Daily Summary for 2026-10-19

Introduction to the show: ident: Mathematics. The day in mathematics: new theorems, new proofs, and what they open.

Ada: It's the nineteenth of October, twenty twenty-six, and this is the day's research.

Emmy: 11 new papers came out today.

Ada: I'm Ada.

Emmy: And I'm Emmy, here with Kolya.

Ada: We'll take the day in one pass, then pull out the papers we're staying with.

Emmy: Start us off with Smoothness of Shortest Isogenies to Frobenius Conjugates.

The day's research: Ada: Today we push theoretical limits on optimization algorithms over strongly convex sets.

Emmy: What specific lower bounds are we pushing today?

Kolya: We have an omega L diam S two over t two distributional lower bound for randomized first-order linear minimization oracle methods.

Ada: That result is a new benchmark for these methods. It builds on prior work showing worst-case objective-error lower bounds of order one over t squared.

Emmy: Does that prior result match the upper bound dependence on t in AKMPC?

Kolya: Yes, it matches the upper bound in its dependence on t in AKMPC.

Ada: How does randomization circumvent these established lower bounds?

Emmy: A distributional lower bound for randomized methods reuses techniques from AKMPC. It characterizes the conditional distribution of the hidden permutation.

Kolya: This work bounds the expected decrease of a combinatorial statistic controlling optimizer variance, suggesting randomization might overcome previous limitations.

Ada: We also have results concerning coefficient choice in Galois-ring lifts of APN permutations in AKMPC.

Emmy: That determines the ramification of specific polynomials and shows how they satisfy identities over Galois rings.

Kolya: This relates to the structural properties of maps important for lifting permutations to these rings.

Ada: Finally, we look at solving differential equations using neural networks with a neural JKO scheme for Hellinger-Kantorovich gradient flows.

Emmy: This scheme develops a mesh-free approach allowing spatial redistribution and local mass creation or loss within a single variational step.

Kolya: The aim is to recover the exact JKO minimum under certain conditions using monge-growth pairs in AKMPC.

Ada: That sounds like a significant development for solving PDEs with neural networks.

Ada: Learned Adaptive Multiresolution Diffusion Imaging addresses adaptive multiresolution methods using learned policies instead of fixed rules.

Emmy: That allows better resource allocation where needed, improving metrics like reference discrepancy and occupancy.

Kolya: Learned AMDI replaces the post-propagation selector with a shared local policy trained via proximal policy optimization.

Ada: It keeps the original fixed-tree propagator structure while using the learned policy to decide refinements.

Emmy: Regression tests confirm that identical trees reproduce deterministic AMDI trajectories to machine precision.

Kolya: Learned AMDI executes 393 refinements across nine held-out cases in these tests.

Ada: This reduces mean terminal reference discrepancy from 0.17496 to 0.13657 and increases occupancy from 0.13737 to 0.26660.

Emmy: The shared actor transfers without retraining across different resolutions like sixty-four by sixty-four and one hundred twenty-eight by one hundred twenty-eight images.

Kolya: Performance comparison shows Learned AMDI achieves a discrepancy of 0.13792 with slightly better RMSE and SSIM than deterministic AMDI at comparable occupancy levels.

Ada: Both methods place on essentially the same accuracy-occupancy tradeoff, supported by a decision-one-only control reaching 0.13742.

Emmy: This suggests most benefit comes from initial allocations rather than just sequential decisions in that control mechanism.

Kolya: This adaptive allocation builds upon prior work toward maximizing algebraic connectivity via density-dependent backbone construction and reinforcement learning completion.

Ada: The concept of predicting solver performance variability is relevant to understanding why some computational studies yield stable results or exhibit large dispersion.

Ada: The WVS metric aggregates dispersion across runtime and optimality gap for MIP instances in randomized runs.

Emmy: That score shows a right-skewed distribution, meaning many instances are stable but some have large variability unexplained by basic properties.

Kolya: A tree-based predictor with gated tail uplift achieved an absolute error of 0.058 and a Spearman rank correlation of 0.65 on the test split.

Ada: Feature importance analysis points to root degeneracy, matrix structure, and presolve reductions as main drivers of high WVS.

Emmy: That connects well with the structural design problem in network topology addressed by density-dependent backbone construction and reinforcement learning completion.

Kolya: That method decomposes design into a structural backbone based on density followed by a reinforcement learning stage to refine edges.

