{"id":"b4debafc-b088-4069-a2b4-d5671d2b3b32","arxiv_id":"2507.20487","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A direct proof that the equal-time multipoint formula for the KPZ fixed point reproduces the extended Airy kernel determinant that defines the parabolic Airy process.","lead":"Two mathematical formulas that supposedly describe the same random curve in the Kardar-Parisi-Zhang universality class are shown to agree exactly. The proof introduces a generalized integration identity that could simplify other computations in random matrix theory and KPZ.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"I read the paper as proving the equivalence of two formulas for the same process, not as introducing a new process. The central claim holds if Theorem 1.1 is correct in the equal-time case, if the contour and series interchanges in Section 2 are valid, and if the operator identity in Section 3 is correct. I checked the operator identity in detail: with S=(I+B)^{-1}, det(I+L)=det(I-SA)=det(I+B-A), and the combination B-A indeed reproduces the conjugated extended Airy kernel, with B contributing exactly the i<j part. The cancellations behind Lemma 2.3 are genuine, the residue cascade enforcing n1>=...>=nm is internally consistent, and the prefactors in Lemma 2.4 match the 1/k_i! in the final Fredholm expansion. The main residual risk is the black-box theorem from [Liu22a] and the cited interchange of summation and integration; the reader identified the same dependency. Since both are external or referenced rather than internally contradicted, I do not find a load-bearing flaw, and the ACCEPT verdict with moderate confidence remains appropriate.","tokens_in":83,"tokens_out":37679,"duration_ms":937334,"concrete_test":"For m=2, directly compute both sides of Lemma 2.3 for n=(1,0) and n=(0,1): perform the H_0 z-integral, the v-integral residue cascades, and verify that \\hat D_{0,1}=0 and \\hat D_{1,0} equals the right-hand side of (2.23). This isolates the z-integral simplification and the n-sum interchange on which Proposition 2.1 rests.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The proof that (1.10) equals (1.1) is internally coherent: Lemma 2.3 reduces the z-integrals by cancelling the z_i^{-n_{i+1}} poles against the -z/(1-z) v-contour coefficients, Lemma 2.4 organizes the resulting integrals into the Fredholm determinant (2.7), and Section 3 factors the kernel through the nilpotent upper-triangular operator B, giving det(I+B-A), which is the conjugated extended Airy kernel after the i<j orientation switch. The weakest point is external: Theorem 1.1 from [Liu22a] is used as a black box, and the interchange of the n-sum with the z-contour integrals in (2.20)-(2.21) is referred to [LZ25] rather than reproved. If either failed, Proposition 2.1 and the final equality would fail. Nothing in the text indicates such a failure, and the algebraic reductions are consistent, including the combinatorial prefactors.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that the equal-time multipoint distribution formula for the KPZ fixed point with narrow-wedge initial condition obtained in [Liu22a], specialized to τ_i = 1 and α_1 < ... < α_m, coincides with the extended Airy kernel Fredholm determinant (1.1) that defines the parabolic Airy process. The proof evaluates the z-contour integrals in the [Liu22a] formula (Lemma 2.3), reorganizes the remaining u-integrals using a generalized Andreief identity (Lemma 2.4), obtains a new contour-integral Fredholm determinant (Proposition 2.1), and then factorizes that kernel through L2(R+) and a strictly upper-triangular nilpotent operator B to identify the result with the conjugated extended Airy kernel. The paper also records several intermediate formulas for the parabolic Airy process and proves a generalized Andreief identity that is independent of the process.","tokens_in":17876,"tokens_out":38867,"duration_ms":357599,"significance":"If the main theorem holds, the paper closes a gap between two independent published formulas for the same object and provides a direct algebraic verification that the equal-time [Liu22a] formula indeed reproduces the original definition of the parabolic Airy process. The proof is largely self-contained modulo Theorem 1.1, which is a published result used explicitly as an input, and the algebraic reductions in Lemmas 2.3, 2.4, and Section 3 are coherent. The generalized Andreief identity is proved in full, and the paper