{"id":"434f290b-32d6-4996-9fc2-b73b2fcf9164","arxiv_id":"2608.14393","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For two weighted exponential systems in L^2[0,1], the paper characterizes when every interleaving of the two systems remains complete and gives a sign condition on the ratio of the weights that forces every interleaving to be a frame.","lead":"This paper studies what happens when you build a frame by mixing two families of exponential functions, choosing some frequencies from one family and the rest from the other. The authors give a condition under which every possible mixing stays complete, and another condition under which every mixing stays a frame.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof of Theorem 3.7 uses a false inner-product identity for complex weights; the theorem is repairable by conjugating h, but the same error makes Theorem 3.2 false as stated.","rationale":"I read the paper as claiming a complete characterization of woven completeness (Theorem 3.7), a woven-frame theorem (Theorem 3.17), and transfer to regular translates and critical-density Gabor systems. The frame theorem's perturbation argument is plausible, and the operator estimate in Theorem 3.17 can be made rigorous after correcting the norms to |C|^2,|D|^2. The main gap is the systematic mishandling of complex conjugation. In Theorem 3.7 the proof as printed contains a false identity, but the theorem is salvageable by taking p = \\bar h, so the central claim is probably correct. The same error in Theorem 3.2 is not salvageable by a one-line change: the stated condition 'fg constant' is simply wrong for complex generators, as f=g=e_1 demonstrates. Corollary 4.1 also applies Proposition 3.4 without its L^\\infty hypothesis. These are fixable through a careful revision, not a rejection of the central framework, so the reader's CONDITIONAL verdict remains appropriate.","tokens_in":15434,"tokens_out":36787,"duration_ms":332567,"concrete_test":"Re-derive the converse of Theorem 3.7 with p := \\bar h instead of p := h, and verify that orthogonality <h, f e_n>=0 gives \\widehat{p f}(-n)=0 for n\\in I, hence p f \\in \\overline{\\mathrm{span}}^{L^1}\\{e_m: m\\in (-I)^c\\}, and similarly p g \\in \\overline{\\mathrm{span}}^{L^1}\\{e_m: m\\in -I\\}; if this derivation is valid, the theorem's statement is correct and only the proof needs revision. Separately, test Theorem 3.2 with f=g=e_1: since W(f,g,J)=E(e_1) for every J, it is a woven orthonormal basis, although f g = e^{4\\pi i t} is not constant; this falsifies the stated if-and-only-if and confirms that the conjugation error is substantive.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central characterization (Theorem 3.7) is not rigorously proved as written for complex-valued f,g. In the converse, the paper sets p=h and uses the chain 0 = <h, f e_n> = <pf, e_n> = <pf, e_{-n}>. With the stated inner product, <h, f e_n> = \\int h \\overline{f e_n} = \\widehat{h\\bar f}(n), so the chain is only correct for real-valued f, or after replacing h by \\bar h and tracking conjugate Fourier coefficients. The theorem statement itself appears to survive: choosing p = \\bar h, the same Fourier-coefficient argument yields exactly the stated spectral-splitting condition. The issue is not purely cosmetic, however: Theorem 3.2 makes the same fg versus f\\bar g substitution and is false in the stated complex form; for f=g=e_1, W(f,g,J)=E(e_1) is a woven orthonormal basis for every J, yet fg=e^{4\\pi i t} is not constant. Corollary 4.1 additionally invokes Proposition 3.4 without verifying its L^\\infty hypothesis. A revision that states the conjugation convention, corrects the p=\\bar h step, and fixes Theorem 3.2 and Corollary 4.1 is needed before the results are fully trustworthy.