{"id":"6b6699d9-1646-4195-9565-a299d5aced36","arxiv_id":"2501.07966","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For grids (p/3^n)Z+(r/3^n) with p rational and r irrational, the Peano curve's horizontal coordinate x has quadratic variation 3^{floor(-log_3 p)}p(1-(3/4)3^{floor(-log_3 p)}p)t.","lead":"The horizontal component of the Peano curve has quadratic variation along certain natural partitions that depends on the scale of the partition, unlike Brownian motion. The paper proves an explicit formula for that variation and shows the component has a local time whose crossing-count normalization is not smooth.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Local time via crossings is proven only along the subsequence c=3^{-n}p; the paper does not justify passing from this subsequential limit to the stated c->0+ limit in (56).","rationale":"The reader's weakest assumption focuses on the dependence of (49) on the unpublished coauthored preprint [BDL23]. That is a legitimate external-validity risk. However, the most concrete and internal load-bearing gap is that even granting (49), Section 3.1 only establishes the normalized-crossing weak limit along the geometric subsequence c=3^{-n}p, while the theorem states a limit for all c->0+. The step from a subsequence to the full limit is not justified by any argument in the manuscript, and the self-similarity of the Peano curve does not trivially supply it because phi(3c)=3*phi(c). This is a fixable but real logical gap in the advertised local-time representation. The other issues (non-coprime choice of p',q' in Theorem 2.3, missing extension of the dyadic local-time construction to all t in Theorem 3.1) are also present, but the subsequential-versus-full-limit issue is the one most directly tied to the paper's central claim about normalized interval crossings. The reader's verdict of CONDITIONAL remains appropriate; no change is needed.","tokens_in":19993,"tokens_out":45263,"duration_ms":392069,"concrete_test":"Take p=1 and g=1. Compute phi(c)*integral(n_{z,c}(x,[0,1]))dz for c_n=3^{-n} and for intermediate widths c'_n=1/(2*3^n) for several n, using the self-similar recursive structure of the Peano curve to evaluate crossing counts. If the two sequences do not converge to the same limit, (56) fails. Alternatively, re-read Section 3.1 and locate any explicit estimate bounding the difference between consecutive widths; if no such argument appears, the passage from subsequential convergence to the full c->0+ limit is missing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3.1, equation (57) is asserted as a limit for all c->0+, but the derivation starts from (49), which is a limit along the fixed geometric sequence c_n=3^{-n}p. The text rewrites the resulting convergence with C_p=C_{3^{-n}p}, so one obtains convergence only for c=3^{-n}p, not for arbitrary c. No interpolation argument is supplied between consecutive powers of 3: the manuscript does not show that phi(c)*integral(n_{z,c}g)dz and phi(3^{-n}p)*integral(n_{z,3^{-n}p}g)dz have the same limit as c ranges over [3^{-n-1}p,3^{-n}p]. Scaling alone does not fill the gap, since phi(3c)=3*phi(c) even though C_{3c}=C_c. Thus the advertised characterization of local time as a weak limit of normalized interval crossings for all widths is unproven; only subsequential convergence along the special grids is established. This concern is internal: it remains even if the external theorem [BDL23] used for (49) is entirely correct.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":20216,"tokens_out":33531,"duration_ms":284951,"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":[{"comment":"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.","section":"§3.1, Eqs. (56)–(57)"},{"comment":"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.","section":"§3.1 / Theorem 3.1"}],"minor_comments":[{"comment":"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.","section":"Theorem 2.3 statement"},{"comment":"The line 'ν1, ν1 ∈ [-N,N]' should read 'ν1, ν2 ∈ [-N,N]'.","section":"§2.3, second case"},{"comment":"The sentence 'It is is smaller from (43) by1' contains a typo and is grammatically tangled; please rephrase.","section":"§2.3, after Eq. (44)"},{"comment":"The mode of convergence in t (pointwise, uniform, or weak-star) is not specified; please clarify.","section":"§3.1, Eq. (49)"},{"comment":"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.","section":"§3.1, dependence on [BDL23]"},{"comment":"The phrase 'These two features distinct the horizontal component' should read 'distinguish'.","section":"Abstract"},{"comment":"The simulation results are described as available upon request; a public repository would strengthen reproducibility.","section":"Introduction, simulation table"}],"recommendation":"major_revision","confidential_remarks":"The quadratic-variation theorem appears sound and self-contained; the formula matches the simulations. The main obstacle is the unproven interpolation in Section 3.1; if the authors can either prove the full limit or honestly weaken the claim, the paper may be acceptable. The dependence on [BDL23] is a second concern but less central: if that preprint fails, the crossing-count representation collapses, though the density existence in Theorem 3.1 would survive. I recommend major_revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The worthwhile part of this paper is Theorem 2.3: the horizontal component of the Peano curve has quadratic variation along Lebesgue partitions for grids (p/3^n)Z + (r/3^n), and the limit is an explicit, p-dependent linear function of t. The modular-counting proof is intricate and, as far as I can tell, sound. No parameters are fitted; the result follows from the self-similarity equations. That is a genuine deterministic counterexample to the Brownian intuition that quadratic variation along Lebesgue partitions should be grid-independent. The recursive construction of the local time density in Section 3 is also explicit and gives L_t^z bounded by 1; that part looks like a reasonable occupation-density argument.