Mantis
Discovery

Question

How much information from outside a local neighborhood is needed to solve an elliptic equation accurately? The paper studies a fixed Dirichlet Laplacian on a grid, where changing the input in one place can change the solution far away.

Results

At each grid point, the solver receives nearby input values and a continuous summary of the entire input. To keep relative energy error below a fixed tolerance for every input, the summary needs a number of real coordinates proportional to the total grid size divided by the volume reached locally. The lower bound allows nonlinear encoding and reconstruction. A linear construction attains the same order.

Experiments

For known linear measurements, the argument also produces two inputs that the solver cannot distinguish. In the Poisson experiments, a model using 16 sine measurements had mean energy error of 6.1% on smooth inputs and 67.0% on white-noise inputs at its training resolution.

Theorems 2.1–2.2; Sections 3–6.

Method

The proof chooses source and observation regions too far apart for local information to connect them. Correlations in the Green matrix measure the dependence that remains. The Borsuk–Ulam theorem then limits how much of that dependence a continuous encoding can preserve. Polynomial approximation with a low-rank correction gives the matching upper bound.

Scope

The bound concerns a fixed operator and accuracy over every right-hand side. It counts real coordinates computed from the input, not trained parameters, bits, or runtime. None of the tested learned initial guesses reduced median total solve time on the largest timed grid.

Question

A perfect matching pairs every vertex with one neighbor. The matching cofactor matrix counts how many perfect matchings remain after each possible pair of vertices is deleted. Can the graph itself determine the signs of this matrix’s eigenvalues?

Results

The main theorem applies to line graphs of connected, simple graphs with maximum degree three and a positive even number of edges. In a line graph, the original edges become vertices. After the prescribed bridge detachments, the number of nonempty pieces equals the number of positive cofactor eigenvalues. A second factorization determines the negative and zero counts.

Consequences

The positive count can therefore be found in linear time without constructing the matrix or enumerating matchings. A separate theorem classifies the cofactor matrices of connected strongly regular graphs of even order with least adjacency eigenvalue −2.

Theorems 1.1–1.2; 3.4–3.5.

Method

A bridge is an edge whose removal disconnects a graph. Direct each bridge toward the side with an odd number of edges, then replace its other endpoint with a leaf. Matrix congruences show that the eigenvalue counts add across the resulting pieces. Cuts consisting of two edges determine the counts within each piece.

Scope

The two classifications have different hypotheses. Counterexamples show that the conclusion does not extend to every graph with least adjacency eigenvalue −2. The strongly regular classification uses exact integer computations for its exceptional cases.

Question

Given a finite set, how large a subset can always be chosen so that no two different unordered pairs have the same sum? This is the Sidon property. Pairing a number with itself is included.

Results

For an N-element set in a torsion-free abelian group, the guaranteed size has leading term 2√N/(3√3), with a correction that is small relative to √N as N grows. The leading constant is approximately 0.3849, twice Bailleul and Riblet’s. The paper also bounds the size of cyclic codes whose pair sums must stay a prescribed distance apart, and classifies equality.

Consequences

The code bound also identifies a limit of this method. Arguments using only the set size, this collision probability, and a fixed error-tolerant code cannot improve the leading constant further.

Theorems 1.1, 1.3 and 1.5.

Method

The construction rounds the original set into a cyclic group. Rounding changes an additive relation by at most one bucket. A code with sufficiently separated pair sums tolerates this error, avoiding the loss of half the input in earlier arguments. In the stated one-dimensional random-rounding model, the probability that two distinct inputs land in the same bucket is exactly the reciprocal of the number of buckets.

Scope

The result is asymptotic and does not give the exact largest Sidon subset of a particular finite set. The conjectured leading constant of 1 remains unresolved. The collision calculation uses the specified one-dimensional random model; the rounding-error bound is deterministic and holds in every dimension.

Question

Can a game table use less storage while still answering exactly whether a win is possible within a move budget? An interval can answer some budgets immediately, leaving only the budgets inside it to be checked by search.

Results

For finite alternating games with irreversible budget resets, the paper proves an endpoint test that accepts exactly the correct signs and distance intervals on any retained subset. Bounds for omitted positions are recovered from their boundary data without increasing interval width, even when the omitted region contains cycles. The paper also gives matching bounds on checkpoint density and worst-case reconstruction depth.