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies woven systems of weighted exponentials W(f,g)={\n  {f e_n}_{n in J} union {g e_n}_{n in J^c} : J subset Z\n} in L^2[0,1]. The main claims are: a complete characterization of when every weaving is complete (Theorem 3.7); a sufficient condition for every weaving to be a frame, in fact a Riesz basis, when E(f) and E(g) are frames and g/f has a constant sign (Theorem 3.17); and a characterization of when every weaving is an orthonormal basis (Theorem 3.2). The paper also provides counterexamples and transfers the results to systems of regular translates and critical-density Gabor systems. The central idea is to reduce the approximation properties of all weavings to spectral-splitting conditions on the Fourier coefficients of pf and pg for a weight p, and to use operator-factorization arguments for the frame result.","tokens_in":15712,"tokens_out":35509,"duration_ms":284450,"significance":"If correct, Theorem 3.7 is a strong and essentially parameter-free spectral characterization of woven completeness for weighted exponential systems, and Theorem 3.17 gives a clean sufficient condition for woven frames from sign consistency of the ratio of the generators. The counterexamples are instructive and the applications to translates and Gabor systems are natural. However, the manuscript as written contains systematic errors in complex-valued inner-product manipulations. These errors affect the proof of the central characterization and make Theorem 3.2 false as stated. The results are likely repairable, but a careful revision is required before the paper can be considered trustworthy.","major_comments":[{"comment":"Theorem 3.2 is false as stated for complex-valued f and g. With the paper's inner product, cross-orthogonality of a woven orthonormal basis gives int f \\bar{g} e^{2pi i (m-n)t} dt = 0, not int f g e^{2pi i (m-n)t} dt. Consequently the 'only if' direction concludes that fg is constant, which is not a necessary condition. For example, take f=g=e_1. Then W(f,g,J)=E(e_1) for every J, so W(f,g) is a woven orthonormal basis, yet |f|=|g|=1 and fg=e_2 is not constant. The correct necessary and sufficient condition is that f/g (equivalently f\\bar{g}) is constant a.e.; the proof must be corrected accordingly.","section":"Section 3.2, Theorem 3.2"},{"comment":"The proof of the central characterization uses the identity \\langle p, f e_n\\rangle = \\langle pf, e_n\\rangle. Under the stated inner product, \\langle p, f e_n\\rangle = \\widehat{p\\bar{f}}(n), whereas \\langle pf, e_n\\rangle = \\widehat{pf}(n). These are not equal for complex-valued f and p. The argument works if one sets p = \\bar{h} in the converse (and h = \\bar{p} in the forward direction) and tracks the conjugated Fourier coefficients; the statement of the theorem itself appears to survive that change. As written, however, both directions of the proof are invalid. Since Theorem 1.2 and Examples 3.9 and 3.15 rely on Theorem 3.7, this is a load-bearing issue that must be fixed.","section":"Section 3.3, proof of Theorem 3.7"},{"comment":"Theorem 2.2(b) states that the biorthogonal system of E(f) is E(1/f). For the standard inner product, the biorthogonal sequence is E(1/\\bar{f}), since \\langle f e_m, (1/\\bar{f}) e_n\\rangle = \\delta_{mn}. The stated version holds only for real-valued f. Lemma 3.11 repeats the same error when it identifies W(1/f,1/g) as the duality partner of W(f,g); the correct reciprocal is 1/\\bar{f}, 1/\\bar{g}. Because Lemma 3.11 is used in the proof of Proposition 3.12 and Corollary 3.14, those results also need to be revisited for complex-valued generators.","section":"Section 2, Theorem 2.2(b); Section 3.3, Lemma 3.11"},{"comment":"Corollary 4.1 invokes Proposition 3.4 to conclude woven completeness of W(Phi_f, Phi_h). Proposition 3.4 has an explicit L^infty hypothesis, and the fiberizations Phi_f and Phi_h need not be bounded under the assumptions of the corollary (only Phi_g \\neq 0 a.e. is assumed). Either an L^infty assumption must be added, or a version of Proposition 3.4 valid for unbounded nonnegative weights must be proved. Additionally, Proposition 3.4 is stated without proof, and the indicated proof (by analogy with the second half of the proof of Theorem 3.2) inherits the fg versus f\\bar{g} problem for complex generators.","section":"Section 4.1, Corollary 4.1"}],"minor_comments":[{"comment":"The product is fg = e^{2pi i (1/5+1/6)t} = e^{2pi i (11/30)t}, not e^{2pi i t/30}. The conclusion that fg is not constant remains valid.","section":"Example 3.5"},{"comment":"The scaling argument at the beginning uses 'min(C,D)' and 'max(C,D)' for general nonzero scalars; the frame inequalities should use |C|^2 and |D|^2. Also, the sentence 'By choosing C large enough that \\|g/(Cf)\\|_\\infty<1, we may assume that \\|g/f-1\\|_\\infty<1' is incorrect: making g/(Cf) small does not make g/f close to 1. The correct scaling