Papers

Full-support local randomizers can incur unnecessary reporting distortion on ordered alphabets. We study randomized response whose output is restricted to a radius-$t$ neighborhood of the private input. Input-dependent supports create privacy leakage at their boundaries, so ordinary likelihood-ratio calibration is no longer sufficient. Under approximate local privacy restricted to input pairs at distance at most $H$, we derive the exact hockey-stick defect of the mechanism. We then solve the joint design of the truthful probability, off-truth probability, and support radius. The fixed-radius optimum is available in closed form, while the global optimum requires only a one-dimensional integer search. We characterize both the privacy floor and its inverse, namely the smallest defect attainable under a reporting-distortion budget. We further prove that stringent approximate privacy forces the support size to scale as $\Omega(H/\delta)$. Deterministic evaluations illustrate the resulting tradeoff.

It is shown that certain finite partially ordered sets arising from unit interval determinantal facet ideals satisfy the $\lambda$-chain condition. In addition, their applications to commutative algebra are briefly discussed.

In this short note we give a Khintchine-type inequality involving non-independentrandom variables. The basic setting is inspired by Breuer-Major theorem and the proof is based on a hypercontractivity argument.

A multiple-access channel transmits messages from several senders to one receiver. For quantum outputs, the receiver must distinguish states depending on all messages, while tests for different decoding errors may not commute. In this paper, we give a self-contained account of simultaneous decoding, in which one joint measurement recovers the messages at the established achievable rates using independent codebooks. The proof combines Sen's tilted-subspace method with Cheng's analysis of the pretty-good measurement, a standard measurement for distinguishing quantum states. Auxiliary labels close to a common reference state separate the geometric directions associated with different errors. A Gram-matrix bound then allows separately chosen binary tests to certify the performance of one measurement. The labels are removed from the completed decoder, leading to a deterministic packing theorem that applies to finite state ensembles, including unequal priors. We develop the two-sender case first, with worked examples and figures, before treating any fixed number of senders. Sharper finite-error guarantees for the same decoder are developed separately, with sharp bounds for two specified steps of the analysis rather than a claim of globally optimal decoding. Applications recover known achievable rates for unassisted and entanglement-assisted classical communication, distributed classical compression with quantum side information, and coherent-state communication through a beamsplitter or passive multiport device. The common proof makes the geometric mechanism and its quantitative consequences available without repeating the construction for each application.

Small area estimation (SAE) traditionally borrows strength across domains through a prespecified working model. We introduce small area learning (SAL), a learner-agnostic framework that formulates SAE as a multi-target semi-supervised learning problem and treats outcome prediction as a nuisance task. Areas may differ through unrestricted covariate shift and structured conditional shift, with information shared through common and area-specific components. We develop standard and cross-fitted estimators for low-complexity and flexible learners, unifying synthetic, model-based, and fully residual-corrected estimation. For area means, we establish consistency and asymptotic normality and provide variance estimators and confidence intervals. We extend the theory to settings with only aggregated population covariates and characterize the resulting aggregation bias. SAL reveals a bias–variance trade-off: partial residual correction can reduce mean squared error, whereas full correction yields estimators robust to working-model misspecification, potentially at higher variance. Simulations yield RMSE reductions of 34%–49% relative to direct estimation, with the best configuration varying across settings. An empirical application favours a cross-fitted estimator using a flexible learner under the chosen validation criterion. The numerical studies also illustrate how shrinkage can produce overly optimistic confidence intervals at the expense of strong modeling assumptions.

