{"id":"cf251258-155e-4df9-ad42-60e39c20d277","arxiv_id":"2601.10665","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For waves with varied coherence, the supremum and infimum of the coherence-spectrum set give provably optimal single-spectrum bounds for any linear measurement.","lead":"When a wave's coherence can vary within a known set, the tightest universal measurement bounds come from the set's mathematical supremum and infimum spectra. This paper proves that and provides an algorithm to compute those spectra.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central sup/inf optimality argument is sound; its cited inputs (Eq. 11, complete-lattice property) are standard and checkable.","rationale":"The reader identified Eq. (11) and the complete-lattice property as the weakest assumptions. I agree these are the load-bearing premises of the paper, but I do not find them suspect: Eq. (11) is a direct consequence of the Ky Fan/von Neumann trace formula and the partial-sum characterization of majorization, and the complete-lattice property is well established in the cited literature. The paper's proof of Eqs. (27)-(29) is a straightforward and correct application of these premises. The numerical demonstrations are consistent with the theory. I therefore see no significant objection to the central universal-bound claim. The conditional aspects of the reader's verdict — missing convexity statement in the abstract, Eq. (8) interval representation, algorithm details for n>3, and unproven O(N^-2) convergence — are presentation/completeness issues that do not affect the main optimality result, so the verdict should remain unchanged.","tokens_in":19712,"tokens_out":32740,"duration_ms":295757,"concrete_test":"Independently verify Eq. (11) numerically and analytically: for n=3..6, generate random spectra x,y∈Δ↓_n and random Hermitian O, compute the intervals [x·λ↑(O), x·λ↓(O)] and [y·λ↑(O), y·λ↓(O)], and check that interval inclusion for all sampled O occurs iff x≺y. Also re-derive the converse direction from the trace/Hölder inequality without invoking Ref. [43]. A single counterexample would invalidate the optimality of λ_sup/λ_inf; otherwise the central claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I reviewed the proof of the central claim, Eqs. (27)-(29). Given the equivalence Eq. (11) and the completeness of (Δ↓_n, ≺), the optimality of λ_sup and λ_inf follows rigorously: any spectrum yielding a universal outer bound is an upper bound of Λ, hence lies above λ_sup in majorization, so {o}_sup is the smallest such single-spectrum bound; the inner-bound argument is symmetric. The two cited inputs are indeed the load-bearing assumptions. Eq. (11) is not proved in this paper, but it is independently derivable: for any Hermitian O with sorted eigenvalues a, the interval {o}_x is [x·rev(a), x·a], and requiring {o}_x⊆{o}_y for all a is equivalent to the partial-sum inequalities defining x≺y. The complete-lattice property of (Δ↓_n,≺) is a standard result (Refs. [79-81]). I could not identify an internal inconsistency or a counterexample. Secondary issues — the abstract's omission of compactness/convexity in the location theorem, the union-as-interval representation in Eq. (8), and the lack of a rigorous convergence proof for the polygon approximation — do not undermine the main optimality claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a majorization-based theory for bounding linear observables of partially coherent waves when the input coherence spectrum varies over a set Λ ⊆ Δ↓_n. For compact Λ it proves that the exact outer and inner bounds of an observable o = tr(ρO) over Λ are attained by maximal and minimal elements of Λ (Eqs. (21)–(22)). It then introduces the supremum λ_sup and infimum λ_inf of Λ in the majorization order, which exist because (Δ↓_n, ≺) is a complete lattice, and proves that these two spectra give universal, and optimal, single-spectrum bounds: for every Hermitian O, {o}_inf ⊆ ⋂_k {o}_k ⊆ ⋃_k {o}_k ⊆ {o}_sup, and any other spectrum with a universal outer (inner) bound lies above λ_sup (below λ_inf), so the bounds cannot be tightened (Eqs. (27)–(29)). An algorithm for computing λ_sup and λ_inf by polytope bracketing is presented, and the geometric location theorem states that for compact convex Λ with nonempty interior these extrema lie at singular boundary points or outside Λ. Numerical simulations for waveguide transport and n = 3, 4 examples illustrate the results.","tokens_in":20029,"tokens_out":21486,"duration_ms":215116,"significance":"The core optimality result is significant and, given the two cited inputs (Eq. (11) and completeness of the majorization lattice), rigorously established. It replaces scalar coherence measures, which fail for n ≥ 3, with a parameter-free, measurement-independent pair of spectral bounds. The proof of Eqs. (27)–(29) is clean and does not rely on fitting or ad hoc assumptions. The numerical demonstrations are consistent with the theory and include a physically relevant diffusive-waveguide setting. The paper's strengths are its conceptual clarity and the fact that the key optimality claims are checkable from standard majorization facts.","major_comments":[],"minor_comments":[{"comment":"The location theorem is proved in Appendix G under the assumptions that Λ is compact, convex, and has nonempty interior (and n ≥ 3), but the abstract and introduction state it without these qualifications. Please restrict the claim or state the assumptions clearly where the theorem is advertised.","section":"Abstract and Sec. VI"},{"comment":"The union and intersection of the intervals {o}_k are written as single intervals. This is true because every {o}_k contains tr(O)/n, so the union and intersection are themselves intervals with endpoints attained by compactness. A sentence explaining this would prevent an apparent gap.","section":"Sec. II, Eqs. (8)–(9)"},{"comment":"The algorithm is called