REVIEW 2 major objections 7 minor 7 references
Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve)
T0 review · 2 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read Theorem 2.3 proves that the horizontal component of the Peano curve has quadratic variation along the Lebesgue partitions for grids $3^{-n}p\mathbb{Z}+3^{-n}r$, with a limit $3^{\lfloor-\log_3 p\rfloor}p(1-\tfrac34\,3^{\lfloor-\log_3…
desk verdict The quadratic-variation side is a solid, parameter-free derivation; the local-time crossing-count representation has a real gap, since the limit is only shown along one geometric subsequence. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The engine is a one-step recursion for the quadratic variation along a grid. From the nine self-similarity relations (3)-(11), the paper obtains $[x]^{c,r}_1=\tfrac13([x]^{3c,3r}_1+[x]^{3c,3r-1}_1+[x]^{3c,3r-2}_1)$ plus indicator correction terms of size $c^2$. Iterating through $k$ steps resolves $[x]^{p/3^n,r/3^n}_1$ into a sum over $3^k$ subgrid copies plus an exact geometric series; when $\theta$ is irrational, every correction term contributes. The remaining task is counting how many of the $3^k$ fractional parts $\{\theta-(3^nq'/(3^kp'))M\}$ fall below a threshold, done with a modular-inverse argument from [HW75]. The count contributes $A-A^2$ and the geometric series adds $\tfrac14 A^2$, where $A=3^{\lfloor-\log_3 p\rfloor}p$, giving the asserted $A(1-\tfrac34 A)t$. Monotonicity in $t$ then extends the limit from times $i/9^m$ to all $t\in[0,1]$. For local time, the recursive construction (53) builds $L_t$ on the $9^{-N}$ grids from the same self-similarity data and yields the bound $0\le L_t(z)\le1$.
What would settle it
A concrete check: compute $[x]^{3^{-n},3^{-n}\sqrt{2}}_1$ and $[x]^{2\cdot3^{-n},2\cdot3^{-n}\sqrt{2}}_1$ for $n=1,\dots,12$; Theorem 2.3 predicts convergence to $1/4$ and $1/3$, respectively, and a failure to approach these numbers would contradict the theorem. Separately, test (49) with $g=\mathbf{1}_{[1/3,2/3]}$: the left side should converge to $\int_0^t\mathbf{1}_{[1/3,2/3]}(x_s)\,ds$, and if it does not, the crossing-count representation fails.
Extended reading notes
Core claim
The central claim is Theorem 2.3: for $x$ defined by (1), with $p=p'/q'$ where $p'$ and $q'$ are positive integers not divisible by $3$, and with $r=\theta p$ where $\theta$ is not a multiple of any $3^m/p'$, one has $\lim_{n\to+\infty}[x]^{3^{-n}p,3^{-n}r}_t = 3^{\lfloor-\log_3 p\rfloor}p(1-\tfrac34\,3^{\lfloor-\log_3 p\rfloor}p)t$ for every $t\in[0,1]$. The value depends on the grid scale $p$, so the quadratic variation is not a universal constant of the path. The paper also proves, in Theorem 3.1, that the occupation measure of $x$ with respect to Lebesgue measure has a density $L_t(z)$ taking values in $[0,1]$, satisfying the occupation formula, and it identifies this local time as the weak limit of normalized crossing counts integrated against continuous test functions.
Load-bearing premise
The load-bearing premise is that the quoted result from the preprint [BDŁ23] — that normalized numbers of interval crossings of a continuous path converge weakly to occupation-measure integrals — is true and applies to $x$; if it fails, the crossing-count representation of the local time collapses, even though the density in Theorem 3.1 might still exist.
Editorial extensions
If this is right
- For $p=1$ the predicted limit is $t/4$, while for $p=2$ it is $t/3$; the same path has different quadratic variations for different grid parameters.
- The occupation measure of the horizontal component is absolutely continuous with respect to Lebesgue measure, and its density is bounded by $1$.
- The local time is recoverable from pure crossing counts of small intervals $[z-c/2,z+c/2]$, with a normalization factor that is not a smooth function of $c$.
- These properties separate the deterministic Peano component from Wiener-process trajectories, the standard stochastic model in finance, where Lebesgue-partition limits and local-time normalizations are smooth and grid-independent.
Reading between the lines
- A natural next step, not taken in the paper, is to classify the rational-$r$ cases excluded by Theorem 2.3; the paper's own simulation table hints at other finite limits, such as $0.3125t$ for $c_n=1/(2\cdot3^n)$ and $r_n=1/(4\cdot3^n)$.
