{"id":"9f160841-f7b9-4dfa-9147-ea77b0ad13ef","arxiv_id":"2608.01155","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"OWNS-S evaluates the same rational spectral projector as OWNS-R but as a partial-fraction sum, eliminating the recursive error amplification that limits OWNS-R at high approximation order.","lead":"This paper introduces OWNS-Summation (OWNS-S), a new way of evaluating the one-way Navier-Stokes projection operator that replaces a recursive product of resolvent factors with an additive sum, preventing multiplicative rounding-error growth. If it holds up, it lets boundary-layer stability computations run at much higher approximation orders, or in parallel, and reach streamwise resolutions that the parabolised stability equations cannot.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"OWNS-S's 'pure convergence parameter' claim is not yet fully supported: weight-computation conditioning at large N is unanalysed.","rationale":"The reader's strongest claim is that OWNS-R's large-N breakdown is caused by multiplicative floating-point propagation and that OWNS-S removes it. The controlled experiment in §4.1.1 is well designed: identical auxiliary poles from the same polynomial solve, same weights in exact arithmetic, only the evaluation structure changed. The observed ~40-order-of-magnitude difference in the accumulated-stage norms strongly supports the mechanism, and the marching results at N=60 and beyond corroborate it. I do not see a flaw in that core comparison. The reader's weakest_assumption focused on the C-term neglect in the first-order pencil, which is an acknowledged modeling approximation and important for physical validity, but it is not the central numerical claim about evaluation structure. My concern is instead with the paper's key corollary that N is a pure convergence parameter in OWNS-S. The paper itself identifies the weight calculation as a potential source of ill-conditioning in §3.1 but provides no scaling analysis or systematic experiment at large N. Since the central claim is precisely that the recursive product's failure mode is removed, the conditioning of the additive evaluation's weights is a load-bearing, unproven step. This warrants keeping the verdict conditional rather than moving to accept, but it does not overturn the demonstrated mechanism. Hence UNCHANGED, with partial agreement with the reader's choice of weakest assumption.","tokens_in":42237,"tokens_out":23881,"duration_ms":253732,"concrete_test":"For the synthetic incompressible pencil of §4.1.1, compute at N = 50, 100, 200, 400: (i) the condition number of the Cauchy-like matrix D in Eq. (19) for the least-squares weight fit, (ii) the maximum |w_k| from the residue formula (17), and (iii) the projection-quality metrics E_idem, E_retain, E_leak from §6.2. If E_idem begins to rise above its saturation level, or if max|w_k| and cond(D) grow faster than polynomially in N, then OWNS-S has its own large-N ceiling and the 'pure convergence parameter' claim fails. If the metrics remain flat to N=400 despite mild conditioning growth, the claim is supported at least in this representative configuration.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that N becomes a pure convergence parameter depends on the additive evaluation remaining well conditioned as N grows. Section 3.1 explicitly acknowledges that the residue-weight formula (17) degrades when auxiliary poles cluster and when weights acquire large alternating signs, and that the least-squares system (18)-(19) becomes severely ill-conditioned for densely distributed parameters. The synthetic experiment in §4.1.1 measures the norm of the partial-sum operator (28), not the conditioning of the weight computation, and reports results to N=100 with a passing mention of N≈300. If the weight computation itself develops a flow-dependent ill-conditioning at larger N, OWNS-S will also acquire an upper bound on N, so the claim that the additive evaluation removes the OWNS-R failure mode and turns N into a purely ordinary convergence parameter is stronger than the evidence presented. The tested configurations support the central mechanism, but the 'pure convergence parameter' corollary is an extrapolation from a limited set of well-behaved parameter placements.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces OWNS-Summation (OWNS-S), an additive partial-fraction evaluation of the rational spectral projector used in one-way Navier–Stokes marching, in place of the recursive product evaluation of OWNS-R. With identical poles and weights the two evaluations are equivalent in exact arithmetic, but differ in floating-point arithmetic because OWNS-R propagates each resolvent-solve error through all later stages. The central diagnostic experiment in §4.1.1 uses identical auxiliary poles from the same polynomial solve and shows the accumulated recursive-product norm growing to roughly 10^40 at N=100 while the summation norm remains bounded. The method is then validated on incompressible, hypersonic, and transonic boundary layers, including a swept-wing transonic case in which the disturbance spectrum reorganizes from subsonic to supersonic topology during the march. A greedy, paired parameter-selection procedure with an analytic candidate pool is also introduced, avoiding eigen-decomposition during the march.","tokens_in":42480,"tokens_out":3358,"duration_ms":39930,"significance":"If the central claim holds, the paper makes a practically important contribution: it