convergent with an O(N^{-2}) trend, but Appendix F only presents numerical fits; no proof is given for general compact convex sets. Please either provide a proof for the claimed convergence rate or soften the language to numerical evidence.","section":"Sec. V and Appendix F"},{"comment":"The algorithm is described and demonstrated for two-dimensional convex hulls in Δ↓_3. The paper's scope is n-mode waves; please clarify the extent to which the algorithm generalizes to n > 3.","section":"Sec. V"},{"comment":"In Proposition 2, 'singular boundary point' is defined via the normal cone after assuming convexity. Please make that definition and the convexity assumption explicit in the main-text statement of the location theorem.","section":"Appendix G"},{"comment":"Equation (11) is load-bearing for the central optimality argument. It is cited to Ref. [43]; a short proof or derivation in an appendix would make the paper more self-contained. This is a presentation issue rather than a correctness concern.","section":"Sec. III, Eq. (11)"}],"recommendation":"minor_revision","confidential_remarks":"The only substantive weakness is the gap between the claimed convergence analysis of the algorithm and the numerical evidence in Appendix F; this can be fixed without changing the central theorem. Equation (11) comes from the authors' own earlier work, but it is independently derivable and is not a circularity concern. The abstract's omission of the convexity/compactness assumptions for the location theorem should be corrected before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a genuine result, not a repackaging. The central claim—that the majorization supremum and infimum of a set of coherence spectra give optimal universal bounds for all Hermitian measurements—is proved correctly. The two load-bearing inputs, Eq. (11) and the completeness of the majorization lattice, are standard and independently checkable, so the construction rests on firm ground. I checked the logic in Eqs. (27)–(29) and the geometric location propositions in Appendix G; they are coherent.\n\nWhat's actually new: the framing of universal outer/inner bounds via sup/inf spectra, the optimality proof, and the location theorem. The entropy counterexample for n≥3 is nice pedagogically and shows why scalar orders fail. The numerical demonstrations match the theory, which is good to see, though they are illustrative rather than confirmatory.\n\nSoft spots, in order of importance:\n\n1. The abstract overstates the location theorem. The theorem requires compactness, convexity, and nonempty interior; the abstract drops those assumptions. That's a fix, not a fatal flaw.\n\n2. Eq. (8) writes the union of ranges as a single interval. For connected Λ and continuous dependence this is fine, but the paper doesn't state the condition. If Λ is disconnected or non-convex, the union need not be an interval; the outer bound should be the convex hull of the union. The main inclusions still work if you interpret \"outer bound\" as the smallest interval covering the union, but the notation should be clarified.\n\n3. The algorithm for computing sup/inf of an infinite set is only validated for n=3. Appendix E explicitly constructs inscribing/circumscribing polygons for 2D sets in Δ_3. For n>3, there is no analogous algorithm. The paper should either extend the algorithm or explicitly scope the claim.\n\n4. No code or data accompanies the paper. The convergence claim O(N^-2) is empirical. A short code release or a rigorous convergence proof for the polygon approximation would remove the last bit of doubt.\n\nNone of these undercuts the main theorem. The paper is an honest, careful application of known majorization tools to a physical problem, and the result is useful for anyone designing systems with partially coherent sources. I would send it to peer review; it deserves a serious referee.","headline":"The sup/inf construction is correct and the optimality argument holds; the paper's real value is turning per-measurement optimization into two universal spectra, with a few scoping and presentation issues to fix.","tokens_in":20484,"tokens_out":2260,"would_cite":true,"duration_ms":23628,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"For any set of partially coherent waves, two extremal coherence spectra—the set's supremum and infimum under the majorization order, which may lie outside the set—give optimal universal bounds: every measurement's achievable range is squeez","keywords":["coherence spectra","majorization","universal bounds","partial coherence","supremum and infimum","wave coherence","Hermitian measurements","complete lattice"],"falsifier":"Perform a dense random search over Hermitian measurement operators O for the n=3 elliptical-disk set of Fig. 3, evaluating the exact achievable range {o}_k for every element k of a fine grid inside the set. If any measured value lies outside the supremum interval [λ_sup·λ↑(O), λ_sup·λ↓(O)]—or, for the inner bound, if the infimum interval fails to be contained in the intersection of all achievable ranges—then Eq. (27) is false. The paper's examples show agreement but do not exhaust the search space.","tokens_in":19613,"feed_emoji":"🌊","tokens_out":5490,"duration_ms":48559,"temperature":0.7,"pith_summary":"The paper asks how to bound the physical response of a wave when its coherence can vary over a set of possible spectra. It shows that scalar measures like entropy fail for three or more modes, because the majorization order is only partial, so there is generally no single most- or least-coherent element. The key move is to relax the search to the whole order: the set's supremum and infimum—the least upper and greatest lower bounds under majorization—exist uniquely even when they are not themselves achievable spectra. For every linear measurement, the achievable range of any member of the set is contained in the supremum range and contains the infimum range; and