{"id":"96bd9209-dcad-49d5-aa47-9cfa0795f332","arxiv_id":"2501.13311","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The p-widths of (RP^2, g_std) are ω_p = 2π floor((1+sqrt(1+8p))/4), achieved by Z2-invariant polynomial sweepouts.","lead":"This paper computes the exact p-widths of the real projective plane with its standard round metric. These are a sequence of min-max lengths, and it is the second surface after the sphere for which the full sequence is known.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.3 (strict monotonicity of ω_p(g_μ) on RP^2) is asserted without proof and is load-bearing for the counting argument; if it fails, Section 4 cannot conclude L=R.","rationale":"The central claim is an adaptation of Chodosh–Mantoulidis to RP^2, and the paper contains several terse steps: the proof that F_d is a D(d)-1-sweepout is justified only by an RP^1 restriction, the application of [CM23, Thm 1.2] to arbitrary homotopy classes is not fully justified, and the μ bound in Section 4 has a small factor-of-two issue. These are fixable presentation problems: the polynomial sweepout property is standard in this circle of ideas, and choosing μ < 1/(4(d+1)) would repair the bound. Proposition 3.3, however, is genuinely load-bearing and unproved. The proof of Theorem 1.6 needs exactly D(d+1)-1 distinct width values for p = 1,...,D(d+1)-1; strict monotonicity is the only stated source of that cardinality. Since the standard metric has equal widths on each block, the strict inequalities for g_μ are a nontrivial perturbative fact, and the even-multiplicity lemma changes the admissible values in a way that may not be a 'minor notational change' from S^2. Thus the appropriate verdict remains CONDITIONAL: the result is plausible and likely correct, but the proof has an identifiable gap that must be filled.","tokens_in":9303,"tokens_out":29075,"duration_ms":287819,"concrete_test":"Independently re-derive Proposition 3.3 for RP^2 by writing out the adaptation of [CM23, Prop 6.11] line-by-line, explicitly tracking the even-multiplicity condition from Lemma 3.2(2) and the role of H_1(RP^2; Z_2) ≠ 0. If any step relies on H_1(S^2; Z_2) = 0 or on the absence of a parity constraint, then Proposition 3.3 is not established and the counting argument in Section 4 collapses.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Proposition 3.3 is the linchpin of the counting argument. Section 4 needs #L = D(d+1)-1 = |R| for the two sets to coincide, and the only stated justification is the unproved assertion that ω_p(g_μ) < ω_{p+1}(g_μ) for every μ ∈ (0, μ1). This is not a cosmetic detail: at g_std the widths are constant on each block D(d),...,D(d+1)-1, so the strict inequalities are a special property of the perturbation g_μ, and on RP^2 Lemma 3.2(2) adds the parity constraint n1+n2+n3 ≡ 0 mod 2. The paper says [CM23, Prop 6.11] carries over 'with only minor notational changes', but gives no proof. If any consecutive pair of widths coincides, then #L < |R| and the equality L=R, hence the final formula for ω_p(g_std), does not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes the p-width spectrum of (RP^2, g_std), proving ω_p = 2π⌊(1+√(1+8p))/4⌋ for all p ∈ N+, with the widths achieved by sweepouts descending from Z_2-invariant polynomials on S^2. The proof follows the Chodosh–Mantoulidis strategy for S^2: one first perturbs g_std to ellipsoid quotients g_μ with exactly three simple geodesics of lengths π, π+μ, π+2μ and a rigidity theorem for short geodesics (Theorem 3.1); one then classifies the least areas LAP(Π) of all p-sweepout homotopy classes, showing they are sums of the three geodesics with even total multiplicity or else ≥ 2πf(p)+2 (Lemma 3.2); and finally a counting argument compares the number of distinct p-widths below a cutoff with the number of admissible lattice values, forcing equality of the two sets (Section 4). Passing μ→0 gives the stated formula for g_std.","tokens_in":9478,"tokens_out":45564,"duration_ms":408050,"significance":"If