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REVIEW 4 major objections 5 minor 33 references

The p-widths of $RP^2$

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Every p-width of the standard real projective plane is 2π times the integer part of (1+√(1+8p))/4.

desk verdict 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. read the letter →

arxiv 2501.13311 v3 pith:J3JAQ5FA submitted 2025-01-23 math.DG

Jared Marx-Kuo This is my paper · ORCID
classification math.DG MSC 53C4249Q2053C22
keywords p-widthsmin-maxtheoryrealprojectiveplanevolumespectrumgeodesicsZ2-invariantpolynomialsellipsoidalmetrics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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 $\mathbb{RP}^2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [Proposition 3.3] 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.
  2. [Section 4, first paragraph] 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.
  3. [Section 4, inclusion direction] 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.
  4. [Lemma 3.2(2)] 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.
minor comments (5)
  1. [Throughout] 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.
  2. [Lemma 3.2(2)] 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).
  3. [Section 3.2] 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.
  4. [Section 3.2, F_d sweepout check] 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.
  5. [Section 4, final paragraph] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the p-width computation for RP^2 is an adaptation of external results, not a derivation whose output is baked into its inputs.

full rationale

Walking the derivation chain, the inputs are: (i) the external CM23 theorem that p-widths are achieved by unions of closed geodesics (Theorem 1.2), (ii) the external Morse/CM23 construction of good metrics g_mu on RP^2 with three short geodesics of lengths pi, pi+mu, pi+2mu (Theorem 3.1), and (iii) a genuinely new upper-bound sweepout by Z_2-invariant polynomials whose Crofton/Bezout estimate gives M(F_d) <= 2*pi*d. The final formula is then obtained by a counting argument comparing the set L of width values with the set R of admissible geodesic combinations below the upper bound. No constant is fitted from the target values, no normalization is defined in terms of omega_p, and no uniqueness theorem from the author's own prior work is invoked. The self-citations [MK24] and [MKSS24] appear only in the introduction for context and are not load-bearing. The only notable fragility is Proposition 3.3, which asserts strict monotonicity omega_p(g_mu) < omega_{p+1}(g_mu) on RP^2 by saying the CM23 S^2 proposition 'holds for RP2 instead of S2, with only minor notational changes' and gives no proof; this is a completeness/correctness risk for the counting step, but it is an external imported result, not a circular one, because the proposition is not derived from the claimed formula and does not restate any fitted input. Thus the derivation is not circular and the correct circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the standard Almgren-Pitts min-max machinery and on several external results from [CM23] and [Mor34]. No free parameters are fitted; the auxiliary parameter μ is a perturbation that goes to zero. The main unproved input is the RP^2 version of [CM23, Prop 6.11] (strict monotonicity), which the paper asserts without proof. No new entities are introduced.

assumptions (6)
  • standard math Almgren-Pitts min-max theory and the isomorphism Z_n(M;Z2) ≃ RP∞
    Used in Section 2 to define p-sweepouts, p-widths, and the cohomology condition Φ^*(λ^p) ≠ 0.
  • standard math Continuity of p-widths under C∞ convergence of metrics ([IMN18, Lemma 2.1])
    Needed in Sections 3.2 and 4 to pass from the perturbed metrics g_μ to the standard metric g_std.
  • standard math Theorem 1.2 from [CM23]: p-widths of a surface are achieved by unions of primitive closed geodesics
    Used in Lemma 3.2 part 2 to represent LAP(Π) as a sum of geodesic lengths, though the paper applies it to arbitrary homotopy classes without further justification.
  • standard math Morse's geodesic classification on ellipsoids ([Mor34]) and generic regularity from [CM23, Cor 5.35]
    Basis for Theorem 3.1, which constructs perturbed metrics on RP^2 with only three short simple geodesics γ1, γ2, γ3.
  • domain assumption Proposition 3.3: strict monotonicity ω_p(g_μ) < ω_{p+1}(g_μ) for RP^2, asserted to follow from [CM23, Prop 6.11]
    The paper gives no proof for RP^2; the counting argument in Section 4 depends on having exactly D(d+1)-1 distinct widths.
  • standard math Bezout's theorem and the Crofton formula for length on S^2
    Used in Section 3.2 to bound M(F_d([a])) ≤ 2π d via the number of intersections with great circles.

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Pith. "Pith review of The p-widths of $RP^2$." pith.science (2026). https://pith.science/paper/J3JAQ5FA

@misc{pith2026250113311,
  author       = {Pith},
  title        = {Pith review of: The p-widths of $RP^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J3JAQ5FA}},
  note         = {Machine review of arXiv:2501.13311}
}
abstract

We compute the p-widths, $\{\omega_p\}$, for the real projective plane with the standard metric.

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