We prove several regularity theorems concerning Lipschitz viscosity solutions to the minimal surface system. The main lemma (i.e. “Decoupling Lemma") says that the component functions of such solutions satisfy the homogeneous Pucci inequalities. This allows us to apply the regularity theory for scalar viscosity solutions for elliptic PDE component-wise to a viscosity solution for the system. As a result of the Decoupling Lemma, we deduce that 2D Lipschitz viscosity solutions to the system are analytic in the interior, and obtain a quantitative partial regularity theorem. We also prove a geometric Harnack inequality for Lipschitz solutions. The Harnack inequality yields a strong maximum principle and half-space Liouville theorem. Another significant consequence of the lemma is a boundary regularity theorem for Lipschitz viscosity solutions. Notably, our theorem does not require any density assumption at the graph boundary, and we do not assume stationarity. This is used to prove a classification theorem for the singular set of a 4D stationary solution on a bounded domain. We also prove an $\varepsilon$-regularity theorem extending a result of the author. To conclude, we survey open problems. As an example, we have included a Bernstein-type theorem for 3D homogeneous entire Lipschitz viscosity solutions. Taken together, our results suggest that viscosity methods are a useful tool for studying the regularity of minimal graphs.

We study particle approximations of maximum mean discrepancy (MMD) flows with the Coulomb kernel on the torus $\mathbb{T}^d$, $d\ge2$, driven by thermal noise of strength $\tau\ge0$. For smooth, strictly positive initial and target densities and i.i.d. initial positions, we prove that the expected Wasserstein-1 distance between the empirical measure and the mean-field solution is at most of order $N^{-1/d}$ for $d\ge3$ and $\sqrt{(\log N)/N}$ for $d=2$, uniformly in time and in the temperature $\tau\ge0$, including the deterministic case $\tau=0$. At each fixed $\tau>0$, we also prove propagation of chaos in total variation for any $k$-fold marginal distributions of particles, uniformly in time, with a bound of order $\sqrt{k/\tau}$ times the empirical convergence rate. Our proof uses the modulated free-energy method of Bresch, Jabin, and Wang to reduce the particle convergence estimate to a relaxation estimate that controls the time integral of the mean-field velocity gradient uniformly in $\tau\ge0$. Near zero temperature, where the smoothing effect of the noise degenerates, we obtain this estimate by recentering the dynamics at thermal equilibrium and using the damping generated by the Coulomb interaction.

We consider the problem of computing Taylor approximations to the optimal feedback law and value function for infinite-horizon finite-dimensional nonlinear optimal control problem—also known as Al'brekht's method. The scalability of existing Al'brekht-based approaches to computing approximate solutions to the Hamilton-Jacobi-Bellman equation associated with the optimal control problem has been severely limited by computational resources. The computational time issue has been partially mitigated by recent works leveraging tensor-structure to develop efficient numerical methods. However, RAM requirements still have remained a major hindrance to scaling beyond state dimensions of a few hundred or low thousands. We propose an acceleration to compute the expensive higher-order terms in a reduced space, mitigating the curse of dimensionality. A fast and scalable Matlab implementation is provided open-source for general purpose use, with the ability to accommodate arbitrary polynomial nonlinearities in the control-affine state equation, as well as polynomial state-dependence in the cost function, and the ability to handle systems in generalized form with a non-identity mass matrix, such as finite element models. Combined with the exploitation of sparsity and modern low-rank Riccati solvers, this permits computing approximations to polynomial feedback laws based on Al'brekht's method in hundreds of thousands of states.

In Fermi's 1926 note, the expression assigned to the probability that all partial sums remain below a threshold is the complementary crossing probability. We obtain the correct probability from the same diffusion equation and locate the inconsistency in the final integration. We then state the limit for independent increments under the Lindeberg condition. A counterexample with smooth densities shows that small individual variances do not suffice to justify this limit. The probabilistic results are classical; the purpose of this note is to reconcile the stated event with the printed calculation and to make the sufficient hypotheses for the discrete limit explicit.

Given a surface of genus $g \geq 2$, its inf spectrum is the ordered multiset of inf lengths of curves on the surface, the infimum being over all hyperbolic metrics. We show that the inf spectrum has exponential growth with coefficients that only depend on g, improving the earlier double exponential upper bound. The proof involves developing a number of tools including the process of "thickening" a surface and the notion of model surfaces. These tools may be of independent interest.

