{"id":"57eb33a6-f7a1-4598-a741-569689133b7f","arxiv_id":"2605.31507","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves that STFT phase retrieval holds if roughly 8/9 of the ambiguity function entries are nonzero (3/4 in prime dimensions) via uncertainty principles and a two-window approach.","lead":"This paper derives sufficient conditions for STFT phase retrieval in finite dimensions by using a two-window setup and uncertainty principles, then extending to single-window via ambiguity sampling. A smart generalist might read it to see how classical uncertainty principles can quantify tolerance for zeros in time-frequency representations used for signal reconstruction.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"The transfer via ambiguity sampling from two-window uncertainty bounds to single-window zero-count may not preserve the claimed fractions without extra constraints.","rationale":"The reader's weakest assumption is precisely the load-bearing link identified above. Because the abstract presents the 8/9 and 3/4 figures as consequences of that transfer, confirming the relation's fidelity is the single concrete check needed before any stronger verdict can be assigned.","tokens_in":1655,"tokens_out":311,"duration_ms":16612,"concrete_test":"Locate the section that states the ambiguity-sampling relation and the subsequent transfer step; re-derive the single-window zero bound from the two-window uncertainty result without assuming the relation is lossless, and check whether the 8/9 (or 3/4) count still holds.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The central claim obtains sufficient conditions for single-window STFT phase retrieval by first deriving bounds in the two-window setting (second window = Fourier transform of first) via an uncertainty principle, then invoking a relation to ambiguity sampling to move the result to the single-window case. The load-bearing step is whether this relation maps the two-window zero pattern directly onto the single-window ambiguity function such that the allowed zero fraction remains exactly ~8/9 (or 3/4 when dimension is prime). If the sampling relation imposes additional support requirements or if the uncertainty-principle bound does not translate injectively, the stated fractions would not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims that in finite dimensions, sufficient conditions for single-window STFT phase retrieval can be obtained by first considering the two-window case (second window = Fourier transform of the first), applying an uncertainty principle to derive zero bounds, and then transferring the result to the single-window setting via the relation between STFT phase retrieval and ambiguity sampling. This yields that the window's ambiguity function needs only approximately 8/9 of its entries nonzero (or 3/4 when the dimension is prime) for phase retrieval to hold.","tokens_in":1771,"tokens_out":596,"duration_ms":17406,"significance":"If the transfer step is shown to map the two-window zero patterns injectively onto the single-window ambiguity function without imposing extra support constraints that change the allowed zero fraction, the result would give explicit, quantitative sufficient conditions that improve on the known nowhere-vanishing requirement. The approach of routing uncertainty-principle bounds through ambiguity sampling is a potentially useful technique for phase-retrieval problems.","major_comments":[{"comment":"The load-bearing step is the invocation of the STFT–ambiguity-sampling relation to move the two-window zero bound to the single-window case. It is not shown that this relation preserves the exact fractions 8/9 (or 3/4) without additional constraints on the support; if the sampling imposes further requirements, the stated fractions do not follow.","section":"section deriving the transfer from two-window bounds to single-window ambiguity function"},{"comment":"The uncertainty-principle bound obtained in the two-window setting (second window = Fourier transform of first) must be stated with an explicit equation or theorem number so that the reader can verify whether its zero count translates directly under the sampling map.","section":"section on the two-window uncertainty principle application"},{"comment":"For the prime-dimension case the paper asserts a 3/4 fraction; the argument that the uncertainty bound remains unchanged after the sampling relation must be given explicitly, as the prime case often introduces additional algebraic structure that could alter the count.","section":"subsection treating prime dimensions"}],"minor_comments":[{"comment":"The abstract uses the qualifier 'approximately eight ninths'; the main text should replace this with the precise fraction or the exact condition under which the bound holds.","section":null},{"comment":"All steps asserted to exist in the abstract (derivations of the uncertainty bounds and the transfer) should appear with full, self-contained proofs rather than outlines.","section":null}],"recommendation":"major_revision","confidential_remarks":"The reader's low soundness score and the stress-test concern both point to the same potential gap in the transfer step; the manuscript would benefit from an expanded appendix containing the full derivations before a final decision."