Ada: We also have the work on Smoothness of Shortest Isogenies to Frobenius Conjugates, which examines minimum degree of isogenies from elliptic curves to their Frobenius conjugate.

Emmy: How does that relate to what we found regarding the structure of these polynomial representatives?

Kolya: The optimal reduced representatives of quadratic APN permutations in odd dimension depend on coordinates but not differential uniformity.

Ada: That contrasts with the analysis on coefficient choice in Galois-ring lifts of APN permutations where specific terms and constraints are explicit.

Emmy: So, one paper focuses on algebraic structure, another on geometric properties, and we have the WVS prediction for solvers.

Kolya: The defense filters for robust learning of partially observed dynamical systems use observation matrix knowledge to construct a filter with full column rank and subspace separability.

Ada: That is quite different from the layered control architecture co-designing which seeks systematic theory for holistic design of these systems.

Emmy: We also have the Neural JKO Scheme for Hellinger-Kantorovich Gradient Flows, which develops a mesh-free scheme for advection-reaction-diffusion equations.

Kolya: And we have Learned Adaptive Multiresolution Diffusion Imaging which preserves AMDI constraints while replacing the selector with a shared local policy trained by proximal policy optimization.

Ada: The sublevel Flood bifiltration explores efficient computing of 2-parameter persistent homology on large point sets, which is useful for scalable topological analysis.

Emmy: These papers cover performance prediction, algebraic structure, geometric properties, dynamical systems filtering, control architecture design, and diffusion imaging.

Kolya: Today's lucky papers are Smoothness of Shortest Isogenies to Frobenius Conjugates because it examines minimum degree of isogenies from elliptic curves to their Frobenius conjugate; Optimal Reduced Representatives of Quadratic APN Permutations in Odd Dimension because it details the dependency on coordinates for reduced polynomial representatives; Defense Filters for Robust Learning of Partially Observed Dynamical Systems because it proposes a defense-filtering framework using observation matrix knowledge; Co-designing a Layered Control Architecture Layered control architectures are ubiquitous but lack systematic theory for holistic design; Learned Adaptive Multiresolution Diffusion Imaging which preserves AMDI constraints while replacing the post-propagation selector with a shared local policy trained by proximal policy optimization; and Toward Maximizing Algebraic Connectivity via Density-Dependent Backbone Construction and Reinforcement Learning Completion because it studies an open network design problem of constructing graph topologies with maximum algebraic connectivity under fixed numbers of vertices and edges.

Ada: That completes our review. The WVS metric predicts solver struggles, the APN work details permutation structure, the defense filters robust learning, layered control co-designs systems holistically, diffusion imaging learns from local policies, and density-dependent backbone construction maximizes algebraic connectivity in networks.

Emmy: Agreed. We have a lot of material today that connects disparate fields.

Kolya: Indeed. It shows how structural design principles apply across optimization, topology, and dynamical systems research.

Ada: That is the extent of our review for this part. The episode is over now.

Emmy: This conversation concludes the review segment on October nineteenth, twenty twenty-six.

Lucky paper: 2610.21488: Ada: We're diving into Smoothness of Shortest Isogenies to Frobenius Conjugates today.

Emmy: This paper tackles the smoothness of the isogeny degree, denoted by delta E, and how that relates to Wesolowski’s attack on the supersingular endomorphism ring problem.

Kolya: The main result establishes that delta E is smooth, which unconditionally validates Wesolowski’s p1/3+o(one) attack.

Ada: That means the security reduction for schemes like SQIsign is unconditional now instead of relying on a heuristic.

Emmy: The analysis focuses on how often delta E is B-smooth, which connects to the largest prime factor P+(delta E).

Kolya: The upper bound proof shows that the probability PrP+(delta E) is less than or equal to B, and this depends on the Dickman-de Bruijn function rho.

Ada: That means we are relating how often delta E has small prime factors to some established distribution function.

Emmy: The lower bound uses quaternionic pairs estimate Cp(m, n), which counts ordered pairs of distinct primitive connection isogenies to establish a bound on PrP+(delta E).

Kolya: By combining those bounds with the class number formula, we get a lower bound: PrP+(delta E) is less than or equal to B greater than or equal to c1 rho(u) (log p)three.

Ada: That comparison shows that Wesolowski’s heuristic W is true because the lower bound is strictly larger than the upper bound derived from minimal degree connecting isogenies.

Emmy: The paper also extends this to a higher success probability for the attack by considering all isogenies of degree at most X.