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with root system $\Phi$. A subset $S\subseteq\Phi$ is closed if $\alpha,\beta\in S$ and $\alpha+\beta\in\Phi$ imply $\alpha+\beta\in S$. For finite and affine Kac-Moody algebras, this closure property is known to extend from pairs to arbitrary finite sums that are roots. We prove that every closed subset of the root system of any symmetrizable Kac-Moody algebra is closed under arbitrary finite sums that are roots. The proof uses real-root strings and simultaneous separation of positive imaginary roots, together with an addition lemma for fundamental imaginary roots. As an application, we extend a necessary condition for wideness of regular subalgebras from finite and affine types to all symmetrizable Kac-Moody algebras: the symmetric closure of the roots associated to a wide regular subalgebra is the entire root system.

We prove sharp residual stability estimates for the critical Poincaré-Sobolev equation near sums of one positive hyperbolic solution and one projected Euclidean bubble in dimensions at least four. The estimates are uniform in the relative positions of the centers. The sharp rate is linear except in dimension six, where it contains a square root of a logarithm. For nonnegative functions, the same estimates hold under an energy constraint near the sum of the two profile energies for \(\lambda\) outside a finite set, and an additional mass constraint removes this restriction. For arbitrary finite numbers of profiles, we identify and realize all nonnegative Palais-Smale levels. We characterize the parameters at which different pairs of profile numbers have the same energy and show that the limiting energy and mass determine both profile numbers.

The set \(P_{n}\) consists of the characteristic vectors of pedigrees, which are combinatorial encodings in one-to-one correspondence with Hamiltonian cycles. For \(n \geq 4\) and \(\tau_{n} = \binom{n}{3}-1\), the Membership Problem for the Pedigree Polytope (M3P) asks whether a given vector \(X \in \mathbb{Q}^{\tau_{n}}\) belongs to \(\mathrm{conv}(P_n)\). This paper gives a complete account, with the full proof statements and machine-verified Lean~4 certificates, of the necessary and sufficient (\NS) condition established in Chapter~5 of Pedigree Polytopes: New Insights on Computational Complexity of Combinatorial Optimisation Problems (Arthanari, 2023): membership of $X$ in $\mathrm{conv}(P_n)$ is characterised by the feasibility, at every stage $k=4,\dots,n-1$, of a recursively constructed layered network, together with, when needed, a multicommodity flow problem $\mathrm{MCF}(k-1)$ achieving its maximum possible flow $z_{\max}$. We present the construction, the \NS{} condition itself, and the proofs of this characterisation, together with the resulting polynomial-time algorithm for M3P. Both directions of the \NS{} condition — sufficiency and necessity (the multicommodity-flow feasibility argument of the book's Chapter~5) — are machine-verified in Lean~4/Mathlib4 with zero unresolved sorrys in either dependency chain. The \NS{} characterisation therefore follows directly from the two machine-checked implications. The complete Lean~4 project is publicly available. A companion introductory article (arXiv:2507.09069) presents the same material for a broader readership, without the Lean~4 verification; the present paper adds full statements, proof sketches, and the complete machine-checked certificates.

The extension complexity $\operatorname{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that projects onto it. Motivated by the complexity of neural networks, Hertrich and Loho introduced the virtual extension complexity $\operatorname{vxc}(P)$ of a polytope $P$, which is the minimum of $\operatorname{xc}(Q) + \operatorname{xc}(R)$ among all polytopes $Q$ and $R$ such that $P+Q=R$. We show the following results pertaining to virtual extension complexity. First, we resolve the core open problem on the topic by showing the virtual extension complexity can be exponentially smaller than extension complexity. Second, we show there are matroid base polytopes with exponential virtual extension complexity by adapting a semi-algebraic counting argument by Hrubeš that extends the counting arguments of Rothvoss. Finally, we show that the maximum virtual extension complexity of a polygon with $n$ vertices is $\Theta(\sqrt{n})$. This resolves an analogue of a question on extension complexity of Fiorini, Rothvoss, and Tiwary in 2011 in the case of virtual extension complexity.

