REVIEW 5 major objections 5 minor 9 references
S-invariant and S-multinvariant functions and some symmetry groups of algebraic sieves
T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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 $N$.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (5)
- [§4.1, Eq. (4.5)] 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.
- [§3.1, Proposition 3.23] 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).
- [§7, Lemma 7.1 and Proposition 7.6] 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.
- [§8, Proposition 8.5] 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.
- [§8, Proposition 8.11] 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.
minor comments (5)
- [Throughout] 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.
- [§3.1] 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).
- [§3.1, after Prop. 3.14] The reference to 'Theorem 3.14' should presumably be to Theorem 3.12; there is no Theorem 3.14 in the paper.
- [§8, Remark 8.2] 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.
- [§8, Example 4] 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.
Circularity Check
No significant circularity: the main isomorphism is derived by explicit computation, and the Goldbach statements are labeled conjectural reformulations.
full rationale
The central structural result, Corollary 3.24, is obtained by direct stabilizer and normalizer calculations for the dihedral action on Z_{2n} in Section 3.1, with the odd and even cases handled separately; no parameter is fitted to the target and no prior result by the same author is invoked. The Section 4 action is defined by transporting the regular D_m-action through an explicitly chosen bijection f on Q_k and by the sign homomorphism on the symmetric complement; although the paper does not spell out the verification, the action is a valid group action and the orbit of 2q_k equals Q_k by equivariance. The Goldbach-related claims in Section 9 are explicitly labeled 'Strong Conjecture', 'Weak Conjecture', and conditional propositions; they reformulate Goldbach's conjecture in the language of sieves rather than deriving it from a loaded assumption, and the paper explicitly warns about the {1, N-1} caveat. There is no self-citation chain, no imported uniqueness theorem, no ansatz smuggled in by citation, and no fitted input renamed as a prediction. The real weaknesses—such as the omitted verification of the group action, the garbled proof of Proposition 8.5, and typographical issues—are correctness or exposition risks, not circularity.
Assumptions & free parameters
assumptions (4)
- standard math Standard group theory facts: orbit-stabilizer theorem, first isomorphism theorem, and the structure of Aut(D_n) for dihedral groups.
- domain assumption The action of D_{floor(N/q)-1} on Z_N in Eq. (4.5) is a valid group action and produces exactly the displayed orbits Q_k.
- ad hoc to paper Assumptions (I) and (II) in Section 3 hold for the cases under study.
- ad hoc to paper The GN-cyclotomic and quasi-mono-orbital classifications in Section 8 correctly isolate the cases where A_N can be determined.
Cite this review
Pith. "Pith review of S-invariant and S-multinvariant functions and some symmetry groups of algebraic sieves." pith.science (2026). https://pith.science/paper/HFFNFQOG
@misc{pith2026241117168,
author = {Pith},
title = {Pith review of: S-invariant and S-multinvariant functions and some symmetry groups of algebraic sieves},
year = {2026},
howpublished = {\url{https://pith.science/paper/HFFNFQOG}},
note = {Machine review of arXiv:2411.17168}
}
abstract
In this article we introduced algebraic sieves, i.e. selection procedures on a given finite set to extract a particular subset. Such procedures are performed by finite groups acting on the set. They are called sieves because there are certain sets of numbers which, with appropriate groups, can select, for example, a set of primes, think of the famous Eratosthenes sieve. In this article we have given a general definition of algebraic sieves. And we also introduced the notion of invariant and multi-invariant functions, certain permutations on the sieve set which, in the invariant case, commute with the action of a given sieve-selecting group and the automorphism of that group, and multi-invariants which commute with all groups and their respective automorphisms. By means of such functions we have given symmetries on such sieves. In particular, we studied certain groups of symmetries of invariant functions. Then, using such notions, we studied a particular example, the Goldbach sieve, where the selector groups are dihedral groups and the selected set consists of the primes and, in some cases, the numbers $1$ and $N-1=p$, with $N$ being even and p prime, which satisfies the Goldbach conjecture for $N$. We have shown that one of these symmetry groups is isomorphic to a subgroup or affinity group of the ring of integers modulo N with N an integer even $\mathbb{Z}_{N}$.
Reference graph
Works this paper leans on
-
[1]
Mach ` ı, A. (2007) Gruppi. Una introduzione a idee e metodi della Teoria dei Gru ppi. Springer Edition
work page 2007
-
[2]
R.D Carmichael, Introduction to the theory of groups of fi nite order, Dover, New York, 1956
work page 1956
-
[3]
D.Gorenstein, Finite groups, Harper and Row, New York, 1 968
-
[4]
M.Hall, The theory of groups, McMillan, New York, 1959
work page 1959
-
[5]
J.J.Rotman, An introduction to the theory of groups, 4th ed.Springer, Berlin-New York, 1995
work page 1995
-
[6]
H.Zassenhaus, The theory of groups, Chelsea, New York, 1 958
-
[7]
Vinogradov, I. M. (2003), ” VI Primitive roots and indice s”, Elements of Number Theory, Mineola, NY: Dover Publications, pp. 105–121, ISBN 978-0-486-49530-9
work page 2003
-
[8]
G. H. Hardy and E. M. W right, An introduction to the theory of numbers, Oxford University Press, 1979. ISBN 0198531702
work page 1979
Show all 9 references
-
[9]
Glyn Harman, ”Eratosthenes, Legendre, Vinogradov and b eyond”, Sieve Methods, Exponential Sums, and Their Applications in Number Theory, ed. G. R. H. Greaves, G. Harman, M. N. Huxley; London Mathematical Society Lecture Note Series 237, Cambridge University Pres s (1997) ISBN ...
1997
Reviewed August 12, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.