K. Fathi

Exact Selection and Deletion of Balanced Pairs

A finite poset unfolds into many linear extensions: total orders faithful to its comparisons. Motivated by the 1/3–2/3 conjecture, this working-paper project asks how a balanced pair can be selected exactly, how its comparison law changes under deletion and reinsertion, and when...

Sharp Coefficient Bounds and Constant-Term Fibers in the Order-Five Nonnegative Inverse Eigenvalue Problem

The order-five nonnegative inverse eigenvalue problem asks which real monic quintics occur as characteristic polynomials of nonnegative matrices. This two-paper project gives a constructive characterization through directed-cycle optimization. The main paper proves that every rea...

Digit Polygons and Prime Signatures for n^2 + 1

Problem and scope This mathematical working paper studies the arithmetic information carried by digit polygons: the closed ordered walks formed from consecutive digits of repeating expansions. Its central family is N = n² + 1 with even n ≥ 2. The paper separates exact geometric c...

Digit polygons and their higher-dimensional lifts

Problem and framework A periodic base-b digit word defines a polygon by taking consecutive digits as coordinates and a family of higher-dimensional delay lifts by taking longer blocks. This working paper studies the affine dimension of those lifts and the arithmetic effect of cha...

Potential Stability and Signed Nests in Negative-Edge Trees

This working paper investigates when a prescribed tree sign pattern admits a real matrix whose eigenvalues all have strictly negative real parts. The underlying graph is a tree, and each edge corresponds to two nonzero matrix entries of opposite signs. Diagonal entries may be pre...

Signed Currents and Structural Matching in Möbius Shells

Signed Currents and Structural Matching in Möbius Shells studies an exact arithmetic matching problem associated with the Möbius and Mertens functions. Odd squarefree integers in a dyadic shell are grouped according to their largest prime factor. The resulting prime-window capaci...

Finite Cores and Eventual Optimality of Ulam Anticodes

The Ulam distance measures how many symbols must be deleted from each of two permutations to obtain a common ordered subsequence. An anticode is a family of permutations whose pairwise distances are bounded by a prescribed diameter. The Devlin-Douhovnikoff conjecture predicts tha...

Odd multiplicity and log-concavity in Fibonacci quotients

This paper studies two questions about Fibonacci quotients: the occurrence of prime divisors of odd multiplicity at semiprime indices, and the log-concavity of the coefficients of q-Fibonomial polynomials. It combines exact arithmetic identities, congruence methods, effective Dio...

Odd multiplicity and log-concavity in Fibonacci quotients

This paper studies two questions about Fibonacci quotients: the occurrence of prime divisors of odd multiplicity at semiprime indices, and the log-concavity of the coefficients of q-Fibonomial polynomials. It combines exact arithmetic identities, congruence methods, effective Dio...

Grounded Operators and Finite Certificates: Quantitative accessibility, signed extinction, and evolutionary stability

This paper develops a mathematical framework connecting accessibility in random fitness landscapes, coexistence and extinction in ecological systems, and stability in evolutionary games. It identifies the operators and finite certificates shared by these problems while specifying...

Grounded Operators and Finite Certificates: Quantitative accessibility, signed extinction, and evolutionary stability

This working paper develops mathematical connections between accessibility in random fitness landscapes, extinction and coexistence in ecological systems, and evolutionary stability in population games. Its organizing tools are grounded operators, positive quadratic forms, and fi...

Anonymous Causal Decks and Cosmological Histories: Reconstruction, Identifiability and Quantitative Stability

This paper develops a unified mathematical theory connecting the reconstruction of finite causal orders with the recovery of an entire homogeneous cosmological history. It studies what can be inferred from causal relations alone when event coordinates and calibrated times are uno...

Anonymous Causal Decks and Cosmological Histories: Reconstruction, Identifiability and Quantitative Stability

This paper develops a unified mathematical theory connecting two inverse problems: reconstructing a finite causal order from anonymous event-deletion observations, and inferring a homogeneous cosmological history from the statistics of that order. The common setting is a fixed ca...

Grounded Operators and Finite Certificates: Quantitative accessibility, signed extinction, and evolutionary stability

This working paper develops mathematical connections between accessibility in random fitness landscapes, extinction and coexistence in ecological systems, and evolutionary stability in population games. Its organizing tools are grounded operators, positive quadratic forms, and fi...

Finite Certificates and Signed Structure in Ecological Coexistence

This paper develops structural criteria and exact certificates for stable coexistence in signed generalized Lotka–Volterra ecological networks. The framework fixes intrinsic growth signs, positive self-regulation, and the exact pattern of beneficial, harmful, and absent interacti...

Explicit Sierpiński and Riesel repunits and repdigits, and the bounded-factor problem

This paper develops explicit infinite families of Sierpiński and Riesel repunits and repdigits, together with rigorous reductions for the bounded-factor problem for decimal repunits. A Sierpiński number is a positive odd integer K for which K·2ⁿ + 1 is composite for every positiv...

Sharp slab obstructions and minimum repairs in C2 combinatorial metamaterials

This working paper develops an exact mathematical theory of prescribed defect patterns in C2 combinatorial mechanical metamaterials. It studies finite rectangular assemblies of cubic building blocks with three allowed orientations and free boundary orientations. The central quest...

Certifying What Radiation-Defect Observations Determine: Completion certificates, tungsten data and continuous topology models

This paper develops a framework for determining which conclusions about radiation defects are justified by incomplete and uncertain observations. Its central question is whether a specified feature is shared by every defect configuration admitted by a declared observation and com...

Certifying What Radiation-Defect Observations Determine: Completion certificates, tungsten data and continuous topology models

This paper develops a framework for determining which conclusions about radiation defects are justified by incomplete and uncertain observations. Its central question is whether a specified feature is shared by every defect configuration admitted by a declared observation and com...

Certifying What Radiation-Defect Observations Determine: Completion certificates, tungsten data and continuous topology models

This paper develops a mathematical and computational framework for determining which defect characteristics are forced by incomplete and uncertain observations. Its central question is whether every configuration allowed by a declared observation and physical model agrees on a sp...