In this paper we prove that the classical thermoelastic Timoshenko and Bresse systems with thermal coupling acting only on the bending moment are not exponentially stable under full homogeneous Dirichlet boundary conditions if the equal wave speed condition is not satisfied. Combined with their well-known converses, our results provide complete characterizations of exponential stability in terms of the coefficients of each system. To the best of our knowledge, these characterizations were previously known only under mixed boundary conditions. The proof is based on the introduction of a boundary correction term in the standard trigonometric ansatz so that the full homogeneous Dirichlet conditions are satisfied.

We study the $p$-Bures-Wasserstein distance $D_p(A,B)=(\mathrm{tr}\,A^p+\mathrm{tr}\,B^p-2\|A^{1/2}B^{1/2}\|_p^p)^{1/2}$ on positive semidefinite matrices; it equals the Bures-Wasserstein distance for $p=1$, and the Hilbert-Schmidt distance for $p=2$. We prove that $D_p$ is a metric for every $1\le p\le2$, by proving a stronger 2014 conjecture of Lin and van den Driessche on $3\times3$ block matrices. Our main tool is a partition inequality for Schatten norms, proved by complex interpolation. We also characterize Fréchet means for $D_p$ by a fixed-point equation and prove geometric convergence of the associated iteration.

We prove the geometric halves of Hilbert's Nullstellensatz and of the real Nullstellensatz for submodules of finitely generated modules and for coherent sheaves, complementing our earlier result that semiprime (resp. real) submodules are intersections of prime (resp. real prime) submodules. First, we pass from prime to maximal submodules: over a Jacobson ring, a prime submodule $P$ of $M$ is an intersection of maximal submodules whenever $M/P$ is a torsionless $R/(P:M)$-module, in particular whenever $M/P$ is finitely generated (the latter case is due to Maani Shirazi and Sharif). Consequently, if $R$ is a finitely generated algebra over an algebraically closed field and $M$ is finitely generated, the smallest semiprime submodule containing a submodule $N$ of $M$ consists of all $f\in M$ such that, for every point $a$, the value $f(a)$ (i.e. the image of $f$ in the fibre $M/\mathfrak{m}_aM$) lies in $N(a)$ (i.e. the image of $N$ in $M/\mathfrak{m}_aM$). We prove the real analogues over real Jacobson rings; for finitely generated algebras over a real closed field they are determined by the real points. The module classifications and directional vanishing formulas yield both Nullstellensätze for coherent sheaves on schemes of finite type over a field. We also prove compatibility of semiprimeness, reality and both radicals with arbitrary localization, and the resulting stalk criteria.

Given $1 < p < \infty$ and a function $m \colon \mathbf{N}_0 \to \mathbf{C}$, we provide a characterization of radial Schur multipliers $M(x,y) = m(d(x,y))$ on the Schatten $p$-class of a tree $(X,d)$ equipped with its graph metric. That is, the operator norm of the radial Schur multiplier $S_M$ is controlled by the norm of the Fourier multiplier $T_m$ on the torus: \[ \big\| S_M \colon S_p(X) \to S_p(X) \big\|_{\mathrm{cb}} \lesssim \frac{p^4}{(p-1)^2} \, \big\| T_m \colon L_p(\mathbf{T}) \to L_p(\mathbf{T}) \big\|_{\mathrm{cb}}, \] with both quantities being comparable when the tree $X$ has infinite diameter. This is a rather unexpected inequality, which necessarily fails at the endpoints $p=1,\infty$ due to the work of Haagerup, Steenstrup and Szwarc. As an application, we construct the first examples of completely $S_p$-bounded Herz-Schur multipliers in the free group that are unbounded as Fourier multipliers. This result disproves Fourier-Schur transference—a longstanding open problem since the work of Neuwirth and Ricard in 2011. In line with this, our main result also implies the surprising fact that free balls Schur multipliers are uniformly $S_p$-bounded, in sharp contrast to their Fourier behavior. In addition, we disprove Fourier-Schur transference over the algebraic groups $\mathrm{PGL}_2(\mathbf{Q}_q)$ and $\mathrm{PSL}_2(\mathbf{R})$ with $\mathrm{K}$-biinvariant counterexamples, the latter involving additional geometric and probabilistic techniques.

