Papers

A holographic code assigns a state-dependent entropy remainder to every boundary region, but such remainders need not be the extremal lengths of any one bulk metric. We study this common-source question for the proto-area of a calibrated recovery, the entropy of a boundary region minus that of the matter recovered from it. At zeroth order the reference values must match the lengths of one reference geometry in a fixed renormalization scheme, and this matching also determines the entropy-to-length factor. On a graph with a fixed minimum-cut chamber the data derivatives must lie in the image of the cut-incidence matrix, whose left null vectors are witnesses. On the hyperbolic disk the first variation of renormalized length is the rank-two geodesic X-ray transform, with closed range in an adapted data space, and the geodesic displacement, fixed by a forced Jacobi equation, enters at second order. The resulting normal Hessian gives a gauge-invariant criterion, necessary and sufficient, for a regular pair of response coefficients to come from one metric two-jet. For the constant-curvature family generated by a boundary reparametrization the Hessian has a nonzero rank-four mode, so the image of the length map is curved. For a fixed recovery the proto-area Hessian is minus the Bogoliubov-Kubo-Mori information lost in recovery, which the Petz map decomposes orthogonally at a product reference, and which, on a finite-dimensional space of logical directions with positive minimum retained fraction, is at most a sharp multiple of the retained information. Geometric realization requires this loss to reproduce the normal length Hessian on every interval. Two codes show that exact regional area operators can fail to sew into one graph geometry, and that a matched reference with vanishing first response can fail at second order. All criteria refer to one recovery branch and one extremal-surface branch.

We study the variation in moduli of hyperplane sections of a hypersurface $V(f)\subseteq \mathbf P^n$ with at most isolated singularities. Using the Milnor algebra $M(f)$, we give an infinitesimal quotient criterion for the hyperplane-section map $\Phi(f):(\mathbf P^n)^*\dashrightarrow M(d,n-1)$ to be generically finite onto its image. The passage from the infinitesimal quotient to the coarse moduli space is justified by a local GIT slice argument. Our approach gives a Jacobian-algebraic extension of the Beauville–Patel–Riedl–Tseng theory from smooth hypersurfaces to hypersurfaces with isolated singularities. In the smooth case it recovers the Lefschetz criterion and, using recent weak Lefschetz results, gives generic finiteness for $n\geq 3$ in the range $d\geq n+2$. In the singular case a new obstruction appears: a linear Jacobian syzygy, equivalently, for non-cones, a positive-dimensional projective automorphism group. After this obstruction is excluded, maximal infinitesimal variation is governed by the injectivity of the critical Lefschetz map $\ell:M(f)_{d-1}\to M(f)_d$. We apply the criterion to plane curves, surfaces in $\mathbf P^3$, and hypersurfaces admitting singular hyperplane sections, obtaining new criteria involving nodal sections and an application to the Schoen quintic threefold.

The $(p,q)$-deformed generalized Weyl algebra is generated by variables $X, Y$ and $Z_p$ which satisfy the commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, with $s\in \mathbb{N}_0$ and where $p^s=1$. We investigate the problem of normal ordering arbitrary words in these letters with the help of Young diagrams, and we treat certain special cases explicitly. In particular, the connection to generalized Stirling numbers is considered in detail.

We argue that second order logic is a weaker form of set theory, despite the fact that the former is formalized in a second order langauge and the latter in a first order language. Along the way, we review the history of interactions between second order logic and set theory, as well as some modern trends.

We prove a family of anisotropic Caffarelli-Kohn-Nirenberg interpolation inequalities involving higher-order fractional derivatives and weights of the form $|x'|^{\theta_1}|x|^{\theta_2}|x_n|^{\theta_3}$.

