{"id":"bb3d4a95-427c-484f-b116-3c910de90c8d","arxiv_id":"2501.03010","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Brownian 2π/3-cone excursions contain a growth-fragmentation process whose law is exactly X_{3/2}, yielding new Brownian proofs of SLE6 and LQG boundary length properties.","lead":"A Brownian motion inside a 120-degree wedge leaves behind a hidden branching collection of pieces, and this paper proves that the pieces evolve exactly like a known growth-fragmentation process. The same result transfers, through a known encoding, to the boundary-length process of SLE6 explorations of a √(8/3) quantum disc.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The branching identification in §5.3, on which Theorem 1.1 rests, is only sketched: the two claims that every fragment lies in the lineage of Z* and that the children of Z* are conditionally independent are asserted with details deferred to ADS22.","rationale":"The reader's weakest assumption identifies exactly the load-bearing point: Section 5.3's two claims are the bridge between the Brownian construction of Z and the growth-fragmentation X_{3/2}. My independent reading confirms this is the least secure part of the argument. The proof of Theorem 5.4, which identifies Z* as the driving process, is substantially detailed, and the analytic ingredients in Sections 3 and 4 are developed at length. But Theorem 1.1 is not a direct corollary of Theorem 5.4 alone; it requires showing that the full collection Z(a) is exactly the cell system generated by Z*. That is precisely what the two sketched claims in Section 5.3 are supposed to establish. The text's own language ('we feel free to only sketch', 'we feel free to skip the details') indicates that the authors are aware these steps are not fully written. The adaptation from ADS22 is not automatic because the present setting involves two-dimensional cone excursions and the coupled Bismut description of Theorem 3.16, rather than the half-plane setting of ADS22. In particular, the conditional independence of children after ranking and after conditioning on the locally largest branch is a delicate point that deserves a full proof. I do not see a separate internal inconsistency in the main analytic results; the concern is that the branching step is under-proved. Therefore the reader's CONDITIONAL verdict is appropriate, and I would not change it. No ad hominem or overstated language is intended: this is a request for a complete proof of a central claim.","tokens_in":57426,"tokens_out":28952,"duration_ms":258916,"concrete_test":"Write out a complete proof of equation (5.8) starting from Theorem 3.16, making explicit: (a) how the children of Z* before time a are generated by the forward and backward cone excursion point processes in the Bismut construction; (b) how the ranking by descending displacement interacts with the conditioning on the event {S((a-b)-)>1/2 S(a-b)}; (c) why the conditional law of each child excursion given its displacement is P^{z_i}; and (d) why no extra dependence or Jacobian factor appears when passing from the Bismut Brownian motions back to the excursion measure n. If the derivation can be completed without additional assumptions, the concern is resolved; if it requires a new argument, Theorem 1.1 is not established by the present text.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 identifies the embedded process Z with the growth-fragmentation X_{3/2}. The proof in Section 5.3 reduces this to two claims: (i) almost surely every fragment of Z lies in the lineage of the locally largest branch Z*, and (ii) the children of Z* are conditionally independent with laws P^{z_i}, equation (5.8). The text says 'we feel free to only sketch the arguments' for (i) and 'we feel free to skip the details to avoid cumbersome technical work' for (ii). These are not peripheral steps: the whole identification of Z with the cell system driven by Z* depends on them. The sketch of (i) uses the assertion that 'the locally largest excursions are always in Z*', which is close to the claim being proved, and the open/closed argument for the set A is only described at a high level. The sketch of (ii), via Bismut's description, invokes two Poisson point processes of forward and backward cone excursions and asserts that, conditional on the displacements z_i, the child excursions are independent with laws P^{z_i}. The ranking by descending size, the coupling between the two Brownian motions in Theorem 3.16, and the conditioning on the event that the branch follows the locally largest evolution are not written out. Since the paper does not provide a full version or a precise theorem in ADS22 with identical hypotheses, the central claim cannot be fully verified from the manuscript as it stands. This is a genuine gap, not a mere stylistic choice.