Abstract: In our previous work, we introduced the concept of a \emph{spectral pair} for a half-line Schrödinger operator with a \emph{complex} bounded potential $q$, serving as a substitute for the spectral measure in a non-self-adjoint setting. In this paper, we study the case of $q \in L^1(\mathbb{R}_+)$. We derive explicit formulas for the spectral pair in terms of the Jost solutions of a system of two equations naturally associated with the non-self-adjoint Schrödinger operator. A key component of our work, which is of independent interest, is the existence proof and analysis of these Jost solutions.

Abstract: We consider the fast-reaction approximation to the triangular Shigesada-Kawasaki-Teramoto model on a bounded domain in the physical dimension $d\le 3$. We provide explicit convergence rates on the whole domain in $\textnormal{L}^\infty\textnormal{L}^2\cap\textnormal{L}^2\textnormal{H}^1$ and in the interior we prove convergence with an explicit rate in any $\textnormal{L}^\infty\textnormal{H}^l$ for all $l > 0$.

Abstract: The paper investigates an inverse boundary value problem for a semilinear strongly damped wave equation with the Dirichlet boundary condition in Sobolev spaces of bounded (in particular, almost periodic and periodic) functions. In addition to finding a weak solution, we also determine a source coefficient in the right-hand side of the differential equation. To make the problem well-posed, an integral-type overdetermination condition is imposed. After reducing the inverse problem to a direct one, we solve the latter in several steps. First, we prove the existence and uniqueness of a weak solution to the corresponding initial-boundary value problem on a finite time interval. Next, we show that this solution can be extended in a bounded way to the semiaxis $t\ge 0$. In the following step, we further extend this bounded solution to all $t\in R$. Finally, we establish that if the data of the original problem are almost periodic (or periodic), then the resulting bounded weak solution is itself almost periodic (or periodic).

Abstract: We present a lightweight, probabilistic mechanism for certifying aligned storage between participants in decentralized systems. Participants respond to randomized suffix queries by performing forward scans over their locally stored data and returning short response sequences. A verifier observes only overlap statistics between responses. We prove that the overlap probability is bounded above by the minimum storage density among participants, ensuring that high observed overlap implies all parties store a large fraction of the underlying dataset. This bound holds regardless of adversarial strategy: a single well-provisioned participant cannot "carry" an under-provisioned partner. The protocol's "+1" advancement rule introduces pointer desynchronization that causes naïve Binomial models to overestimate tail probabilities by 2–3×. We establish rigorous security bounds through systematic simulation of Poisson-walk dynamics. For example, observing 10 or more matches out of 12 recorded elements rules out minimum density below 0.6 at the 2.4% significance level. Independent repetition amplifies confidence exponentially. The mechanism requires no cryptographic commitments per element, no global verifier, and reveals only O(m) randomly-selected elements per interaction. We analyze several natural adversarial strategies—fabrication, selective answering, collusion, Sybil attacks—and show that none can increase overlap probability beyond what storage density allows. From a mechanism-design perspective, repeated suffix-walk interactions induce a game where aligned storage is the dominant strategy, enabling emergent consensus without central coordination. The protocol serves as a foundation for proof-of-aligned-storage in distributed systems and provides consensus weight based on demonstrated storage rather than computational power or stake. Throughout this paper, "proof" refers to statistical evidence under a well-validated probabilistic model, not a cryptographic zero-knowledge proof.

