All works

math.DS — Dynamical Systems

The orphan theorem: degenerate perpendicular orbits in rational right triangles

Contributed by Joshua Boorstein

Let $T$ be a right triangle with acute angle $\alpha$, legs $L_1$, $L_2$ and hypotenuse $H$, and suppose $T$ is rational: $2\alpha/\pi = P/Q$ in lowest terms. Shoot a beam perpendicular to the leg $L_1$ from each of its points. The leg is partitioned into branches — maximal intervals of launch heights sharing a combinatorial type — and we prove that the partition is finite, with at most $2Q + 1$ branches, which cover the leg but for finitely many points. The first-return map $J$ of the beam to $L_1$ is an involution, and on each branch it does one of two things: it carries the branch onto a distinct branch of equal width by a reflection, or it fixes the branch pointwise. A pointwise-fixed branch — a whole interval of trajectories that hit a side head-on and retrace — is an orphan. Theorem A. There is no orphan branch when $P$ is even and at most one when $P$ is odd, with no hypothesis on the direction; when $P$ is odd and $Q$ is a prime power, or twice or four times a prime, the orphan exists. Non-existence is a parity count on the velocity set of the unfolded beam. Uniqueness is an Euler count on the genus-zero quotient of the unfolded surface, which admits minimal components. Existence is arithmetic: on a folded disc, an absent orphan forces $1$ to be a signed sum of doubled heights of images of the right-angle vertex, which is impossible at the centres named. Running the direction count and the Euler count for each side shows that at most two of the three sides carry a retracing cell, at most one each, the two ends of one band of double normals. On the unfolded surface — the holonomy double cover of a genus-zero quadratic differential $Q(P-2, Q-P-2, -1^Q)$ — the images of the right-angle vertex are Weierstrass points, the leg is the quotient by the hyperelliptic involution of a geodesic segment through one of them, and the first-return map is read in that quotient.

Author comments: First version. Companion papers in preparation: "The necklace of a rational right triangle" and "No arrivals". Code: https://doi.org/10.5281/zenodo.24037744

math.QA — Quantum Algebra

Positivity of Canonical Basis Multiplication in Modified Quantum Groups

Contributed by Qiyue Tang

We prove the multiplication part of Lusztig’s positivity conjecture for symmetric Cartan data.

math.AG — Algebraic Geometry

The tangent line at a marked point on $M_{0,n}/\mathfrak S_{n-1}$, after a table of Grothendieck

Contributed by Michel Hua

Let $L_n$ be the line bundle on $M_{0,n}$ whose fibre is the tangent line to $\mathbb P^1$ at the last marked point, and $L$ the tensor product of the tangent lines at all of them. We show that, in characteristic zero, the Picard group of the quotient stack $[M_{0,n}/\mathfrak S_{n-1}]$ is cyclic of order $(n-1)(n-2)$ for every $n \ge 3$, generated by $L_n$, and that $L$ has order $n-1$ in $\operatorname{Pic}[M_{0,n}/\mathfrak S_n]$. For even $n \ge 6$ the group was known, through the integral Chow ring computed by Edidin and Hu; the odd case does not seem to be in print. The question comes from a table in Grothendieck’s working notes, which gives these orders for four points and only guesses for five and six. The proof takes a few lines once the stack is written as monic squarefree polynomials modulo the affine group. An elementary version, with explicit trivialisations and fixed configurations, is formalised in Lean.

math.CO — Combinatorics

A bigon-free dichotomy for pseudoline arrangements, after a note of Grothendieck

Contributed by Michel Hua

A pseudoline of an arrangement in the real projective plane is bigonal when it carries exactly two vertices of the arrangement. In an undated folder of working notes from the early 1980s, Grothendieck states that every arrangement without bigonal pseudolines is either four pseudolines in general position or contains one of three configurations: five pseudolines in general position, a pencil of three and a pencil of two without the line joining their centres, or the complete quadrangle. The notes derive this from a statement that is false as written, and they give no independent proof. We give a short proof, together with the ordering of the pseudolines that the notes ask for, and we deduce that for n ≥ 6 the near-pencils are the only arrangements of n pseudolines containing none of the three configurations. The dichotomy has been checked by an exhaustive enumeration of all arrangements of at most nine pseudolines, and it is formalised in the proof assistant Lean for a combinatorial model of arrangements, generalised maps of Euler characteristic one.

math.CO — Combinatorics

Counterexamples to Source–Sink Recursions for Denominator Vectors

Contributed by Qiyue Tang

Reading and Stella’s source–sink recursions for denominator vectors, Conjectures 2.7 and 2.8, fail for an infinite family of four-vertex exchange matrices. The member realized by an unpunctured annulus with one marked point on one boundary and three on the other also refutes their Theorem 2.11.

math.RT — Representation Theory

Singleton Dominance Regions in Affine Cluster Algebras

Contributed by Qiyue Tang

For an extended exchange matrix of affine type, the dominance region of every point in an extended (g)-vector cone consists of that point alone.

math.RA — Rings and Algebras

The L’vov–Kaplansky Conjecture for Standard Polynomials

Contributed by Y. T. Miah

The L'vov–Kaplansky conjecture asks whether the image of a multilinear polynomial on a full matrix algebra is a vector subspace. For the unnormalized standard polynomial $s_d$ over an algebraically closed field $K$ of characteristic zero, the sets of individual matrix values are determined in every degree and order. The image of $s_{2r-1}$ is zero when $n<r$ and all of $M_n(K)$ when $n\ge r$; the image of $s_{2r}$ is zero when $n\le r$ and $\mathfrak{sl}_n(K)$ when $n\ge r+1$. The odd-degree proof uses explicit linear slices and Jordan forms. The even-degree argument combines a uniform nilpotent construction and an additive transfer for admissible cores with generic-nullity results of Brassil–Reichstein and Lin–Tang–Zhao, extending a method of Qiyue Tang to obtain actual polynomial values.

math.DS — Dynamical Systems

Globally periodic rational difference equations of arbitrary order

Contributed by John Rudge

A rational difference equation of order $k$ with nonnegative coefficients is globally periodic if every solution with positive initial data repeats with a common period. Five such equations have long been known, among them the Lyness equation $x_{n+2}=(1+x_{n+1})/x_n$, and a problem of Gasull asks whether every globally periodic equation of this kind is obtained from one of the five by rescaling and unfolding. The answer was known to be yes for $k\in\{1,2,3,4,5,7,9,11\}$. Here I show that it is yes at every order. The proof rests on two ideas. The first is a tropical degeneration: global periodicity forces an associated min-plus recursion to be periodic, and this is refuted, for every choice of coefficients at once, by an unbounded solution of the min-plus recursion linearised along a periodic orbit. The second is the $\sigma$-bump, a linearised solution whose existence reduces to a combinatorial condition on residues modulo $2k$; in the hardest case that condition comes down to a lemma on three-term progressions under unit multipliers modulo $2k$. The proof, including its finite computations, has been formalised in the Lean proof assistant.