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a growth-fragmentation process Z from the excursions of a correlated planar Brownian motion in a cone of angle 2π/3, and proves that Z has the law of the Bertoin–Curien–Kortchemski growth-fragmentation X_{3/2} driven by the positive 3/2-self-similar Markov process with Lamperti exponent Φ_{3/2} (Theorem 1.1). Via the mating-of-trees encoding, this is translated into the statement that the branching total quantum boundary length process of a space-filling SLE_6 exploration of a √(8/3)-quantum disc has the law of X_{3/2} (Theorem 1.2). Along the way the paper develops a Bismut description of backward cone excursions, proves an explicit joint law for the displacement and duration of 2π/3-cone excursions (Proposition 4.1), derives target-invariance properties (Propositions 1.5 and 4.11), and gives a pathwise Brownian construction of the spectrally positive 3/2-stable process conditioned to stay positive (Theorem 4.16). The main probabilistic ingredients up to Section 5.2 are proved in detail, but the final proof of Theorem 1.1 in Section 5.3 relies on two branching claims that are only sketched and deferred to [ADS22].","tokens_in":57700,"tokens_out":2692,"duration_ms":29803,"significance":"If the main theorem is correct, this is a substantial result: it gives an exact, purely Brownian realization of the growth-fragmentation process X_{3/2} and, through the mating-of-trees theorem, describes the branching total boundary length process of SLE_6 explorations of the √(8/3)-quantum disc. The paper contains several independently valuable explicit computations, including the joint law of displacement and duration of 2π/3-cone excursions (Proposition 4.1), which solves Le Gall's question for the corresponding Lévy measure, and the pathwise construction of the conditioned 3/2-stable process (Theorem 4.16). The derivations in Sections 3 and 4 are largely self-contained and use only Brownian excursion theory and Lévy process techniques, with no fitted parameters beyond the normalization of local time. The main caveat is that the proof of the central branching theorem is not fully contained in the manuscript.","major_comments":[{"comment":"The claim that almost surely every fragment of Z lies in the lineage of the locally largest branch Z* is load-bearing for Theorem 1.1, but the proof is only sketched. The open/closed argument for the set A uses the assertion that 'the locally largest excursions are always in Z*', which is essentially the statement being proved. A fully detailed proof is needed, in particular to justify that the local largest evolution inside e_t^b coincides with the branch Z* rather than merely being consistent with its definition.","section":"Section 5.3, first claim after Theorem 5.5"},{"comment":"The conditional independence of the children of Z* with laws P^{z_i} is the second load-bearing claim of Theorem 1.1, and the paper explicitly says 'we feel free to skip the details to avoid cumbersome technical work'. The proof sketch invokes two Poisson point processes of backward and forward cone excursions under Bismut's description, but it does not verify that the ranking by descending size, the coupling between the two Brownian motions from Theorem 3.16, and the conditioning on the event that the branch follows the locally largest evolution are compatible with the Poisson structure. These details are essential and cannot be replaced by a reference to a different setting without a precise transfer statement.","section":"Section 5.3, equation (5.8)"},{"comment":"The text says that both remaining claims 'are adapted from [ADS22]', but no theorem in [ADS22] with matching hypotheses is stated. Since the present setting involves 2π/3-cone excursions and a correlated two-dimensional Brownian motion, rather than the half-plane excursions of [ADS22], the cited arguments do not automatically apply. The authors should either state and prove a precise transfer result or include full proofs of both claims.","section":"Section 5.3, first paragraph"},{"comment":"The uniqueness and well-definedness of the time t* of the locally largest fragment is asserted by reference to the 'topological arguments presented in [ADS22, Section 2.5]'. Since Z* is the driving process of the entire cell system in Theorem 5.5, this assertion is load-bearing. A proof or a precise statement of the cited result with verifiable hypotheses should be included.","section":"Section 5.2, definition of t*"}],"minor_comments":[{"comment":"In the proof of Proposition 3.15, 'week