\n\nThe soft spots are real. The advertised claim in (56) and (57), that local time is a weak limit of normalized crossing counts as the interval width c goes to 0, is not proven for all c. What is proven via (49) is convergence along the subsequence c = 3^{-n}p. The paper then writes the limit with C_{3^{-n}p} = C_p, defines φ(c) = c/C_c, and asserts (57) for all c→0+. But φ(3c)=3φ(c), and there is no relation between crossing counts at width 3c and at width c that would let you interpolate between powers of 3. So the stated limit for arbitrary c is unjustified; only subsequential convergence along the special grids is established. This is internal and remains a gap even if the external theorem [BDL23] used for (49) is correct.\n\nThere is a second, smaller gap in Theorem 3.1: the local time is constructed on the dyadic times k/9^N, and the passage from those to all t is skipped. A monotone or compactness argument might close it, but it is not written. Also, the assumptions in Theorem 2.3 (p', q' not divisible by 3, θ not a multiple of any 3^m/p') are stated hastily, and the proof has typos. These are fixable issues, not fatal.\n\nThe citation of [BDL23] is defensible if that preprint is correct, but it is a coauthor's unpublished paper used as a black box, so the crossing-count part is fragile. Still, the quadratic variation theorem stands on its own and deserves referee time. The paper is useful for people working on path regularity and deterministic local times; the finance motivation is window dressing.\n\nSend it to peer review. The referee should ask for the missing interpolation argument for the c→0 limit and for the extension of the local time to all times. Those look repairable without changing the main quadratic-variation result.","headline":"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.","tokens_in":20745,"tokens_out":3114,"would_cite":true,"duration_ms":30534,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["26A27","28A80","60J55"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["horizontal component","Peano curve","quadratic variation","Lebesgue partitions","local time","occupation measure","interval crossings","regularisation by noise"],"falsifier":"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.","tokens_in":19824,"feed_emoji":"📐","tokens_out":14829,"duration_ms":123497,"temperature":0.7,"pith_summary":"The paper establishes a precise variation theory for a deterministic fractal path: the horizontal component of the Peano curve has quadratic variation along the Lebesgue partitions generated by grids of the form $3^{-n}p\\mathbb{Z}+3^{-n}r$, and the limiting value is $3^{\\lfloor-\\log_3 p\\rfloor}p(1-\\tfrac34\\,3^{\\lfloor-\\log_3 p\\rfloor}p)t$, which changes with the rational grid parameter $p$. This contrasts with Brownian motion and semimartingales, where the quadratic variation along Lebesgue partitions is the same for every grid. The same argument produces a local time for the path: the occupation measure of $x$ has a Lebesgue density bounded between $0$ and $1$, and that density is the weak limit of normalized numbers of interval crossings, with a normalization that is not a smooth function of the interval width. These two features make the Peano component a concrete deterministic instance of regularisation by noise.","feed_headline":"Proved: grid choice changes Peano curve's quadratic variation","feed_subtitle":"A deterministic square-filling path has a well-defined variation limit that changes with the partition's grid parameter.","key_machinery":"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 $0\\le L_t(z)\\le1$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the definition of the Peano curve and the nine self-similarity relations (3)-(11) that drive the quadratic-variation recursion.","marker":"[Sag94]"},{"why":"Provides the Theorem 2.2 and Remark 3.9 results that convert truncated variation into the crossing-count convergence used for the local-time representation.","marker":"[BDŁ23]"},{"why":"Theorem 57 establishes the modular inverse used to count the integer shifts satisfying the fractional-part inequalities in the proof of Theorem 2.3.","marker":"[HW75]"},{"why":"Introduces Lebesgue partitions for Brownian motion and gives the grid-independent quadratic variation that the Peano component is contrasted with.","marker":"[CLJPT81]"}],"fun_headline_variants":["One path, many quadratic variations","Peano curve's variation limit depends on grid scale","Deterministic Peano path has local time, unlike Wiener","Grid choice changes Peano curve's quadratic variation","Peano curve: variation and local time distinct from Wiener"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["One path, many quadratic variations","Peano curve's variation limit depends on grid scale","Deterministic Peano path has local time, unlike Wiener","Grid choice changes Peano curve's quadratic variation","Peano curve: variation and local time distinct from Wiener"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000361,"raw_usage":{"total_tokens":1943,"prompt_tokens":933,"completion_tokens":1010,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":936}},"tokens_in":549,"tokens_out":1010,"duration_ms":9464,"temperature":1.0,"reasoning_tokens":936,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:32:04.225375+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}