_N to Goldbach's conjecture. The main theorem is supported by explicit computations in the odd and even dihedral cases, and several worked examples are provided.","tokens_in":37902,"tokens_out":13539,"duration_ms":114684,"significance":"If the main structural theorem is correct, the paper gives a clean and nontrivial description of the full symmetry group of a natural permutation action of the dihedral group, and it introduces a promising algebraic framework for studying selection sieves. The Goldbach connection is explicitly framed as a reformulation and as conjectures rather than as a proof, which is appropriate. The paper contains machine-checkable explicit calculations in Section 3.1 and several falsifiable structural predictions in Sections 7 and 8. The main weakness is that several load-bearing proofs in Sections 7 and 8 are presented in a garbled and incomplete way, which prevents the results from being fully verified as written.","major_comments":[{"comment":"The action of D_{floor(N/q_k)-1} on Z_N defined in Eq. (4.5) is asserted but not verified to be a group action. In particular, the dihedral relations are not checked on the complement of Q_k, nor is the mixed case (one element in Q_k and one outside) treated. This is load-bearing because the identification of A_N with the Goldbach pair set depends on the orbit of 2q_k being exactly Q_k and on the action being well defined. The authors should add a short verification of the group action, including the homomorphism on the complement.","section":"§4.1, Eq. (4.5)"},{"comment":"The proof for even n is incomplete at the final step. After showing that Assumption (II) fails, the text states 'we already know that <T_n> is normal in \\hat{Aut}(D_n)_S' and then asserts '\\hat{Aut}(D_n)_S = <T_n><f_ν>' without demonstrating the latter equality. Since Assumption (II) is false, the earlier semidirect decomposition theorem (Theorem 3.12) cannot be invoked, so a direct generation argument is needed. This gap affects Corollary 3.24 and hence the interpretation of Section 7's group G_N as a subgroup of Aff(Z_N).","section":"§3.1, Proposition 3.23"},{"comment":"The proofs of Lemma 7.1 and Proposition 7.6 contain statements that are unclear or contradictory. In the proof of Proposition 7.6, the line 'Orb_{D_N}(0) ∈ A' cannot be right, since the orbit of 0 under the dihedral action is the set of even residues, which is contained in the covering rather than in the complement A_N; the intended statement appears to be that the orbit is not in A_N. Similarly, the argument in Lemma 7.1 that w_i ∈ U(Z_{2n}) omits the key step. These propositions are essential for the structural description of G_N in Section 8, so they must be rewritten carefully.","section":"§7, Lemma 7.1 and Proposition 7.6"},{"comment":"The proof of Proposition 8.5 is garbled. It begins with 'If A_N = ∅ we will have that -r is a multiple number in Z for every prime number r', which is not meaningful under the assumption A_N = ∅, and the subsequent derivation that U(Z_N) = {1} is not justified. Since Proposition 8.5 is the basis for the classification of GN-cyclotomic numbers and for the statement of Proposition 8.7, the proof needs to be rewritten with a clear chain of implications.","section":"§8, Proposition 8.5"},{"comment":"The proof of Proposition 8.11 contains a substantial flaw: it claims 'Orb_{D_[N/q]}(2q) ⊃ U(Z_N)', but an orbit is a subset of Z_N and cannot contain the unit group in the manner stated; the conclusion U(Z_N) = <q> also does not follow from the preceding text. Since Proposition 8.11 is used to assert that m.o. numbers such as 90 and 120 give Goldbach sieves, this proof must be corrected or replaced.","section":"§8, Proposition 8.11"}],"minor_comments":[{"comment":"The manuscript contains numerous typographical errors, including 'soubgroup', 'biunivocal', 'if' for 'of', and inconsistent notation for the complement of a set (sometimes A with an overline, sometimes A without). A careful copyedit is needed.","section":"Throughout"},{"comment":"The symbol T_k is used both for an automorphism of D_n and for a translation on Z_{2n}; these should be distinguished (e.g., use α_k for the automorphism and T_k for the translation).","section":"§3.1"},{"comment":"The