convergence' should read 'weak convergence'.","section":"Proposition 3.15"},{"comment":"The computation of E^{P^z}[ζ] = ∥z∥_1^2/√3 is referred to as 'a back-of-the-envelope calculation'; since this identity is used in the martingale argument, it would be better to display the short derivation from Proposition 4.1.","section":"Section 5.4, proof of Theorem 5.6"},{"comment":"There is a typographical inconsistency in the definition of ~g_t(b), where the tilde symbol is rendered on the wrong side of the variable; this should be cleaned up for readability.","section":"Section 1.1, equation (1.4)"},{"comment":"In the statement of Proposition 5.2, 'spectrally negative3 2–stable' is missing spacing and correct exponent formatting; similar formatting issues occur in several other displayed statements.","section":"Section 5.1, Proposition 5.2"},{"comment":"The absolute continuity argument for Y(a) is somewhat compressed; in particular, the transition from the density of (Ξ'(a), V(a)) to the joint density F and the subsequent integration over x,y,z could be expanded for clarity, though the mathematical content appears sound.","section":"Section 4.2, Step 1 of Proposition 4.8"}],"recommendation":"major_revision","confidential_remarks":"The two sketched claims in Section 5.3 are not merely technical: they are the backbone of the identification of Z with X_{3/2}. Given that one of the authors is also an author of [ADS22], it should be feasible to supply full proofs or a precise reduction. I would not recommend rejection, but the manuscript in its current form cannot be accepted without closing this gap."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick read: this is a substantial paper with a real new theorem—the growth-fragmentation embedded in 2π/3 Brownian cone excursions is X_{3/2}—and a lot of genuinely new excursion theory along the way. I agree with the conditional verdict. If correct, it connects Brownian cones to the Bertoin–Curien–Kortchemski growth-fragmentation and, via mating-of-trees, to SLE6 on the √(8/3)-quantum disc. The paper is honest about what is proved and what is assumed: the LQG corollaries are explicitly conditional on the external mating-of-trees theorem.\n\nWhat is genuinely new and good: the explicit joint law of displacement and duration under n (solving Le Gall's question for α=π/3, ν=3/2), the Bismut description for backward cone excursions, the Brownian proof of target-invariance, and the pathwise construction of the 3/2-stable process conditioned to stay positive. These are proved in detail; I spot-checked Proposition 4.1 and Lemma 4.3, and the computations are there. The paper is also well-written and gives a clean account of the growth-fragmentation background.\n\nThe soft spot is exactly where the reader and stress-test put it: Section 5.3. Theorem 1.1 rests on the claim that the cell system driven by the locally largest fragment Z* reproduces all of Z, and that the children of Z* are conditionally independent with laws P^{z_i}. The text says 'we feel free to only sketch the arguments' and 'we feel free to skip the details'. These are not peripheral; without them the identification of Z with X_{3/2} does not go through. The open/closed argument for the set A is plausible but the key step—'the locally largest excursions are always in Z*'—is close to the claim itself. The Poisson point process argument for the children is also only sketched. So as the manuscript stands, the central theorem is not fully verified. I do not think this is fatal: the adaptation from ADS22 is likely to work, and the rest of the paper gives me confidence. But the gap is real and needs to be filled or replaced by a precise pointer to a theorem with matching hypotheses.\n\nWho gets value: people working in excursion theory, growth-fragmentations, and SLE/LQG. The Brownian results are of independent interest. My recommendation: send to peer review. A serious referee should spend time on Section 5.3 and ask the authors to either complete the proofs or state exactly which result in ADS22 covers these hypotheses. The paper deserves revision, not rejection.","headline":"Strong new Brownian excursion theory with an exact growth-fragmentation identification, but the final branching step is only sketched and must be completed before Theorem 1.1 is fully established.","tokens_in":58311,"tokens_out":3093,"would_cite":true,"duration_ms":30032,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J65","60G51","60J80","60G52","60J55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the nested-growth process extracted from a Brownian excursion in a $2\\pi/3$ cone has the law of the growth-fragmentation process $X_{3/2}$, and hence describes