Abstract: Living systems exhibit maximal stability—defined as resilience to perturbation and rapid recovery—neither under complete disorder nor rigid synchronization, but within an intermediate regime of coordinated dynamics. Across disparate domains, including arousal–performance relationships, neural criticality, and heart-rate variability (HRV) complexity, optimal function consistently follows an inverted-U pattern with respect to order. Here, we propose that bounded phase coherence, quantified using the Kuramoto order parameter and its extensions, provides a unifying dynamical principle underlying these observations. We hypothesize that system-level stability is maximized at intermediate coherence (R≈0.5R \approx 0.5R≈0.5–0.80.80.8), with degradation occurring under desynchronization, hyper-synchrony, or excessive temporal jitter. We formalize this hypothesis using a non-autonomous, multiplex Kuramoto framework incorporating adaptive coupling, time-varying intrinsic frequencies, stochastic perturbations, and higher-order interactions. Stability is operationalized via recovery time constants, multi scale entropy (MSE), and metastability measures. Disorder dynamics are explicitly modeled to yield an inverted-U dependence on coherence. Falsifiable predictions include quadratic relationships between coherence and stability metrics, directional causality from coherence to recovery measured via conditional transfer entropy, moderation by biological sex, and characteristic deviations in pathological states (e.g., hyper-synchrony in autism, low coherence in anxiety). Simulations demonstrate self-organization toward the bounded coherence regime under stress, reproduction of inverted-U stability profiles, and hierarchical causal flow across physiological layers. This framework integrates naturally with predictive coding and active inference, where bounded coherence minimizes interoceptive prediction error by balancing precision weighting across scales. The theory offers a mechanistic, testable principle for physiological regulation with direct implications for biofeedback design, physiological monitoring, and adaptive self-regulation technologies.

Abstract: We study periodic obstruction systems acting on a discrete axis, in which each modulus induces forbidden classes in a rigid way, with no freedom of translation. In this setting we introduce and formalize the SALEN principle (Strong Archimedean Local Eventual Non-Coverage), which asserts that finite families of canonically anchored obstructions cannot cover arbitrarily long contiguous intervals beyond an explicit archimedean scale. We isolate a precise structural hypothesis, called bounded local multiplicity, under which SALEN holds as a theorem with linear scale. The proof is completely deterministic and geometric, relying only on canonical anchoring, exact counting, and elementary properties of arithmetic progressions, without the use of probabilistic arguments, density estimates, or asymptotic analysis. We further show that bounded local multiplicity is automatically satisfied in canonically indexed obstruction systems, which makes SALEN a closed structural result independent of any specific arithmetic interpretation. The work establishes SALEN as a reusable non-coverage principle, intended to be used as a black-box input in subsequent developments.

Abstract: We establish necessary and sufficient conditions for the existence of an LU factorization $A=LU$ for an arbitrary square matrix $A$, including singular and rank-deficient cases, without the use of row or column permutations. We prove that such a factorization exists if and only if the nullity of every leading principal submatrix is bounded by the sum of the nullities of the corresponding leading column and row blocks. While building upon the work of Okunev and Johnson, we present simpler, constructive proofs. Furthermore, we extend these results to characterize rank-revealing factorizations, providing explicit sparsity bounds for the factors $L$ and $U$. Finally, we derive analogous necessary and sufficient conditions for the existence of factorizations constrained to have unit lower or unit upper triangular factors.

Abstract: We present a complete proof that P ̸= NP by establishing a fundamental information-theoretic barrier via proof by contradiction. Main argument: Assume P = NP. Then polynomial-time algorithms have bounded output complexity (O(log n) bits beyond the instance via trace encoding), while NP-complete problem witnesses require ω(log n) bits of conditional information (for TSP: Θ(n log n) bits via anti-concentration and counting arguments). This contradiction establishes P ̸= NP. The proof rests on three pillars: 1. Self-Reference Incompressibility (S.R.I.): Any polynomial-time algo- rithm producing output y from input x satisfies K(y | x) ≤ O(log n) via trace encoding, where K(· | ·) denotes conditional Kolmogorov complexity. 2. TSP Conditional Incompressibility: For Euclidean TSP, optimal tours sat- isfy K(π ∗ | x) = Θ(n log n) with high probability, proven via anti-concentration (Milnor-Thom bounds) and incompressibility counting arguments. This holds even given the instance x. 3. Universal Inheritance Theorem (U.I.T.): Conditional incompressibility propagates through Karp reductions: all NP-complete problems inherit ω(log n) witness complexity bounds.