all claims are justified, the paper adds RP^2 as the second surface (after S^2 in [CM23]) whose full p-width spectrum is known explicitly, and it complements the spectral rigidity theorem of Ambrozio–Marques–Neves [AMN24]. Strengths of the manuscript include the parameter-free formula, the explicit algebraic achieving sweepouts, the clean adaptation of the parity constraint from H_1(RP^2; Z_2), and a transparent statement of which results are imported from [CM23]. The main caveat is that the paper's central counting argument rests on an unproved strict monotonicity assertion (Proposition 3.3) that is load-bearing, so I cannot certify the proof in its present form.","major_comments":[{"comment":"Proposition 3.3 asserts ω_p(RP^2,g_μ) < ω_{p+1}(RP^2,g_μ) for every μ ∈ (0, μ1), stating that [CM23, Prop 6.11] carries over 'with only minor notational changes,' but no proof is given. This is load-bearing: Section 4 needs the set {ω_1(g_μ),...,ω_{D(d+1)-1}(g_μ)} to have exactly D(d+1)-1 elements to conclude L = R, and the block identification (2π(d+1) ≤ ω_p(g_μ) ≤ (2π+4μ)(d+1) for p ∈ {D(d),...,D(d+1)-1}) depends on the strict increase. Strict monotonicity is a special property of the perturbed metrics rather than a general fact, since at g_std the widths are constant on each block, and on RP^2 the admissible lattice is restricted by the parity condition n1+n2+n3 ≡ 0 mod 2 (Lemma 3.2(2)), a structural difference from the S^2 case. The paper needs either a complete proof of strict monotonicity under the parity constraint or a precise citation of a statement covering RP^2; as written, the counting argument in Section 4 does not go through if any two consecutive widths coincide.","section":"Proposition 3.3"},{"comment":"The proof fixes 0 < μ < min(μ1, 1/[2(d+1)]). The count |R| = Σ_{j=1}^{d+1}(4j+1) requires every admissible lattice value with n1+n2+n3 = 2(d+1) to lie in (0, 2π(d+1)+1]; the largest such value is 2π(d+1)+4(d+1)μ, which is below the cutoff only if 4(d+1)μ < 1, i.e., μ < 1/[4(d+1)]. With the stated bound 1/[2(d+1)], the largest lattice values can exceed 2π(d+1)+1, so |R| would be strictly smaller than (d+1)(2d+5) and the equality L = R would fail. This is easily repaired by replacing 1/[2(d+1)] with 1/[4(d+1)] (or any smaller positive constant), but the estimate as written is incorrect.","section":"Section 4, first paragraph"},{"comment":"The sentence 'R ⊆ L follows directly from lemma 3.2' states the wrong inclusion. Lemma 3.2(2) (together with part (1) and ω_p = inf_Π LAP(Π)) shows that each width ω_p(g_μ) with p ≤ D(d+1)-1 is a lattice value below 2π(d+1)+1, i.e., L ⊆ R. The opposite inclusion R ⊆ L is precisely what the cardinality computation is meant to establish. The conclusion L = R still follows from the corrected inclusion L ⊆ R combined with |L| = |R|, so the argument is repairable by swapping the labels in that sentence, but the statement as written is false.","section":"Section 4, inclusion direction"},{"comment":"The decomposition LAP(Π) = Σ_j a_j ℓ_{g_μ}(σ_j) is justified by a reference to 'theorem 1.2,' but Theorem 1.2, as stated in the introduction, concerns the global p-width ω_p(M,g), not the least-area value LAP(Π) of an arbitrary F-homotopy class Π ⊆ P^F_{p,m}. What is needed here is the statement that each such homotopy class has its Almgren–Pitts min-max value realized by a union of primitive closed geodesics. Please provide the precise CM23 statement covering LAP for individual homotopy classes, or add a sentence explaining that the proof of Theorem 1.2 applies verbatim to each class.","section":"Lemma 3.2(2)"}],"minor_comments":[{"comment":"There are several typos and display errors: 'eixsts' (Lemma 3.2), 'analgoous' (§3.2), 'nothing g_μ → g_std' (should be 'noting' in Section 4), and the subscript/superscript formatting in equation (3.1) and nearby text is garbled.","section":"Throughout"},{"comment":"In the statement of Lemma 3.2(2), the lattice set is written with misplaced parentheses: '({...: (n1,n2,n3) ∈ N^3}\\{0}, n1+n2+n3 ≡ 0 mod 2)' — the parity condition should sit inside the set braces. The convention N = {0,1,2,...