We recover a scalar $c_0$-approximation criterion for Banach spaces canonically embedded into $\ell^\infty$ by a countable norming family. It identifies the dual of the associated $c_0$-subspace with the prescribed atomic predual precisely when bounded coordinatewise approximation by $c_0$-elements is available. The formulation is a self-contained scalar version of the atomic predual and two-stars framework of D'Onofrio–Greco–Perfekt–Sbordone–Schiattarella. We then apply the criterion to Lipschitz-free spaces. The weak-star density of Lipschitz functions with bounded support is due to Aliaga–Pernecká–Petitjean–Procházka; in proper spaces this yields a canonical quotient from the bidual of the norm-closed span of compactly supported Lipschitz functions onto $\Lip_0(M)$. The main concrete application concerns countable proper subsets $K\subset [0,\infty)$ with the Euclidean metric. The connected components $(a_j,b_j)$ of the complement of $K$ give gap coordinates \[ \Delta_j f=\frac{f(b_j)-f(a_j)}{b_j-a_j}. \] We prove that, in the infinite case, these coordinates identify $\Lip_0(K)$ with $\ell^\infty$, that the corresponding $c_0$-subspace coincides with Dalet's predual $S(K)$, and hence that \[ S(K)^*\cong \F(K),\qquad S(K)^{**}\cong \Lip_0(K) \] canonically and isometrically. This gives a coordinate $c_0$ realization, for a non-discrete class with accumulation points, of the preduality covered abstractly by Dalet's theorem for countable proper metric spaces. No new proof of Dalet's theorem in full generality is claimed.

We prove a sharp fixed-volume product inequality for the first $N$ nonzero Steklov eigenvalues of bounded Lipschitz domains in $\mathbb R^N$. More precisely, if $N\ge2$ and $\Omega\subset\mathbb R^N$ is a bounded Lipschitz domain, then $$ \prod_{j=1}^N \sigma_j(\Omega)\le \frac{\omega_N}{|\Omega|}, $$ where $0=\sigma_0(\Omega)<\sigma_1(\Omega)\le\sigma_2(\Omega)\le\cdots$ are the Steklov eigenvalues of $\Omega$, and $\omega_N$ denotes the volume of the unit ball in $\mathbb R^N$. This extends the convex-domain theorem of Henrot, Philippin, and Safoui to arbitrary bounded Lipschitz domains, and in particular settles the remaining higher-dimensional case of a problem posed by Henrot.

This paper establishes an equivalence between the Riemann hypothesis and the association, together with uniform $L^2$-boundedness, of a moderate net in a Colombeau-type algebra built from polynomially damped Báez-Duarte sums. The regularization is performed by multiplicative (Mellin) convolution, which respects the dilation symmetry of the Beurling functions and guarantees that every approximant lies in the $L^2$-closure of the Beurling space. The equivalence is unconditional under the Riemann hypothesis: it uses only the qualitative convergence of Báez-Duarte, Mazur's theorem, and the classical Nyman–Beurling criterion. As a separate quantitative refinement, we prove that under two additional hypotheses on the non-trivial zeros of $\zeta$ (simplicity and separation), the damping error admits a power-law bound with an explicit constant. This refinement is independent of the equivalence and is not used in its proof. The exponential damping $e^{-k\eps^2}$ is discussed as an open problem.

We show that two finite isoclinic groups of the same order have isomorphic conjugacy quandles. We also show that the converse hold under some assumptions on the groups. It happens that isoclinsim of group of the same order is equivalent to having isomorphic quandles for groups of order $n<128$. We construct two non isoclinic groups of order $128$ having isomorphic conjugacy quandles.

An improved formulation of the recently developed cell-centered nodal integral method (MCCNIM) is proposed for the numerical solution of the nonlinear Burgers' equation. The improved scheme, referred to as RCCNIM, reformulates the nonlinear convection term prior to discretization by evaluating the convective velocity using cell-averaged values from the previous time level, rather than the present-time approximation used in the original MCCNIM formulation. This reformulation leads to a fully algebraic system whose coefficients depend only on known quantities and are therefore evaluated once per time step. As a result, the proposed method significantly reduces computational cost while retaining the accuracy of the original MCCNIM. The performance of RCCNIM is assessed through systematic numerical comparisons with MCCNIM for the nonlinear Burgers' equation. The results demonstrate that RCCNIM achieves comparable accuracy with improved computational efficiency, indicating its potential for extension to more complex nonlinear fluid flow problems.