Consequences

The chess implementation stores 26 complete pawnless classes in 7,585,632 bytes, 61.64% below the specified Syzygy WDL/DTZ inventory. It preserves all five outcome categories and exact values for queries beginning with a fresh move counter.

Abstract; Sections 3–7. Measurements reported in the manuscript.

Method

Each endpoint test must examine a legal move before using stored shortcuts, so an incorrect label cannot pass by referring to itself. Two boundary calculations, using the smallest and largest allowed distances, recover the bounds for omitted positions. A query outside an interval is answered immediately; one inside it expands legal moves.

Scope

The storage result uses the existing paired grids, not the general grid optimizer tested elsewhere in the paper. Query time remains a cost: the reported 99th-percentile latency near a threshold is 133 milliseconds. The executable certificate is not a proof-assistant formalization.

Question

What can a compressed regression dataset preserve? Matching one fitted coefficient vector does not ensure that numerical optimization is well-conditioned, or that the same subset will work after the regularization strength or responses change.

Results

The paper characterizes all fits attainable by reweighting a fixed subset. With full feature rank and exactly one more observation than the feature dimension, the best attainable loss reduces to a convex quadratic program. It also gives necessary and sufficient conditions for a fixed weighted subset to reproduce the entire ridge-regression path, meaning the fits at every positive regularization strength.

Experiments

On 16 configurations from three tabular data sources, reweighting improved every starting condition number, by a median factor of 6.46. It usually increased sensitivity to new labels. The paper also gives new worst-case loss bounds for some intermediate subset sizes.

Theorems 3.1–3.3, 4.1–4.2 and 5.1.

Method

The interpolation hyperplanes, each describing the coefficients that fit one observation exactly, determine which fits are attainable. Constraints on residuals let weights change without changing the current fit. To preserve every ridge fit, the subset must match the response cross moment and the covariance’s action on the relevant Krylov subspace.

Scope

Features remain fixed, selected observations retain their original responses, and weights are nonnegative. Feature computation and the initial full-data fit are still needed. Better conditioning does not by itself preserve the fit under changed responses.

Question

An ascent sequence starts at zero. Each later entry is at most one plus the number of earlier rises. How many sequences of a given length remain when an increasing pattern followed by a repeated value is forbidden?

Results

For every fixed nonempty family of the specified return patterns, each using at least three distinct values, the logarithm of the count is n log n − 2n log log n, up to an error proportional to n. The paper also proves that these counts cannot satisfy a fixed-order linear recurrence with polynomial coefficients. Simpler pattern families have explicit rational generating functions.

Consequences

The method also applies to related weak ascent and inversion sequences, with their own formulas. For some previously studied classes, it changes the growth rate inferred from finite enumerations without disputing the enumerated coefficients.

Theorems 1.1–1.2; Sections 2–4.

Method

Connect consecutive occurrences of each value. Coloring the resulting repetition graph limits how much information is needed to recover a sequence from one retained occurrence per value. Counting the retained permutations gives an upper bound. Appending new, distinct values to suitable prefixes gives a lower bound of the same order.

Scope

The main formula gives logarithmic asymptotics, not an exact count at each length. The forbidden family remains fixed as the sequence length grows. Ascent sequences, weak ascent sequences, and inversion sequences have different membership rules and stated bounds.

Question

The permanent adds the products obtained by choosing one entry from every row and column. With row and column totals fixed, where can this function have a minimum, maximum, or saddle point?

Results

The stationary-point equations have algebraic length sixteen for every compatible choice of totals. For generic totals, the sixteen points are distinct and both Galois groups are the full symmetric group S₁₆. Every nondegenerate real parameter choice has between four and sixteen real stationary points.

Consequences

The fixed algebra supports exact computation even when stationary points coincide. In one positive-marginal family, the paper locates the transition where an interior minimum and a saddle appear. It also extends the degree calculation to quadratically coupled 3 × 3 permanents.

Theorem 1.1; Proposition 4.2.

Method

Fourier coordinates give four monic quadratic equations with a fixed sixteen-element basis for the quotient algebra. Factorizations of one exact specialization determine the arithmetic group. A parameter choice where precisely two stationary points meet supplies the transposition needed for the geometric group.

Scope

Sixteen counts multiplicity, not sixteen distinct positive matrices at every parameter choice. At equal unit totals, all sixteen points are real but only the uniform matrix is strictly positive. Boundary optima require separate analysis.