} should also be stated explicitly, since the counting in Section 4 uses triples with zero entries such as (2(d+1),0,0).","section":"Lemma 3.2(2)"},{"comment":"The paper does not explicitly verify that the algebraic families F_d satisfy the no-concentration-of-mass condition of Definition 2.2; this is standard for algebraic sweepouts in the CM23 framework, but a sentence confirming it would make the upper bound complete.","section":"Section 3.2"},{"comment":"The verification that F_d is a (D(d)-1)-sweepout checks only that the restriction to a copy of RP^1 is a 1-sweepout; the implicit step is that Φ*(λ) ≠ 0 in H^1 forces Φ*(λ^{D(d)-1}) ≠ 0 in H^{D(d)-1}(RP^{D(d)-1}; Z_2) via the ring structure H*(RP^N; Z_2) = Z_2[λ]/(λ^{N+1}). Stating this explicitly would improve readability.","section":"Section 3.2, F_d sweepout check"},{"comment":"The phrase 'by induction and the fact that the widths are increasing' is unnecessary for the displayed bounds 2π(d+1) ≤ ω_p(g_μ) ≤ (2π+4μ)(d+1): given L = R and Proposition 3.3, those bounds follow directly from the structure of R (the D(d)-1 values below 2π(d+1) are taken by the first D(d)-1 widths, and the remaining 4d+5 widths are exactly 2π(d+1), 2π(d+1)+μ, ..., 2π(d+1)+4(d+1)μ). Removing the induction reference would clarify the argument.","section":"Section 4, final paragraph"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a competent and readable adaptation of the S^2 computation in Chodosh–Mantoulidis, and the claimed formula passes basic sanity checks: the Weyl asymptotics ω_p ~ π√(2p) are consistent with the area-1/2 scaling of RP^2 relative to S^2, the parity constraint correctly encodes H_1(RP^2; Z_2), and the result is compatible with the AMN24 rigidity theorem. The main obstacle is Proposition 3.3: the paper gives no proof of strict monotonicity for the perturbed RP^2 metrics, and this assertion carries the entire counting argument. I would recommend asking the author to supply the full proof (or a precise reduction to the CM23 statement), and to fix the μ bound and the inclusion direction in Section 4. If the proof of Proposition 3.3 fails in the RP^2 case, the claimed formula would be in doubt; otherwise the remaining issues are local. The novelty relative to [CM23] is incremental but real, and the paper fits the scope of a differential-geometry journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read, off the record.\n\nThe result is real. The explicit formula ω_p = 2π floor((1+√(1+8p))/4) for (RP^2, g_std) is new, and it makes RP^2 the second closed surface, after S^2, with a known full p-width spectrum. The two genuinely new ingredients are the parity constraint on min-max geodesics (even total multiplicity, coming from H_1(RP^2;Z_2)) and the use of Z2-invariant polynomial sweepouts to get the upper bound. The Bezout/Crofton count and the overall architecture are clear adaptations of Chodosh–Mantoulidis, and the paper is not circular: [CM23] is used as an external benchmark, and [MK24] only appears in the introduction.\n\nThe soft spots are real but localized.\n\n(1) Proposition 3.3 — strict monotonicity of ω_p(g_μ) on RP^2 — is asserted with 'minor notational changes' and no proof. This is load-bearing: Section 4 needs exactly D(d+1)−1 distinct width values to force L = R. At g_std the widths are constant on each block, so this is not a cosmetic perturbation property; it's the thing the perturbation is for. A referee should ask for the transferred proof, or at least a precise statement of why the parity constraint doesn't obstruct the [CM23] argument.\n\n(2) Lemma 3.2(2) applies Theorem 1.2 to the LAP of an arbitrary F-homotopy class, but Theorem 1.2 is stated for global p-widths. The needed statement is plausible and probably follows from the same machinery, but the cited theorem does not say it.