Let $p\ge7$ be a prime with $p\equiv3\pmod4$ and let $\chi$ be the Legendre symbol modulo $p$. We prove that $\det[x+\chi(j-k)]_{0\le j,k\le(p-7)/2}=\floor{(p-2)/3}^{2}x$ in $\mathbb{C}[x]$, which settles Conjecture~3.4 of Zhi-Wei Sun. The truncated matrix is a corner of Chapman's Legendre-symbol matrix $C$, and its determinant can be expressed through a handful of entries of $C^{-1}$ and of $C^{-1}\one$. Those entries are in turn read off from Vsemirnov's cyclotomic factorization of $C$ with the help of Schur's Pfaffian identity.

We investigate the local geometry of generic smooth surfaces in $\mathbb R^4$ via the contact with $2$-planes and the associated apparent contour. We study the $\mathcal A$-singularities of parallel projections of such surfaces along planes to transverse planes. When the projection plane does not contain an asymptotic direction, we show that, at hyperbolic or elliptic (respectively, parabolic) points, there exist up to ten (respectively, seven) tangent directions determining planes along which the projection exhibits $\mathcal A$-singularities of type butterfly or worse. Moreover, we prove that the locus of points where the discriminant of the equation defining these directions vanishes generically forms regular curves in the hyperbolic and elliptic regions, and isolated points in the parabolic set of $M$. When the projection plane contains an asymptotic direction, we establish connections between the singularities of parallel projections, orthogonal projections to hyperplanes, and height functions. We further study the apparent contour associated with the parallel projection. When the projection is a fold, we prove a Koenderink-type theorem relating the Lipschitz–Killing curvature of the surface to the curvature of the apparent contour. Moreover, we show that elliptic or inflections points of the surface give rise to vertices of the apparent contour, whereas hyperbolic and parabolic points give rise to inflections. These phenomena are then characterized in terms of the singularities of the projection. In the non-fold case, we show that the apparent contour admits singularities of type $(t^k,t^\ell)$-cusp associated with the singularities of the projection.

Network-constrained unit commitment (UC) problems must be solved repeatedly to ensure power system reliability, yet face computational challenges from the growing system scale and fast response requirements. This letter proposes a quantum restricted Boltzmann machine (qRBM) method to accelerate UC solution by predicting solution patterns via quantum-hardware-compatible energy scoring and recovering optimal solutions through affine mapping. This method reduces the computation time for UC by 99.96%. Compared to existing quantum methods, it cuts the number of qubits needed by 276 times.

We extend the one-dimensional lonely set method to two dimensions for the purpose of studying the chromatic number of integer distance graphs in two dimensions. Given a lacunary sequence of displacement vectors in $Z^{2}$, we use a lacunary matrix theorem given by Broderick, Fishman and Kleinbock, to prove the existence of a satisfactory multiplier vector. We then give an explicit geometric colouring argument. This proves that any integer distance graph generated by a lacunary sequence of vectors in two dimensions has finite chromatic number.

Classical uncertainty sets for robust optimisation impose fixed geometric shapes that cannot represent the complex dependencies present in real-world data. We propose Generative Robust Optimisation (GRO), a framework in which a deep generative model defines the uncertainty set as the image of a neural network decoder over a calibrated latent set, naturally accommodating nonlinear correlations, asymmetry, and multimodality. A five-point evaluation framework (reconstruction fidelity, distribution matching, latent regularity, robust relevance, and computational tractability) provides systematic, model-agnostic criteria for assessing any neural network-based uncertainty set. We instantiate this framework with a Wasserstein Adversarial Autoencoder employing Gaussian mixture model-guided training for latent regularity and constraint-consistency regularisation for robust relevance. Restricting the decoder to ReLU activations enables exact worst-case verification through mixed-integer programming embedding. Extensive experiments on a production planning problem across six uncertainty distributions and six generative architectures, together with a multi-period facility location study, validate the framework and demonstrate that systematic attention to all five criteria yields uncertainty sets that are simultaneously expressive, well-calibrated, and optimisation-tractable.