math.CO — Combinatorics

Height-bounded mutation to acyclic quivers on four vertices

Contributed by Qiyue Tang

We prove Conjecture 5.5 of Armstrong-Williams, Hirst, Jackson and Lee.

math.CO — Combinatorics

Complexity of Mutation-Acyclicity and Matrix Mutation Equivalence

Contributed by Qiyue Tang

Mutation-acyclicity of integral skew-symmetrizable matrices belongs to NP. Mutation equivalence belongs to promise-NP when at least one input is mutation-acyclic. For pairs of mutation-cyclic matrices, mutation equivalence is RE-complete and strongly NP-hard. Both results hold even for connected skew-symmetric matrices.

math.RA — Rings and Algebras

The Lu–Wenzel Conjecture

Contributed by Qiyue Tang

We prove the Lu–Wenzel conjecture for real and complex matrices.

cs.DS — Data Structures and Algorithms

A certified exponent of 0.48758 for worst-case Subset Sum, and limits of the alias–checksum analysis

Contributed by Jennifer Voitle

For more than fifty years, since Horowitz and Sahni's meet-in-the-middle algorithm of 1974, the best exponent for worst-case Subset Sum was $1/2$. In October 2026 that barrier fell twice within a week, with OpenAI's $2^{0.49n}$ algorithm and, independently, Li and Ye's $2^{(1/2 - 1/450000)n} \approx 2^{0.4999978n}$, and improvements have followed quickly since: within three weeks A. Kapila's modification of the OpenAI approach ("Base") reached 0.4877, a retuning of Base by JoshuaQFEX reached 0.487676, and a refinement by Y. Bills reached 0.4876723. Here we push this line of work further and examine what limits it. Conditional on the cited non-numerical proofs, we give a certified parameter point with threshold $T=0.4875754036$: for every fixed $T'>T$, the uniform randomized algorithm uses $O(2^{T'n})$ word operations on inputs of polynomial bit length, with success probability at least $2/3$. In particular, it runs in time $O(2^{0.4875755n})$. A single asymmetric filter replaces Base's family of 43 filters. Independent interval computations certify all required numerical conditions, including both filtering moments and their quadrature errors. We prove a lower bound on the first filtering moment, valid for all positive continuous densities, that excludes every parameter choice with $\tau\le0.484$ in the original OpenAI proof's sufficient budget. Three algebraic bounds explain the costs at our improved point. Separately, an enumeration phase that inserts at least $2^{(1-\eta-o(1))n}$ alias classes, spending a counted step on each, and uses checksum gap at most $\Delta$ needs exponent at least $(1-\Delta)/2$. This is a conditional operation-count bound; a global upper bound on achievable checksum capacity remains open. Finally, we isolate a small-layer statement that would yield threshold $0.4873407528$, prove combinatorial special cases, and identify the remaining questions.

Author comments: 30 pages, 2 figures. The main result is conditional on the written proofs of OpenAI (4 Oct 2026) and A. Kapila (commit 44c34961); its numerical conditions are certified in interval arithmetic. The companion file SECTION_T_v3.pdf (full proofs of the ternary alias step, Appendix A) is included with the source. Code, certificates, verification scripts and a timestamped research record: https://github.com/jv1000/subset-sum. AI use is described in Section 11.

math.CO — Combinatorics

Four-cycle supersaturation and polarity stability at $q^2+q$ vertices

Contributed by Zhenyu Sun

Let $m_0=\frac12q(q+1)^2-q$. For every even $q\ge1600$, we prove that a graph on $q^2+q$ vertices with at least $m_0+1$ edges contains at least $\lceil(q-1)/12\rceil$ four-cycles. For powers of two in this range, $m_0$ is the extremal number and a polarity construction has exactly $q-1$ four-cycles at $m_0+1$ edges. These bounds determine the order of magnitude of the minimum count on this sequence. We also prove a stability theorem: for every nonnegative integer $b$ and every even $q\ge10^{18}(b+1)^{16}$, each $C_4$-free graph on $q^2+q$ vertices with $m_0-b$ edges is a spanning subgraph of an orthogonal polarity graph with one point deleted. The required degree bounds follow from the edge count. As a consequence, a graph on $q^2+q$ vertices with $m_0+1$ edges has at least $q-1$ four-cycles if an edge set of size $t\ge1$ meets every four-cycle and $q$ is even with $q\ge10^{18}t^{16}$. The counting proof uses a codegree commutator; the stability proof combines degree-deficit concentration with incidence completion and polarity extension.

Author comments: 64 pages.

math-ph — Mathematical Physics

The Hawking-Page transition at weak coupling

Contributed by Anthropic

In anti-de Sitter space a black hole forms through a first-order phase transition, discovered by Hawking and Page. In $\mathcal{N}=4$ super Yang-Mills theory on $S^3$ this is deconfinement at strong coupling. Does the transition remain first order at weak coupling, where there is no classical geometry? The free theory has a transition at its Hagedorn temperature, and it was suggested in 2003 that the two could be continuously connected, either by a single first-order line or through a pair of continuous transitions at weak coupling, which have no counterpart in classical gravity. One number decides: the sign of the quartic coefficient $b$ in the effective action of the Polyakov loop. It first appears at three loops and, to our knowledge, has not been computed in the two decades since. We find $b=-1.48\times10^{-3}\,\lambda^2$, where $\lambda$ is the 't Hooft coupling. The transition is first order. It lies slightly below the Hagedorn temperature, and its three saddle points appear, exchange dominance and disappear in the same order as thermal anti-de Sitter space and the small and large black holes. The structure of the Hawking-Page transition is thus already present in perturbation theory. The computation is Hamiltonian, and almost every number in it was obtained in two independent ways. It reproduces the two-loop Hagedorn temperature known from integrability and the exact results available in closed subsectors.

math.RT — Representation Theory

Disproof of Xi’s conjecture on the representation dimension of idempotent corners

Contributed by Qiyue Tang

We disprove Xi’s conjecture on the representation dimension of idempotent corners. Over every field and for each integer (L\ge3), we construct an algebra of representation dimension (L+1) with a corner of representation dimension (2L-1).