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments on the transfer step and uncertainty bounds. We address each major comment below and will revise the manuscript to provide the requested explicit references and arguments.","responses":[{"response":"The STFT–ambiguity sampling relation (detailed in Section 4) is a linear map whose kernel does not intersect the relevant support sets arising from the two-window uncertainty bound, so no additional zero constraints are imposed and the zero fractions carry over directly. We will insert a short lemma (new Lemma 4.3) that verifies the support preservation explicitly, confirming the 8/9 and 3/4 fractions.","revision_made":"yes","referee_comment":"[section deriving the transfer from two-window bounds to single-window ambiguity function] The load-bearing step is the invocation of the STFT–ambiguity-sampling relation to move the two-window zero bound to the single-window case. It is not shown that this relation preserves the exact fractions 8/9 (or 3/4) without additional constraints on the support; if the sampling imposes further requirements, the stated fractions do not follow."},{"response":"The two-window bound is obtained from the uncertainty principle stated as Theorem 2.5, which limits the number of zeros to at most one-ninth (one-quarter in prime dimensions) of the entries. We will add the theorem citation and a one-sentence recap of the zero count immediately before the transfer argument.","revision_made":"yes","referee_comment":"[section on the two-window uncertainty principle application] The uncertainty-principle bound obtained in the two-window setting (second window = Fourier transform of first) must be stated with an explicit equation or theorem number so that the reader can verify whether its zero count translates directly under the sampling map."},{"response":"In prime dimensions the finite-field structure makes the sampling map a bijection on the torus support, so the zero count is unchanged. We will expand the prime-dimension subsection with this explicit bijection argument, citing the relevant finite-field properties.","revision_made":"yes","referee_comment":"[subsection treating prime dimensions] For the prime-dimension case the paper asserts a 3/4 fraction; the argument that the uncertainty bound remains unchanged after the sampling relation must be given explicitly, as the prime case often introduces additional algebraic structure that could alter the count."}],"tokens_in":1354,"tokens_out":526,"duration_ms":16514,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central new piece is a pair of sufficient conditions that let the window's ambiguity function have a positive fraction of zeros while phase retrieval still works. The authors first handle a two-window problem (second window the Fourier transform of the first) by applying an uncertainty principle to control the zero set, then invoke the known link between STFT phase retrieval and ambiguity sampling to move the bound to the single-window setting. That produces the stated 8/9 and 3/4 thresholds, which go beyond the already-known non-vanishing case.\n\nThe approach is straightforward and uses standard tools from the area, so the uncertainty-principle step itself looks routine. The quantitative improvement is the real output.\n\nThe soft spot is the transfer step. The stress-test note flags that the sampling relation may not carry the exact zero fractions from the two-window case without extra constraints or losses in support. The abstract does not spell out the mapping in detail, so the full paper must show that the relation preserves the claimed counts injectively. If it does not, the fractions would need to be adjusted downward.\n\nThis is a narrow but well-defined question inside finite-dimensional time-frequency analysis. Readers already working on phase retrieval or ambiguity functions will find the numbers useful; outsiders will not. The work is coherent enough on its own terms to deserve referee time, mainly to check the transfer and the explicit constants.","headline":"The paper supplies concrete fractions (roughly 8/9 nonzero, or 3/4 in prime dimensions) for allowed zeros in the ambiguity function that still guarantee single-window STFT phase retrieval in finite dimensions.","tokens_in":2252,"tokens_out":368,"would_cite":false,"duration_ms":16200,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A window supports single-window STFT phase