is to replace f by f/C with C chosen so that Cg/f lies between 1-epsilon and 1+epsilon.","section":"Proof of Theorem 3.17"},{"comment":"The notation F is ambiguous: Section 2 defines F as the Fourier transform on L^2(R), while the proof of Theorem 3.17 uses F to denote the Fourier-series transform on [0,1] (elsewhere denoted by S). The norm bound \\|F M_{(g-f)/f} F^{-1} P_{J^c}\\| \\le \\|(g-f)/f\\|_\\infty is valid for the Fourier-series unitary, but the notation should be clarified.","section":"Proof of Theorem 3.17"},{"comment":"There are several typos: 'eahcn' for 'each' in the proof of Theorem 3.7; 'straightfoward' in Proposition 3.3; 'crtical' in Section 2; 'resuls' in Section 4; 'iff /g' in the abstract should be 'if f/g'. In the proof of Theorem 3.7, the displayed chain should end with \\langle pg,e_{-n}\\rangle, not \\langle pf,e_{-n}\\rangle.","section":"Throughout"},{"comment":"The claim \\tau f \\in \\overline{\\mathrm{span}}\\{E(f,2Z)\\} cites the preprint [21], which is a self-citation. Since this is a nontrivial step, the authors should either provide a proof or confirm that the preprint is publicly available and accepted.","section":"Example 3.16"}],"recommendation":"major_revision","confidential_remarks":"The recurring fg versus f\\bar{g} conjugation error appears in several places: Theorem 2.2(b), Theorem 3.2, Lemma 3.11, and the proof of Theorem 3.7. This suggests a systematic notational or conceptual slip rather than an isolated typo, and the editor may want to ask the authors for a comprehensive correction. The self-citation [21] in Example 3.16 is used for a load-bearing claim and is worth checking before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the quick take.\n\nThe paper asks a good question—when do arbitrary interleavings of two weighted exponential systems stay complete or stay frames—and the headline frame result (Theorem 3.17) is genuine: if E(f) and E(g) are frames and g/f has constant sign, then every weaving is a Riesz basis. The operator proof is short and works. The characterization in Theorem 3.7 is the right thing to aim for, and its statement appears to be correct.\n\nBut there is a systematic sign bug. The paper repeatedly uses ⟨p, f e_n⟩ = ⟨pf, e_n⟩, which under the stated inner product is only true when f is real-valued. The correct expression is \\widehat{p\\bar f}(n). This is not cosmetic: Theorem 3.2 is false as stated (f=g=e_1 gives a woven ONB while fg is not constant), and Proposition 3.4 is also false as stated (f=e_1, g=e_{-1} has fg=1>0, but the weaving with J={0} misses e_{-1}). Theorem 3.7's proof as written is invalid for complex-valued generators, though the statement survives if you conjugate the annihilator. The stress-test identifies Theorem 3.2 and Theorem 3.7, but it misses Proposition 3.4.\n\nThe abstract also overstates the frame result: it omits the framedness assumption on E(f), E(g). And the translate applications have gaps: Corollary 4.1 cites Proposition 3.4 without the L∞ hypothesis, and Corollary 4.2's hypothesis is written as an L∞ norm bound, which does not by itself give the pointwise lower bound a frame needs.\n\nBottom line: the core idea is sound and the paper is worth a serious referee, but right now the complex-valued statements need a careful revision. I would not cite Theorem 3.2 or Proposition 3.4 in their current form; after the fg vs f\\bar g fix, the paper would be a solid specialized contribution.","headline":"The paper has a good question and a correct frame theorem, but a systematic fg vs f\\bar g bug makes Theorem 3.2 and Proposition 3.4 false as stated and breaks the proof of Theorem 3.7.","tokens_in":16213,"tokens_out":26150,"would_cite":false,"duration_ms":197615,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["42C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two weighted exponential systems are wovenly complete exactly when their spectra cannot be split into complementary halves.","keywords":["weighted exponentials","woven frames","woven completeness","Gabor systems at critical density","regular translates","Riesz bases","Fourier coefficients","spectral splitting"],"falsifier":"Take f(t)=$e^{{2pi i t}}$, p(t)=$e^{{-2pi i t}}$, and n=0. Then <p, f e_0> = $integral_0^{1}$ $e^{{-2pi i t}}$ overline{$e^{{2pi i t}}$} dt = 0, while <pf, e_0> = $integral_0^{1}$ $e^{{-2pi i t}}$ $e^{{2pi