We study the approximate channel simulation problem in the normal approximation regime. Our starting point is the fact established by Cao et al.~\cite{Cao_channel_simulation}, that the minimal codebook size $M_\epsilon^*(W)$ for a fixed error tolerance $\epsilon>0$ is closely related to the $\epsilon$-smooth channel max-information, $I_{\max}^\epsilon(W)$. Using this, we refine earlier large-blocklength approximations to $M_\epsilon^*(W^{\otimes n})$ for discrete memoryless channels. Specifically, we provide tight third-order expansions for Gallager-symmetric channels and demonstrate that the third-order term is channel-dependent. As in corresponding channel coding expansions, we identify a substantial rate gap between singular and nonsingular channels. Notably, the direction of this gap is reversed relative to channel coding: The third-order term is $-\frac{1}{2}\log n$ for singular channels but $O(\log\log n)$ for nonsingular channels.

We study stationary discrete-time mean-field games with finite state and action spaces under the average-reward criterion, with an entropy-regularized objective and population-dependent dynamics and rewards. Under a minorization condition, the soft Q-operator is a span-seminorm contraction for the full transition kernel, so relative value iteration does not require the minorizing measure. When the Lipschitz constant of the kernel is below the minorization mass and the temperature exceeds an explicit threshold, the equilibrium fixed-point operator is a contraction: an equilibrium exists, its population distribution is unique, and a fixed-point iteration converges to it geometrically. We then learn the equilibrium model-free from a simulator, combining soft Q-learning at a fixed population distribution with an empirical transition kernel. We prove a finite-sample bound for soft Q-learning in the span seminorm, with temperature-independent constants, and derive convergence rates for the learned population distribution and policy. A congestion-aware routing game illustrates the results.

Let $R$ be a commutative ring. A submodule $N$ of an $R$-module $M$ is semiprime if $m\in(N:m)M$ implies $m\in N$. It is known that every semiprime submodule of $R^n$ is an intersection of prime submodules, and Aryapoor extended this to finitely generated modules. We show that no finiteness assumption is needed: every semiprime submodule of an arbitrary $R$-module is the intersection of the prime submodules containing it. The proof is short and uses neither coordinates nor presentations; its key step is that a nonzero element $q$ of a semiprime module $Q$ satisfies $sq\notin\p Q$ for all $s\notin\p$, where $\p$ is any minimal prime over the annihilator of $q$. We define an $R$-module $M$ to be real if $\sum_i m_i\otimes m_i=0$ in $M\otimes_RM$ implies $m_i=0$ for all $i$, and a submodule $N\subseteq M$ real if the module $M/N$ is real. We prove the corresponding result for real submodules: every real submodule of an arbitrary module is the intersection of the real prime submodules containing it. On $R^n$ our notion of realness coincides with the coordinate definition used in earlier work on real Nullstellensätze for free modules.