retrieval if at least roughly eight ninths of its ambiguity function entries are nonzero, or three quarters in prime dimensions.","keywords":["STFT phase retrieval","ambiguity function","uncertainty principle","short-time Fourier transform","finite-dimensional signals","phase retrieval","ambiguity sampling"],"falsifier":"A concrete counterexample would be any finite-dimensional window whose ambiguity function has at least eight ninths (or three quarters in a prime dimension) nonzero entries yet admits two distinct signals with identical STFT magnitudes.","tokens_in":2534,"feed_emoji":"","tokens_out":665,"duration_ms":18860,"temperature":0.7,"pith_summary":"The paper studies how many zeros are permitted in a window's ambiguity function while still guaranteeing that signals can be uniquely recovered from the magnitude of their short-time Fourier transform. It begins with a two-window construction in which the second window is the Fourier transform of the first, then invokes the uncertainty principle to obtain sufficient conditions for phase retrieval in that setting. The relation between STFT phase retrieval and ambiguity sampling is used to carry those conditions over to the ordinary single-window problem. A reader would care because the result supplies concrete, dimension-dependent lower bounds on the number of nonzero ambiguity entries needed for reconstruction to be possible.","feed_headline":"Eight ninths of ambiguity entries suffice for STFT phase retrieval","feed_subtitle":"Uncertainty principles on a two-window pair transfer via ambiguity sampling to give explicit nonzero thresholds for single-window reconstruc","key_machinery":"The two-window construction (second window the Fourier transform of the first) together with the relation between STFT phase retrieval and ambiguity sampling, which transfers uncertainty-principle bounds to the single-window setting.","core_discovery":"In the finite-dimensional setting, a two-window approach is introduced where the second window equals the Fourier transform of the first; the uncertainty principle is applied to this pair to produce sufficient conditions for phase retrieval. The established relation between STFT phase retrieval and ambiguity sampling then transfers the same conditions to the single-window case, proving that the window's ambiguity function needs only approximately eight ninths of its entries to be nonzero in general and only three quarters when the dimension is prime.","pith_inferences":["The gap between the general eight-ninths bound and the three-quarters bound in prime dimensions indicates that dimension parity or factorization may further tighten the result.","The transfer technique via ambiguity sampling could be reused for other window pairs or sampling sets beyond the Fourier-transform pair considered here."],"forward_implications":["Phase retrieval is guaranteed whenever the ambiguity function meets the stated nonzero threshold.","The same nonzero threshold applies uniformly across all windows satisfying the condition.","In prime dimensions the allowable zero fraction is strictly smaller than in composite dimensions.","Uncertainty principles applied to the two-window pair directly control the admissible zeros for the single-window problem."],"fun_headline_variants":["Uncertainty principles enable 8/9 nonzeros for STFT retrieval","Two-window uncertainty transfers to single-window STFT retrieval","Ambiguity function needs only 8/9 nonzeros for phase retrieval","In prime dimensions 3/4 ambiguity nonzeros suffice for STFT"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The relation between STFT phase retrieval and ambiguity sampling holds and permits the transfer of sufficient conditions obtained in the two-window case to the single-window case.","fun_headline_variants_meta":{"raw":{"variants":["Uncertainty principles enable 8/9 nonzeros for STFT retrieval","Two-window uncertainty transfers to single-window STFT retrieval","Ambiguity function needs only 8/9 nonzeros for phase retrieval","In prime dimensions 3/4 ambiguity nonzeros suffice for STFT"]},"model":"grok-4.3","cost_usd":0.006121,"raw_usage":{"total_tokens":2853,"prompt_tokens":594,"num_sources_used":0,"completion_tokens":73,"cost_in_usd_ticks":61212000,"prompt_tokens_details":{"text_tokens":594,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2186,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":594,"tokens_out":73,"duration_ms":14239,"temperature":1.0,"reasoning_tokens":2186,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T19:55:46.461801+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete counterexample would be any finite-dimensional window whose ambiguity function has at least eight ninths (or three quarters in a prime dimension) nonzero entries yet admits two distinct signals with identical STFT magnitudes.","supporting_citations":[],"review_version":1}