reference to 'Theorem 3.14' should presumably be to Theorem 3.12; there is no Theorem 3.14 in the paper.","section":"§3.1, after Prop. 3.14"},{"comment":"The statement '|G26| = {2, 4, 6}' is imprecise; it should be '|G26| divides 2, 4, or 6' or '|G26| ∈ {2, 4, 6}' depending on intent. The surrounding sentence is also hard to parse.","section":"§8, Remark 8.2"},{"comment":"In the case N = 10, the sentence 'we are in the equals of the type N = 2p with p prime and in this case p = 10 already proved above' contains an obvious typo: p = 10 should be N = 10 or p = 5.","section":"§8, Example 4"}],"recommendation":"major_revision","confidential_remarks":"The main structural theorem (Corollary 3.24) appears to be correct and is the strongest part of the paper. However, the manuscript as written is not yet ready for publication: the proof of Proposition 3.23 for even n has a genuine gap, and Sections 7 and 8 contain several proofs that are too garbled to be checked. These issues are local and fixable, so I would not reject the paper, but the revision must be substantial. The paper would also benefit from professional copyediting and from a short verification of the group action in Section 4.1. The Goldbach conjectures are clearly labeled as such, and the connection is honest; the paper should not be judged on proving Goldbach, but on the correctness of its algebraic claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The cleanest result is Corollary 3.24: for the dihedral action on Z_{2n}, the group of automorphism-twisted equivariant permutations is the affine group Aff(Z_{2n}). That computation is done honestly, with the n odd/even cases handled separately. The Goldbach sieve construction is genuinely new as a group-action encoding of Goldbach pairs, but it is a reformulation, not a proof, and the paper is honest about that.\n\nWhat is actually new: the explicit normalizer identification for D_n on Z_{2n}, and the sieve whose complement is the Goldbach pair set. The general theory of S-invariant functions is mostly a repackaging of standard twisted equivariance and normalizer extensions, but the dihedral example and the sieve structure are original.\n\nThe soft spots are real but mostly expository. The action in Section 4, Eqs. (4.4)-(4.5), is asserted rather than verified; the stress-test note shows it is well-defined, so the gap is in the presentation, but a referee should demand the verification. Proposition 8.5 is garbled and needs rewriting. Assumptions (I) and (II) in Section 3 are ad hoc, and their necessity claim in Theorem 3.13 is not well motivated. The later sections are a pile of criteria and examples with uneven justification. Typos and broken cross-references are everywhere.\n\nThe Goldbach conjectures are clearly labeled as conjectures and are equivalent to Goldbach by design. No circularity, but also no new number-theoretic evidence. The value is in the group-theoretic reformulation, not in a path to a proof.\n\nWho is this for: group theorists working on permutation actions or normalizers, and anyone curious about algebraic reformulations of Goldbach. I would send it to a serious referee, but with a request for major revision. The core isomorphism holds up, and the sieve construction deserves a careful audience, but the paper is not in publishable shape yet.","headline":"A rough but real paper: the dihedral normalizer computation is solid and the Goldbach sieve is a novel reformulation, but the presentation needs major revision before it can be trusted by a casual reader.","tokens_in":38388,"tokens_out":2227,"would_cite":false,"duration_ms":21622,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20B25","20F28","11P32"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the symmetries of the Goldbach algebraic sieve form a subgroup of the affine group modulo $N$, and that for the dihedral action the full invariant permutation group is exactly that affine group.","keywords":["algebraic sieve","S-invariant function","S-multinvariant function","dihedral group action","affine group","Goldbach sieve","symmetry group","Goldbach conjecture"],"falsifier":"For an explicit pair such as $N=30$, $q=7$, write down the set $\\{\\pm(2+m)7\\bmod 30\\}$, compute the two