{"id":"21f9b947-289d-4663-8f47-d159fc8103a5","arxiv_id":"2505.06735","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a conducting cylindrical waveguide, the skyrmion number is preserved during propagation when the transverse electric field is everywhere nonzero, and TE1n/TM1n modes act as topologically stabilizing modes that can enforce this condition.","lead":"This paper shows that optical skyrmions, tiny swirling patterns of light polarization, can keep their topological identity as they travel through a conducting waveguide if the field never vanishes. It introduces a classification of waveguide modes as stabilizing, weakly stable, or unstable, and shows that adding a stabilizing mode can protect other modes from modal dispersion.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central theorem's hypothesis 'E0 everywhere non-zero' is not tied to the propagation coordinate; coefficients that give a non-zero field at z=0 can develop zeros at later z, so the homotopy argument is missing a uniform-in-z condition.","rationale":"In good faith, the paper's homotopy argument is sound when the transverse electric field is non-zero throughout the full propagation cylinder, and the boundary condition in a conducting waveguide does fix the boundary polarization profile. The reader's weakest assumption about quantization via canonical extension is also concerning, but it is repairable by a direct relative-homotopy argument: for the boundary curve on the equator traced twice, the inside skyrmion number is an integer (a reference map gives charge 1, and any two maps with the same boundary differ by an integer). I therefore do not treat quantization as the most load-bearing issue. The genuinely load-bearing gap is that the theorem's hypothesis is stated in terms of coefficients and a single field E0 without any explicit all-z requirement or a quantitative dominance bound. The abstract claims that presence of a stabilizing mode guarantees protection despite modal dispersion, but presence alone does not prevent destructive interference at later z; point 3's '|c| sufficiently large' is the needed condition and it is not quantified or verified in the numerical demonstrations. This is an addressable gap rather than a fatal flaw, and it supports the reader's CONDITIONAL verdict. The concrete test directly checks whether the specific demonstrated superpositions actually satisfy the uniform nonvanishing condition over a beat period; if they do not, the central claim as written is false, and if they do, the paper still needs to state the condition explicitly.","tokens_in":9324,"tokens_out":35117,"duration_ms":372139,"concrete_test":"Take the TE11+TE21 superposition of Fig. 4c (A11=1, B21=1, all other coefficients 0) and compute min_{r,theta,z} |E_t(r,theta,z)| over the disk and over one beat period z in [0, 2*pi/|k11-k21|], where k11 and k21 are the propagation constants of TE11 and TE21 (Eq. 6). Repeat for the 10 random perturbed coefficient sets used in Fig. 4. If the minimum is zero for any case, the theorem as stated is false and the paper must add a uniform-in-z nonvanishing hypothesis or an explicit dominance threshold. If the minimum is always positive, the numerical examples are consistent, but the gap in the stated theorem remains.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's main claim (Section 1, after Eq. 1) is that if the superposition coefficients make E0 everywhere non-zero, then the skyrmion number is preserved along the waveguide. The preceding homotopy argument is valid only if the transverse electric field is non-zero on the whole cylinder (every z), because a zero punctures the domain and lets the integral change. But the hypothesis is phrased in terms of the coefficients and a single field E0 (Eq. 1 has no z-dependence), and no condition on propagation length or coefficient magnitudes is stated. For two modes with different propagation constants, the relative phase changes with z; even if |E(z=0)|>0 everywhere, destructive interference can produce a zero at some z, after which topological protection fails. Point 3 gives a sufficient dominance condition (|c| large) but the abstract and Fig. 4 claims do not state a threshold. The classification of TE1n/TM1n as 'topologically stabilizing' rests on nonvanishing at r=0 at one cross-section, not on a uniform bound over the propagation interval. Thus the central claim is under-specified: either E0 must be assumed non-zero for all z (which is not checked in the numerics), or a dominance bound must be proven.