We study the behavior of the $k$-th-order nonhomomorphicity of Boolean functions under restriction to affine subspaces. For odd $k\ge3$ and $0\le t<n$, we introduce the $t$-restriction $k$-th-order nonhomomorphicity $P_{k,t}(f)$, defined as the minimum of $P_k(f_{\mid A})$ over affine spaces of codimension at most $t$. For a direct sum of $m$ monomials of degrees at least two, we determine $P_{k,t}$ exactly. When $t<m$, a minimum is obtained by setting one variable equal to zero in each of the $t$ monomials of lowest degree. When $m\le t<n$, the minimum is zero. The proof combines multiplicativity of even Walsh moments with a contraction argument for affine restrictions. We establish a general formula for the Walsh transform of a Boolean function restricted to an affine subspace of arbitrary codimension, yielding an exact spectral characterization of the $t$-restriction normalized nonlinearity $\rnnl_t(f)$. As odd $k$ tends to infinity, $(1-2P_{k,t}(f))^{1/(k+1)}$ converges to $1-2\rnnl_t(f)$. We further prove that every non-affine function has a hyperplane restriction with strictly smaller odd-order nonhomomorphicity. Finally, we determine $\rnnl_1(T_{d,n})$ exactly for all threshold functions with $n\ge3$. This proves the two threshold-function conjectures left open in~\cite{SergeAnaPante2025}, including the previously unresolved odd-majority case, where a minimizing hyperplane is $x_1+x_2=0$.

We propose k-Hessian equations as auxiliary equations for degenera- tions of Kähler classes to nef classes of numerical dimension k. As proof-of-concept results, we provide a uniform mean-value inequality and uniform à priori estimates for Hessian equa- tions.

We study the greedy sequence of positive integers introduced by Slater and Vélez. They proved that it is a permutation of the positive integers but left open whether its sequence of consecutive absolute differences is also a permutation. We prove that it is.

We initiate a systematic operadic approach to anti-pre-Lie algebras. We study the operad controlling anti-pre-Lie algebras, its relationship to the operad of Novikov algebras and its cyclic structure. We establish that the Lie bracket of an anti-pre-Lie algebra satisfies the standard identity of degree five. We also establish a quadratic maximality property of the anti-pre-Lie operad with respect to this identity. We then show that the anti-pre-Lie operad and its Koszul dual are not Koszul and give a full description of the latter. Finally, we identify the Lie elements in the Koszul dual operad with the operad of metabelian Lie algebras.

Belief propagation (BP) is attractive for quantum low-density parity-check (QLDPC) decoding because of its low complexity and low latency, but short cycles and degeneracy can limit its error-correction performance. Reinforcement-learned sequential scheduling (RL-S) substantially improves this performance by adapting the variable node (VN) update order, although Q-table storage and heap-based selection introduce additional overhead. We propose bucket RL-S (B-RL-S), which quantizes learned Q-values into integer priorities and schedules VN updates using dynamically maintained first-in, first-out buckets. Without retraining or changing the BP message updates, B-RL-S reduces policy storage and refreshes only priorities affected by local residual syndrome changes. A fixed number of priority levels enables constant-time VN selection and queue relocation. For the $[[288,12,18]]$ bivariate bicycle code over the depolarizing channel at $p=0.05$, B-RL-S with eight priority levels achieves approximately one fifth the block error rate (BLER) of flooding BP followed by order-10 ordered-statistics decoding. The respective caps are $1000$ sequential iterations for B-RL-S and $1000$ flooding iterations for the BP stage of the reference decoder. These results highlight strong error correction with inexpensive scheduling and no additional post-processing stage.

Based on a semigroup approach, we prove quantitative uniform-in-time propagation of chaos for two systems of interacting particles with singular interaction kernels $K$ of type $L^k$, $k>d$, and a contractive Lipschitz confinement force field. First, we consider a system with moderate interaction, which means that the particles interact through the mollified kernel $K\ast V^N$, for some approximation of identity $V^N = N^{d\alpha} V(N^\alpha \cdot),\, \alpha>0$. Assuming that $\|K\|_{L^k}$ is sufficiently small, we obtain that for any $\varepsilon>0$ the uniform-in-time rate is of order $N^{-\min\left(\alpha,\frac12-\frac{d\alpha}{2}-\varepsilon\right)}$, plus an exponentially small in time contribution of the initial condition. Second, for the classical, non-mollified particle system we establish uniform-in-time propagation of chaos with rate $N^{-\min\left(\alpha,\frac12-\frac{d\alpha}{2}-\varepsilon\right)} + \|K-K\ast V^N\|_{L^{k}}$, plus an exponentially small in time contribution of the initial condition. The method relies on the regularity properties of the solution of the associated nonlinear Fokker-Planck equation combined with estimates of the Duhamel formulation of the mollified empirical measure, together with partial Girsanov transforms for the non-mollified system of particles.