the branching $\\mathrm{SLE}_6$ boundary length process…","keywords":["growth-fragmentation","Brownian cone excursions","SLE6","Liouville quantum gravity","quantum disc","stable Lévy processes","mating of trees","positive self-similar Markov processes"],"falsifier":"A direct check of the conditional-independence equality asserted in Section 5.3 for the children of the locally largest fragment would settle the main theorem: simulate a $2\\pi/3$ Brownian cone excursion, build the fragments up to a small time $a$, and test whether the joint law of the sub-excursions created by the jumps of $Z^\\star$ factorizes as the product of the laws $P^{z_i}$; any violation would break the identification of $Z$ with $X_{3/2}$.","tokens_in":57171,"feed_emoji":"📐","tokens_out":15536,"duration_ms":138760,"temperature":0.7,"pith_summary":"This paper establishes that a purely Brownian object — the nested family of cone excursions cut out by a planar Brownian path that stays inside a cone of angle $2\\pi/3$ until exiting at the apex — is an exact instance of a previously studied growth-fragmentation process. The process records, at each level of a local time, the total lengths of the intervals into which the excursion has split, and the main theorem identifies its law with $X_{3/2}$, the growth-fragmentation driven by a positive self-similar Markov process of index $3/2$. Because this same correlated Brownian motion is the standard 'mating of trees' encoding of a $\\sqrt{8/3}$-quantum disc decorated with space-filling $\\mathrm{SLE}_6$, the result gives an explicit law for the total quantum boundary length along every branch of an $\\mathrm{SLE}_6$ exploration. The proof is carried out entirely with Brownian excursion theory, so the Liouville quantum gravity statement is a corollary rather than an input. The payoff is that an abstract growth-fragmentation law is realized pathwise by Brownian motion and simultaneously governs the quantum boundary length process.","feed_headline":"Brownian cone paths realize the 3/2 growth-fragmentation","feed_subtitle":"The same process drives branching SLE6 boundary lengths on a pure-gravity quantum disc, proved with Brownian motion alone.","key_machinery":"The load-bearing object is the nested cone-excursion interval construction. For each time $t$ in the excursion, the time-reversed past $e_{t,-}$ has its own forward cone-free local time $\\varsigma_t$; setting $g_t(b)=t-\\tau_t(b)$ and taking $d_t(b)$ to be the first simultaneous running infimum of the future that falls below the whole past interval produces the family of disjoint intervals whose total lengths form $Z(a)$. Two tools carry the identification with $X_{3/2}$: the uniform-time description of the backward cone excursion measure, which gives the total cone-free local time a Lebesgue measure marginal and couples the past and future Brownian paths through a single stopping rule, and a martingale change of measure that turns the spectrally negative $3/2$-stable process conditioned to be absorbed at $0$ into the locally largest fragment $Z^\\star$ with driving Lévy exponent $\\Phi_{3/2}$. The remaining step is a branching argument: every fragment lies in the lineage of $Z^\\star$, and the children of $Z^\\star$ are conditionally independent copies started from their jump sizes.","core_discovery":"The central claim is Theorem 1.1: for an excursion $e$ with law $P^z$ in the $2\\pi/3$ cone, the process $Z$ built from nested intervals has the law of the growth-fragmentation process $X_{3/2}$ started from the total length of the starting point. At every time $t$ of the excursion, the time-reversed past has its own forward cone-free local time; the value $b$ of that local time selects an interval $(g_t(b), d_t(b))$ whose right endpoint is the first time at which both coordinates of the future path reach a running minimum below the whole past trajectory. As $b$ decreases, these intervals split, and $Z(a)$ is the multiset of the total lengths of their displacements. The authors show that this multiset evolves as a Markovian self-similar population, identify its driving Lévy process through the branch that follows the locally largest fragment, and conclude that its law is that of $X_{3/2}$. Through the mating-of-trees correspondence, this becomes the statement that the branching total boundary length process of a space-filling $\\mathrm{SLE}_6$ exploration of a $\\sqrt{8/3}$-quantum disc is $X_{3/2}$.","pith_inferences":["A natural next step, not pursued in the paper, is to track the pair of left and right boundary lengths rather than only their sum; the techniques here suggest that the branching structure for general $\\gamma$ should be a two-dimensional