removes an unknown, flow-dependent upper bound on the approximation order N that limits OWNS-R, turns N into a convergence parameter in the ordinary sense, and exposes a parallel structure with per-station critical path reduced to 1+ceil(N/n_t) solves. The controlled comparison in §4.1.1 is a genuine strength: the poles are held fixed while only the evaluation changes, so the observed ~40-order-of-magnitude differential identifies the recursive product rather than the auxiliary poles as the source of instability. The replication of the qualitative behaviour across incompressible, hypersonic, and transonic configurations is also persuasive. However, two load-bearing points need further support before the strongest claims can be accepted: the conditioning of the weight computation in OWNS-S at large N, and the validity of constructing the projector from the first-order pencil only while the second-order C term is retained in the march.","major_comments":[{"comment":"The claim that N in OWNS-S is a 'pure convergence parameter' is stronger than the evidence presented. Section 3.1 explicitly states that the residue formula (17) degrades when auxiliary poles cluster and when weights acquire large alternating signs, and that the least-squares system (18)-(19) becomes severely ill-conditioned for densely distributed parameters. The synthetic experiment in §4.1.1 measures the accumulated-stage norm (28), not the conditioning of the weight computation or the backward error of the weighted sum; it is reported to N=100, with a passing mention of N≈300. If weight computation itself develops flow-dependent ill-conditioning at larger N, OWNS-S will also have an upper bound on N. Please report condition numbers of D, weight norms, or residual/backward errors versus N across the three test configurations, or explicitly limit the 'pure convergence' claim to the ran","section":"§3.1, §4.1.1"},{"comment":"The projector is constructed from the first-order pencil (B,A) only, while the marched equation retains the second-order C d^2/dx^2 term. The authors acknowledge that C does not commute with the projection operator and that the construction is therefore not an exact spectral splitting of the second-order system. This is load-bearing for the physical validity of the one-way march: if the neglected C term contributes to the upstream–downstream partition at low Reynolds number or under strong non-parallel effects, the projected march could miss or misfilter modes. The statement that 'its neglect has not been observed to affect' the separation is an empirical assertion, not a demonstrated bound. Please quantify the neglected contribution, for example by comparing against a projection built from the full quadratic pencil in a representative low-Reynolds-number or strongly non-parallel case, o","section":"§2.1"}],"minor_comments":[{"comment":"The PSE reference is quoted at n_x=150 and described as under-resolved, while OWNS-S is step-converged at n_x=5000 and reproduces PSE when run at the coarse step. This is plausible, but the discussion would be clearer if the PSE reference were also shown at a step where it is converged in its own resolution study, or if the resolution limitation were documented with a separate PSE step-size test.","section":"§6.5.2"},{"comment":"The color/line legend entries are dense and some are distinguished only by marker style; please add a short description in each caption of which curve corresponds to which variant, especially for 'Greedy-Heuristic Nc=100/200/300'.","section":"Captions, Figures 4, 6, 23, 24"},{"comment":"The abstract states that N 'becomes a pure convergence parameter' without the caveats given in §3.1. Please soften the abstract to reflect that the evidence covers the tested placement families and that weight conditioning at very large N is a separate question.","section":"Abstract and §1"},{"comment":"Some references are typeset inconsistently (e.g., 'Badcock and Mughal, 2026a,b', 'Gushchin and Fedorov, 1990' spacing). A final formatting pass would help.","section":"References"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper identifies the recursive product evaluation, not the pole set, as the dominant source of the large-N breakdown in OWNS-R, and replaces it with an additive partial-fraction sum. The evidence for that mechanism is genuinely good. Section 4.1.1 holds the poles fixed from the same polynomial solve and shows the OWNS-R accumulated norm reaching ~1e40 at N=100 while the OWNS-S norm stays bounded. That is a clean, controlled diagnostic, and it is reproduced across incompressible, hypersonic, and transonic marches. The paper also ships a workable greedy parameter-selection strategy that avoids eigendecompositions, which matters for practitioners. Credit where due: the authors are explicit that additive filter evaluation is not new in itself, and they frame their contribution as its bearing on one-way marching. The self-citation and prior-work citations are normal; nothing circular here.