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript claims to establish the first theory of topological protection of optical skyrmions in conducting waveguides. It argues that when the transverse electric field of a waveguide mode superposition is everywhere nonzero, the skyrmion number integral is quantized and remains constant during propagation because propagation is a homotopy of the polarization field. It introduces a classification of solutions into topologically stabilizing, weakly stable, and unstable modes, identifies TE1n and TM1n as candidate stabilizing modes, and supports the claims with numerical phase diagrams for TE11, TE21, and their superpositions, including propagation along z. It further invokes a generalized skyrmion number from an unpublished preprint to show that a multi-valued topological charge can be preserved even when the usual skyrmion number fails, in two numerical examples.","tokens_in":9533,"tokens_out":4224,"duration_ms":40612,"significance":"If correct, this is a useful conceptual step toward using polarization topology in multimode waveguides, and the proposed classification of modes could guide experimental designs. The numerical simulations are a strength: the paper provides quantitative phase diagrams, tests robustness to coefficient perturbations, and demonstrates a concrete mechanism by which a stabilizing mode changes the topology of a superposition. The generalized-skyrmion examples are also valuable as a demonstration. However, the central theorem's hypothesis is under-specified with respect to the propagation coordinate, and the quantization argument rests on an unpublished framework; these issues prevent the paper from fully establishing the advertised claims.","major_comments":[{"comment":"The condition that 'E0 is everywhere non-zero' is not tied to the propagation coordinate z. Equation (1) is written without explicit z-dependence, but the actual fields in Eq. (3) contain factors exp(ik_n z) and exp(iκ_n z), so the relative phases between modes change with z. Non-vanishing at a single cross-section does not imply non-vanishing for all z; destructive interference between modes with different propagation constants can create a zero at a later z. The homotopy argument requires E0(r,θ,z) ≠ 0 for every z in the propagation interval, or a dominance threshold such as |c| > C that guarantees this uniformly. The paper states no such threshold in the abstract or in the propagation claims associated with Fig. 4. This is load-bearing because a zero punctures the domain and can change the skyrmion number.","section":"Section 1, Eq. (1) and the three numbered statements"},{"comment":"The quantization of the skyrmion number integral is asserted via a 'canonical extension' of the boundary polarization field to a compactifiable S^2-valued function, citing refs. [23] and [30]. For a conducting waveguide whose boundary curve is the equator traced twice, this extension is not constructed and the homotopy invariance is not proved. Since the subsequent statement that a nonzero field cannot change its skyrmion number under propagation depends on this quantization, the paper should either provide a self-contained derivation for the waveguide geometry or state precisely which theorem from [30] applies and what assumptions it requires.","section":"Section 1, first paragraph"},{"comment":"The claim that adding a sufficiently large multiple of a nonzero mode to any solution yields a field that is everywhere nonzero is stated without proof. It is plausible if the stabilizing mode has a positive lower bound on its magnitude over the compact cross-section, but this lower bound and the threshold on |c| are not given. Also, the identification of TE1n and TM1n as 'topologically stabilizing' is based only on non-vanishing at r = 0; Fig. 1 shows that TE11 itself can develop zeros for |B11| ≈ 1, so the classification needs a quantitative criterion rather than the m ≠ 1 observation.","section":"Section 1, numbered statement 3 and the paragraph on TE1n/TM1n"},{"comment":"The preservation of the generalized skyrmion numbers in Fig. 5 is asserted on the basis of visual inspection of the boundary curves and the framework of ref. [30]. No definition of the generalized skyrmion number is given in this paper, and no numerical value is computed from an integral; the reported values (0, −2, −4) appear to be inferred from the component structure rather than from a calculation. The reader cannot verify the claim without consulting an unpublished preprint. A self-contained definition and a computation recipe are needed.","section":"Section 2 and Fig. 5, generalized skyrmion numbers"}],"minor_comments":[{"comment":"The caption writes 'ImB11' in panel b, but the panel varies B21; this should be 'ImB21'. The word 'skrymion' in the caption should be 'skyrmion'.","section":"Fig. 2 caption"},{"comment":"The caption