Question

Does a small first variation of the Gaussian logarithmic Sobolev deficit imply that a distribution is close to a Gaussian? A small amount of distant mass can carry a substantial second moment even when the first variation is small.

Results

At fixed Fisher information and a cap on the second moment, the paper determines the infimum of the squared first-variation norm and all attainment cases. It identifies the Gaussian limit and moment lost to infinity along minimizing sequences. Integrating along Ornstein–Uhlenbeck flow bounds Wasserstein distance, Fisher information, and the deficit with constants independent of dimension.

Consequences

The flow estimates need no covariance or Poincaré assumption and include singular initial measures. Centered isotropic Gaussians attain equality. The work also identifies the Wasserstein metric slope at regular densities.

Theorems 1.1, 4.3 and 7.1.

Method

Remainder identities relate the first variation to derivatives of the log density. Separated Gaussian mixtures prove sharpness and account for mass escaping to infinity. Diffusion identities give the time-integrated estimates and sharp constants.

Scope

No fixed observation interval suffices for the full flow estimate. The instantaneous bound depends on dimension. The extension to closed weighted manifolds assumes positive weighted Ricci curvature.

Question

How large a complete graph can be obtained by deleting edges and vertices and contracting connected vertex sets? Hadwiger’s conjecture asks whether its order is always at least the number of colors needed for the original graph.

Results

For graphs of independence number at most two, the paper transfers the gap between chromatic number and complete-minor size to one odd-order induced subgraph. Fractional clique covering number at most four gives a gap of at most one; a value strictly below 31/8 gives Hadwiger’s inequality. The Higman–Sims complement has an explicit complete minor of order 56.

Consequences

The certificate checks connectivity and all 1,540 required branch adjacencies. Averaging its images under graph symmetries gives bounds for weighted clique inflations. Induced subgraphs of the Higman–Sims complement have a chromatic gap of at most two.

Theorems 1.1–1.3; Appendix.

Method

A construction between join factors avoids adding a separate error for each factor. Fractional clique-cover estimates then convert order bounds into chromatic bounds. The finite minor uses twelve single-vertex and forty-four edge branch sets.

Scope

Hadwiger’s conjecture remains unresolved for the full class and for every Higman–Sims inflation. The threshold 31/8 is strict. The supplied minor is not claimed to be the largest possible one.

Question

How much history must an engine retain when repeated visits permit a draw and quiet moves consume a finite reserve? Answering one query and maintaining the value after every legal move are different problems.

Results

For one supplied history, the relevant visits reduce to a fixed draw boundary in the current strongly connected component and one bit for a claim at the current vertex. Online monitoring needs no persistent history when H ≤ 2k − 2, where H is the capacity and k the repetition threshold. Above that threshold, the paper proves matching memory bounds and identifies two exceptional one-bit cases.

Consequences

The results specify when different histories can safely share a game-value calculation. All capacity profiles can be represented and computed without checking every possible capacity separately.

Theorems 3.2, 5.2 and 7.2.

Method

Alternating play limits how often a position can recur before the budget expires. Counts that cannot reach the repetition threshold can be discarded permanently. Legal continuations distinguish the histories needed for the lower bound. Separate algorithms compute how winning requirements change with capacity.

Scope

The history results require alternating ownership and no reset on a directed cycle. Later capacity-profile results permit reset cycles and positive integer costs. A value monitor must handle every legal move, including moves that decline a win or draw claim.

Question

What is the shortest set that blocks every line through a unit square? The barrier may be disconnected and may extend outside the square.

Results

Two sharp inequalities combine lengths of connected pieces with distances from their convex hulls to the corners still to be reached. After adjoining the corners, a shorter closed barrier must have either four separate corner components or one component through two adjacent corners and two further components. The latter arrangement has quantitative separation requirements.

Consequences

The restrictions narrow the configurations that could improve the classical barrier. The argument applies to closed finite-length sets without assuming boundedness or finitely many components.

Theorems 1.1–1.2; Appendix A.

Method

The proof compares connected sets with Steiner trees and verifies inequalities using balanced vector loads. It retains distances between individual component hulls. Exact arithmetic checks all 105 six-leaf topologies and all 45 five/four-leaf pairs used in the proof.

Scope

The unrestricted minimum remains unresolved. The classical construction has length √2 + √6/2, approximately 2.639. The paper neither proves that value optimal nor improves the unrestricted numerical lower bound.

Question

Can a covariance estimate have small absolute error and still be unreliable? An error that looks small overall can overwhelm a direction whose variance is much smaller.