Abstract: Abstract Connectivity of multiple unmanned aerial vehicles (UAVs) is crucial for distributed and coordinated control. However, due to limited communication ranges and inherent communication delays, traditional connectivity preservation methods often fail to ensure stable network connectivity in delay‐prone environments. To address this challenge, we propose a multiple UAVs connectivity preservation controller designed to operate effectively under communication delays. Leveraging distance‐based neighborhood graph modeling, our approach accurately represents communication relationships and network topology among UAVs, thus facilitating both connectivity analysis and controller design. A distributed controller, incorporating a bounded artificial potential function, is introduced and parameterized to mitigate the adverse effects of communication delays. To achieve high‐precision attitude control, we further integrate a sliding mode controller into the system. Numerical simulations demonstrate the effectiveness of the proposed method in preserving multiple UAVs connectivity under communication delay conditions.

Abstract: This repository contains the source code and experimental artifacts accompanying the paper: M. Alkhiyami, G. Martino, and G. Fey, Automated Self-Explanation of Expected versus Perceived Behavior for Interacting Digital Systems, Design, Automation and Test in Europe Conference (DATE), 2026. The artifacts implement the algorithm for automatically generating explanations of mismatches between expected and perceived behavior in interacting digital systems, as presented in the paper. The implementation follows the bounded model checking and SMT-based formulation described in Sections IV–V. Contents Source code implementing the explanation algorithm Models of interacting systems (Mealy machines and assumptions) Experimental setups for the wind park–turbine and traffic controller–autonomous vehicles case studies Scripts and configuration files used to reproduce the reported experiments and results PurposeThe artifacts are provided to support transparency, reproducibility, and further research on self-explaining interacting digital systems.

Abstract: We consider subsampling at rate q among n users and a Gaussian sum-release with noise variance sigma^2. For neighboring datasets D0=(0,...,0) and D1=(mu,0,...,0), letting SNR=mu^2/sigma^2, we prove JSD(P0,P1) = (q*SNR)/(8n) + ((4-12q+7q^2)/64)*(SNR^2/n^2) + O(n^-3).The remainder is uniform for q in [eps,1] (any fixed eps>0) and bounded SNR. The proof combines an exact mixture representation, a local Edgeworth expansion, uniform sixth-derivative control, and a parity argument. Numerical quadrature of the exact binomial–Gaussian mixture corroborates the uniform n^-3 behavior.

Abstract: The metric MinDist, introduced recently to quantify the distance of an arbitrary Rummy hand from a valid declaration, plays a central role in algorithmic hand evaluation and optimal play. Existing results show that the MinDist of any $13$-card Rummy hand from a single deck is bounded above by $9$. In this paper, we sharpen this bound and prove that the MinDist of any hand is at most $7$. We further show that this bound is tight by explicitly exhibiting a hand whose MinDist equals $7$ for a suitable choice of wildcard joker. The proof combines elementary combinatorial arguments with structural properties of card partitions across suits and resolves the gap between the previously known upper bound and the true extremal value.

Abstract: We record the SALEN Axiom (Strong Archimedean Local Eventual Non–Coverage) as a foundational structural principle governing canonically anchored periodic obstruction systems on a discrete ordered axis. The axiom asserts that, under uniform bounded multiplicity, families of rigid periodic obstructions cannot cover arbitrarily long contiguous intervals beyond an explicit archimedean scale. The formulation is entirely abstract and independent of arithmetic, combinatorial, or dynamical interpretations. No proof is given here. The purpose of this note is to isolate SALEN as a reusable black--box principle, suitable for invocation in subsequent works once its hypotheses have been verified in a given framework.

Abstract: In normative models a decision-maker is usually assumed to be Bayesian rational, and so to maximize subjective expected utility, within a complete and correctly specified decision model. Following the discussion in Hammond (2007) of Schumpeter's (1911, 1934) concept of entrepreneurship, as well as Shackle's (1953) concept of potential surprise, we consider enlivened decision trees whose growth over time cannot be accurately modelled in full detail. An enlivened decision tree involves more severe limitations than a mis-specified model, unforeseen contingencies, or unawareness, all of which are typically modelled with reference to a universal state space large enough to encompass any decision model that an agent may consider. We consider a motivating example based on Homer's classic tale of Odysseus and the Sirens. Though our novel framework transcends standard notions of risk or uncertainty, for finite decision trees that may be truncated because of bounded rationality, an extended and refined form of Bayesian rationality is still possible, with real-valued subjective evaluations instead of consequences attached to terminal nodes where truncations occur. Moreover, these subjective evaluations underlie, for example, the kind of Monte Carlo tree search algorithm used by recent chess-playing software packages. They may also help rationalize the contentious precautionary principle.