i t}}$ dt = 1, so the identity used in the proof of Theorem 3.7 fails for complex-valued generators. A corrected theorem must either restrict f and g to real-valued functions or replace that step; testing any complex-valued pair against the spectral-splitting condition would show whether the stated characterization still holds.","tokens_in":15241,"feed_emoji":"🔀","tokens_out":9668,"duration_ms":82906,"temperature":0.7,"pith_summary":"This paper asks when every interleaving of two systems of weighted exponentials in $L^{2}$[0,1] keeps the same approximation property as the original systems. It proves a complete characterization of woven completeness: the two generators must be nonzero almost everywhere, and there must be no nonzero weight p whose products pf and pg have Fourier spectra confined to complementary index sets. It also proves that if both systems are frames and the ratio g/f has a constant strict sign, then every interleaving is a frame, indeed a Riesz basis. The results transfer to systems of regular translates and to Gabor systems at critical density, and several counterexamples show that natural-looking sufficient conditions are not necessary.","feed_headline":"Woven exponentials: completeness reduces to a spectral-splitting test","feed_subtitle":"A new theorem decides when every mix of two weighted exponential systems stays complete in L²[0,1].","key_machinery":"The weaving W(f,g) is the collection of sets {f e_n}_{n in J} union {g e_n}_{n in J^c} over all J subseteq Z; the question is whether every such set spans $L^{2}$[0,1]. The completeness characterization is carried by the Fourier-coefficient map on $L^{1}$: if a product pf lies in the closed span of a family of exponentials, its Fourier coefficients vanish on the complementary index set, and Lemma 3.6 converts span membership into coefficient vanishings. The frame result is carried by writing the synthesis operator of a weaving as M_f $S^{{-1}}$(I + (M_f $S^{{-1}}$)^{-1} M_{g-f} $S^{{-1}}$ P_{J^c}) and controlling the perturbation by ||(g-f)/f||_infty < 1, which forces every weaving to be a Riesz basis. A reciprocal-weight duality (Lemma 3.11) connects woven completeness of W(f,g) to that of W(1/f,1/g).","core_discovery":"The central theorem (Theorem 3.7) states that W(f,g) is wovenly complete in $L^{2}$[0,1] if and only if f and g are nonzero almost everywhere and no nonzero p in $L^{2}$[0,1] admits a subset J of Z with pf in the $L^{1}$-closed span of {e_n}_{n in J} and pg in the $L^{1}$-closed span of {e_n}_{n in J^c}. In other words, woven completeness fails exactly when some weighted combination of the two generators splits the exponential spectrum into two complementary halves. A second theorem (Theorem 3.17) says that when E(f) and E(g) are frames and g/f is strictly positive or strictly negative almost everywhere, every weaving is a frame — and the argument shows it is a Riesz basis. The paper further shows that disjoint Fourier spectra for f and g imply the woven system is not wovenly complete, and that woven completeness is equivalent to woven $\\ell^2$-minimality when the constituent systems are frames.","pith_inferences":["The spectral-splitting obstruction suggests a quantitative refinement: if the overlap between the spectra of pf and pg is bounded below rather than merely nonzero, one might expect uniform frame bounds across all weavings; this is a natural next question the paper does not address.","Because the proof of Theorem 3.7 as written identifies inner products with Fourier coefficients without conjugation, a repaired version likely needs f and g real-valued; the applications to regular translates produce real generators, so the scope of the result may survive intact there.","The L^1 closure in the characterization is worth probing: replacing L^1 by L^p in the span condition would test how much of the theorem is genuinely about Fourier-coefficient vanishings rather than about the ambient norm.","Example 3.5 shows the strict-sign condition of Theorem 3.17 is sufficient but not necessary, so a weaker geometric condition on arg(g/f) may characterize woven frames."],"forward_implications":["If f and g have disjoint Fourier spectra (b_f(n)b_g(n)=0 for every n), then W(f,g) is never wovenly complete.","Whenever g/f is strictly positive or strictly negative a.e. and E(f), E(g) are frames, every weaving