\n\n(3) In Section 4, the displayed condition 0 < μ < min(μ1, 1/2(d+1)) is too weak for the count. The top value for j = d+1 is 2π(d+1)+4(d+1)μ, which needs to be below 2π(d+1)+1, so the second constant should be 1/4(d+1), not 1/2(d+1). This looks like a typo, but it should be fixed.\n\nNone of this makes me doubt the main theorem; all three are fixable and the counting argument is otherwise coherent. The paper is for people working on p-widths or min-max on surfaces, and it gives [AMN24] its explicit numerical target. I would send it to a competent referee, and I'd want the referee to push on Proposition 3.3 specifically. I would not cite the formula in my own work until that gap is patched, but I expect it to be.","headline":"The RP^2 width formula is new and likely right, but strict monotonicity of the perturbed widths must be proved before the counting argument closes.","tokens_in":10045,"tokens_out":8397,"would_cite":false,"duration_ms":80612,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C42","49Q20","53C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every p-width of the standard real projective plane is 2π times the integer part of (1+√(1+8p))/4.","keywords":["p-widths","min-max theory","real projective plane","volume spectrum","geodesics","Z2-invariant polynomials","ellipsoidal metrics"],"falsifier":"Compute $\\omega_3(g_\\mu)$ and $\\omega_4(g_\\mu)$ on the ellipsoidal metrics $g_\\mu$ for small $\\mu$: if they are ever equal, the strict monotonicity used in the counting argument fails. Alternatively, find on some $g_\\mu$ a closed geodesic shorter than the threshold that is not an iterate of the three axial geodesics, which would contradict Theorem 3.1 and break the lower-bound classification.","tokens_in":9076,"feed_emoji":"📐","tokens_out":9659,"duration_ms":85356,"temperature":0.7,"pith_summary":"The paper computes the full p-width spectrum of the real projective plane with its standard round metric. The p-widths are a geometric analogue of Laplacian eigenvalues: the p-th width is the infimum, over p-parameter families of curves, of the largest length swept out. The result is that for every positive integer $p$ one has $\\omega_p = 2\\pi\\lfloor(1+\\sqrt{1+8p})/4\\rfloor$, so the spectrum consists of long plateaus of constant value. This matters because it makes $\\mathbb{RP}^2$ only the second surface whose p-widths are known exactly, and it gives a concrete target for the rigidity statement that equality of all p-widths with $\\mathbb{RP}^2$ forces isometry.","feed_headline":"RP^2 p-widths computed: 2π floor((1+√(1+8p))/4)","feed_subtitle":"The standard projective plane is only the second surface whose full volume spectrum is known.","key_machinery":"The carrying device is the family of ellipsoidal metrics $g_\\mu$ near the standard metric, for which every sufficiently short closed geodesic is an iterate of one of three simple axial geodesics of lengths $\\pi$, $\\pi+\\mu$, $\\pi+2\\mu$. The upper bound uses level sets of polynomials $f(x,y)+z g(x,y)$ with $f$ even and $g$ odd, which are $\\mathbb{Z}_2$-invariant and therefore descend to $\\mathbb{RP}^2$; Crofton's formula and Bézout's inequality bound their mass by $2\\pi d$. The lower bound counts the number of distinct sums $n_1\\pi+n_2(\\pi+\\mu)+n_3(\\pi+2\\mu)$ with nonnegative integers and even total multiplicity, and matches that count to the number of widths in each plateau.","core_discovery":"The central claim is Theorem 1.6: for $(\\mathbb{RP}^2, g_{std})$, one has $\\omega_p = 2\\pi\\lfloor \\tfrac14(1+\\sqrt{1+8p})\\rfloor$ for all $p\\in\\mathbb{N}^+$, and the equality is achieved by sweepouts built from $\\mathbb{Z}_2$-invariant polynomials on $S^2$. The proof perturbs the standard metric to nearby ellipsoidal metrics whose only short geodesics are three axial ones with lengths $\\pi$, $\\pi+\\mu$, $\\pi+2\\mu$, computes an upper bound from polynomial level sets, and obtains a matching lower bound by counting how many distinct