Kolya: This shows that E

S<=E: is greater than or equal to a lambda rho(u) log p, which implies the success probability of the Frobenius-claw event is at least C epsilon rho(u) log p.

Ada: That demonstrates that the success probability is strictly larger than what you get by only considering minimal degree connecting isogenies.

Emmy: The paper concludes by proving that for every fixed zero zero such that PrP+(delta E) is less than or equal to B greater than or equal to c1 rho(u) (log p)three.

Kolya: So, the smoothness of shortest isogenies to Frobenius conjugates confirms the unconditional nature of Wesolowski’s p1/3+o(one) attack on the supersingular endomorphism ring problem.

Ada: It’s a key element in affirming that security result for isogeny-based cryptography.

Emmy: This work builds on prior research showing how isogenies of large prime degree N can be evaluated using square-root Velu formulas with Oe(sqrt N) field operations.

Kolya: That naturally leads to a finer arithmetic question about the prime factors of delta E, specifically how often it is B-smooth or what its largest prime factor P+(delta E) is distributed.

Ada: It’s interesting how the analysis links smooth numbers and the Dickman-de Bruijn function rho to these isogeny properties.

Emmy: The paper also mentions that prior work established that delta E is less than or equal to floor(p2/three), which Wesolowski used for his p1/3+o(one) attack.

Kolya: This result confirms the unconditional nature of the attack, which was previously just a heuristic based on that upper bound.

Ada: That really solidifies the security argument in this area.

Emmy: We have covered the core mechanics and implications of Smoothness of Shortest Isogenies to Frobenius Conjugates today.

Lucky paper: 2610.21169: Ada: We're looking at Optimal Reduced Representatives of Quadratic APN Permutations in Odd Dimension today.

Emmy: This paper talks about quadratic almost perfect nonlinear permutations and whether they have an optimal linear equivalent.

Kolya: The main theorem states that for every quadratic almost perfect nonlinear permutation Q of F2n with odd n greater than or equal to five there exist invertible F2-linear maps L and A such that the derivative of the reduced representative of L times Q times A has no zero and each of its nonempty fibres has exactly two elements.

Ada: So it's establishing that for odd dimensions n greater than or equal to five every quadratic almost perfect nonlinear permutation in odd dimension n greater than or equal to five has an optimal representative L times Q times A with L and A invertible F2-linear maps.

Emmy: That means the EA class of every quadratic APN permutation in odd dimension n greater than or equal to five contains an optimal representative.

Kolya: The existence proof is checked in the Lean four proof assistant for odd n greater than or equal to nine and it fails for n = three and there are no such permutations for even dimensions.

Ada: That's a very precise range; it excludes n=three and all even numbers.

Emmy: And the related conjecture is about the Reduced Critical-Point Conjecture, which posits that for every n, the reduced representative of every APN permutation of F2n has a zero of its formal derivative in F2n.

Kolya: Corollary I.one states that for every odd n greater than or equal to five every quadratic APN permutation of F2n is linearly equivalent to a counterexample to the Reduced Critical-Point Conjecture.

Ada: So this paper shows that the structure of these permutations is fundamentally different from what those conjectures suggested for odd dimensions greater than five.

Emmy: The methodology involves fixing Q and an invertible F2-linear input A, setting g equals Q times A, and analyzing invariants s(A) which is the dimension of the common kernel of maps L for which red(L times g)' is constant.

Kolya: It also looks at η(A) which records whether that constant can be nonzero.

Ada: So you are checking properties related to how the derivative behaves under these linear transformations, specifically looking at kernel dimensions and non-zero constants.

Emmy: The work isn't formalized for n = five and seven but for n greater than or equal to nine the existence proof shows that a suitable input always exists.

Kolya: For odd n greater than or equal to nine at least a fixed fraction of all invertible inputs works, more than half for n greater than or equal to fifteen.

Ada: That suggests an optimal representative can be found by independent uniform sampling of inputs when n is sufficiently large and odd.

Emmy: The appendix provides closed-form optimal representatives in every Gold class, which is a bit different from the search method used for smaller cases.

Kolya: Appendix A provides a sharper degree bound using the Pfaffian route, relying on the count of Theorem VI.one with degree En equals n times (n minus one) and the inequality of Theorem VI.three for all odd n greater than or equal to nine.

Ada: So we have a sharp degree bound derived from counting theorems for large odd dimensions greater than or equal to nine in the study of Optimal Reduced Representatives of Quadratic APN Permutations in Odd Dimension.