any spectrum that claims to give universal bounds is dominated by (or dominates) these extrema, so the bounds are the tightest possible from one spectrum. A reader should care because this gives a measurement-independent certificate: two spectra summarise the worst- and best-case response of an entire family of waves.","feed_headline":"Two extreme coherence spectra set the tightest universal bounds","feed_subtitle":"The sup and inf of a coherence set—even when outside it—govern worst- and best-case responses for any linear measurement.","key_machinery":"The central object is the majorization order on coherence spectra: for ordered probability vectors x,y ∈ Δ↓_n, x ≺ y iff all partial sums of x are ≤ those of y. The load-bearing identity is Eq. (11): λ↓(ρ1) ≺ λ↓(ρ2) iff, for every Hermitian O, the range of possible values of tr(ρ O) for ρ1 is a subset of that for ρ2. The paper exploits the fact that (Δ↓_n, ≺) is a complete lattice, so any subset Λ has a unique least upper bound (supremum) and greatest lower bound (infimum), even if these lie outside Λ. These two spectra then serve as single 'most coherent' and 'least coherent' stand-ins that dominate or are dominated by every element, allowing universal bounds for all measurements.","core_discovery":"The central claim is that, for any compact set Λ of coherence spectra (ordered eigenvalue vectors of density matrices), the supremum λ_sup and infimum λ_inf of Λ in the majorization order satisfy for every Hermitian measurement O: {o}_inf ⊆ ⋂_k {o}_k ⊆ ⋃_k {o}_k ⊆ {o}_sup, where {o}_k is the interval of achievable values for spectrum k. Moreover, if any spectrum λ_u yields universal outer bounds, then λ_sup ≺ λ_u, and if λ_l yields universal inner bounds, then λ_l ≺ λ_inf; hence no single spectrum can produce tighter universal constraints. The proof hinges on the equivalence between majorization and containment of achievable ranges for all measurements (Eq. 11), and on the complete-lattice p","pith_inferences":["The same machinery should apply to any resource theory where a partial order captures universal constraints on linear functions—for example, entanglement or thermodynamic resource theories—so the sup/inf of a set in the appropriate order could yield optimal universal bounds there too.","A testable extension: for a fixed measurement O, the gap between the supremum bound and the exact outer bound measures how much is lost by insisting on a single spectrum; the paper's examples suggest the gap can be small, and it would be interesting to bound this gap as a function of the geometry of Λ.","The paper's claim that sup/inf lie outside smooth Λ implies that for smooth families, no single physically realisable state attains the universal bound; experiments would need to approach it asymptotically, which could be probed by sampling dense subsets of Λ.","The dependence of the algorithm's convergence rate on boundary smoothness (O(N^-2) in examples) might be generalised to higher dimensions, where inscribing/circumscribing polytopes would have more complex convergence behaviour."],"forward_implications":["For any family of partially coherent waves, one can precompute the supremum and infimum spectra once and then bound any observable without re-optimising per measurement.","Scalar measures such as entropy are insufficient for n≥3; the majorization-based sup/inf replace them as the correct notion of 'most/least coherent' for bounding purposes.","The tightness result means the universal bounds cannot be improved by any single spectrum; improvements would require abandoning single-spectrum bounds or using more than one spectrum.","The geometric classification tells designers that for smooth sets of coherence spectra, the extremal bounding spectra are not physically realisable inputs—they are virtual limits—so exact worst/best cases are approached, not attained, within the set.","The algorithm gives a practical route to compute these bounds for closed infinite sets via convex hull and polygon approximations with O(N^-2) convergence."],"fun_headline_variants":["Coherence spectra extremes set optimal universal bounds","Sup and inf coherence spectra: the tightest universal constraints","Waves with varied coherence: extreme spectra bound all measurements","Single spectra can't beat sup and inf for universal bounds"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole construction rests on the premise that one coherence spectrum majorizes another exactly when every linear measurement's achievable range for the first is contained in that for the second; if that equivalence has any exception, the universal bounds and their optimality claim collapse.","fun_headline_variants_meta":{"raw":{"variants":["Coherence spectra extremes set optimal universal bounds","Sup and inf coherence spectra: the tightest universal constraints","Waves with varied coherence: extreme spectra bound all measurements","Single spectra can't beat sup and inf for universal bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1103,"prompt_tokens":639,"completion_tokens":464,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":383,"completion_tokens_details":{"reasoning_tokens":412}},"tokens_in":383,"tokens_out":464,"duration_ms":5684,"temperature":1.0,"reasoning_tokens":412,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T10:14:18.241808+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Perform a dense random search over Hermitian measurement operators O for the n=3 elliptical-disk set of Fig. 3, evaluating the exact achievable range {o}_k for every element k of a fine grid inside the set. If any measured value lies outside the supremum interval [λ_sup·λ↑(O), λ_sup·λ↓(O)]—or, for the inner bound, if the infimum interval fails to be contained in the intersection of all achievable ranges—then Eq. (27) is false. The paper's examples show agreement but do not exhaust the search space.","supporting_citations":[],"review_version":1}