generators defined in equations (4.4)-(4.5) on every residue outside that set, and check the dihedral relation $\\sigma\\rho=\\rho^{-1}\\sigma$ together with the claim that the orbit of $2q$ is exactly that set; a single failure would break the identification of the sieve's complement with Goldbach pairs.","tokens_in":37422,"feed_emoji":"🔢","tokens_out":13324,"duration_ms":103785,"temperature":0.7,"pith_summary":"This paper builds a group-theoretic language for number sieves. It defines an algebraic sieve as a partition of a finite set into a union of group orbits and its complement, and calls a permutation S-invariant when it intertwines the action of the selecting group with an automorphism of that group. The main result is that for the dihedral action on $\\mathbb{Z}_{2n}$, the S-invariant functions are exactly the affine maps $x\\mapsto ux+a$ with $u$ a unit, so their group is $\\mathrm{Aff}(\\mathbb{Z}_{2n})$. Applied to the Goldbach sieve, this says the sieve's symmetry group is a subgroup of the affine group of $\\mathbb{Z}_N$. If the paper's conjectures are right, the size and shape of that subgroup determine whether every even $N$ is a sum of two primes.","feed_headline":"Dihedral sieve symmetries reduce to affine maps modulo N","feed_subtitle":"Symmetry group of the Goldbach sieve is a subgroup of the affine group; its size ties to Goldbach's conjecture.","key_machinery":"The central object is the group $\\widehat{\\mathrm{Aut}}(G)_S$ of $S$-invariant functions: permutations $f$ of the sieve set $X$ satisfying $f(g\\cdot x)=\\varphi(g)\\cdot f(x)$ for some automorphism $\\varphi$ of $G$ that sends each orbit stabilizer to another stabilizer. The proof decomposes this group into the normal subgroup generated by functions that move orbit representatives through normalizer elements, the group of induced permutations on the set of orbits, and the action of $\\mathrm{Aut}(G)_S$; Theorems 3.12 and 3.13 give the exact semidirect factorization under two assumptions that are verified for the dihedral action. For $G=D_n$ acting on $\\mathbb{Z}_{2n}$, this machinery produces translations $T_a:x\\mapsto x+a$ and unit multiplications $f_u:x\\mapsto ux$, whose combined group is the affine group. The Goldbach sieve is the same machinery applied to a list of dihedral groups chosen so that the complement of the orbit covering is the set of residues representing Goldbach pairs.","core_discovery":"For the action of the dihedral group $D_n$ on $\\mathbb{Z}_{2n}$ by $\\rho^k\\cdot x=x+2k$ and $\\sigma\\cdot x=-x$, the paper studies $S$-invariant functions: permutations $f$ for which there is an automorphism $\\varphi$ of $D_n$ with $f(g\\cdot x)=\\varphi(g)\\cdot f(x)$ for all $g,x$, where $S$ is the list of point stabilizers. The central discovery is that every such function is an affine map $x\\mapsto ux+a$ with $u\\in\\mathrm{GL}(\\mathbb{Z}_{2n})$, so the group $\\widehat{\\mathrm{Aut}}(D_n)_S$ is isomorphic to $\\mathrm{Aff}(\\mathbb{Z}_{2n})=\\mathbb{Z}_{2n}\\rtimes\\mathrm{GL}(\\mathbb{Z}_{2n})$ (Corollary 3.24). The paper reaches the same structure for both odd and even $n$. It then defines the Goldbach sieve on $\\mathbb{Z}_N$ by selecting dihedral groups so that the complement of the covering is the set of residues representing Goldbach pairs, and shows that the sieve's symmetry group $G_N$ is a subgroup of $\\mathrm{Aff}(\\mathbb{Z}_N)$; the final sections give numerical criteria and two conjectures connecting $G_N$ to Goldbach's strong conjecture.","pith_inferences":["The same $S$-invariant machinery could be applied to other selecting families, such as cyclic or symmetric groups, to compute symmetry groups of other algebraic sieves; the dihedral case suggests the result will again be a subgroup of the affine group of the underlying ring.","A computational scan of $G_N$ for even $N$ up to a few thousand using the criteria in Section 8 could test the Strong Conjecture: one non-cyclotomic $N$ with $G_N$ larger than $\\mathbb{Z}_2$ or $V$ would refute it while leaving the Weak Conjecture untouched.","The identification of the orbit of $2q$ with the set of signed multiples of $q$ in the Goldbach sieve construction is the point most worth checking by direct computation; if it fails