is not just a frame but a Riesz basis.","For a frame generator f, the pair (f, 1/f) gives a woven Riesz basis.","The same criteria decide when every interleaving of regular translates, or of Gabor systems at critical density, remains complete or a frame.","Under the frame assumption, woven completeness and woven l2-minimality coincide."],"supporting_citations":[{"why":"supplies Theorem 2.2, the characterization of completeness, minimality, Bessel, and frame properties of weighted exponentials that the paper builds on.","marker":"[15]"},{"why":"introduces woven frames and the weaving construction that motivates the paper's questions.","marker":"[2]"},{"why":"Kadets' 1/4 theorem is used in Example 3.5 to exhibit woven completeness when the sign condition fails.","marker":"[17]"},{"why":"gives the shift-invariant subspace criterion used in Corollary 4.1 for systems of regular translates.","marker":"[3]"},{"why":"provides the Zak transform and Feichtinger algebra background used to transfer results to critical-density Gabor systems.","marker":"[13]"},{"why":"supplies the translate-system incompatibility result used in Example 3.16 to build an exact pair whose weaving is not minimal.","marker":"[21]"}],"fun_headline_variants":["Woven exponentials: complete iff spectra never split","Woven completeness fails iff spectra split complementarily","Strict sign of g/f turns every weave into a Riesz basis","Spectral split is the only barrier to woven completeness"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the main theorem relies on a step that treats the product f times p as if no complex conjugation were needed; this is only valid when f is real-valued, an assumption the theorem never states.","fun_headline_variants_meta":{"raw":{"variants":["Woven exponentials: complete iff spectra never split","Woven completeness fails iff spectra split complementarily","Strict sign of g/f turns every weave into a Riesz basis","Spectral split is the only barrier to woven completeness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001395,"raw_usage":{"total_tokens":5694,"prompt_tokens":1047,"completion_tokens":4647,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":663,"completion_tokens_details":{"reasoning_tokens":4581}},"tokens_in":663,"tokens_out":4647,"duration_ms":30672,"temperature":1.0,"reasoning_tokens":4581,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-27T21:15:15.466738+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take f(t)=$e^{{2pi i t}}$, p(t)=$e^{{-2pi i t}}$, and n=0. Then <p, f e_0> = $integral_0^{1}$ $e^{{-2pi i t}}$ overline{$e^{{2pi i t}}$} dt = 0, while <pf, e_0> = $integral_0^{1}$ $e^{{-2pi i t}}$ $e^{{2pi i t}}$ dt = 1, so the identity used in the proof of Theorem 3.7 fails for complex-valued generators. A corrected theorem must either restrict f and g to real-valued functions or replace that step; testing any complex-valued pair against the spectral-splitting condition would show whether the stated characterization still holds.","supporting_citations":[{"cited_title":"Heil,A Basis Theory Primer, Expanded Edition, Birkh¨ auser, Boston, 2011","cited_arxiv_id":null,"evidence_quote":"supplies Theorem 2.2, the characterization of completeness, minimality, Bessel, and frame properties of weighted exponentials that the paper builds on."},{"cited_title":"Bemrose, P","cited_arxiv_id":null,"evidence_quote":"introduces woven frames and the weaving construction that motivates the paper's questions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Kadets' 1/4 theorem is used in Example 3.5 to exhibit woven completeness when the sign condition fails."},{"cited_title":"de Boor, R","cited_arxiv_id":null,"evidence_quote":"gives the shift-invariant subspace criterion used in Corollary 4.1 for systems of regular translates."},{"cited_title":"Gr¨ ochenig,Foundation of Time-Frequency Analysis, Birkh¨ auser, Boston, 2001","cited_arxiv_id":null,"evidence_quote":"provides the Zak transform and Feichtinger algebra background used to transfer results to critical-density Gabor systems."},{"cited_title":"Operations that are incompatible with certain systems of translates in $L^2(\\mathbb{R})$","cited_arxiv_id":"2508.16529","evidence_quote":"supplies the translate-system incompatibility result used in Example 3.16 to build an exact pair whose weaving is not minimal."}],"review_version":1}