This paper is motivated by a recent result of Pitman and Yakubovich stating that a partially exchangeable, stationary (PES) random partition of $\mathbb{N}$ is exchangeable. This echoes an earlier theorem of Kallenberg on the equivalence of spreadability (contractability) and exchangeability for infinite partitions. We revise the hierarchy of symmetries with a focus on partitions of finite sets $[n]$, and ask about the extent to which these relaxed symmetry types differ from exchangeability, framing the question in terms of the geometry of the polytope of distributions defined by the symmetry constraints. We show that single-orbit exchangeable partitions remain extreme among spreadable and PES distributions, and that $n=5$ is the first case where non-exchangeable extreme partitions occur, causing the associated polytopes to deviate from a simplex structure.

We calculate the conditional expectation of $\prod_{j=1}^{n-1}H_{k_{j}}% (X_{j})$ given $X_{n}=z,$ if random vector $(X_{1},\ldots,X_{n})^{T}$ has multivariate normal distribution and $H_{n}(x)$ denotes $n$-th Hermite polynomial. This expectation is a polynomial in $z$ of order $\sum_{j=1}% ^{n-1}k_{j}$. Our formula has an iterative form with respect to $n$. We also present some auxiliary observations concerning the expansion of the density of the $n$-dimensional normal distribution in the series of the Hermite polynomials. Mostly concerning the properties of the coefficients of this expansion. To perform these calculations, we give a few auxiliary formulas concerning Hermite polynomials and multivariate normal distributions. We apply this result to obtain exact, simple forms of these expansions for $n=2$ and $3$, thus looking at known results from a different perspective. We believe that the results can be very useful in Probability Theory, in particular in the Theory of distributions, Statistics, and the Theory of Stochastic Processes, in particular Gaussian ones.

We develop a risk-aware information theory by replacing expectation with expectiles, introducing expectile entropy, divergence, and mutual information. These quantities exhibit behaviors impossible under Shannon's risk-neutral framework, including negative divergence under risk-seeking regimes and a fundamental separation from classical mutual information. In multiuser systems, the framework naturally induces a mean-field-type game theory of information exchange, where achievable rate regions become endogenous to heterogeneous risk-sensitivity indices. Our results reveal that Shannon information alone cannot quantify the extreme risks driving advanced machine intelligence, establishing a foundation for risk-aware communication, learning, collective intelligence, and safe autonomous systems.

Evolution partial differential equations (PDEs) describe time-dependent physical systems governed by differential laws and arise widely across science and engineering. In recent years, operator learning has emerged as a powerful and efficient paradigm for solving evolution PDEs by learning mappings between infinite-dimensional function spaces, enabling solution prediction without explicit time-step integration. In this work, we propose PI-DOSnet, a physics-informed operator learning framework built upon DOSnet and operator splitting. Unlike purely data-driven operator learning methods, PI-DOSnet incorporates physical constraints during training, allowing it to operate even in the absence of paired input-output data. Once trained, PI-DOSnet performs long-time inference of PDE solutions through an iterative strategy. We analyze the linear stability and approximation error of PI-DOSnet and demonstrate its accuracy, efficiency, and robustness through multiple numerical experiments. Moreover, for the Allen–Cahn equation, PI-DOSnet achieves energy stable solutions even with a large time-step size.

