{"id":"28bc7958-1e23-46bb-9fdc-65f2cd3818c5","arxiv_id":"2505.06757","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In Z^2, every solution of a finite-tile convolution equation with periodic right-hand side can be replaced by a periodic solution, whether the solution is a set or integer-valued; this settles the multi-tiling PTC in Z^2 and implies decidability.","lead":"This paper proves three variants of the periodic tiling conjecture, including the hardest case: in the square grid Z^2, every multi-tiling by a finite tile at any level has a periodic multi-tiling. The results also yield algorithms that decide whether certain tiling equations are solvable, a question with roots in Wang's work on decidability.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The §9.4 reduction to finite linear predicates uses a false density claim: the orbit is not dense in the stated torus because Φ(e0)=1, leaving a repairable gap in the proof of Theorem 1.9.","rationale":"The reader correctly identified the structure theorem and slicing lemma as the load-bearing foundation, and no counterexample to the main theorem was found. However, a careful reading of the final rationalization step in §9.4 reveals a concrete false assertion about the orbit being dense in the full torus; the stated predicates may not contain the real solution Φ. This is a real gap in the written proof of Theorem 1.9, though it is repairable by removing the redundant e0 coordinate. A second, less central defect is the overstrong 'linearly independent' wording in Theorems 3.2 and 3.4, whose proofs only establish pairwise non-collinearity; all subsequent uses appear to require only the pairwise property, so this does not undermine the main result. Because the paper's central claim is very likely correct but the manuscript as written contains these issues, a conditional acceptance is appropriate.","tokens_in":37224,"tokens_out":61677,"duration_ms":590183,"concrete_test":"Rewrite §9.4 by replacing (R/Z)^{d+1}×(R/Z)^{d+1} with (R/Z)^d×(R/Z)^d and omitting the i=0 term from F_{x0} and S'_{w,b}; verify that the real injective Φ satisfies the corrected predicates and that Proposition 7.6 still applies. Separately, run the construction of Theorem 3.2 with f=δ_0+δ_{e1}+δ_{e2}+δ_{e1+e2} and a generic bounded integer solution a: if the resulting W is {e1,e2,e1+e2}, the theorem's 'linearly independent' wording must be weakened to 'pairwise non-collinear'.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 1.9 hinges on the reduction in §9.4 of the infinite constraints (9.6) and (9.12) to locally definable predicates. The argument claims that the orbit {((y1Φ(e_i) mod 1)_{i=0}^d, (y2Φ(e_i) mod 1)_{i=0}^d) : y1,y2∈Z} is dense in (R/Z)^{d+1}×(R/Z)^{d+1}. This is false: since Φ∈Hom_1(Q^{d+1},R), one has Φ(e0)=1, so the i=0 coordinates are identically 0 mod 1, and the orbit is dense only in the subtorus with r0=s0=0, i.e. (R/Z)^d×(R/Z)^d. Consequently the predicates Λ_{x0} and Λ'_{w,b}, as written with universal quantification over all r0,...,rd,s0,...,sd, need not contain the actual injective Φ; the spurious i=0 term in F_{x0} depends on free real variables that never occur in the original tiling constraint. Proposition 7.6 would then be applied to a predicate not known to be satisfied by Φ. The repair is direct: drop the e0 coordinate (equivalently fix r0=s0=0) and work on the d-dimensional torus; the density claim and the local definability then go through. As written, however, this is a genuine gap in the central proof. A related statement issue is that Theorem 3.4(i) asserts W is linearly independent, while the proof of Theorem 3.2 only yields pairwise non-collinear primitive directions (e.g. {e1,e2,e1+e2}); the later arguments appear to use only pairwise non-collinearity, so this should be corrected rather than being fatal.