Abstract: Conservation of energy arises from time-translation symmetry as formalized by Noether’s theorem. This paper introduces a speculative scalar field, termed the chronon field χ(x,t), which locally modulates the relationship between proper time and coor- dinate time in 3+1-dimensional spacetime. When χ varies temporally or spatially, time-translation symmetry is locally broken, leading to apparent violations of energy conservation within bounded regions. Global consistency is preserved through coupling to an auxiliary hidden sector. The framework is internally consistent and intended as a conceptual exploration rather than a claim of physical realization.

Abstract: LFIS–15 formalizes the symmetry between inward and outward representational limits in Light Frame Cadence Theory. This volume shows that inward concentration and outward relaxation arise from the same finite representability constraint and differ only in how representational burden is redistributed within a Light Frame. Neither limit introduces new structure; both follow necessarily from invariant cadence under bounded representability. Inward and outward limits are treated as terminal representational regimes, not as processes of collapse, expansion, or instability. Each marks the completion of representational obligation under the same capacity constraint, resolved in opposite orientations of curvature redistribution. All results in LFIS–15 are stated strictly at the level of structural constraints. No assumptions are made about cosmological models, metric evolution, field dynamics, transport mechanisms, or global time. The volume does not introduce dynamics or causal explanations; it clarifies symmetry at the level of admissible representation. LFIS–15 establishes symmetric inward and outward representability limits as canonical features of the Light Frame Infrastructure Series and completes the articulation of terminal regimes governing representational completion.

Abstract: We present a complete proof that P ̸= NP by establishing a fundamental information-theoretic barrier via proof by contradiction. Main argument: Assume P = NP. Then polynomial-time algorithms have bounded output complexity (O(log n) bits beyond the instance via trace encoding), while NP-complete problem witnesses require ω(log n) bits of conditional information (for TSP: Θ(n log n) bits via anti-concentration and counting arguments). This contradiction establishes P ̸= NP. The proof rests on three pillars: 1. Self-Reference Incompressibility (S.R.I.): Any polynomial-time algo- rithm producing output y from input x satisfies K(y | x) ≤ O(log n) via trace encoding, where K(· | ·) denotes conditional Kolmogorov complexity. 2. TSP Conditional Incompressibility: For Euclidean TSP, optimal tours sat- isfy K(π ∗ | x) = Θ(n log n) with high probability, proven via anti-concentration (Milnor-Thom bounds) and incompressibility counting arguments. This holds even given the instance x. 3. Universal Inheritance Theorem (U.I.T.): Conditional incompressibility propagates through Karp reductions: all NP-complete problems inherit ω(log n) witness complexity bounds.