math-ph — Mathematical Physics

The spinless modular bootstrap at large central charge

Contributed by Anthropic

The spinless modular bootstrap bounds the dimension $\Delta_1$ of the lightest primary other than the vacuum in a two-dimensional conformal field theory, using only modular $S$ invariance of the torus partition function at zero angular potential. For unitary, compact theories with central charge $c>1$ and no conserved currents beyond the Virasoro algebra, these bounds read $\Delta_1\le c/\Lambda+o(c)$ at large $c$. The best slope $\Lambda$ was known only to lie between $8.503$ and $12$, where the bound would reach the BTZ black hole threshold. We show that it is exactly $\Lambda_V=12\pi^2/(\pi^2+4\,\mathrm{arcsinh}^2 1)\approx9.1266$. No linear functional in a class that contains all finite combinations of derivatives improves on $c/\Lambda_V$ by more than $O(c^{1/3}\log^{2/3}c)$, and an explicit functional gives $\Delta_1<c/\Lambda_V+0.483\,c^{1/3}+O(1)$ for all sufficiently large $c$. The value has an electrostatic origin: as a function of a complexified dimension, $c^{-1}\log\lvert g\rvert$ of a functional $g$ is a two-dimensional potential with positive charges at the zeros of $g$. Positivity forces $g$ to fall as fast as the Cardy density of states grows, and the maximum principle bounds the potential by an explicit harmonic function, which fixes $\Lambda_V$. The field of this potential is the spectral curve of a mean-field theory of the optimal functionals, and its charge density is the distribution that the zeros of any functionals attaining $\Lambda_V$ must follow. We also conjecture the coefficient of the $c^{1/3}$ correction to the optimal bound, supported by numerics up to $c=2048$.

math.RA — Rings and Algebras

The L'vov--Kaplansky conjecture for multilinear polynomials of degrees four to six

Contributed by Shengyong Pan

Let $K$ be an algebraically closed field of characteristic zero. We determine the images of ordinary multilinear polynomials of degrees four, five and six on matrix algebras. For $n\geq3$, a nonidentity has image $\mathfrak{sl}_n(K)$ when its cyclic class is zero and $M_n(K)$ otherwise. In these degrees the only nonzero identity exception is the standard degree-six line on $M_3(K)$. The proof starts with proper sextics and treats arbitrary sextics by unit restrictions and local trace lifting. An image-preserving product substitution then gives degrees four and five as consequences. We also give cyclic and matrix lifting results applicable in arbitrary degree. The finite calculations are given by explicit matrix-unit identities.

cs.FL — Formal Languages and Automata Theory

Degree Transport and a Smaller Cubic Coefficient for Reset Thresholds

Contributed by SUNGBIN HONG

We prove that every synchronizing complete deterministic finite automaton on $n\ge2$ states has reset threshold at most $\frac{8242}{49923}n^3+O(n^2)$. The cubic coefficient improves on Shitov's $8257/49923$ by $5/16641$. The main ingredient is a degree-transport inequality for graphs of state pairs mergeable by short words. It relates a long gap between consecutive minimum-rank word lengths to a large corank achievable in at most $n-1$ letters. Combining this relation with a discrete dual of Shitov's counting argument gives the improvement. An appendix makes the quadratic remainder explicit for every $n\ge2$. The argument applies to arbitrary finite alphabets and does not prove the quadratic Černý conjecture.

math.NT — Number Theory

Four exponentials at CM and Beta periods

Contributed by Shawn Krisman

For every nonzero eigen-period $P$ of a CM abelian variety over $\overline{\mathbb{Q}}$, we show that the four exponentials conjecture and its strong form hold at $t\in\overline{\mathbb{Q}}^\times P^{\pm1}$ and $t\in\overline{\mathbb{Q}}^\times(P/\pi)^{\pm1}$; these include Beta values such as $t=\Gamma(1/5)^2/\Gamma(2/5)$. For an elliptic curve $E$ over $\overline{\mathbb{Q}}$ with complex multiplication, we also show that $e^{\pi^2}$ and $\wp_E(1/\pi)$ are not both algebraic. Further results concern two CM elliptic curves: for example, $\pi,\varpi,\Omega_\rho,\varpi\Omega_\rho$ are linearly independent over $\overline{\mathbb{Q}}$, where $\varpi$ and $\Omega_\rho$ are periods of $y^2=4x^3-4x$ and $y^2=4x^3-4$. These are explicit consequences of published theorems of Waldschmidt (1988) and Roy–Waldschmidt (1993), together with classical results of Baker, Schneider, Masser and Wüstholz; the contribution is the statements and the proofs written out at these periods. Cases at specific periods, such as that $2^{\varpi}$ and $3^{\varpi}$ are not both algebraic, are checked in Lean 4 and reported separately.

math.NT — Number Theory

A Lean formalization of Waldschmidt's subgroup theorem with period term for groups with elliptic factors

Contributed by Shawn Krisman

We formalize in Lean 4 with Mathlib Waldschmidt's algebraic subgroup theorem with period term (Theorem 4.1 of his 1988 Durham paper) for $G=\mathbb{G}_{\mathrm{a}}^{d_0}\times\mathbb{G}_{\mathrm{m}}^{d_1}\times E_1\times\dots\times E_m$, with $E_j$ elliptic curves with algebraic invariants, under the additional hypothesis $2N\ge d_1+2d_0+\kappa$. The proof uses only Lean's three standard axioms. Along the way we give what we believe are the first formal proofs of Schneider's theorems on elliptic functions and of the Schneider--Lang criterion, and the first Lean library for the Weierstrass $\sigma$ and $\zeta$ functions. Restricted forms of the theorem cover some vector extensions and the Jacobian of $y^5=x(1-x)$. A companion certificate uses these to prove explicit cases of the four exponentials conjecture: for example, $2^{\varpi}$ and $3^{\varpi}$ are not both algebraic, where $\varpi$ is the real period of $y^2=4x^3-4x$, and likewise $2^{B(1/5,1/5)}$ and $3^{B(1/5,1/5)}$.

math.CO — Combinatorics

Foregger's Conjecture on Minimum Permanents of Nearly Decomposable Doubly Stochastic Matrices

Contributed by Yair Lavi

We prove Foregger's conjecture: every nearly decomposable doubly stochastic matrix $A$ of order $n\ge2$ satisfies $\operatorname{per}A\ge 2^{1-n}$. Equality holds exactly for matrices obtained from $\tfrac12(I_n+P_n)$ by independent row and column permutations, where $P_n$ is a single-cycle permutation matrix. In fact, for every fixed nearly decomposable support whose bipartite graph is not a single cycle, the minimum permanent on its closed doubly stochastic face is strictly larger than $2^{1-n}$.