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes three variants of the periodic tiling conjecture. First, for f*a=0 with f finitely supported and a bounded integer-valued, existence of a nonzero integer solution implies existence of a nonzero periodic integer solution (a result attributed to Tim Austin), together with decidability of this problem via Theorem 1.3, vanishing sums of roots of unity, and Szmielew's decidability theorem. Second, in Z^2, every bounded integer-valued solution to f*a=g with g periodic can be replaced by a periodic integer-valued solution (Theorem 1.8). Third, in Z^2, if f*1_A=g has an indicator-function solution with g periodic integer-valued, then it has a periodic indicator-function solution (Theorem 1.9), in particular settling the periodic tiling conjecture for multi-tilings of Z^2 at constant level. The proof uses the dilation lemma and a structure theorem that decomposes solutions into a periodic part plus finitely many singly periodic parts, followed by a rationalization procedure: real parameters occurring in an equidistributed description are replaced by rational parameters using locally definable linear predicates and a nondegeneracy result (Proposition 7.6).","tokens_in":37569,"tokens_out":25268,"duration_ms":243093,"significance":"If the proofs are completed as intended, this is a major contribution to the periodic tiling conjecture and to the decidability of tiling problems. Theorem 1.9 goes substantially beyond the known k=1 case in Z^2 (Bhattacharya; Greenfeld--Tao) by covering multi-tilings and periodic right-hand sides, despite the existence of non-weakly-periodic higher-level tilings. The paper also gives a clean decidability result for f*a=0 in all finitely generated abelian groups, and for multi-tilings in Z^2. The methods are notable: the systematic use of the dilation lemma and structure theorems, the retraction to Q, and the model-theoretic/local-definability rationalization in Section 7 are elegant and are likely to be reused. The proofs are detailed and the imported tools (Mann's theorem, Szmielew's theorem, Krylov--Bogolyubov, Weyl equidistribution) are cited. The main caveat is the false equidistribution claim in Section 9.4, which is localized and repairable but currently leaves a gap in the proof of Theorem 1.9.","major_comments":[{"comment":"The proof that the injective map Φ lies in the predicates Λ_x0 and Λ'_{w,b} uses a false density claim. The text states that the orbit {((y1 Φ(e_i) mod 1)_{i=0}^d, (y2 Φ(e_i) mod 1)_{i=0}^d) : y1,y2 ∈ Z} is dense in (R/Z)^{d+1}×(R/Z)^{d+1}, and similarly that {(Φ(e_i) m mod 1)_{i=0}^d : m ∈ Z} is dense in (R/Z)^{d+1}. Since Φ ∈ Hom_1(Q^{d+1},R), we have Φ(e_0)=1, so for integer y1, y2, m the i=0 coordinates are identically 0 mod 1. The correct statement is that the first orbit is dense in {0}×T^d × {0}×T^d and the second in {0}×T^d. Consequently the predicates Λ_x0 and Λ'_{w,b}, as defined with unrestricted r_0,s_0,p'_0, are not shown to contain Φ, and the application of Proposition 7.6 to these predicates is not justified. This is a load-bearing gap in the proof of Theorem 1.9, because the final rationalization step requires a locally definable predicate satisfied by Φ. The repair is direct: drop the e0 coordinate (equivalently fix r_0=s_0=0 and p'_0=0) and work on the d-dimensional torus; then the orbits are dense in T^d×T^d and T^d respectively, and the subsequent local-definability and quantifier-elimination arguments go through. The manuscript should be revised to make this restriction explicit in equations (9.15) and (9.17) and in the definitions of Λ_x0 and Λ'_{w,b}.","section":"§9.4"}],"minor_comments":[{"comment":"The wording 'linearly independent primitive elements' is incorrect for Z^2: the proof of Theorem 3.2 only yields pairwise non-collinear primitive directions, and Corollary 8.6(ii) allows three elements such as {e1,e2,e1+e2} in W. The later arguments use only pairwise non-collinearity, so the statement should be corrected to 'pairwise non-collinear' rather than 'linearly independent'.","section":"Theorem 3.4(i) / Definition 3.3"},{"comment":"In the definitions of F_x0 and S'_{w,b,a,x}, the notation \\tildeΦ(α_{w,x0,i}) is undefined because α_{w,x0,i} is a scalar and \\tildeΦ is defined on Q^{d+1}; these occurrences should be α_{w,x0,i} \\tildeΦ(e_i).","section":"§9.4"},{"comment":"The question states that the periodic solution a_p should lie in ℓ∞(Z^2,Z)_p, but the context is Z^d; this should be ℓ∞(Z^d,Z)_p.","section":"Question 1.12"},{"comment":"After the main density repair is made, the sentence identifying the torus with [0,1)^{d+1}×[0,1)^{d+1} should be changed to [0,1)^d×[0,1)^d (and correspondingly in the treatment of (9.17)), to match the reduced coordinates.","section":"§9.