We study edge ideals of sequentially Cohen-Macaulay co-chordal graphs through the maximal-clique structure of their chordal complements. After making the required construction order explicit, we derive a closed formula for the graded Betti numbers of a co-chordal graph whose complement is a $(d_1,\ldots,d_e)$-tree, and we characterize the Cohen-Macaulay case by its Betti sequence. The general formula is then specialized to split and threshold graphs, prime ideal graphs of finite rings, nilpotent graphs of products of finite chain rings, and zero-divisor graphs of finite chain rings. For products of chain rings, we characterize when the nilpotent graph is threshold and compute its Betti numbers in that range. Finally, among co-chordal zero-divisor graphs of $\mathbb{Z}_n$, we determine exactly when the quotient is sequentially Cohen-Macaulay: this occurs for $n=p^a$, $n=2p$, and $n=2p^2$, with $p$ odd in the last two cases.

In this paper, we extend the writhe polynomial invariant from virtual knots to multi-virtual knots. Several questions asked in [16] have been answered.

This paper presents two new theoretical results for Hybrid High-Order (HHO) methods applied to elliptic problems. First, we establish $hp$-error estimates for the HHO discretization of the Poisson problem that achieve optimal approximation rates with respect to both the mesh size $h$ and the polynomial degree $k$. These results improve upon previous analyses of hybrid methods, whose convergence estimates were suboptimal in $k$. Second, building on these estimates, we develop and analyze a non-inherited $p$-multigrid solver for the statically condensed HHO system. We prove results that improve upon the corresponding theory available for other non-conforming methods and constitute, to the best of our knowledge, the first rigorous convergence analysis of a $p$-multigrid algorithm for HHO discretizations.

We construct asymmetric quantum CSS codes with transversal \(CCZ\) gates from algebraic expander codes \cite{KT26}. For every fixed \(m\ge 3\), our growing-alphabet codes have length \(N\), dimension \(\Theta(N)\), and distances \[ d_X=\Theta(N), \qquad d_Z=\Theta(N^{1/m}). \] Moreover, the \(Z\)-stabilizer space has an explicit generating set of weight \(O(N^{1/m})\). We build on the algebraic puncturing framework of Golowich and Guruswami \cite{GG24}, which turns classical codes with the required Schur-product and distance conditions into CSS codes with transversal \(CCZ\). However, applying the framework directly to the algebraic expander codes runs into their small dual distance, and therefore produces only sublinear dimension. Our main technical step is a refined puncturing theorem in which the global dual-distance assumption is replaced by a condition only on the selected puncturing set. We also reduce the alphabet to a fixed prime field using a projective-multiplicity version of multiplication-friendly codes. The resulting fixed-prime-field CSS code triples, of length \(n\), still have transversal \(CCZ\) gates. Their dimension is \(\Theta(n/(\log n)^4)\), with distances \[ d_X=\Omega\!\left(\frac{n}{(\log n)^4}\right), \qquad d_Z=\Omega\!\left(\frac{n^{1/m}}{(\log n)^{4/m}}\right), \] and the \(Z\)-stabilizer generating set remains sublinear.

We develop an abstract regularity framework for a class of two-dimensional nonlinear elliptic systems, including almost harmonic maps. The approach combines a Campanato-type iteration scheme with a Caccioppoli-type estimate and identifies general assumptions under which local Hölder continuity follows. More precisely, we prove that any class of admissible pairs $(u,f)$ that is stable under rescaling and satisfies an oscillation-decay property consists of locally Hölder continuous maps. The resulting Hölder exponent is explicit and matches the classical Morrey–Campanato threshold determined by the Lebesgue integrability of the source term $f$. The framework is purely analytic and avoids the use of $\mathcal{H}^1$–$\mathrm{BMO}$ duality, Wente's inequality, moving frames, and conformal uniformization. We illustrate the flexibility of the framework through several classes of elliptic systems. As a first example, we recover local Hölder continuity for almost harmonic maps \[ -\Delta u=|\nabla u|^2u+f \] into $\mathbb{S}^n$ with $L^q$-integrable tension fields by means of a direct argument independent of the classical harmonic map regularity theory. We next consider systems of the form \[ -\Delta u=\Omega\cdot\nabla u+f, \] showing that the analytic condition $\operatorname{div}\Omega\in L^q$ for some $q>1$ is sufficient to ensure regularity (for classical harmonic maps, $\operatorname{div}\Omega=0$). We further apply the framework to...