math.DG — Differential Geometry

HOLOMORPHIC SECTIONAL CURVATURE OF FLAG MANIFOLDS VIA KÄHLER REDUCTION

Contributed by Ching Hsien Lee

We give an alternative proof that every invariant Kähler metric on a compact generalized flag manifold has positive holomorphic sectional curvature. Invariant Kähler metrics on unitary partial flag manifolds are realized as reductions of Hermitian vector spaces at compact moment-map levels contained in spheres. A curvature identity for linear Kähler quotients gives an explicit positive lower bound. For an inclusion of compact connected Lie groups, we prove that the induced inclusion between adjoint orbits through the same Lie algebra element is holomorphic, isometric, and totally geodesic for the Kähler structures determined by an invariant inner product and its restriction. This reduces the general case to unitary flags. We also obtain a strict estimate for the second fundamental form of the anticanonical projective embedding

math.CO — Combinatorics

On a counting conjecture for binary rich words

Contributed by Max Glass

Let $R_2(n)$ denote the number of binary rich words of length $n$. We prove that $\limsup_{n\to\infty}\frac{\log R_2(n)}{\sqrt{n}\log n}=\infty$. Consequently, $R_2(n)$ is not $O(n^{C\sqrt{n}})$ for any fixed $C>0$, disproving the conjectured upper bound $O((n/g(n))^{\sqrt{n}})$ with $g(n)\to\infty$. The proof combines independent permutation families through an ordered concatenation construction and a binary gap encoding.

Author comments: Formatting correction to the web abstract: repaired math delimiters and removed hard line breaks. The paper and mathematical results are unchanged. 4 pages. Lean 4 formalization and paper source: https://github.com/glasss13/rich-word-counting.

math.CO — Combinatorics

The constant Steiner kernel is not optimal at large girth

Contributed by Matthew Kwan, Huy Tuan Pham

In the paper "nonlinear lower-tail large deviations at criticality", we bound the number of order-$n$ Steiner triple systems with no Pasch configuration. In doing so, we show that a certain variational problem is minimised by a constant kernel. Using ChatGPT 6 Pro, we also discovered that the analogous variational problems for Steiner triple systems of girth at least $g$, for large $g$, are *not* minimised by the constant kernel, and we included a brief remark to this effect. In this note (not intended for publication), we present the details of this proof.

math.AP — Analysis of PDEs

Anomalous regularization of passive scalars up to the Obukhov-Corrsin exponent

Contributed by Scott Armstrong, Vlad Vicol

We consider the advection-diffusion equation $\partial_t\theta+\mathbf{b}\cdot\nabla\theta-\kappa\Delta\theta=0$ on the two-dimensional torus $\mathbb{T}^2$, where $\mathbf{b}$ is a divergence-free velocity field constructed as in arxiv:2305.05048, H\"older continuous with an exponent $\alpha<\nf13$ and built from shear flows alternating in direction at infinitely many scales. We prove anomalous regularization: for every $s$ below the Obukhov--Corrsin exponent $\frac12(1-\alpha)$, with $\mathbf{b}$ depending on $s$, the solutions are bounded in~$C^{0,s}(\mathbb{T}^2)$ at positive times by a constant times the $L^2(\mathbb{T}^2)$ norm of the initial data, uniformly in the diffusivity $\kappa$ in a union of bands accumulating at $0$. As a consequence, we obtain anomalous dissipation at a uniform rate: solutions dissipate a fixed fraction of their variance in unit time, so that the variance decays exponentially, with constants independent of $\kappa$, of the initial datum and of the initial time. The proof combines iterated quantitative homogenization with large-scale regularity: at each scale the solution is compared with a solution of the homogenized equation of that scale. Since the derivatives consumed by each comparison are regenerated at every scale, the H\"older norm is carried from the coarsest scale to the finest by induction. Each of our main results, together with its proof, is fully formalized in the Lean 4 proof assistant.

math.QA — Quantum Algebra

The Haagerup center at central charge eight via conformal nets

Contributed by Cédric ARHANCET

Let $S\subset V_{E_6\oplus A_2}$ be the unitary vertex operator algebra of central charge eight with currents at levels $(3,13)$ constructed in our previous work. An exact presentation of a full corner of its Zhu algebra gives strong rationality, the twelve simple modules and the lattice branching rules. Using these results, we identify the representation category of its conformal net with the Haagerup center $Z(\mathcal{H}_3)$. We construct a holomorphic diagonal extension of $\mathcal {A}_S\otimes \mathcal{A}_S$ and apply the vacuum trace class theorem of Henriques and Tener to prove trace class heat operators in every sector of $\mathcal{A}_S$. Reconstruction gives a unitary holomorphic vertex operator algebra containing $S\otimes S$ as mutual commutants. The coset theorems of Gui then give the complete unitarity of $S$ and a braided equivalence $\mathrm{Rep}(S) \cong Z(\mathcal{H}_3)$, so that $S$ is a Haagerup algebra. Using results of Dong, Jiao and Xu, we also identify the Huang--Riesen algebra with the twin of $S$ and describe their common fixed point algebra using the intersection of the two binary lattices of discriminant $-39$.

Author comments: 73 pages

math.CO — Combinatorics

Decidability of Mutation-Acyclicity

Contributed by Qiyue Tang

Mutation-acyclicity is decidable for all finite integral skew-symmetrizable exchange matrices. Mutation equivalence up to relabeling is decidable when at least one of the two inputs is mutation-acyclic.

math.CV — Complex Variables

Schmeisser’s Conjecture in Large Degree

Contributed by Qiyue Tang

We prove Schmeisser’s conjecture for all sufficiently large degrees.

math.AG — Algebraic Geometry

The four-dimensional Hessian conjecture for potentials of degree at most five

Contributed by David Niedbala Giraudin

The Hessian conjecture HC_n asserts that if f in C[x_1,...,x_n] has det Hess(f) in C^×, then ∇f is a polynomial automorphism. It is known for n ≤ 3, false for n ≥ 5, and open for n = 4, where it implies the plane Jacobian conjecture. Ni proved HC_4 for potentials of degree at most four, and the author treated degree five when the top form depends on exactly three linear forms (hexagon:2611.00134). We prove HC_4 for every potential of degree at most five. The remaining top forms are a binary form or a fifth power of a linear form. Both cases are reduced, by weighted layers of the Hessian determinant and exact computer algebra over Q, to four elementary mechanisms: a Schur complement in one variable (which reduces to HC_3), a partial Legendre transform in two variables (which reduces to HC_2), a "dead direction" lemma (which reduces to HC_2 and the plane Jacobian conjecture in degrees at most (3,4), known by Moh), and explicit triangular inverses. Five self-contained verifiers are supplied as ancillary files. The paper has not been refereed.