for some $N$, the classification results on $\\widehat{\\mathrm{Aut}}(D_n)_S$ would remain true but their arithmetic consequences would need a different sieve definition."],"forward_implications":["For every $n$, every permutation of $\\mathbb{Z}_{2n}$ that intertwines the dihedral action with an automorphism of $D_n$ is of the form $x\\mapsto ux+a$; no other permutation can be a symmetry of the dihedral sieve.","The symmetry group $G_N$ of the Goldbach sieve is a subgroup of $\\mathrm{Aff}(\\mathbb{Z}_N)$, so all symmetries of the Goldbach residue set are compositions of translations and unit multiplications.","For $G_N$-cyclotomic $N$, the paper computes $G_N$ explicitly, e.g. $G_{12}\\cong(\\mathbb{Z}_2)^3$, $G_{18}\\cong\\mathbb{Z}_3\\rtimes\\mathbb{Z}_6$, and $G_{24}\\cong\\mathbb{Z}_4\\rtimes(\\mathbb{Z}_2)^3$.","If the Strong Conjecture holds, every even $N$ that is not $G_N$-cyclotomic has symmetry group $\\mathbb{Z}_2$ when $4\\nmid N$ and the Klein four-group $V$ when $4\\mid N$.","If the Weak Conjecture holds for an $N$ with $1\\notin A_N$ or $\\{1,N-1\\}\\not\\subset A_N$, then Goldbach's strong conjecture holds for that $N$."],"supporting_citations":[{"why":"Supplies the presentation of $\\mathrm{Aut}(D_n)$, the automorphisms $T_k$ and $\\varphi_\\nu$, and the composition rules used to classify the $S$-invariant functions.","marker":"[1]"},{"why":"Supplies Bertrand's postulate, used in Proposition 8.7 to prove that a $G_N$-cyclotomic sieve has nonempty Goldbach complement and to rule out the exceptional case.","marker":"[8]"},{"why":"Provides the Eratosthenes sieve, the motivating numerical sieve whose selection of primes the algebraic-sieve definition is designed to abstract.","marker":"[9]"}],"fun_headline_variants":["Dihedral sieve symmetries are affine maps modulo N","Goldbach sieve symmetries live in affine group","S-invariant functions on dihedral sieves are affine","Symmetry of Goldbach sieve reduces to affine maps","Dihedral actions yield affine symmetry groups for sieves"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The link to Goldbach's conjecture rests on the unverified assertion that each auxiliary dihedral group action moves exactly the intended set of signed multiples of a prime and leaves the rest of $\\mathbb{Z}_N$ fixed; if that assertion fails, the sieve's complement need not be the Goldbach-pair set.","fun_headline_variants_meta":{"raw":{"variants":["Dihedral sieve symmetries are affine maps modulo N","Goldbach sieve symmetries live in affine group","S-invariant functions on dihedral sieves are affine","Symmetry of Goldbach sieve reduces to affine maps","Dihedral actions yield affine symmetry groups for sieves"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000183,"raw_usage":{"total_tokens":1391,"prompt_tokens":1096,"completion_tokens":295,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":712,"completion_tokens_details":{"reasoning_tokens":219}},"tokens_in":712,"tokens_out":295,"duration_ms":3567,"temperature":1.0,"reasoning_tokens":219,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:28:03.558006+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For an explicit pair such as $N=30$, $q=7$, write down the set $\\{\\pm(2+m)7\\bmod 30\\}$, compute the two generators defined in equations (4.4)-(4.5) on every residue outside that set, and check the dihedral relation $\\sigma\\rho=\\rho^{-1}\\sigma$ together with the claim that the orbit of $2q$ is exactly that set; a single failure would break the identification of the sieve's complement with Goldbach pairs.","supporting_citations":[{"cited_title":"(2007) Gruppi","cited_arxiv_id":null,"evidence_quote":"Supplies the presentation of $\\mathrm{Aut}(D_n)$, the automorphisms $T_k$ and $\\varphi_\\nu$, and the composition rules used to classify the $S$-invariant functions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Bertrand's postulate, used in Proposition 8.7 to prove that a $G_N$-cyclotomic sieve has nonempty Goldbach complement and to rule out the exceptional case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Eratosthenes sieve, the motivating numerical sieve whose selection of primes the algebraic-sieve definition is designed to abstract."}],"review_version":1}