This paper is concerned with an inverse parameter identification problem governed by a second-kind elliptic variational inequality involving a non-differentiable boundary functional. The objective is to reconstruct an unknown Robin-type coefficient from boundary measurements while preserving the intrinsic nonsmooth structure of the underlying model. The inverse problem is formulated as a constrained optimization problem, where the cost functional measures the discrepancy between the computed and observed data and is stabilized by Tikhonov regularization. To overcome the difficulties associated with the nonsmooth variational inequality, we reformulate the state problem as a constrained minimization problem and employ an augmented Lagrangian framework combined with the alternating direction method of multipliers (ADMM). Fenchel duality is used to treat the non-differentiable boundary term without introducing smoothing approximations. We establish identifiability of the unknown coefficient and prove existence and convergence results for the proposed approximation scheme. Numerical experiments performed on a classical benchmark problem confirm the theoretical analysis and demonstrate that accurate parameter reconstruction can be achieved with reduced computational effort. The proposed methodology provides a robust framework for inverse problems arising in applications such as heat transfer, frictional contact, and corrosion detection.

Federated learning (FL) is an emerging distributed machine learning paradigm that enables local devices to jointly train a global model while keeping data decentralized and private. We propose a variance-reduction based algorithm, VRA-FedSGD, for FL in the presence of heavy-tailed gradient noise and communication noise, where these noises are prevalent in large-scale machine learning over wireless networks and Internet of Things deployments. VRA-FedSGD employs a momentum variance reduction technique together with a nonlinear mapping to mitigate heavy-tailed gradient noise, and uses a variance-reduced aggregation mechanism to suppress heavy-tailed communication noise. In the mean sense, VRA-FedSGD achieves a convergence rate of {\small$\mathcal{O}\left(K^{-(p-1)/(2p-1)}\right)$} for nonconvex objective functions, where $p$ is the tail index of heavy-tailed noise. In the almost sure sense, VRA-FedSGD achieves a convergence rate of $\tilde{\mathcal{O}}\left(K^{-(1-1/(p-\epsilon))}\right)$ for strongly convex objective functions, where $\epsilon$ is an arbitrarily small constant. Simulated experiments on a logistic regression problem with real-world data verify the effectiveness of VRA-FedSGD.

This work proposes an iterative code-aided detection, decoding, and channel estimation scheme for metasurface-based holographic MIMO systems employing stacked intelligent metasurfaces (SIM-RIS) and their fully connected counterparts (BD-SIM-RIS). A novel channel estimation strategy is developed by exploiting low-density parity-check (LDPC) coding in the uplink, enabling both pilot and parity bits of the encoded packet to contribute to the iterative refinement of the channel. In addition, closed-form expressions for the metasurface parameter design are derived and incorporated into an alternating-optimization (AO) procedure. Numerical results demonstrate substantial gains in normalized mean square error (NMSE) and bit error rate (BER), with particularly strong improvements observed for BD-SIM-RIS architectures.

Ramanujan's third-order mock theta function $f(q)$ admits the well-known Appell-Lerch series representation: \[ \sum_{n=0}^{\infty}\frac{q^{n^2}}{(-q;q)_n^2}=\frac{2}{(q;q)_{\infty}}\sum_{n=-\infty}^{\infty}\frac{(-1)^nq^{\frac{3}{2}n^2+\frac{1}{2}n}}{1+q^n}. \] In this paper, we establish a natural generalization of this classical identity by utilizing the theory of harmonic Maass forms. Furthermore, we prove analogous Eichler-Selberg type relations for another third-order mock theta function $\omega(q)$. The method presented in this paper can be extended to study other classes of mock theta functions.