\n\nSoft spots, in proportion. The C-term neglect in the projector is acknowledged but load-bearing: the projection is built from the first-order pencil while the march retains the second-order term. The authors say it has not been observed to matter, but that is empirical, not a guarantee, and near strong non-parallel effects or low Reynolds number it could bite. The second soft spot is the stress-test concern, which lands partially: the synthetic experiment measures the norm of the partial-sum operator, not the conditioning of the weight computation. Section 3.1 itself admits the residue formula and least-squares system degrade when poles cluster or parameters are dense, so the claim that N is a pure convergence parameter is stronger than the evidence shown. It may be true in the tested regimes, but the upper bound, if any, is flow-dependent and unquantified. Third, the transonic SWiFT case has no independent DNS or global-solver baseline; the OWNS-S fine-step result is compared against a PSE reference that the authors argue is under-resolved. That argument is plausible and step-convergence helps, but it is not independent confirmation. Finally, code and data are not public, so independent replication is not possible.\n\nNone of these sink the central claim. The additive-versus-recursive mechanism is convincingly demonstrated, and the paper is honest about its assumptions. The audience is boundary-layer stability and one-way marching specialists, plus anyone who relies on PSE and wants a step-size floor removed. It deserves a serious referee and likely a revise-and-resubmit that asks for more analysis of weight-conditioning at large N, an independent benchmark for the transonic case, and ideally a code release. Send it out.","headline":"A well-controlled experiment shows the additive OWNS-S evaluation removes the recursive-product instability that caps OWNS-R order; the mechanism is convincing, though the 'pure convergence parameter' claim outruns the conditioning analysis.","tokens_in":43059,"tokens_out":1658,"would_cite":true,"duration_ms":19878,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65F15","65F60","76E05","76M20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Evaluating the one-way Navier–Stokes projector as an additive partial-fraction sum instead of a recursive product removes the large-order numerical breakdown, turning the approximation order into an ordinary convergence parameter.","keywords":["Boundary-Layer Instability","Parabolised Stability Equations","One-Way Navier-Stokes","OWNS-Summation","OWNS-Recursion","spectral projection","floating-point error","transonic boundary layer"],"falsifier":"The paper's own synthetic experiment is the sharpest check: with identical poles, the accumulated Frobenius diagnostic $\\rho^R_k$ for the recursive product should grow to about $10^{40}$ at $N=100$ while $\\rho^S_k$ for the summation stays bounded; reproducing this on any pencil would confirm the mechanism. To test the $C$-term assumption, compute the retained/leaked eigenvalue sets for the full quadratic pencil $\\mathbf{B} - i\\alpha\\mathbf{A} - \\alpha^2\\mathbf{C}$ on a strong non-parallel or low-$R$ case and compare with the first-order-pencil projector; any mismatch in the directional classif","tokens_in":42022,"feed_emoji":"🌊","tokens_out":4989,"duration_ms":50157,"temperature":0.7,"pith_summary":"This paper argues that the large-$N$ breakdown of the OWNS-R one-way marching method is not caused by its pole placement but by the way it evaluates the rational approximation: as a recursive product of resolvent factors, in which each stage's rounding error is multiplied through all later stages. The paper proposes OWNS-Summation (OWNS-S), which evaluates the same rational approximation as an additive partial-fraction sum, with every resolvent solve acting on the same input state. In exact arithmetic the two are the same operator; in floating point the sum stays bounded while the product grows to about $10^{40}$ in an accumulated Frobenius diagnostic at $N=100$. If correct, the result makes $N$ a genuine convergence parameter, lets the $N$ solves run in parallel, and removes the need to keep the order small, so practitioners can certify convergence by raising $N$ as in ordinary discretisation studies. The paper supports this with projection-quality metrics and spatial marches through incompressible, hypersonic, and transonic boundary layers, including a continuous march through a subsonic-to-supersonic spectral reorganisation.","feed_headline":"Summed resolvents lift the N ceiling in one-way marching","feed_subtitle":"OWNS-S evaluates the same projector additively, so the approximation order converges normally and the solves run in parallel.","key_machinery":"The rational approximation of the spectral projector, $P_N^+ = w_0 I + \\sum_{k=1}^N w_k (B - i\\beta_k^* A)^{-1} A$, evaluated additively rather than as the product $\\frac{1}{c+1}\\prod_{k=1}^N (B-i\\beta_k^* A)^{-1}(B-i\\beta_k^- A)$. The auxiliary poles $\\beta_k^*$ come from the polynomial identity of Eq. (14), and the weights from residues or a least-squares fit; the additive evaluation is what decouples the solves and stops multiplicative rounding-error amplification.","core_discovery":"At fixed auxiliary poles, OWNS-S is shown to reproduce OWNS-R's projected operator in exact arithmetic but to have fundamentally different floating-point error propagation. In OWNS-R, the error committed at stage $k$ is carried through the $N-k$ remaining stage operators, so the accumulated diagnostic $\\rho^R_k$ grows roughly to $10^{40}$ at $N=100$; in OWNS-S each solve's error enters only one weighted term, so $\\rho^S_k$ stays bounded. The paper therefore identifies the recursive evaluation---not the poles---as the dominant source of the instability that previous work attributed to pole-solving rounding errors and recommended extended precision for. With the same poles, OWNS-S remains accu","pith_inferences":["If the multiplicative