math.DS — Dynamical Systems

Normal Cone Is All You Need: Unsoftened Symplectic Integration Of The Newtonian N-Body Problem Across Collisions In Finite Steps

Contributed by Jeremy Nusa

In the classical formulation of the Newtonian $N$-body problem, symplectic geometry and collision singularities appear fundamentally irreconcilable: fixed-step symplectic integrators preserve the Poincaré integral invariants on smooth orbits only by softening the gravitational potential or breaking down at close encounters, whereas classical Sundman and Levi-Civita regularizations resolve collisions only by deforming the global time flow or altering the canonical phase-space structure. In this work, we show that this dichotomy is not an intrinsic obstruction of Hamiltonian mechanics, but an artifact of working on the unblown Cartesian configuration space. By lifting the dynamics to the standard cotangent bundle $T^*\mathbf{Q}[N]$ of the real oriented Fulton–MacPherson compactification $\mathbf{Q}[N] = \operatorname{Bl}_{\Pi(N)}(M/\mathbb{R}^3)$ and applying the Calaque–Safronov shifted cotangent deformation to the normal cone $\mathcal{D}_Z \mathbf{Q}[N] \to \mathbb{A}^1$, we construct an exact, unsoftened ($\varepsilon = 0$) Hamiltonian splitting covered by the combinatorial atlas of $(2N-3)!!$ rooted binary Jacobi trees. Along the normal-cone degeneration, the interaction decomposes canonically into commuting Kepler–Sundman leaf flows, synchronous anti-Zeno supernodes, and a scale-invariant quadrupolar-screened apex potential of global $\mathcal{C}^{1,1}$ regularity. Endowing the exceptional collision divisors with McGehee's unitary elastic bounce involution ($\mathcal{B}_C^2 = \mathrm{Id}$) renders the leaf flows globally complete and time-reversible across binary collisions in a finite number of physical steps, thereby lifting the Sheng–Suzuki negative-timestep barrier for high-order symmetric compositions. In the bi-graded Poisson–Lie algebra $\mathcal{L}_{n,m}$ associated with this splitting, we establish a Universal Double Obstruction Theorem at order six: momentum-parity alternation prevents any pure configuration-space corrector from cancelling the leading secular error, while a non-vanishing Hall cohomology class in $H^1(D_A, \mathcal{L}_{5,2})$ creates an incompressible $\mathcal{O}(\varepsilon^2 h^4)$ quadratic tidal floor. We overcome the parity obstruction by introducing a torque-free cotangent corrector $K_4 \in \mathcal{L}_{4,1} \cap \mathcal{P}_1$, realized via a palindromic Lie conjugator $\mathcal{Z}_4^{\mathrm{BM4}}(h)$ that vanishes naturally as $\mathcal{O}(r^{7/2})$ at collisions (scheme $\mathrm{PBM}_4$). Whenever the trajectory changes local Jacobi chart or dyadic step-size level, a non-commutative covariant gauge connection $T_{A\to B}^{(\mathrm{PBM4})}$ closes the Čech 1-cocycle on the atlas groupoid and eliminates interfacial shadow-energy jumps. Accompanied by a complete, axiom-free Lean 4 formalization and validated up to 116-bit quadruple precision against the REBOUND integrator suite (IAS15, TRACE, MERCURIUS, and WHFast), the resulting architecture preserves all Galilean Lie–Bargmann invariants to machine precision across secular, collisional, and chaotic multi-body regimes.

math.NT — Number Theory

Irrational even beta values in shifted intervals

Contributed by twinheterodox

For every positive odd integer $m$, we prove that at least one of $\beta(m+1),\beta(m+3),\ldots,\beta(19m-1)$ is irrational, where $\beta$ is the Dirichlet beta function. More generally, for each positive integer $B$ we give an explicit constant $C_B$ such that at least $B$ even beta values with indices between $m+1$ and $C_Bm-1$ are irrational, for every positive odd $m$. Specific parameter choices give at least two irrational values among $\beta(2),\beta(4),\ldots,\beta(70)$, at least three through $\beta(186)$, and at least four through $\beta(360)$.

Author comments: Version 2 improves the uniform one-value endpoint from 67m-1 to 19m-1 and sharpens the general bound for any prescribed irrationality count. Specific parameter choices give at least two irrational values through beta(70), three through beta(186), and four through beta(360). The 20-page revision includes comparisons with v1, worked Gram and prime-dilation examples, and an independent audit of the underlying argument and exact certificates.

math.NT — Number Theory

Saouter's primality criterion in Lean, a ladder constant, and a 25 rung route to odd Goldbach up to $10^{20}$

Contributed by Steve Hubbard

This note gives Lean proofs, with no hypothesis, of the primality criterion of Saouter's paper, Lemma 1 for every even $F$, Theorems 1 and 2, and it records the reading of the constant on page 865 under which the formula agrees with the rest of the paper; separately, given the published even Goldbach computation to $4\cdot 10^{18}$ as an explicit hypothesis, a ladder of 25 Proth primes reaches $10^{20}$ in the Lean kernel. The binary Goldbach conjecture remains open and this note makes no progress on it. The ternary Goldbach theorem for all odd numbers is Helfgott's (arXiv:1312.7748), and it is not formalized here and not used.

Author comments: 8 pages. Version 3 of the note, first deposit on Hexagon.

math.CO — Combinatorics

Undecidability of Quiver Mutation Equivalence

Contributed by Qiyue Tang

We prove that quiver mutation equivalence up to relabeling is undecidable even for one fixed source quiver. We also prove strong NP-hardness.

math.NT — Number Theory

An explicit prime gap under the Riemann hypothesis, formalized in Lean with its prime power input made elementary

Contributed by Steve Hubbard

We report on a Lean 4 formalization of Theorem 2 of Dudek, 2015, as numbered in arXiv v2, which states that if the Riemann hypothesis is true, then for every real $x \ge 2$ there is a prime in the interval $(x - (4/\pi)\sqrt{x}\log x, x]$. In the formalization, Theorem 2 is proved assuming the Riemann hypothesis and one cited explicit zero-counting bound of Hasanalizade, Shen and Wong, arXiv:2107.06506. Four cited inputs of the printed proof are replaced in the formal proof. The two that carry the argument are these: first, the prime power contribution $\psi - \theta$ is required only across the moving window $(x - h, x + h]$ and is proved outright in Lean by an elementary counting argument with no cited input and without assuming the Riemann hypothesis; second, the sum over nontrivial zeros is bounded from the cited Hasanalizade, Shen and Wong zero count for $T \ge 70$ by trading slack between the head and tail sums at the split point $T = 2x/h$. The Riemann hypothesis is open and this note makes no progress on it.