Results

In a fixed finite Gibbs model, the energy levels at which new feature directions appear determine the asymptotic covariance scales. The final such level gives a necessary and sufficient threshold for relative recovery from independent samples. For distributions on matroid bases and for hard-core models, the paper also gives relative matrix bounds from model structure.

Experiments

At one expected occurrence of the rarest necessary item in the reported experiment, sample covariance was singular in 36.7% of replicates. Increasing that expected count from one to 100 reduced median relative error from 1 to 0.080. The hard-core approximation retained relative accuracy on growing strips at low activity.

Theorem 2.1; Proposition 2.2; Sections 3–5.

Method

Samples must reach configurations outside each smaller affine span. The probability of missing the last necessary span gives the lower bound; matrix concentration gives sufficiency. Single-element exchanges and cluster expansions give deterministic approximations when probabilities are known.

Scope

The sampling result assumes independent observations of a fixed finite model. Deterministic approximations use specified probabilities. Exact Newton was faster in the small conditional-prediction experiment.

Question

Can a reduced plasma model discard unresolved states on the assumption that they all decay quickly? Predicting a field or flux accurately does not require resolving the entire particle distribution, but a uniform decay claim does.

Results

For the linear electrostatic gyrokinetic model studied here, slow-particle subspaces have arbitrarily small transport residuals and field response. In the undriven case, the absorbing generator has the closed left half-plane as its spectrum; the periodic generator has the imaginary axis. Two weak-collision models admit quantitative approximate-eigenvector estimates.

Consequences

These states can remain difficult to detect even when finitely many observations are added. Under the stated parity and coupling assumptions, the paper derives necessary limits on feedback that would enforce a decay rate uniform in collision strength.

Abstract; Sections 2–7.

Method

The proof localizes particle packets in velocity and chooses their phase to cancel transport frequency and magnetic drift. Shrinking the support reduces the field response. Related constructions use trapped orbits and passing particles near the trapped–passing boundary. Derivative estimates bound the effect of weak collisions.

Scope

The conclusions concern the displayed linear operators, a coercive energy norm, and the full particle distribution. They remain compatible with accurate field and flux predictions. The weak-collision estimates concern approximate eigenvectors, not identified collisional eigenvalues.

Question

Which part of an initial kinetic perturbation remains after its oscillatory component is averaged away? The electric field changes the inner product that determines the stationary projection.

Results

A positive field equation determines the stationary response without assuming a kinetic spectral gap. Time averages converge for each fixed state, but not uniformly over all states when nonzero frequencies approach zero. A constant drift detuning removes the stationary space, so long-time and zero-detuning limits need not commute.

Consequences

In the specified 25-dimensional orbit discretization, ordinary averaging instead of the energy projection changed the stationary field by 23.2%. The calculation separates the choice of projection from quadrature and approximation errors.

Theorem 2.2; Sections 5–10.

Method

Projection in the polarization-energy inner product gives a field equation suitable for variational approximation and residual error bounds. For one vanishing-polarization limit, the paper constructs an evolution on a constrained continuous-velocity space and proves a square-root error bound for prepared initial data.

Scope

The percentage is not a certified continuum value. The singular-limit bound concerns one coercive regularization and constrained initial data. The slow effective evolution is unitary and does not represent decay of the kinetic norm.

Question

When assigning different precisions to network weights, which calibration measurements help: scalar losses, output vectors, or direct comparisons of complete assignments?

Results

On 25 small trained networks, output scoring reduced normalized allocation regret by 61.0% under rounding. After the specified GPTQ-style reconstruction update, the reduction was 6.7%, with an interval including zero. The comparisons separate approximation bias, sampling variance, and assignments omitted from the shortlist.

Experiments

Cross-tensor Fisher terms helped under rounding but hurt selection after reconstruction in the main comparison. Lower disagreement with the full-precision model did not consistently improve task accuracy. The theory gives examples of information omitted by restricted calibration measurements.

Sections 2–5; release README.

Method

Methods use the same measured evaluations and fixed quantized candidates. Exhaustive offline evaluation supplies the reference optimum. Output directions retain cancellation between perturbations. Higher-order scalar profiles can reduce approximation bias but spread a small budget across too many measurements, increasing sampling error.

Scope

The study uses three datasets and 37 networks below 73,000 parameters. Exhaustive allocation does not scale to many tensors. The release’s executable reproduction artifacts remain incomplete.