width values those geodesic lengths can produce under an even total multiplicity constraint.","pith_inferences":["The same polynomial-sweepout and parity-counting mechanism may extend to other $\\mathbb{Z}_2$ quotients or orbifolds of the round sphere, though the paper does not claim this.","The plateau boundaries $D(d)=(d+1)(2d+1)$ look like a combinatorial count of $\\mathbb{Z}_2$-invariant polynomial degrees, so one could test whether similar counts govern widths of other symmetric surfaces.","A direct numerical min-max computation for the first few values of $p$ on the ellipsoidal metrics would independently verify the plateau endpoints and the exact value of each width."],"forward_implications":["For every $d$, all widths in the block $D(d)=(d+1)(2d+1)$ through $D(d+1)-1$ share the same value $2\\pi(d+1)$, so the p-width spectrum is highly degenerate.","The widths grow asymptotically like $\\sqrt{2}\\pi\\sqrt{p}$, consistent with the surface Weyl law for the volume spectrum.","The realizing min-max geodesic networks must have total multiplicity divisible by 2, reflecting the nontrivial $\\mathbb{Z}_2$ homology of $\\mathbb{RP}^2$.","Since the computed spectrum is exact, it supplies the full isospectral target for the rigidity theorem: any surface whose p-widths match these values is isometric to the standard $\\mathbb{RP}^2$."],"supporting_citations":[{"why":"Supplies the two-sphere computation, the ellipsoidal metric perturbation, and the counting argument that this paper adapts to $\\mathbb{RP}^2$.","marker":"[CM23]"},{"why":"Establishes that on ellipsoids close to the round sphere every sufficiently short geodesic is an iterate of one of three axial geodesics, which underlies Theorem 3.1.","marker":"[Mor34]"},{"why":"Introduces p-widths as the min-max analogue of the Laplacian spectrum, the object computed here.","marker":"[Gro06]"},{"why":"Provides the modern sweepout definitions, no-concentration condition, and existence framework for p-widths used in the proof.","marker":"[MN17]"},{"why":"Proves p-width spectral rigidity for $\\mathbb{RP}^2$, making this explicit computation the isospectral target.","marker":"[AMN24]"},{"why":"Gives the Weyl law for volume spectra, the asymptotic benchmark against which the computed growth rate is checked.","marker":"[LMN18]"}],"fun_headline_variants":["RP^2 p-widths: 2π⌊(1+√(1+8p))/4⌋ for all p","RP^2 is only 2nd surface with known p-width spectrum","Explicit formula gives every p-width of RP^2","RP^2 p-widths: a single formula, all p"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that for the nearby ellipsoidal metrics the p-widths are strictly increasing in $p$, a fact imported from the two-sphere case with only minor notational changes and no proof; the counting argument needs exactly $D(d+1)-1$ distinct width values to force the lower bound.","fun_headline_variants_meta":{"raw":{"variants":["RP^2 p-widths: 2π⌊(1+√(1+8p))/4⌋ for all p","RP^2 is only 2nd surface with known p-width spectrum","Explicit formula gives every p-width of RP^2","RP^2 p-widths: a single formula, all p"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001697,"raw_usage":{"total_tokens":6597,"prompt_tokens":694,"completion_tokens":5903,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":310,"completion_tokens_details":{"reasoning_tokens":5814}},"tokens_in":310,"tokens_out":5903,"duration_ms":45304,"temperature":1.0,"reasoning_tokens":5814,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:17:22.790734+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\omega_3(g_\\mu)$ and $\\omega_4(g_\\mu)$ on the ellipsoidal metrics $g_\\mu$ for small $\\mu$: if they are ever equal, the strict monotonicity used in the counting argument fails. Alternatively, find on some $g_\\mu$ a closed geodesic shorter than the threshold that is not an iterate of the three axial geodesics, which would contradict Theorem 3.1 and break the lower-bound classification.","supporting_citations":[],"review_version":1}