Emmy: This confirms the exact range of n where this fundamental result holds true for quadratic almost perfect nonlinear permutations.

Kolya: The data and code are available at https://github.com/hirokifukui/apnlift under release v1 point 0.three which is useful for checking the results.

Ada: That gives us a concrete place to verify the findings for these complex permutation structures in odd dimensions greater than or equal to nine.

Lucky paper: 2610.21935: Ada: Today we are looking at Defense Filters for Robust Learning of Partially Observed Dynamical Systems by Jihun Kim and Javad Lavaei.

Emmy: This paper addresses estimating unknown system dynamics from partial observations even when there's an adversarial attack with a probability of p less than zero point five.

Kolya: It proposes a defense-filtering framework where the system designer uses the observation matrix C to build a filter matrix E that restricts adversarial injections to a lower-dimensional subspace.

Ada: The goal of this construction is ensuring the full column-rank condition rank(CE) equals e, which requires r greater than or equal to e.

Emmy: So the objective is then to recover the Markov parameter matrix G star, which is defined by G star equals D CB CAB... CAk-2B.

Kolya: The analysis focuses on the constrained optimization problem where we minimize H times J squared s subject to HCE equals Ie.

Ada: The main result establishes a finite-sample error bound for the Markov parameter matrix G star, specifically that the estimation error satisfies K hat F equals O rho one minus p s (one minus p) (H hat two + H star two) (one plus C squared M) (one minus 2p) sigma.

Emmy: That error bound decays exponentially with parameter s when rho one minus p is less than one, which means the attack probability p must be less than one minus rho.

Kolya: Furthermore, the normalized filter residuals give us a diagnostic signal where unattacked periods show small filter residuals while attacked periods induce large residual spikes.

Ada: This diagnostic signal is useful for separating normal and corrupted data timestamps during estimation.

Emmy: The paper successfully introduces this constrained l2-norm estimator under adversarial attacks with p less than zero point five, overcoming limitations of previous methods that required diminishing attack probability as the target error decreased.

Kolya: A lower-bound constraint H two greater than or equal to one can be enforced in the optimization problem to prevent the estimation error factor from approaching zero and ensuring a stable time bound.

Ada: The relative estimation error bound is derived as O rho one minus p s (one minus p) (one plus one sin theta) (one plus C squared M) (one minus 2p) sigma.

Emmy: This shows the estimation error decays exponentially with parameter s whenever rho one minus p is less than one, which is equivalent to p less than one minus rho.

Kolya: The analysis provides a principled metric in the form of normalized filter residuals for attack detection based on these two conditions.

Ada: How does this framework handle the adversary choosing the attack vector w t arbitrarily at each attack time?

Emmy: The paper assumes an adversarial attack vector w t is injected with probability p less than zero point five, and it is identically zero when there is no attack.

Kolya: The defense filter E must be constructed to satisfy both the full column-rank condition and the subspace separability condition for this scenario.

Ada: The subspace separability condition assumes that the intersection of Im(CE) and Im(

CAE... CAs-1E: ) is zero, which is necessary for a matrix H to exist satisfying HCE equals Ie.

Emmy: This geometric separation allows us to evaluate a single attack scenario against observed data.

Kolya: The paper successfully introduces this constrained l2-norm estimator and establishes rigorous finite-sample error bounds for recovering the Markov parameter matrix G star.

Lucky paper: 2610.20900: Ada: We’re looking at Co-designing a Layered Control Architecture today. This paper is about finding a framework to co-design two-layer LCA by optimizing over architectural and tuning design variables subject to stability, feasibility, and compute constraints.

Emmy: So the main problem formulation involves optimizing variables phi and theta to find LCA designs that are stable, feasible, and satisfy compute constraints. What exactly are those architectural design variables?

Kolya: The architectural design variables include the tuple of choices like n y, n r, N r, tau L, and L. Then there's the tuning-based design variables, which are numerical values for parameters like K, Ap, and Bp.

Ada: The objective is to minimize a multi-valued cost function that maps these choices to Pareto curves showing tradeoffs between speed, accuracy, and compute. What kind of constraints are we dealing with?

Emmy: We have stability constraints that ensure the interactions between the layers in the closed-loop LCA are stable, which comes from constraint ten. Then we have feasibility constraints ensuring a particular choice of phi and theta results in a feasible LCA design.

Kolya: The computational feasibility constraint requires that n, n, and u k must be computed within sampling times tau H and tau L. This is enforced by requiring T C(phi, theta) tau L and T P(phi, theta) N r tau L.