says 'As in Figure 4, there is a critical region between the two stable modes'; the reference appears to be to Fig. 1 rather than Fig. 4, and should be corrected.","section":"Fig. 3 caption"},{"comment":"The sentence 'The figure is organized in a similar way to (a)' is unclear because panel (a) is a phase diagram while panels (b)-(d) are propagation plots; the intended comparison should be stated explicitly.","section":"Fig. 4 caption"},{"comment":"The phrase 'which cases the usual skyrmion number to fluctuate' should read 'which causes the usual skyrmion number to fluctuate'.","section":"Fig. 5 caption"},{"comment":"The coefficients A_mn, B_mn, C_mn, D_mn and the mode functions E_TE and E_TM are not fully defined until Methods; a brief definition or a forward reference at the point of Eq. (3) would improve readability.","section":"Main text, Eq. (3)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on two self-cited works, ref. [23] (published) and ref. [30] (unpublished preprint). I recommend that the editor ensure the present paper does not depend on results that are not publicly peer-reviewed, and that the authors be asked to summarize the generalized skyrmion framework self-containedly. The central z-dependence issue is fixable, but it is currently a load-bearing gap in the main theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper has a real idea and some nice numerics, but the central theorem is under-specified in exactly the way the stress-test note says. The field at z=0 being nonzero does not guarantee it stays nonzero as the relative phases of the modes drift. That needs a uniform-in-z condition or a dominance bound.\n\nWhat's actually new: the classification of TE/TM modes into topologically stabilizing, weakly stable, and unstable; the observation that only TE1n and TM1n avoid the r=0 singularity; and the numerical demonstration that a TE11+TE21 superposition can carry a protected skyrmion number of -2 in propagation, with apparent robustness to coefficient perturbations. That last point is a genuine contribution for on-chip structured light. The phase diagrams and the Stokes field evolution are also useful, even if they come without error bars or raw data.\n\nSoft spots, in proportion. The main theorem's hypothesis is ambiguous. Eq. (1) defines E0 with no z-dependence, but the proof of protection along the waveguide needs the transverse field nonzero on the whole cylinder. Two modes with different propagation constants accumulate relative phase; destructive interference can create zeros later even if the z=0 field is nowhere zero. Point 3's dominance bound (|c| large) is a valid sufficient condition, but the abstract and Fig. 4 don't state any threshold. So the abstract overclaims: \"if a stabilizing mode is present\" reads as a general guarantee, but the actual guarantee is conditional on a bound that is never quantified.\n\nThe quantization argument via canonical extension is also heuristic, and it leans on two self-cited papers, one of which (ref 30) is an unpublished preprint. That doesn't make the result wrong, but it makes the generalized-skyrmion section hard to verify independently. No code, no data, no error bars. All of this is addressable.\n\nWho it's for: researchers working on structured light in waveguides and topological photonics. It deserves a serious referee, not a desk reject. I'd ask for a rewrite that states the uniform-in-z condition explicitly, proves or quantifies the dominance bound, and releases the numerical data.\n\nRecommendation: send to peer review, expect major revision.","headline":"Real idea and useful numerics, but the central theorem is under-specified: nonvanishing at z=0 does not imply nonvanishing along the whole waveguide, so the topological protection claim needs a uniform-in-z condition or a dominance bound.","tokens_in":10083,"tokens_out":3402,"would_cite":false,"duration_ms":37242,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["42.79.Gn","42.25.Ja"],"model":"deepseek-v4-flash","headline":"Within a conducting waveguide, a skyrmion's topological charge survives propagation whenever the transverse electric field stays nonzero, and adding a TE1n or TM1n stabilizing mode can guarantee that condition.","keywords":["optical skyrmions","topological protection","conducting waveguides","modal dispersion","skyrmion number","topologically stabilizing modes","generalized skyrmion number","polarization topology"],"falsifier":"Take a cylindrical conducting waveguide, launch a TE11 plus TE21 superposition in a parameter region predicted to be stable, and measure the Stokes-resolved transverse field at several propagation distances; finding the skyrmion number integral changing to a