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is strong and the main results are very likely correct, but the proof of Theorem 1.9 as written contains a genuine gap in Section 9.4: the equidistribution claim is false as stated because the e0 coordinate is constant modulo 1. I agree with the stress-test note that the repair is direct — drop the e0 coordinate and work on the d-dimensional torus — but the authors must actually rewrite the definitions of Λ_x0 and Λ'_{w,b} and the surrounding density argument. The 'linearly independent' wording in the structure theorem should also be corrected to 'pairwise non-collinear'; this is not fatal because only pairwise non-collinearity is used. I would not reject; the issues are localized and fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this. The paper proves three genuinely new variants of the periodic tiling conjecture, most notably Theorem 1.9: any multi-tiling of Z^2 at periodic level (f*1_A = g with g periodic, f finitely supported, A a set) has a periodic multi-tiling counterpart. That settles the level-k PTC in Z^2 and, via the usual compactness argument, makes multi-tilings decidable. Along the way there are two other solid results: Theorem 1.3 (due to Austin, proved in full here) characterizing nonzero integer solutions to f*a=0 by vanishing finite-order Fourier coefficients, and Theorem 1.8 giving periodic integer-valued solutions for arbitrary periodic right-hand sides in Z^2. The proofs are detailed and use the dilation/structure machinery from earlier works, extended with a slicing lemma and a model-theoretic rationalization lemma. This is real mathematics, carefully written.\n\nThe soft spots are real but not fatal.\n\n1. In §9.4 the proof of Theorem 1.9 claims the orbit {((y1 Φ(e_i) mod 1),(y2 Φ(e_i) mod 1)) : y1,y2 ∈ Z} is dense in (R/Z)^{d+1} × (R/Z)^{d+1}. It isn't: Φ(e0)=1, so the 0-th coordinates are identically 0 mod 1. The orbit only fills the subtorus r0=s0=0. As a consequence the locally defined predicates Λ_{x0} and Λ'_{w,b} with universal quantification over all r_i,s_i are not known to contain the actual Φ; the paper does not justify that Φ satisfies them. This is a genuine gap in the main proof. The repair is direct: drop the e0 coordinate (fix r0=s0=0) and repeat the equidistribution argument on (R/Z)^d. The original constraints don't involve free r0/s0. I didn't find any other issue in the rationalization step.\n\n2. Minor statement issue: Theorem 3.4(i) says W is linearly independent, but the proof only gives pairwise non-collinear primitive directions; in Z^2 the equidistributed case later uses three directions, which cannot be linearly independent. The arguments only need pairwise non-collinearity, so this is a wording problem, not a mathematical one.\n\nThe citation pattern is fine; the new results are clearly distinguished from the literature, and Austin gets proper credit.\n\nWho is this for: anyone working on tilings, decidability, or the structure theory of convolution equations. It deserves a serious referee; I'd send it out and ask the referee to check the §9.4 fix. My own view: after the repair, it's a strong accept.","headline":"A strong, significant paper with a repairable gap in the main proof: the §9.4 density claim is false as stated, but the fix appears straightforward.","tokens_in":38192,"tokens_out":4161,"would_cite":true,"duration_ms":37155,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52C23","03B25","37B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every finite-tile multi-tiling of the square grid has a periodic twin.","keywords":["periodic tiling conjecture","multi-tilings","translational tilings","decidability","finitely generated abelian groups","structure theorem","vanishing sums of roots of unity","rationalization"],"falsifier":"Exhibit one finite subset $F$ of $\\mathbb{Z}^2$ and one periodic integer-valued $g$ for which $f * 1_A = g$ has a set solution $A$ but no periodic set solution; equivalently, find a bounded integer solution $a$ of some $f * a = g$ that cannot be decomposed as $\\tilde g - \\sum \\varphi_w$ with the stated one-directional periodicities and periodic slices. Either example would disprove Theorem 1.9 or Theorem 1.8 and is directly checkable by searching over increasing periods.","tokens_in":36989,"feed_emoji":"🧩","tokens_out":7646,"duration_ms":70181,"temperature":0.7,"pith_summary":"The paper proves three variants of the periodic tiling conjecture, each saying that if a tiling-type equation has any solution at all, it also has one that repeats periodically. The headline result, in the plane grid $\\mathbb{Z}^2$, allows the right-hand side to be any periodic integer function and keeps the objects on the left as indicator functions: whenever $f * 1_A = g$ has a set solution $A$, it also has a periodic set solution $A_p$. This settles the conjecture for multi-tilings of $\\mathbb{Z}^2$ at every level $k$, where $1_F * 1_A = k$. Along the way the paper shows that homogeneous integer convolution equations $f * a = 0$ always have periodic integer solutions when they have any integer solution, and that the solvability of such equations is algorithmically decidable; it also gives the decidability of multi-tilings in $\\mathbb{Z}^2$.","feed_headline":"Every finite-tile multi-tiling of the square grid has a periodic twin.","feed_subtitle":"Three variants of the periodic tiling conjecture are proved, including new decidability results for multi-tilings.","key_machinery":"The engine is a structure theorem (Theorem 3.4) for bounded integer solutions on $\\mathbb{Z}^2$: any solution $a$ of $f * a = g$ decomposes as $a = \\tilde g - \\sum_{w \\in W} \\varphi_w$, where $\\tilde g$ is periodic and each $\\varphi_w$ is periodic along a distinct primitive direction $w$, together with a slicing lemma saying that each convolution $(1_{x+\\langle w\\rangle} f) * \\varphi_w$ is periodic. This is derived from the dilation lemma (Lemma 3.1), which says the identity $f * a = g$ is stable under dilating $f$, i.e. $(\\tau_r f) * a = g$ for $r \\equiv 1 \\bmod q$. A second, model-theoretic ingredient is the rationalization Proposition 7.6: a system of linear inequalities with rational coefficients that has a real solution has a rational solution, provided a non-degeneracy condition holds; this is what converts the real coefficients appearing in the \"equidistributed\" parts of a multi-tiling into rational ones, forcing periodicity.","core_discovery":"The central claim, stated as Theorem 1.9, is that for any finitely supported integer-valued $f$ on $\\mathbb{Z}^2$ and any periodic integer-valued $g$, if $f * 1_A = g$ for some subset $A$ of $\\mathbb{Z}^2$, then $f * 1_{A_p} = g$ for some periodic subset $A_p$. In particular, the periodic tiling conjecture is true for level-$k$ multi-tilings $1_F * 1_A = k$ in $\\mathbb{Z}^2$ for every natural number $k$. Two supporting discoveries extend the same principle to weaker settings: the homogeneous equation $f * a = 0$ over the integers admits a nonzero periodic solution whenever it admits any nonzero bounded integer solution (Theorem 1.3, with a proof supplied by Tim Austin), and the non-homogeneous equation $f * a = g$ over the integers in $\\mathbb{Z}^2$ admits a periodic integer solution whenever it has any bounded integer solution (Theorem 1.8). The paper also derives decidability statements from these results.","pith_inferences":["Because Theorem 1.8 reduces solvability of $f * a = g$ to the existence of a periodic integer solution, the decidability question for integer tilings (Question 1.11) now hinges on bounding the period or sup norm of such a periodic solution; an effective bound would close the problem.","The rationalization Proposition 7.6 has a life beyond tilings: any constraint set that is locally a finite boolean combination of rational linear inequalities admits rational witnesses whenever real witnesses exist, so it could serve as a general tool for turning real-parameter aperiodic