- If the quoted crossing-count theorem from [BDŁ23] holds up, the Peano component becomes a deterministic toy model in which the choice of rebalancing grid changes the effective variation; the financial-model motivation is noted in the abstract but not developed.
- The same nine-interval self-similarity structure may transfer to other space-filling curves, yielding a family of deterministic paths with computable, grid-dependent quadratic variation.
Formalized claims in Lean
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Claim #1: The central claim is Theorem 2.3: for $x$ defined by (1), with $p=p'/q'$ where $p'$ and $q'$ are positive integers not divisible by $3$, and with $r=\theta p$ where $\theta$ is not a multiple of any $3^m/p'$, one has $\lim_{n\to+\infty}[x]^{3^{-n}p,3^{-n}r}_t = 3^{\lfloor-\log_3 p\rfloor}p(1-\tfrac34\,3^{\lfloor-\log_3 p\rfloor}p)t$ for every $t\in[0,1]$. The value depends on the grid scale $p$, s
/-- @claim 1 The central claim is Theorem 2.3: for $x$ defined by (1), with $p=p'/q'$ where $p'$ and $q'$ are positive integers not divisible by $3$, and with $r=\theta p$ where $\theta$ is not a multiple of any $3^m/p'$, one has $\lim_{n\to+\infty}[x]^{3^{-n}p,3^{-n}r}_t = 3^{\lfloor-\log_3 p\rfloor}p(1-\tfrac34\,3^{\lfloor-\log_3 p\rfloor}p)t$ for every $t\in[0,1]$. The value depends on the grid scale $p$, s -/ def central_claim : Prop :=
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This paper studies the horizontal coordinate x of the Peano space-filling curve. The authors define the quadratic variation [x]^{c,r}_t along Lebesgue partitions generated by grids cZ+r, and prove (Theorem 2.3) that for c_n=3^{-n}p and r_n=3^{-n}r, with p a positive rational whose numerator and denominator are not divisible by 3 and r such that r/p avoids a certain set of rationals, the limit is C_p t for an explicit p-dependent constant C_p. The proof uses a one-step recursion (27), a k-step recursion (33), and a modular counting argument in Section 2.3. The paper then proves (Theorem 3.1) that the occupation measure of x has a density L^z_t (local time) bounded by 1, via an explicit recursive construction on 9-adic grids. Finally, Section 3.1 claims that this local time is the weak limit of normalized interval-crossing counts with a non-smooth normalization φ(c).
Significance. If the results hold, Theorem 2.3 provides a rare explicit deterministic example where the quadratic variation along Lebesgue partitions depends on the grid parameter p, in contrast to Brownian motion and semimartingales. The quadratic-variation proof is elementary and self-contained, using only the self-similarity of the Peano curve and modular arithmetic, and the formula is corroborated by the simulation table. The local time construction is also explicit and yields the density bound L≤1. The main advertised consequence, that local time is the weak limit of normalized crossing counts for all interval widths, currently rests on an unproven interpolation step.
major comments (2)
- [§3.1, Eqs. (56)–(57)] The passage from (49) to (57) is not justified. Equation (49) is a limit along the geometric subsequence c=3^{-n}p for each fixed p, while (57) asserts a full limit as c→0+ for arbitrary widths. The definition φ(c):=C_c^{-1}c uses the coefficient C_c, but no argument shows that the integrals ∫φ(c)n_{z,c}g dz converge to the same value as c ranges continuously to 0; in particular, writing an arbitrary small c as 3^{-n}p forces p=3^n c to vary with n, and (49) supplies no uniformity over p. Consequently, the advertised statement that local time equals the weak limit of normalized crossing counts for all interval widths is not proven; only subsequential convergence along the grids 3^{-n}p is established. Please either supply an interpolation or monotonicity argument, or weaken (56)–(57) to a sequential statement.
- [§3.1 / Theorem 3.1] The recursive construction defines L^z_t only for t of the form k/9^N. The proof never explains how L^z_t is obtained for arbitrary t∈[0,1], nor why the occupation-measure identity (50) holds there. Since the left side of (51) is continuous in t, a limiting argument is available, but it is absent; without it, the local time in (50) and its use in (56) are not well-defined for all t. Please add the extension argument.
minor comments (7)
- [Theorem 2.3 statement] The condition p=p'/q' with 3∤p'q' excludes rationals such as p=3/2, although the abstract and introduction say p is a positive rational; since powers of 3 can be factored out (Remark 2.1), the statement should be generalized or explicitly restricted.