honestly flags its dependence on convergence details from [Liu22a, LZ25]. No ad hoc parameters are introduced, and the final claim is falsifiable through the explicit equality of two contour-integral formulas.","major_comments":[],"minor_comments":[{"comment":"The equality of det(I+B-A) with the conjugated extended Airy determinant (1.1) is asserted in a single sentence (\"after a conjugation and a simple notation change\"), but this is the main conclusion of the paper. Please spell out the conjugation, for example by setting λ = x - β_i - α_i^2 and θ = y - β_j - α_j^2, and show explicitly how the (i,j) entries of I+B-A combine to reproduce the piecewise definition of K_ext, in particular how the strictly upper-triangular B corrects the i<j entries of A. As written, a reader cannot verify the orientation switch.","section":"Section 3, after Lemma 3.2"},{"comment":"The claimed Hadamard-inequality proof of absolute convergence of the series (2.5) is not immediate, because for i = 1 the kernel K(1,z;1,u) decays only like |u|^{-1} along Γ_L rather than super-exponentially in the second variable. Please either give a trace-class or Hilbert-Schmidt factorization before using the series expansion, or define det(I+K) via the absolutely convergent expression obtained from (2.21) and from the cited estimates in [Liu22a, LZ25], and correct the sentence accordingly.","section":"Section 2.1, after (2.5)"},{"comment":"The derivation of (2.25) suppresses the cancellation between the prefactors (1-z_i^{-1})^{n_{i+1}} and the surviving contour coefficients (-z_i/(1-z_i))^{n_{i+1}} coming from the Γout_{i,R} integrals. Please display this cancellation explicitly; as printed, (2.25) appears to omit the R-contour coefficients, and the reader must reconstruct the analyticity argument by hand.","section":"Lemma 2.3(i), around (2.25)"},{"comment":"The Fredholm determinant in (2.5) is written on the contour Γ_L defined in (2.4), while the proof of Proposition 2.1 in (2.50) uses the contour Γ1_L from Definition 2.2. Please state explicitly that Γ_L may be taken to be Γ1_L, or explain that the determinant is invariant under admissible deformations of the left contour.","section":"Sections 2.1 and 2.3"},{"comment":"In the statement of Lemma 1.2, \"µ be a measure on Γ\" should presumably read \"measure on X\"; the symbol Γ has not been defined at that point.","section":"Lemma 1.2"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is within the scope of the journal and the central mathematical claim appears sound. The requested revisions are local: the final identification with the extended Airy kernel should be written out, and the convergence justifications for the Fredholm determinant series should be corrected or made more precise. I do not see a need for another full round of refereeing after those changes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a careful, honest equivalence proof. The authors show that Liu's equal-time multipoint formula for the KPZ fixed point (Liu22a) reduces, after contour surgery, to the original extended Airy kernel determinant that defines the parabolic Airy process. It does not resolve a long-open question and it does not produce a new physical phenomenon, but it closes a real gap: the two formulas were believed to agree, and now the chain of simplifications is written down in detail.\n\nThe genuinely new pieces are the intermediate contour-integral formulas, especially the operator factorization through the nilpotent matrix B that turns the determinant into det(I+B−A), and the generalized Andreief identity in Lemma 1.2. The proof of that identity is self-contained and correct; the m=1 case is the classical Andreief identity. The reduction from the multi-contour mess to a Fredholm determinant is coherent, and the final conjugation to the extended Airy kernel checks out at the operator level.\n\nThe soft spots are real but minor. The paper takes Theorem 1.1 of Liu22a as a black box; if that theorem failed, everything downstream fails. That is a reasonable starting point, since the theorem is published and the authors state plainly that they are not reproving it, but readers should know the conclusion is conditional on it. The interchange of the n-sum with the z-contours in (2.20)–(2.21) is not proved here and is referred to LZ25. The contour deformations in Lemma 2.3 and Lemma 3.1 are plausible, and I did not find a pole-crossing problem, but a few are sketched rather than fully justified; a referee should ask for those details to be filled in. None of these look load-bearing.