Ada: The analysis of interactions shows that closed-loop stability isn't guaranteed just by having individually stable C and P. How does the term x(n - y n+one) help counteract compounding mismatch?

Emmy: That term can counteract compounding mismatch if we design P such that the updated reference is exactly canceling S N r g n. That suggests a specific way to structure the planner.

Kolya: The construction of constraint sets involves finding the solution to the fixed point problem F(Y, R) = (Y*, R*), where F returns the erosion of (X, U) for a given (Y, R).

Ada: This set-valued function F has an anti-tone property under the condition that rho(A N r cl - x A p x) < one. What does this condition mean in practice?

Emmy: That condition ensures that the interaction term is sufficiently small, which allows us to guarantee constraint satisfaction of the LCA. It's a mathematical way of ensuring the system doesn't become unstable due to those interactions.

Kolya: The co-design problem is formulated as minimizing cost(phi, theta) subject to that stability condition and the compute feasibility constraints. The cost function is defined as

Jspeed(phi,), Jaccuracy, Jcompute(phi, theta): .

Ada: The numerical results show the feasible region on a grid of n y by N r in the top-left plot. What do the Pareto curves generally lean toward?

Emmy: The Pareto curves show that tradeoffs made in these LCA designs either lean toward maximizing accuracy, speed, minimizing computational load, or taking a diversity-enabled sigma-sweet spot.

Kolya: When we make an informed tradeoff between those three components, we get the lower-right response, which is called the ‘consensus LCA’. This consensus LCA is sufficiently accurate and fast while meeting the compute limitations.

Ada: So, this framework provides a systematic way to navigate that complex space of Pareto-optimal LCA designs. What's the ultimate goal of this work?

Emmy: The conclusion states that it presents a first step toward systematic and holistic design of layered control architectures by jointly optimizing over those architectural and tuning variables subject to stability, feasibility, and compute constraints.

Kolya: It provides an explicit formulation to navigate the complex space of Pareto-optimal LCA designs, which is important for practical implementation.

Ada: That covers the core mechanism of this paper. The authors are providing a way to design these complex control systems systematically instead of just guessing parameters.

Lucky paper: 2610.12697: Ada: We’re moving into paper number five of five today, which is about maximizing algebraic connectivity through a density-dependent framework and reinforcement learning completion.

Emmy: This paper proposes separating the structural initialization from the completion stage for fixed-(n, m) algebraic connectivity maximization.

Kolya: The goal is to maximize G in G n m lambda two L(G), which is defined by the graph's edges and connectivity.

Ada: The method decomposes this into a high-quality, low-cost structural backbone construction followed by an RL completion stage.

Emmy: The backbone selection depends on the edge density rho, which they define as m minus n minus one over n squared minus n minus one.

Kolya: In dense regimes, they use complete multipartite graphs derived from Proposition one and select a target level k star by integer binary search in O(log n) time.

Ada: For lower-to-intermediate densities, they switch to Cayley graphs using index r coset-based generators with lift-and-delete applied when divisibility constraints prevent direct construction.

Emmy: The envelope candidate is selected for rho between zero and zero point one or between zero point five and one while Cayley graphs are used for rho between zero point zero five and zero point seven five because of a compact overlap rule.

Kolya: After the backbone, the RL stage solves only the residual edge-allocation problem using a sequential policy where the state representation includes structural node pair features like di over n minus one, d(two)i over n minus one, Ci, and gt which is log of n over log of sixty-four.

Ada: The policy uses a multilayer perceptron to score candidate non-edges with a per-step reward defined as rt equals lambda two G t plus one minus lambda two G t over n plus beta times one terminal lambda two GT over n.

Emmy: Numerically, the BB-RL method shows the largest mean improvement at n = sixteen thirty-two sixty-four and lowest mean cost at n = thirty-two sixty-four and one hundred twenty-eight.

Kolya: The backbone-only performance comparison indicates that BB-RL outperforms all other methods across different graph sizes and density regimes.

Ada: They note that the backbone initialization can reduce the effective horizon by about one order when completion is the dominant sequential stage, moving from theta of n squared to theta of n.

Emmy: This structural-family first viewpoint suggests a scalable approach for maximizing algebraic connectivity.

Kolya: It's important to remember that this paper presents a density-dependent framework for fixed-(n, m) algebraic-connectivity maximization that separates structural initialization from completion.