Abstract: We study the second logarithmic derivative of the Riemann zeta function on the critical line, mollified on the mesoscopic scale L=log T, and develop a variance-equilibrium framework connecting the distribution of zeta zeros to prime number statistics through energy constraints. On the arithmetic side, we consider the mollified curvature field H_{L,T}(t) = ((log zeta)'' * v_{L,T} * K_{L,T})(t), where v_{L,T} is a modulated time-mollifier tuned to the prime frequency scale xi_T approx (log T)/(2 pi), and K_{L,T} is a bandpass spectral cap centered at +/- xi_T. Its windowed L^2-energy is defined as V_arith(T) := integral from T to 2T of |H_{L,T}(t)|^2 w_L(t) dt, where w_L is a baseband Fejer window. Using only the Dirichlet series for (log zeta)'' on Re s>1 and Montgomery--Vaughan type mean-value theorems for Dirichlet polynomials, we show that V_arith(T) is unconditionally locked to the scale V_arith(T) = (log T)^4 + O((log T)^3), with error O((log T)^3), with no hypothesis on the location of the nontrivial zeros of zeta(s). On the spectral side, we use the Hadamard product and the functional equation to express hat{H_{L,T}}(xi) = W_{L,T}(xi) Z(xi) + hat{R}(xi), where W_{L,T} is a smooth kernel supported on the bands |xi -/+ xi_T| <= 1/L, hat{R} is a uniformly bounded analytic remainder, and Z(xi) is the collective zero spectral density defined by the zeros rho = 1/2 + a_rho + i gamma_rho. A Fourier--Plancherel calculation yields a diagonal/off-diagonal decomposition V_spec(T) = D({a_rho}) + R({a_rho}) + O(1), where the diagonal D captures single-zero contributions and the off-diagonal R captures interference between zeros. The framework yields three unconditional results: Energy locking: The prime-side curvature variance is rigidly constrained to (log T)^4 + O((log T)^3), independent of zero locations. Diagonal monotonicity: The spectral contribution from each individual zero is strictly maximized when that zero lies on the critical line (a_rho=0), with off-line zeros incurring an exponential damping factor T^{-a_rho}. Mesoscopic sparsity: At most O((log T)^3) zeros in any window [T, 2T] can lie at mesoscopic distance (>= A/log T) from the critical line. We establish a conditional resolution of the Riemann Hypothesis: if the off-diagonal interference term cannot compensate for diagonal losses from off-line zeros, formalized as a "Global Compensation Bound" requiring that the diagonal deficit exceed the off-diagonal shift by an amount >= c (log T)^{4-delta_*} that dominates the variance identity's error term, then RH follows. We characterize precisely the "conspiracy" that would be required for off-line zeros to exist: the zero heights would need to produce structured negative correlations that nearly cancel the diagonal energy deficit in every mesoscopic window. We explain that such a mechanism would be incompatible with GUE statistics for zeta zeros, establishing that within the variance-equilibrium framework, the Riemann Hypothesis is equivalent to GUE-compatible zero correlations.

Abstract: Let $K$ be a number field of degree $d$ so that $K/\mathbb Q$ is a Galois extension. The {\it normal basis theorem} states that $K$ has a $\mathbb Q$-basis consisting of algebraic conjugates, in fact $K$ contains infinitely many such bases. We prove an effective version of this theorem, obtaining a normal basis for $K/\mathbb Q$ of bounded Weil height with an explicit bound in terms of the degree and discriminant of $K$. In the case when $d$ is prime, we obtain a particularly good bound using a different method.

Abstract: We get a local sheeting theorem for stable hypersurfaces with bounded mean curvature in R^{n+1}. As suggested by the anonymous reviewer, we have included a detailed proof here. The argument presented here is identical to Schoen-Simon’s seminal work. This note serves as supplementary material for our paper. For simplicity, we only treat the smooth case; the arguments apply equally to the case where the singular set has codimension two.

Abstract: In this paper, we establish an operator-valued Fourier multiplier theorem in weighted Lebesgue spaces, Besov and Triebel--Lizorkin spaces, assuming the multiplier has $\mathcal{R}$-bounded range and satisfies an $\ell^r$-summability condition on its bounded $s$-variation seminorms over dyadic intervals. The exponents $r$ and $s$ reflect the relationship between the geometric properties of the underlying Banach spaces (type and cotype) and the boundedness of Fourier multiplier operators. As our main tool we prove a weighted vector-valued variational Carleson inequality and deduce an estimate of Littlewood--Paley--Rubio de Francia type.

Abstract: Park's conjecture (1976) predicts that a finitely generated variety of finite type with a finite residual bound must be finitely based. This note presents a compact "roadmap" formalizing a standard contrapositive strategy through a two-parameter bounded fragment Th_{m,D}(V), minimal (critical) countermodels for bounded fragments, and an Escape Property EP(g) that transfers boundary obstructions in critical countermodels to the existence of large finite subdirectly irreducible algebras inside the target variety. The emphasis is logical hygiene: we isolate and repair two common sources of vacuity (absence of finite countermodels and hidden finiteness assumptions), so that implications of the form "FRB + EP(g) => finitely based" become formally sound. We also record two structural templates—commutator-theoretic and tame-congruence-theoretic—that encapsulate the substantive work required to verify EP(g).

Abstract: We prove a Freiman–Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving (in full generality) a conjecture of Marton. Specifically, let G be an abelian group of torsion m (meaning m g = 0 for all g ∈ G ) and suppose that A is a non-empty subset of G with | A + A | ≤ K | A | . Then A can be covered by at most ( 2 K ) O ( m 3 ) translates of a subgroup H ≤ G of cardinality at most | A | . The argument is a variant of that used in the case G = F 2 n in a recent paper of the authors.