\n\nThe citation pattern is fine. Liu22a is the natural input, the dependence is flagged transparently, and the rest of the references are standard.\n\nWho should read it: people working on integrable probability and the KPZ fixed point who need multipoint formulas or want to convert between the two known formulas. It is not a paper for a general audience. I would send it to a serious referee. My verdict matches the reader's: accept after minor revision, with the convergence details tightened.","headline":"A solid, honest equivalence proof that Liu's equal-time KPZ fixed point formula really is the parabolic Airy process; it deserves peer review, not a desk reject.","tokens_in":18427,"tokens_out":2659,"would_cite":true,"duration_ms":27154,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the equal-time multipoint distribution formula for the KPZ fixed point with narrow-wedge initial condition gives the same parabolic Airy process as the extended Airy kernel Fredholm determinant.","keywords":["parabolic Airy process","KPZ fixed point","multipoint distribution","Fredholm determinant","extended Airy kernel","contour integral","Andreief identity","narrow wedge initial condition"],"falsifier":"Compute both sides for the first non-trivial case, $m=2$, at specific ordered points such as $\\alpha_1=0$, $\\alpha_2=1$, $\\beta_1=\\beta_2=0$: evaluate the Fredholm determinant of $K$ in Proposition 2.1 by truncating its series expansion and evaluate the extended-Airy-kernel determinant in (1.1) by a separate numerical scheme; any discrepancy beyond numerical error would disprove the stated equality.","tokens_in":17526,"feed_emoji":"📐","tokens_out":9610,"duration_ms":89695,"temperature":0.7,"pith_summary":"The paper establishes that two seemingly different formulas for the finite-dimensional distributions of the parabolic Airy process are actually the same: the equal-time multipoint formula obtained from the KPZ fixed point (Theorem 1.1) is shown, after contour-integral rewriting, to equal the original Fredholm determinant with the extended Airy kernel. A reader would care because the parabolic Airy process is conjectured to be a universal limit of models in the KPZ universality class with narrow-wedge data, and a multipoint formula is only usable if it is known to describe the same process as the defining kernel. The proof supplies a chain of explicit identities, evaluating auxiliary integrals by residues, reorganizing determinants through a generalized determinant identity (Lemma 1.2), and factoring the final kernel into upper-triangular pieces. It also yields several new contour-integral formulas for the process along the way.","feed_headline":"Two formulas for the parabolic Airy process proven identical","feed_subtitle":"A contour-integral determinant from the KPZ fixed point equals the extended Airy kernel determinant.","key_machinery":"The machinery is the factorization of the kernel as $L=-A_1(\\sum_{k=0}^{m-1}(-1)^k B^k)A_2$, where $B$ is strictly upper-triangular, meaning it vanishes on and below the diagonal; this makes $B^m=0$ and $\\det(I+B)$ harmless inside the Fredholm determinant, so $\\det(I+L)=\\det(I+B-A)$, and the latter is the conjugated Airy kernel determinant. The generalized Andreief identity (Lemma 1.2) is the combinatorial tool that replaces determinants of single-variable integrals, $\\det(\\int_X A_i(x)B_j(x)\\,d\\mu(x))$, by integrals over products of the block determinants $\\det[B_i(x_j)]_{i,j\\in I_k}$, and it is what lets the paper reorganize the $m$-level contour integrals from the KPZ fixed point formula.","core_discovery":"The central claim is that formula (1.10), the equal-time case of the KPZ fixed point multipoint distribution with narrow-wedge initial condition, coincides with (1.1), the extended Airy kernel Fredholm determinant that defines the parabolic Airy process, for every ordered $m$-tuple $\\alpha_1<\\cdots<\\alpha_m$. The argument passes through an intermediate Fredholm determinant $\\det(I+K)$ (Proposition 2.1) whose kernel $K$ is built from contour integrals of the functions $f_i(w)=e^{-w^3/3+\\alpha_i w^2+\\beta_i w}$. By evaluating the auxiliary integrations, conjugating the kernel, and using $\\det(I+AB)=\\det(I+BA)$, the determinant is reduced to the extended Airy kernel determinant, with the Gaussian factors in (3.16)--(3.17) providing the conjugation and the final identification.","pith_inferences":["The