Author comments: 11 pages. Version 2 of the note, first deposit on Hexagon.

math.AG — Algebraic Geometry

Equivalence of the Four-Variable Hessian and Planar Jacobian Conjectures

Contributed by Qiyue Tang

We prove that the four-variable Hessian conjecture and the planar Jacobian conjecture are equivalent over the complex numbers.

math.NA — Numerical Analysis

Quartic Fekete points on every real simplex

Contributed by Yunpeng Li

We prove that on every real simplex Bos's quartic interpolation nodes uniquely maximize the absolute Vandermonde determinant, up to ordering, among all configurations of the prescribed size. In dimension four the squared Euclidean norm of the cardinal vector exceeds one at the centroid. We bound its excess by products of the cardinal polynomials for tetrahedral face centers. These products also measure correlations between columns of the evaluation matrix, which give the required loss in the determinant. An elementary consequence of the same corrected squared-norm estimate is Bos's fourth-power bound in every dimension, with equality exactly at the nodes. The corrected estimate follows from cubic perturbations, reduction to at most two distinct positive coordinate values, and coefficient monotonicity in their multiplicities. All polynomial bounds are given explicitly. The common base case is the tetrahedral Fej\'er bound previously stated by Panda and Pal in equivalent optimal-design form. We give an independent proof of that bound and determine all its equality points.

math.CV — Complex Variables

Sheil-Small's (1974) covering problem has a positive answer

Contributed by DAVE GOEL

T. Sheil-Small asked whether every self-inversive polynomial maps the unit disc onto a domain that contains a disc whose radius is the largest modulus of the polynomial's coefficients (Problem 4.24 in Hayman and Lingham's problem book). We show that the answer is yes. More generally, every non-constant polynomial whose coefficients have the same moduli when read in reverse order maps the unit disc onto a domain that contains an open disc of this radius. This class contains the self-inversive polynomials and the polynomials whose zeros, with multiplicity, are symmetric under taking reciprocals (the problem's printed definition of self-inversive). The radius cannot be increased. For a polynomial with all its zeros on the unit circle, the largest disc in the image has radius exactly half the maximum modulus on the circle. For these polynomials the question is therefore equivalent to the known bound: every coefficient has modulus at most half this maximum (O'Hara-Rodriguez, Saff-Sheil-Small, Kristiansen). The new case is that of polynomials with zeros off the unit circle. We treat this case with the Schur-Cohn Hermitian form and that bound. The main theorem is also proved in the Lean 4 proof assistant, with that bound taken as given.

Author comments: An expanded version of this paper, with further results, is posted on Zenodo: https://doi.org/10.5281/zenodo.23908408

math.DG — Differential Geometry

EVERY CLOSED SMOOTH $n$-MANIFOLD EMBEDS ISOMETRICALLY IN $\mathbb{R}^{\,n(n+1)/2+28}$

Contributed by Yi Wang, Wenchang Yang

We prove that every closed smooth Riemannian $n$-manifold admits a smooth isometric embedding into $\mathbb{R}^{s_n+28}$, where $s_n=n(n+1)/2$. This improves the previous upper bound $s_n+\max\{n+5,\,2n\}$ to a dimension-independent surplus of $28$.

cs.LO — Logic in Computer Science

Proof-Carrying Modernization: LLM Translations of COBOL Batch Programs, Each Proved Against the Original in Lean

Contributed by Tejasvi Singh Tomar

Large language models translate legacy code quickly, but a translation can differ from the original on inputs that no test exercises. We show that LLM translations of whole, unmodified COBOL programs into Rust and Java can be proved in Lean 4 to behave like our semantics of the originals on every admissible input: a bounded number of well-formed records, among other premises. The COBOL semantics is executable; it computes record layouts, decimal arithmetic and file handling from our front end’s output for the unmodified source. For the eight standalone batch programs of AWS CardDemo covered by that front end, all 16 translations are proved, as equivalences for Rust and as refinements with partial correctness for Java. The statements are generated from the programs except for a few small parts, LLM agents write the proofs, and Lean’s kernel checks them. Before any proof, 35% of the first running translations differed from our semantics of the original on a generated input, and no proof found a difference that this testing had missed.

math.CO — Combinatorics

Contracting a pendant edge can increase the mean subtree order

Contributed by Alexander Kramer

Cambie, Jooken and Wagner conjectured that contracting an edge of a connected graph decreases the mean subtree order by at least 1/3, as it does for trees. We construct unicyclic graphs of order n and maximum degree 3 in which contracting a pendant edge increases the mean subtree order by n/450+O(1), disproving the conjecture. In this family the bound first fails at 481 vertices and contraction first increases the mean at 631 vertices.

math-ph — Mathematical Physics

Era statistics for the vacuum Bianchi IX flow

Contributed by Yi-kun Li

We prove the Gauss statistical law for eras of the exact vacuum Bianchi IX flow with zero cosmological constant. Eras are defined by transverse events along the solution, and their lengths count curvature transitions. Along almost every trajectory, the era phases have the Gauss distribution and every finite block of era lengths has its Gauss frequency. The tail decays inversely with era length; the limiting truncated mean grows logarithmically with the cutoff, while the full mean diverges. One invariant Borel exceptional set is null on every smooth transverse section and in the four-dimensional constraint manifold. The proof combines estimates for exact return branches with absolute continuity of their coding measure. Transverse contraction controls the accumulated phase error within each era, uniformly over unbounded era lengths and arbitrarily unequal small curvature variables. A density with strictly expanding section flux then forces almost every recurrent orbit to enter a statistical set on a small section.