different integer while the transverse electric field remains everywhere nonzero would refute the central claim. Alternatively, an explicit numerical search could look for a coefficient path between two regions of different skyrmion number that avoids any transverse zero, which the theory says is impossible.","tokens_in":9087,"feed_emoji":"🌀","tokens_out":6106,"duration_ms":57837,"temperature":0.7,"pith_summary":"This paper establishes conditions under which the skyrmion number of an optical polarization field survives propagation inside a conducting waveguide. It argues that if the transverse electric field is nowhere zero, the skyrmion number integral is constant along the guide even when modal dispersion reshapes the field, because propagation is then a continuous deformation of a compactifiable map. From this it derives a practical classification: TE1n and TM1n modes can act as topologically stabilizing modes that, when added to a superposition, remove field zeros and make the topological charge robust to small coefficient changes. The paper also shows that when the ordinary skyrmion number is not preserved, a generalized skyrmion number tied to components carved out by boundary and singularity curves can remain protected. If correct, this gives multi-mode waveguides a route to carry topologically protected, high-dimensional information.","feed_headline":"Waveguides carry skyrmions intact when a stabilizing mode is present","feed_subtitle":"Topological polarization patterns resist modal dispersion whenever a nonzero TE or TM mode anchors the field.","key_machinery":"The central object is the skyrmion number integral, the degree of the map from the waveguide cross-section to the Poincaré sphere, computed from the Stokes parameters; its quantization rests on a canonical compactification of the boundary polarization field fixed by the conducting walls. The load-bearing mechanism is the observation that a transverse electric field that is everywhere nonzero makes the polarization map smooth and defined on the whole cross-section, so propagation along the waveguide becomes a homotopy that cannot change the integer. The named tool is the topologically stabilizing mode: a solution, in practice TE1n or TM1n, that is nonzero everywhere across the cross-section and can be added to any superposition to eliminate zeros, thereby enforcing both propagation stability and robustness to coefficient variations. For fields with persistent singularities, the paper invokes the generalized skyrmion number, which assigns separate integer charges to each connected component of the Poincaré sphere carved out by the images of physical boundaries and singularity boundaries, and is preserved as long as that component structure does not split.","core_discovery":"Within a hollow conducting waveguide, the fixed polarization state on the metal boundary lets the interior polarization field be canonically extended to a compactified sphere, so the interior skyrmion number integral is an integer homotopy invariant. The paper's central claim is that this invariant is preserved during propagation whenever the transverse electric field has no zeros: field zeros are the only way the domain gets punctured, and a smooth nonzero field makes propagation a homotopy. Therefore, for a superposition of TE and TM modes, any coefficient set whose transverse field is everywhere nonzero yields a constant skyrmion number along the guide, and nearby coefficient sets with the same property yield the same integer. Because all m not equal to 1 TE and TM modes vanish at the waveguide center, only TE1n and TM1n modes can serve as stabilizing additions that remove zeros; adding one to an unstable state recovers both propagation stability and parameter robustness. The paper also demonstrates numerically stable skyrmion numbers of 1, -1, and -2 in TE11/TE21 superpositions, and shows that when the ordinary skyrmion number varies, a generalized skyrmion number assigned to each connected component on the Poincaré sphere can remain constant or lose only the charge of a disappearing component.","pith_inferences":["If the same reasoning carries to dielectric waveguides, where the boundary polarization is not fixed, the generalized skyrmion number should provide protection over finite propagation distances as long as the boundary-curve topology does not change; an experimental check would be to launch a known TE/TM superposition and measure Stokes-resolved cross-sections at successive propagation distances.","The phase diagrams suggest a testable bifurcation structure: the skyrmion number can only change when a transverse zero crosses the domain, so one could predict critical coefficient surfaces where the