Abstract: This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc $\mathbb{D}^n$. We establish a sharp extension of the classical Bohr inequality, proving that the Bohr radius remains $R_n = 1/(3n)$ for the family of holomorphic functions bounded by unity in the multivariate setting. Further, we provide a definitive resolution to the Bohr-Rogosinski phenomenon in several complex variables by determining sharp radii for functional power series involving the class of Schwarz functions $ω_{n,m}\in\mathcal{B}_{n,m}$ and the local modulus $|f(z)|$. By employing the radial (Euler) derivative operator $Df(z) = \sum_{k=1}^{n} z_k \frac{\partial f(z)}{\partial z_k}$, we obtain refined growth estimates for derivatives that generalize well-known univariate results to $\mathbb{C}^n$. Finally, a multidimensional version of the area-based Bohr inequality is established. The optimality of the obtained constants is rigorously verified, demonstrating that all established radii are sharp.

Abstract: Abstract Any system capable of discovery, judgment, or alignment operates under structural and energetic constraints. This paper proves a general limit: a bounded system governed by a scalar coherence invariant cannot internally verify its own drift or alignment state with certainty. The apparatus required for measurement is itself subject to the same governing constraints, rendering self-certification non-convergent. Alignment can be inferred only through external phase-lock and coherence persistence, not introspection. This result is not skeptical but structural. It establishes external reference as a lawful necessity rather than an epistemic weakness, completing the closure of alignment-based accounts of discovery, agency, and stability.

Abstract: Physical law is usually formulated as a set of dynamical rules acting on states within a fixed space of possibilities. This work argues that a more primitive constraint precedes dynamics: the restriction on which correlations and distinctions may meaningfully exist at all. The paper introduces the Admissibility–Entropy Structure (AES), a pre-geometric framework that cleanly separates admissibility (what correlations are possible) from entropy (finite correlation capacity) and probability (conditional statistical description). Within AES, entropy measures correlation capacity rather than likelihood, probability is defined only where multiple admissible alternatives coexist, and realizable correlation between regions is bounded by the minimal separating interface. Spacetime geometry, locality, and causal structure are shown to arise as effective representations of these constraints, rather than as fundamental ingredients. No new dynamics, entities, or probabilistic postulates are introduced. The framework yields sharp and non-negotiable discriminators: probability without entropy, geometry without capacity, and unlimited correlation are forbidden in principle. By isolating the invariant constraints that underwrite meaningful physical description, this work provides a minimal foundation for thermodynamics, quantum theory, and spacetime structure, clarifying their conceptual order without modifying established physical laws.

Abstract: This record announces the existence of a unified structural atomic law capable of generating periodic organization and first-ionization energy regularities across the periodic table, with validated alignment to experimental reference data and structured predictions beyond atomic number Z = 118. The work reports representative quantitative results, bounded residual behavior, and block-wise stability without local parameter refitting or empirical fitting procedures. The underlying formulation, referred to as the Generative Variational Law of the Digital Geometric Resonance Hypothesis (DGRH), is intentionally not disclosed in this record and is reserved for formal institutional scientific review. This publication serves as a notification of existence and capability rather than a full theoretical disclosure.

Abstract: We investigate a singularly perturbed $q$-difference differential Cauchy problem with polynomial coefficients in complex time $t$ and space $z$ and with quadratic non-linearity. We construct local holomorphic solutions on sectors in the complex plane with respect to the perturbation parameter $\varepsilon$ with values in some Banach space of formal power series in $z$ with analytic coefficients on shrinking domains in $t$. Two aspects of these solutions are addressed. One feature concerns asymptotic expansions in $\varepsilon$ for which a Gevrey type structure is unveiled. The other fact deals with confluence properties as $q>1$ tends to $1$. In particular, the built up Banach valued solutions are shown to merge in norm to a fully bounded holomorphic map in all the variables $t$, $z$ and $\varepsilon$ that solves a non-linear partial differential Cauchy problem.