product is the mechanism, then every OWNS-R implementation---regardless of extended-precision poles or factor ordering---should hit the same size ceiling; a direct test would be to run OWNS-R with the least-squares weights that repair retention, and check whether the ceiling survives.","The additive-vs-recursive distinction is not specific to OWNS: other contour-integral projectors and one-way methods that currently evaluate products of resolvents could inherit the same cure, and the paper leaves the quadratic-pencil projector (including $C$) as an obvious next step to test the first-order-pencil assumption.","Since the $\\beta_k^*$ contour is only an estimate of the spectral separation, the method's reliability on strongly non-parallel or low-Reynolds-number flows likely hinges on whether the neglected $C$ term changes the upstream–downstream partition; computing a quadratic-pencil projector would settle that directly."],"forward_implications":["Approximation order $N$ becomes a convergence parameter in the ordinary sense: it can be raised until the solution stops changing, so converged results can be certified without an unknown ceiling.","The $N$ resolvent solves per station are independent, reducing the per-station critical path to $1+\\lceil N/n_t\\rceil$ on $n_t$ threads and bringing OWNS-S's critical path to PSE-like levels.","The same rational projector used by OWNS-R is evaluated accurately at $N$ where OWNS-R diverges; in the hypersonic case, the two distinct failure modes (parameter-set deficit at low $N$, recursive instability at high $N$) are separated, with OWNS-S surviving both.","OWNS-S can march continuously through a transonic spectral reorganisation from subsonic to supersonic topology at streamwise resolutions below the PSE step floor.","The paired greedy parameter selection removes the need for eigendecomposition during the march, with the analytic candidate pool matching eigenvalue-based selection in the benchmark case."],"supporting_citations":[{"why":"Supplies the OWNS-R recursive product formulation, the polynomial identity (14), and the documented large-$N$ deterioration that this paper re-examines.","marker":"Zhu and Towne (2023)"},{"why":"Reports that the deterioration persists with quadruple-precision poles and provides the greedy parameter-selection framework that this paper adapts with paired candidates.","marker":"Sleeman and Colonius (2026)"},{"why":"Introduced one-way spatial integration for hyperbolic equations and the use of recursion parameters that OWNS-S inherits.","marker":"Towne and Colonius (2015)"},{"why":"Developed the OWNS-P projection formulation and resolvent analysis that motivates the exact projector (7).","marker":"Towne et al. (2022)"},{"why":"Provides the contour-integral projector context (FEAST) that makes the partial-fraction evaluation of resolvents a standard tool.","marker":"Polizzi (2009)"},{"why":"Establishes the rational-approximation/matrix-sign-function background and near-optimal Zolotarev approximants referenced for the rational filter.","marker":"Nakatsukasa and Freund (2016)"},{"why":"Supplies the analytical continuous-branch estimates that the heuristic and greedy parameter placements sample.","marker":"Schmid and Henningson (2001)"},{"why":"Provides the wave-carrier factoring and the heuristic placement formulas used by the OWNS-S march.","marker":"Badcock and Mughal (2026b)"}],"fun_headline_variants":["Summed resolvents remove the N ceiling in one-way marching","OWNS-S: parallel solves, no multiplicative error catch-22","Additive projector tames rounding errors, lifts N limit","One-way marching: sum instead of multiply to dodge instability","No more N barrier: summed resolvents fix rounding errors"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The projector is built from the first-order pencil $(B,A)$ only, while the marched equation keeps the second-order $C\\,\\partial^2_x$ term; the paper assumes this does not affect the upstream–downstream separation, and if that assumption fails in some flow the projection could retain or remove the wrong modes.","fun_headline_variants_meta":{"raw":{"variants":["Summed resolvents remove the N ceiling in one-way marching","OWNS-S: parallel solves, no multiplicative error catch-22","Additive projector tames rounding errors, lifts N limit","One-way marching: sum instead of multiply to dodge instability","No more N barrier: summed resolvents fix rounding errors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0003,"raw_usage":{"total_tokens":1632,"prompt_tokens":873,"completion_tokens":759,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":674}},"tokens_in":617,"tokens_out":759,"duration_ms":8913,"temperature":1.0,"reasoning_tokens":674,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T00:25:32.040774+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The paper's own synthetic experiment is the sharpest check: with identical poles, the accumulated Frobenius diagnostic $\\rho^R_k$ for the recursive product should grow to about $10^{40}$ at $N=100$ while $\\rho^S_k$ for the summation stays bounded; reproducing this on any pencil would confirm the mechanism. To test the $C$-term assumption, compute the retained/leaked eigenvalue sets for the full quadratic pencil $\\mathbf{B} - i\\alpha\\mathbf{A} - \\alpha^2\\mathbf{C}$ on a strong non-parallel or low-$R$ case and compare with the first-order-pencil projector; any mismatch in the directional classif","supporting_citations":[],"review_version":1}