- [§2.3, second case] The line 'ν1, ν1 ∈ [-N,N]' should read 'ν1, ν2 ∈ [-N,N]'.
- [§2.3, after Eq. (44)] The sentence 'It is is smaller from (43) by1' contains a typo and is grammatically tangled; please rephrase.
- [§3.1, Eq. (49)] The mode of convergence in t (pointwise, uniform, or weak-star) is not specified; please clarify.
- [§3.1, dependence on [BDL23]] The proof of (49) relies on Theorem 2.2 and Remark 3.9 of the coauthored preprint [BDL23], which is not yet published. The authors should either restate the needed results or flag this dependency more prominently.
- [Abstract] The phrase 'These two features distinct the horizontal component' should read 'distinguish'.
- [Introduction, simulation table] The simulation results are described as available upon request; a public repository would strengthen reproducibility.
Circularity Check
No significant circularity: the main results are derived from the defining self-similarity of the Peano x-component, with no fitted parameters and no reduction of the conclusions to their inputs.
full rationale
Theorem 2.3 is derived self-containedly. The one-step recursion (27) and the k-step recursion (33) follow from the explicit self-similarity relations (3)-(11) together with the identity [x]_{s,t}^{c,r} = (l-k)c^2 (13). The limit computation in Section 2.3 is a counting argument over the index set M in {0,1,...,3^k-1}; no parameter is fitted, and the claimed value is not fed back into the proof. The extension from t=1 to all t uses the monotonicity of the liminf and limsup of quadratic variations and the limiting values on the dense set {i/9^m}, which is again derived from self-similarity. Theorem 3.1 is also a direct recursive construction of the occupation density via (53), with equality (51) verified by changes of variables using (3)-(11); it does not presuppose the local time it constructs. The crossing representation in Section 3.1 uses (49), imported from [BDL23], a general theorem on local times and normalized interval crossings for continuous paths. That theorem is not tailored to the Peano curve and does not assume Theorem 2.3 or Theorem 3.1; the paper verifies one of its inputs, (48), using its independently proved Theorem 2.3. The involvement of a coauthor of the cited preprint makes this a self-citation, but it is independent mathematical support rather than circular reasoning, because the cited theorem's assumptions do not contain the paper's target conclusion. A separate concern, that (57) is derived from convergence along c=3^{-n}p without explicitly justifying the passage to all c tending to 0, is a correctness or gap risk, not a circularity of the derivation chain.
Assumptions & free parameters
assumptions (5)
- domain assumption Peano curve self-similarity equations (3)-(11) define x and its ranges on each ninth interval.
- domain assumption For c in (1/3,1] the quadratic variation along a Lebesgue partition is determined by condition (40), i.e. [x]^{c,r}_1 is 0, c^2, or 2c^2 based on the position of the grid relative to the range [0,1].
- standard math Existence of modular inverse Q: if q' is coprime to 3^N p', then q' is invertible modulo 3^N p' (Hardy-Wright Theorem 57).
- domain assumption The external theorem [BDL23, Theorem 2.2 and Remark 3.9] relating normalized interval-crossing counts to occupation measure applies to the Peano horizontal component.
- standard math Extension from 3-adic intervals to all Borel sets via monotone class arguments.
Cite this review
Pith. "Pith review of Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve)." pith.science (2026). https://pith.science/paper/QNCFIAS6
@misc{pith2026250107966,
author = {Pith},
title = {Pith review of: Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve)},
year = {2026},
howpublished = {\url{https://pith.science/paper/QNCFIAS6}},
note = {Machine review of arXiv:2501.07966}
}
abstract
We show that the horizontal component of the Peano curve has quadratic variation equal the limit of quadratic variations along the Lebesgue partitions for grids of the form $3^{-n}p\mathbb{Z}+3^{-n}r$, $n=1,2,\ldots$, where $p$ is a rational number, while $r$ is irrational number, but the value of such quadratic variation depends on $p$. This also yields that the horizontal component of the Peano curve is an example of a deterministic function possessing local time (density of the occupation measure) with respect to the Lebesgue measure, whose local time can be expressed as the limit of normalized numbers of interval crossings by this function but the normalization is not a smooth function of the width of the intervals. These two features distinct the horizontal component of the Peano curve from the trajectories of the Wiener process, which is widely used in financial models.
Figures
Figures from the paper (2 more)
Reference graph
Works this paper leans on
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Reviewed August 10, 2026 · model on record in the stance chip above.
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