Author comments: The Lean 4 formalization is also available on GitHub: https://github.com/l46430640-del/ERA-STATISTICS-FORMAL

cs.CC — Computational Complexity

What the ETH refutations do not give: barriers for the counting engines, with machine-checked lemmas

Contributed by Vaibhav Tulsyan

In October 2026 OpenAI and Anthropic each claimed a deterministic subexponential algorithm for counting the solutions of a $k$-CNF; both counting engines sum products of local tables. For SAT itself, Majdoddin bounds the rate of Anthropic's grid recursions with product terms, and Dias and Lima bound OpenAI's clause-product scheme. However, Majdoddin notes that no barrier is known for decompositions whose terms are not products, and neither paper treats the counting problems beyond SAT. We introduce the engine barrier problem: bound from below the shared bits $U$ of every network of a stated class that computes a given count, and the rate of every recursion of a stated class. Our approach puts a rank on every cut of a local network, and on one contraction inside both flattenings of a term. Machine-checked in Lean are: correlated (non-product) decompositions of two rectangles, of the $3\times2$ grid and of the full $3\times3$ grid level have flattening-rank sums at least $N^2$, $N^3$ and $N^8$, while $N^9$ is open; the row cut of the permanent has rank $\binom nk$, which on paper gives $U\ge n\log_2(n/s)$ for row-local networks; algebras in which the permanent is a functional of a product have dimension at least $\binom n{n/2}$; pieces have saving $\rho\ge1/2$ (rank form; Anthropic's observation); and exact colouring filters for Hamiltonicity need $q\ge n-4$ colours. On paper we prove $U\ge(1-o(1))n\log_2n$ for vertex-local networks for the Tutte polynomial off the integer hyperbolas, and $\rho\ge3/8$ for correlated decompositions of full grid levels. We cite $45$ Lean theorems, with standard axioms only. Every barrier is about a named model: in Anthropic's general model no unconditional bound on $U$ exists without separating $\mathrm P$ from $\mathrm P^{\#\mathrm P}$, and an unconditional cap on oblivious algebraic algorithms would separate $\mathrm{VP}$ from $\mathrm{VNP}$.

math.OA — Operator Algebras

Bures Geometry of Projection States

Contributed by Zhenyu Sun, Jilin Wang

We study the Bures metric on states obtained by conditioning a faithful normal state on nonzero projections of a von Neumann algebra. The metric determines the reference state, projection probabilities, orthogonality, and order. For algebras with no type $I_2$ direct summand, surjective isometries are exactly those induced by reference-state-preserving normal Jordan $^*$-isomorphisms; the same conclusion holds for $M_2(\mathbb C)$. A semifinite tracial version determines the trace up to a single positive factor. In finite tracial algebras, we characterize optimal finite projective measurements and express their fidelity deficit as a sum of squared $L^2$ errors. Spectral measurements with at most $n+1$ outcomes approximate fidelity with error $O(n^{-2})$; the interval grid is minimax optimal, and the order is sharp within spectral measurements. We compare the fidelity defect of a common commutative context with approximation by commuting projections, obtain sharp spectral rounding constants, and solve both optimization problems in the algebra generated by two projections. The space of normalized finite-trace projection states is complete; in a finite tracial algebra its rectifiable components are unitary orbits, and within each orbit the intrinsic Bures distance is the root mean square of the principal angles. Finally, a fidelity bound on local unitary orbits characterizes tracial reference states.

Author comments: 38 pages.

cs.LO — Logic in Computer Science

Formalization of Robust Strictly Positive Real Synthesis for Fourth-Order Interval Polynomial Families

Contributed by Tiancheng Xu, Wensheng Yu

That every robustly Hurwitz-stable fourth-order interval polynomial family admits a common strictly positive real (SPR) numerator was conjectured by Anderson et al. and proved by Yu and Wang. In this paper, we present a complete formalization of this classical synthesis theorem in the Lean~4 interactive theorem prover.

math.NT — Number Theory

Degeneracy and the level-one Maass spectrum: a measured verdict on arithmetic quantum chaos

Contributed by Zeng Chunping

We isolate the spectral axis of a seven-world program: the length spectrum of the modular surface, its degeneracy, and the statistics of the level-one Maass spectrum. (1) The true degeneracy of a closed geodesic of discriminant $D$ equals the narrow class number $h^{+}(D)$, i.e. the number of $\rho$-orbits of Zagier-reduced indefinite binary quadratic forms; on all 36 fundamental discriminants $5\le D\le 120$ the equality holds as an exact integer. (2) With the corrected counting (all non-square discriminants $D\equiv 0,1 \pmod 4$, primitive forms, $\rho$-orbits), the prime geodesic theorem has error exponent $1/2$: $N(X)/\mathrm{li}(e^X)=0.9963$ at $X=12$ with $|\mathrm{err}|/e^{X/2}$ bounded and decreasing overall. (3) The level-one Maass spectrum is long-range GOE and short-range non-Wigner: on 1110 even levels, $\Sigma_2(20)=1.203\ [1.02,1.26]$ against Poisson 20 and compatible with GOE 1.049, while the spacing distribution fails the Wigner surmise with a $17\times$ small-gap excess. This is the standard signature of arithmetic quantum chaos: rigidity without local Wigner repulsion. We also record the archimedean parity law $\Gamma^{a}_{2k}=2\Gamma^{a}_{2k+1}\sim r^{-2k}$, synthetic-spectrum pipeline self-certification, and the corrected upper-envelope saturation criterion. Every measured number carries a confidence interval; corrections applied are recorded in an appendix.

math.AP — Analysis of PDEs

Non-Uniqueness of Leray-Hopf Solutions to the Unforced Three-Dimensional Navier-Stokes Equations

Contributed by Tristan Buckmaster, Matei Coiculescu, Chen Li

We prove that the incompressible Navier--Stokes equations in $\R^3$, without external force, admit a continuum of distinct suitable weak solutions in the Leray--Hopf class with the same initial datum. The datum is compactly supported, axisymmetric without swirl, and smooth away from the origin; it lies in $L^{3,\infty}$ and in $L^q$ for every $q<3$, but not in $L^3$. The solutions are axisymmetric without swirl and smooth at positive times, and they satisfy the local and global energy equalities. Any two of them separate in $L^2$ at the rate $t^{1/4+\sigma}$, for a fixed $0<\sigma<1/4$. The proof follows the program of Jia and \v Sver\'ak, with a discretely self-similar background in place of a self-similar one. The background is an exact unforced solution built from a train of thin vortex rings, which we obtain by lifting a periodic row of planar vortices with Gaussian stream function. We construct a growing Floquet mode of its linearization directly, from the angular-two instability of the planar vortex, and conclude by a nonlinear correction and a localization of the datum.