integer jumps and verify that no jump occurs without a zero.","The TE1n/TM1n stabilizing condition may extend to other confining geometries, such as rectangular or ridge waveguides, wherever a mode is nonzero everywhere; engineering such modes could become a design rule for topologically robust integrated photonics.","An implicit consequence is that the topological charge carried by a disappearing component in the generalized setting is genuinely lost, so redundancy or error-correction schemes would be needed if multiple charges are used for data encoding."],"forward_implications":["Multi-mode conducting waveguides can carry topologically protected polarization patterns despite modal dispersion, so information encoded in skyrmion number need not be disrupted by mode beating.","Coupling precision can be relaxed: any perturbation of the mode coefficients that keeps the transverse field nonzero leaves the transported skyrmion number unchanged.","Superposing a topologically stabilizing mode can stabilize otherwise singular modes and create stable skyrmion numbers, such as -2, that neither mode alone can produce.","When the ordinary skyrmion number fails, the generalized skyrmion number can still protect several integer charges simultaneously, raising the information density per field.","Weakly stable single modes remain protected in propagation but not against coefficient changes; only genuinely stabilizing superpositions give both kinds of robustness."],"supporting_citations":[{"why":"Establishes that skyrmion topology is fixed by boundary values and compactification, the basis for assigning an integer charge inside the waveguide.","marker":"[23]"},{"why":"Supplies the generalized skyrmion number framework used when the ordinary skyrmion number is not preserved.","marker":"[30]"},{"why":"Provides the elliptic regularity used to argue the electric field is smooth, so being nonzero implies a smooth polarization map.","marker":"[31]"},{"why":"Documents that free-space propagation does not preserve skyrmion number in general, the contrast motivating the waveguide result.","marker":"[22]"}],"fun_headline_variants":["Skyrmions survive in waveguides with a stabilizing mode","Stabilizing mode keeps optical skyrmions intact in waveguides","Zero-free field preserves skyrmion number in waveguides","Waveguide skyrmions saved by a stabilizing mode","Stabilizing modes lock skyrmion number along waveguides"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the premise that the skyrmion number integral inside the waveguide is quantized through a canonical extension of the fixed boundary polarization to a compactified sphere; if that quantization fails for the waveguide's boundary curve, the conclusion that nonzero fields cannot change their skyrmion number does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Skyrmions survive in waveguides with a stabilizing mode","Stabilizing mode keeps optical skyrmions intact in waveguides","Zero-free field preserves skyrmion number in waveguides","Waveguide skyrmions saved by a stabilizing mode","Stabilizing modes lock skyrmion number along waveguides"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000528,"raw_usage":{"total_tokens":2566,"prompt_tokens":987,"completion_tokens":1579,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":603,"completion_tokens_details":{"reasoning_tokens":1496}},"tokens_in":603,"tokens_out":1579,"duration_ms":12123,"temperature":1.0,"reasoning_tokens":1496,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:36:35.417913+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a cylindrical conducting waveguide, launch a TE11 plus TE21 superposition in a parameter region predicted to be stable, and measure the Stokes-resolved transverse field at several propagation distances; finding the skyrmion number integral changing to a different integer while the transverse electric field remains everywhere nonzero would refute the central claim. Alternatively, an explicit numerical search could look for a coefficient path between two regions of different skyrmion number that avoids any transverse zero, which the theory says is impossible.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that skyrmion topology is fixed by boundary values and compactification, the basis for assigning an integer charge inside the waveguide."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the elliptic regularity used to argue the electric field is smooth, so being nonzero implies a smooth polarization map."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents that free-space propagation does not preserve skyrmion number in general, the contrast motivating the waveguide result."}],"review_version":1}