same residue-and-factorization pattern should apply to the unequal-time KPZ fixed point formula, where time differences enter only through extra Gaussian factors in the $F_i$; proving that would give a direct, limit-free derivation of the alternative multitime formula discussed in the paper.","The strict upper-triangular structure of $B$ suggests a hierarchical decomposition of the $m$-point distribution: each added spatial point contributes a new layer conjugated by Gaussian kernels, which could make asymptotic regimes such as widely spaced $\\alpha_i$ tractable from the formula alone.","Replacing the narrow-wedge initial condition by other initial data in the KPZ fixed point would likely require a different $F_1$, and the proof indicates exactly where the Gaussian-factor structure is indispensable."],"forward_implications":["The two known multipoint formulas for the parabolic Airy process, the defining extended Airy kernel determinant and the equal-time KPZ fixed point formula, can now be used interchangeably.","Proposition 2.1 gives a new contour-integral Fredholm determinant for the process, which is simpler than the original $D$-function expansion and may be better suited to asymptotic analysis.","The intermediate identities recorded as (2.21), Lemmas 2.3 and 2.4, (3.5), and (3.15) are new explicit formulas for the same multipoint distributions.","The generalized Andreief identity stands on its own as a tool for converting Fredholm determinants into block-determinant integrals, independent of the probabilistic context."],"supporting_citations":[{"why":"Provides Theorem 1.1, the multipoint distribution formula of the KPZ fixed point with narrow-wedge initial condition from which the paper starts.","marker":"[Liu22a]"},{"why":"Gives the alternative multipoint formula whose equal-time version is the object the paper compares against.","marker":"[JR21]"},{"why":"Constructs the KPZ fixed point and provides the relation identifying its one-time marginal with the parabolic Airy process.","marker":"[MQR21]"},{"why":"Introduces the Airy$_2$ process and the parabolic Airy process; the definition in (1.1) is taken from this line of work.","marker":"[PS02]"},{"why":"Supplies the classical determinant identity that Lemma 1.2 generalizes and whose proof strategy the lemma follows.","marker":"[And86]"},{"why":"Supplies the Airy-function integral identity used to prove the Gaussian form of $B$ in Lemma 3.2.","marker":"[Oko02]"},{"why":"Used for convergence arguments and contour-deformation techniques, in particular in the proof of part (i) of Lemma 2.3.","marker":"[LZ25]"}],"fun_headline_variants":["Direct proof matches KPZ equal-time formula to Airy kernel","Parabolic Airy multipoint distribution: two formulas, one proof","Contour determinant reduces to extended Airy kernel","Equal-time KPZ distribution equals extended Airy formula","New proof unifies parabolic Airy distribution formulas"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper relies, without reproving it, on the multipoint distribution formula for the KPZ fixed point with narrow-wedge initial condition (Theorem 1.1) and on the absolute convergence of several contour integrals cited to other works; if either gave way, the equality of the two formulas would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Direct proof matches KPZ equal-time formula to Airy kernel","Parabolic Airy multipoint distribution: two formulas, one proof","Contour determinant reduces to extended Airy kernel","Equal-time KPZ distribution equals extended Airy formula","New proof unifies parabolic Airy distribution formulas"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00053,"raw_usage":{"total_tokens":2519,"prompt_tokens":875,"completion_tokens":1644,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":1565}},"tokens_in":491,"tokens_out":1644,"duration_ms":15964,"temperature":1.0,"reasoning_tokens":1565,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:44:47.733915+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides for the first non-trivial case, $m=2$, at specific ordered points such as $\\alpha_1=0$, $\\alpha_2=1$, $\\beta_1=\\beta_2=0$: evaluate the Fredholm determinant of $K$ in Proposition 2.1 by truncating its series expansion and evaluate the extended-Airy-kernel determinant in (1.1) by a separate numerical scheme; any discrepancy beyond numerical error would disprove the stated equality.","supporting_citations":[],"review_version":2}