Author comments: Lean formalization forthcoming at the public repository at https://github.com/mcoiculescu/unforced-leray-hopf-public Exposition will be improved in future versions. We used an adaptation of Scott Armstrong's skill files for the Lean formalization process.

math.CV — Complex Variables

Origin-Centered Mobius Circles, Bernoulli Lemniscates, and Beta–Gamma Dyadic Pairings of the Riemann Zeta Function on the Critical Line

Contributed by Ryan Hackbarth

We investigate the complex geometry of the normalized Dirichlet eta terms $t_n(s) = \frac{1}{1 - 2^{1-s}} \frac{(-1)^{n-1}}{n^s}$ for $s = a + bi$. Whereas individual terms on the critical line $\operatorname{Re}(s) = \frac{1}{2}$ undergo a nearly sixfold modulus oscillation driven by the Euler prefactor $(1 - 2^{1-s})^{-1}$, the two-term partial sum $S_2(s) = t_1(s) + t_2(s)$ traces an exact Mobius circle for every fixed $a \neq 1$ that is centered at the origin $(0,0)$ if and only if $\operatorname{Re}(s) = \frac{1}{2}$, where it reduces to a scaled Blaschke factor $\sqrt{\frac{1}{2}} B(z)$ of constant radius $\sqrt{\frac{1}{2}}$. Factoring $m^{-s}$ across odd-even dyadic pairs $P_m(s) = t_m(s) + t_{2m}(s) = m^{-s} S_2(s)$ eliminates prefactor amplitude oscillations entirely, collapsing precessing annular rosettes into concentric origin-centered circles on the critical line with strictly descending radii $R_m = \sqrt{\frac{1}{2m}}$. Removing all odd-even pairs leaves the complementary multiples-of-$4$ terms $t_{4k}(s) = k^{-s} t_4(s)$: we prove that on the critical line the anchor term $t_4(\frac{1}{2} + ib)$ traces an exact Lemniscate of Bernoulli with foci at $(0,0)$ and $(1,0)$ (uniquely for $\operatorname{Re}(s) = \frac{1}{2}$), pure powers of $2$ ($k = 2^r$) trace closed periodic multi-loop rational curves, and non-powers of $2$ trace precessing lemniscate rosettes, while the dyadic differences $D_{4m}(s) = t_{4m}(s) - t_{8m}(s) = -4^{-s} P_m(s)$ collapse both families into exact origin-centered circles, partitioning the sequence $\{t_n(s)\}_{n=1}^{\infty}$ termwise and decomposing $\zeta(s)$ via analytic continuation. Finally, we establish a pointwise velocity--modulus duality $\left|\frac{dt_4}{d\theta}\right| = \left|t_2\left(\frac{1}{2} + ib\right)\right| = \sqrt{\frac{n}{2}} \left|t_n\left(\frac{1}{2} + ib\right)\right|$ ($\theta = b \ln 2$) and a double-angle Blaschke identity $-z B(z) = e^{i 2\alpha(b)}$ linking the central polar angle $\alpha(b)$ of $t_4(\frac{1}{2} + ib)$ to its fundamental perimeter $L(t_4) = \frac{\Gamma(1/4)^2}{2\sqrt{\pi}}$, thereby parameterizing every individual term $t_n(\frac{1}{2} + ib)$ and every concentric dyadic circle on the critical line in terms of the regularized inverse incomplete Beta function $I_{q(b)}^{-1}(\frac{1}{4}, \frac{1}{2})$. As a geometric consequence, critical-line zeros $\zeta(\frac{1}{2} + ib) = 0$ are reframed as values of $b$ where the resultant of the inner concentric circles lands on the antipode $-P_1(\frac{1}{2} + ib)$ of the initial circle of constant radius.

math.CO — Combinatorics

Vertex splitting, global rigidity and nondegenerate equilibrium stresses

Contributed by James Cruickshank

We prove Whiteley's conjecture that every nontrivial vertex split of a generically globally rigid graph preserves generic global rigidity in $\mathbb{R}^d$, for all $d\geq1$. We give two proofs of the stronger assertion that the split graph is infinitesimally rigid at every generic coincident realisation of the two split vertices. Both proofs use maximum-rank equilibrium stress matrices and then apply the coincident-rigidity criterion of Cruickshank, Jackson and Tanigawa. The second construction also shows that every generically globally rigid graph admits a nondegenerate equilibrium stress of maximum rank at every generic realisation, establishing a conjecture of Jord\'an and Tanigawa.

math.NT — Number Theory

Explicit lower bounds for the number of irrational even beta values

Contributed by twinheterodox

For every even integer $s\ge160$, we prove that at least $\left\lfloor\pi\sqrt{2/135}\,\sqrt{s/\log s}\right\rfloor$ of the numbers $\beta(2),\beta(4),\ldots,\beta(s)$ are irrational, where $\beta$ is the Dirichlet beta function. We also give sharper finite endpoints for specified counts of irrational values

math.FA — Functional Analysis

A modular theory of contractive projections on Banach spaces with applications to noncommutative $\mathrm{L}^p$-spaces

Contributed by Cédric ARHANCET

We develop a differential-geometric approach to contractive projections on complex Banach spaces based on Levi forms of plurisubharmonic functions. We show that if a projection $P$ contracts a $\mathrm{C}^2$ plurisubharmonic function $\Phi$, then, at any point of its range, the Levi form of $\Phi$ defines a tangent Hilbertian geometry for which $P$ becomes an orthogonal projection. When the range is finite-dimensional and the relevant Levi forms are nondegenerate, comparing the Levi geometries associated with different points leads naturally to positive relative operators and associated one-parameter groups. We apply this framework to tracial noncommutative $\mathrm{L}^4$-spaces. Exploiting the cubic form of the duality mapping, we characterize contractive projections by a Jordan triple orthogonality property and prove that a projection on $\mathrm{L}^4(\mathcal{M})$ is $2$-contractive if and only if it is completely contractive. When the ambient von Neumann algebra is finite-dimensional, we identify the relative operator associated with the left and right tangent forms on the range with a restriction of an inverse square root of a modular operator on a linking von Neumann algebra. This entails that the range of any $2$-contractive projection is completely isometric to a rectangular noncommutative $\mathrm{L}^4$-space associated with a $\mathrm{W}^*$-ternary ring of operators. As a consequence, every such projection is contractively decomposable. Finally, for every $1 < p < \infty$ with $p \neq 2$, we prove that every finite-rank $2$-contractive projection on a tracial noncommutative $\mathrm{L}^p$-space is completely contractive. For $p > 2$, we obtain a weighted ternary description of its range in terms of a finite-dimensional $\mathrm{W}^*$-$\mathrm{TRO}$.