constructions into periodic ones.","If the structural picture from Theorem 3.4 extends to $\\mathbb{Z}^d$ for $d \\geq 3$ with periodic right-hand sides, the obstruction to periodicity likely remains concentrated in linear parameters; testing $d = 3$ with a non-constant level function would indicate whether the known high-dimensional failure of the indicator-function conjecture is already visible in the integer-valued setting."],"forward_implications":["There is an algorithm that decides, for any finitely supported $f$ on $\\mathbb{Z}^2$ and periodic integer $g$, whether a set $A$ with $f * 1_A = g$ exists (Corollary 1.10).","There is an algorithm that decides, for any finitely generated abelian group $G$ and finitely supported integer $f$, whether $f * a = 0$ has a nonzero bounded integer solution (Corollary 1.5).","Level-$k$ multi-tilings of $\\mathbb{Z}^2$ by a finite tile $F$ always have periodic counterparts, for every natural number $k$.","In rank-one groups $\\mathbb{Z} \\times H$ with $H$ finite, any integer solution of $f * a = g$ with periodic $g$ can be replaced by a periodic one (Proposition 1.7)."],"supporting_citations":[{"why":"It supplies the proof of Theorem 1.3, the homogeneous integer case that underlies the decidability result.","marker":"[A23]"},{"why":"It established the base periodic tiling conjecture in $\\mathbb{Z}^2$ for level one and supplied the dilation lemma template that the present proofs adapt.","marker":"[B20]"},{"why":"Its structure theorem and slicing lemma for translational tilings are extended here to periodic right-hand sides and multi-tilings.","marker":"[GT21]"},{"why":"It provides the measurable and dilation formulations used in the structure theorem.","marker":"[GGRT23]"},{"why":"Wang's compactness argument converts periodic-solution theorems into decidability of tilings, used for Corollary 1.10.","marker":"[W63]"},{"why":"Szmielew's decidability of the theory of divisible abelian groups makes the root-of-unity criterion for $f * a = 0$ algorithmic.","marker":"[S55]"},{"why":"Its structural result on minimal vanishing sums of roots of unity bounds the order of the characters in the criterion.","marker":"[M65]"}],"fun_headline_variants":["Multi-tilings of Z^2 always have periodic twins","Periodic tiling conjecture holds for multi-tilings","Grid multi-tilings admit periodic solutions","Decidability of multi-tilings from periodic twins","PTC variants: periodic solutions for multi-tilings"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the structure theorem: every bounded integer solution on $\\mathbb{Z}^2$ of $f * a = g$ with fixed periodic $g$ can be written as a periodic function minus finitely many functions each periodic in one direction, with slice convolutions also periodic; if a single solution escaped this description, the periodic-replacement argument would fail.","fun_headline_variants_meta":{"raw":{"variants":["Multi-tilings of Z^2 always have periodic twins","Periodic tiling conjecture holds for multi-tilings","Grid multi-tilings admit periodic solutions","Decidability of multi-tilings from periodic twins","PTC variants: periodic solutions for multi-tilings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000422,"raw_usage":{"total_tokens":2233,"prompt_tokens":1075,"completion_tokens":1158,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":691,"completion_tokens_details":{"reasoning_tokens":1079}},"tokens_in":691,"tokens_out":1158,"duration_ms":9810,"temperature":1.0,"reasoning_tokens":1079,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:34:25.202190+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit one finite subset $F$ of $\\mathbb{Z}^2$ and one periodic integer-valued $g$ for which $f * 1_A = g$ has a set solution $A$ but no periodic set solution; equivalently, find a bounded integer solution $a$ of some $f * a = g$ that cannot be decomposed as $\\tilde g - \\sum \\varphi_w$ with the stated one-directional periodicities and periodic slices. Either example would disprove Theorem 1.9 or Theorem 1.8 and is directly checkable by searching over increasing periods.","supporting_citations":[],"review_version":1}