In any noncommutative Carnot group, for every $0\le \alpha<1$, we show the existence of a $C^{1,\alpha}_\mathrm H$ surface that is unrectifiable with respect to the Euclidean distance. Precisely, the surface has Euclidean Hausdorff dimension higher than its topological dimension. We show that in step-2 Carnot groups the construction gives hypersurfaces of maximal Euclidean Hausdorff dimension, equal to $q-\frac{1+\alpha}{2}$, where $q$ is the topological dimension of the group. Here the range of unrectifiability is sharp, since for $\alpha=1$ the surface is rectifiable in the classical sense. In step-2 Carnot groups, it is also possible to prove that the $(q-\frac12)$-dimensional Euclidean Hausdorff measure of every $C^1_\mathrm H$ hypersurface is zero. These results extend the seminal work by Kirchheim and Serra Cassano in the first Heisenberg group.

The involution numbers $1,1,2,4,10,26,\ldots$ repeat modulo $3$, $5$ and $7$. We explain this, and many similar patterns, by one rule. Many sequences count permutations in which every cycle of length $m$ carries a weight $w_m$ (involutions, derangements, "sets of lists", and many others in the OEIS). Writing $A(n)$ for such a count, the rule is $A(n+k)\equiv A(n)A(k)$ modulo $\mathrm{lcm}(n,k)$. It generalizes our earlier proof of the case modulo $k$ for one such sequence, but now it holds for all weights at once, as a coefficientwise congruence between universal polynomials, and we find the largest modulus that works for the universal polynomials, $nk/\mathrm{rad}(\gcd(n,k))$, and show that nothing larger does. We then show that passing from $k$ to $pk$ behaves like raising to the $p$-th power modulo a high power of $p$, with a sharp exponent for the polynomials, which gives $A(p^r)\equiv(w_1+(p-1)w_p)^{p^{r-1}}$ modulo $p^r$ for odd $p$, and explains why the prime $2$ is special. A family of sequences defined by Lagrange inversion fits the same framework, which proves Bala's conjecture for the whole family (the single sequence A278070 had been proved earlier by an automated system) and the divisibility observed in the entries A251568 and A251569. As further applications we prove OEIS conjectures of Richman, Hanna, van Hoeij and Bala (A227937), and the theorems apply to 101 OEIS sequences, whose congruences are recorded in a supplementary catalogue. Higher-order versions of the content formula are treated in a separate paper.

Information-theoretic generalization bounds usually control dependence on the task sample, whereas selective privacy may constrain only a different, private coordinate. We study i.i.d. pairs $(X,Y)$, where a learner observes $(X^n,Y^n)$, leaks at most $\epsilon_n$ nats about $X^n$, and incurs loss on $Y$. A chain-rule transfer converts this mismatched constraint into the closed-form universal certificate $I(W;Y^n)\leq\min\{nH(Y),nH(Y\mid X)+\epsilon_n\}$. Combining the certificate with mutual-information generalization yields a bound governed by residual task uncertainty and normalized privacy leakage. We derive complementary task-compression and individual-sample refinements, identify a sharp obstruction showing why selective privacy alone need not give a vanishing certificate, and invert the bound to obtain the exact leakage allowance for a prescribed generalization tolerance. We prove exact information-level achievability for deterministic-task and independent-residual source families and show that the resulting $\sqrt{\epsilon_n/n}$ generalization rate is tight up to a universal constant. We also give a fixed-horizon adaptive application in which conditional leakage costs add.