self-similar Markov tree, in line with the authors' conjecture.","The two-Brownian-motion construction of the spectrally positive $3/2$-stable process conditioned to stay positive looks like a two-dimensional analogue of a classical one-dimensional construction; testing whether the same recipe works for other stable indices would show whether the special role of $3/2$ is tied to $\\sqrt{8/3}$ quantum gravity.","Because the proof is entirely Brownian, the construction gives a direct simulation recipe for the growth-fragmentation process and the quantum boundary length process: generate correlated planar Brownian paths and record simultaneous running infima, which could be used to check the law numerically and to estimate quantum disc areas."],"forward_implications":["The branching total quantum boundary length process of a space-filling $\\mathrm{SLE}_6$ exploration of a $\\sqrt{8/3}$-quantum disc has an exact law, $X_{3/2}$, rather than only a scaling-limit description.","The duration of a $2\\pi/3$ cone excursion has an explicit density, which gives the area law for a unit-boundary $\\sqrt{8/3}$-quantum disc and supplies an explicit Lévy measure for Brownian motion subordinated at backward cone times in this case.","For an area-biased quantum disc and a quantum-typical target point, the split of total boundary length between left and right at any exploration time is uniform and independent of the total boundary process.","The branch of the exploration targeting a uniformly chosen time has the law of a spectrally negative $3/2$-stable process conditioned to be absorbed continuously at $0$.","The process $M(n)=3^{-1/2}\\sum_{|u|=n} Z_u(0)^2$ is a uniformly integrable martingale that converges almost surely to the total duration of the excursion, equivalently to the quantum area of the disc."],"supporting_citations":[{"why":"Introduces the growth-fragmentation process $X_{3/2}$ whose law is the stated target of Theorem 1.1.","marker":"[BCK18]"},{"why":"Supplies the cell-system branching argument, adapted in Section 5.3, that every fragment lies in the lineage of the locally largest fragment with conditionally independent children.","marker":"[ADS22]"},{"why":"Provides the mating-of-trees encoding of Liouville quantum gravity by correlated Brownian motion and the forward cone-free local time machinery used to define $Z$.","marker":"[DMS21]"},{"why":"Gives the quantum-disc version of the mating-of-trees theorem that turns Theorem 1.1 into the boundary-length statement for $\\mathrm{SLE}_6$ explorations.","marker":"[AG21]"},{"why":"Contains the martingale change-of-measure lemma used to identify the law of the locally largest fragment $Z^\\star$.","marker":"[LGR20]"},{"why":"Defines backward cone times and the associated stable subordinator; the paper's Proposition 4.1 makes the Lévy measure explicit in the $2\\pi/3$ case.","marker":"[LG87]"}],"fun_headline_variants":["Brownian cone paths give 3/2 growth-fragmentation","Cone excursions encode SLE6 quantum disc splits","Growth-fragmentation X_{3/2} from Brownian cone","Brownian proof for SLE6 boundary length process","Quantum disc: Brownian cones yield 3/2 fragmentation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every fragment of the process descends from the branch following the locally largest sub-excursion, and that the children of that branch are conditionally independent copies with laws determined by their sizes; the paper sketches rather than fully proves these two claims.","fun_headline_variants_meta":{"raw":{"variants":["Brownian cone paths give 3/2 growth-fragmentation","Cone excursions encode SLE6 quantum disc splits","Growth-fragmentation X_{3/2} from Brownian cone","Brownian proof for SLE6 boundary length process","Quantum disc: Brownian cones yield 3/2 fragmentation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000234,"raw_usage":{"total_tokens":1503,"prompt_tokens":959,"completion_tokens":544,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":461}},"tokens_in":575,"tokens_out":544,"duration_ms":5580,"temperature":1.0,"reasoning_tokens":461,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:58:57.998724+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check of the conditional-independence equality asserted in Section 5.3 for the children of the locally largest fragment would settle the main theorem: simulate a $2\\pi/3$ Brownian cone excursion, build the fragments up to a small time $a$, and test whether the joint law of the sub-excursions created by the jumps of $Z^\\star$ factorizes as the product of the laws $P^{z_i}$; any violation would break the identification of $Z$ with $X_{3/2}$.","supporting_citations":[],"review_version":1}