We extend the theory of indefinite Hardy spaces on finite bordered Riemann surfaces to the setting of ramified analytic coverings. Given a finite $n$-sheeted ramified covering $F\colon S_1\to S_2$ of finite bordered Riemann surfaces satisfying a spin-compatibility hypothesis, we construct (i) the direct image of a unitary flat vector bundle $\VxX{1}\otimes \Del{1}$ on the double $X_1$ under $F$, taking full account of the ramification divisor $R_F$ and establishing the extension across the branch locus via a local analysis; (ii) a canonical matrix function $G_2$ encoding the parahermitian structure on $X_2$, together with the induced representation $\chi_2$ of $\piX{X_2}{p_0}$; (iii) an explicit isometric isomorphism $\phi_F\colon H^{2,J_1(p)}(S_1,\VxS{1}\otimes\Del{1}) \xrightarrow{\;\sim\;} H^{2,J_2(p)}(S_2,\VxS{2}\otimes\Del{2})$ between the associated Hardy-Kreĭn spaces, provided that $h^0(X_1,\VxX{1}\otimes\Del{1})=0$ and that the branch locus is disjoint from $\partial S_2$. We then develop the resulting operator theory in terms of vessels and Bezoutian operators. To each object in the category $\mathcal{RH}$ of finite bordered surfaces with unitary flat bundles we attach a triangular vessel whose input and output spaces are the Hardy-Kreĭn spaces on the two surfaces. The Bezoutian of the vessel is expressed as a finite-rank operator on $\mathcal{H}_2$ whose kernel is built from bounded holomorphic point-evaluation functionals in $\mathcal{H}_2$ evaluated at the interior ramification images $F(r_\nu)\in S_2$, consistently with the boundary-transversality hypothesis $\partial S_2\cap B_F=\left\{\varnothing\right\}$. We prove that the assignment $(S,\Vx{},J)\mapsto H^{2,J(p)}(S,\Vx{}\otimes\Delta)$ extends to a covariant functor from $\mathcal{RH}$ (with ramified morphisms) to the category of Kreĭn spaces.

We study exact nonnegative matrix factorization (NMF) of small exact-rank-r matrices through the polyhedral cones of nonnegative preimages of the truncated SVD. Restricting each factor to an r-subset of a cone's extreme rays collapses the factorization constraint to the entrywise nonnegativity of a single r x r witness matrix; witness feasibility is a certificate that an exact size-r NMF exists, decided in one matrix inverse. A single-coupling completeness theorem shows every size-r NMF is representable this way, for every m, and a one-sided relaxation gives a closed-form solver that provably dominates the two-sided witness. Our main result concerns findability: at a fixed search budget, witness recoverability undergoes a sharp conic phase transition whose width collapses to a step as r grows. We rule out a universal-constant explanation (Goemans-Williamson) and a statistical-dimension one: the statistical dimension is flat across the transition, while the intrinsic-volume profile's variance tracks it. The transition is not an existence boundary; by completeness a ray-economical witness always exists and persists past the threshold, so what decays with m is its density among r-subsets. The threshold is thus budget-relative, moving logarithmically as the pool grows, and combinatorial rather than smooth-conic. A two-sided union of one-sided relaxations returns exact machine-precision factorizations; being budget-limited, its recovery obeys the transition we study, and a well-initialized coordinate-descent solver, not subject to this threshold, recovers more at larger sizes. Beyond the exact-rank case, the same coupling certifies the gap regime (nonnegative rank > r): infeasibility of its linear relaxation decidably proves no size-r factorization exists, exhibiting a dual witness. Our contribution is these certificates and the transition they expose, not a faster general-purpose solver.