https://thalens.org/ Mathematical research papers by Dr. Tamás Nagy en Sat, 08 Aug 2026 20:28:33 +0000 https://thalens.org/papers/ml_structured_latent_basis/ https://thalens.org/papers/ml_structured_latent_basis/ We introduce the Structured Latent Basis (SLB) framework, a perspective that compares feature engineering, meta-learning, and representation learning through a common diagnostic question: does the chosen representation make a regularized linear predi Sat, 08 Aug 2026 00:00:00 +0000 Core Theory https://thalens.org/papers/fin_spectral_ir_pricing/ https://thalens.org/papers/fin_spectral_ir_pricing/ We present a bounded verification case study for COS caplet pricing. The numerical component expands the payoff of a log-forward rate, pairs characteristic-function coefficient proxies with raw payoff integrals, and reproduces Black benchmark values in a lognormal special case. Fri, 07 Aug 2026 00:00:00 +0000 quantitative_finance Lean-Verified https://thalens.org/papers/sgd/ https://thalens.org/papers/sgd/ We present a machine-checked source containing 78 named real-arithmetic declarations that occur as local steps in standard analyses of stochastic gradient descent (SGD). This declaration count includes support identities, recovery variants, and direct restatements; it is not a count of novel or publication-credit theorems. Thu, 06 Aug 2026 00:00:00 +0000 optimization and formal methods Lean-Verified https://thalens.org/papers/core_the_latent/ https://thalens.org/papers/core_the_latent/ We define the **Latent** of a smooth system as the basis-free element of a graded Hilbert tensor algebra that completely characterizes the system's distributional, dynamic, and functional properties. Tue, 28 Jul 2026 00:00:00 +0000 Core Theory https://thalens.org/papers/nt_bsd/ https://thalens.org/papers/nt_bsd/ The Birch and Swinnerton-Dyer conjecture predicts that for an elliptic curve $E/\mathbb{Q}$ the Mordell-Weil rank equals the order of vanishing of the Hasse-Weil $L$-function at $s = 1$. Wed, 22 Jul 2026 00:00:00 +0000 number_theory Lean-Verified https://thalens.org/papers/ind_verified_design_certificates/ https://thalens.org/papers/ind_verified_design_certificates/ We present a reusable algebra for machine-checked certificates in declared engineering models, where supplied equations, inequalities, monotonicity directions, and perturbation bounds imply an encoded safety, lifetime, stability, or performance predicate. Mon, 20 Jul 2026 00:00:00 +0000 Formal Verification Lean-Verified https://thalens.org/papers/math_latent_path_integral/ https://thalens.org/papers/math_latent_path_integral/ Saddle-point approximations replace an action by its quadratic part, but transferring a quantitative error statement between applications requires an explicit remainder estimate. Mon, 20 Jul 2026 00:00:00 +0000 mathematics Lean-Verified https://thalens.org/papers/fin_fenton_solved/ https://thalens.org/papers/fin_fenton_solved/ An elementary representation route for sums of correlated lognormals; published with DOI and stated for review. Sun, 19 Jul 2026 00:00:00 +0000 Quantitative Finance Lean-Verified https://thalens.org/papers/ml_self_improvement/ https://thalens.org/papers/ml_self_improvement/ We establish conditional bounds in a stylized antitone threshold model of recursive AI self-improvement. A mode $k$ is model-learnable at budget $N$ when the stipulated predicate $N g(k)\ge 1$ holds, where $g$ is positive and antitone. Sun, 19 Jul 2026 00:00:00 +0000 Formal Verification Lean-Verified https://thalens.org/papers/phy_smale6/ https://thalens.org/papers/phy_smale6/ A proposed route toward finiteness of central configurations for positive masses, with formalized components and explicit assumptions. Sat, 18 Jul 2026 00:00:00 +0000 Formal Verification https://thalens.org/papers/phy_spectral_error_mitigation/ https://thalens.org/papers/phy_spectral_error_mitigation/ Mon, 13 Jul 2026 00:00:00 +0000 Physics https://thalens.org/papers/phy_m_theory_dimensions/ https://thalens.org/papers/phy_m_theory_dimensions/ The Z₃ Ansatz $\sqrt{m_r} = a(1 + b\cos(\theta_0 + 2\pi r/3))$ with $b^2 = 2$ is a parametrization — not a dynamical model — that encodes the Koide mass relation $Q = 2/3$. Sun, 12 Jul 2026 00:00:00 +0000 Formal Verification Lean-Verified https://thalens.org/papers/nt_riemann_hypothesis/ https://thalens.org/papers/nt_riemann_hypothesis/ We present an algebraic framework relating the growth of the zeta moments to Hankel-determinant structure, together with an honest account of where the framework does and does not reach the Riemann Hypothesis. Fri, 10 Jul 2026 00:00:00 +0000 Mathematics Lean-Verified https://thalens.org/papers/phy_exact_3body_solution/ https://thalens.org/papers/phy_exact_3body_solution/ A proposed finite Latent encoding for gravitational three-body trajectories, with convergence and formalization status separated from the main claim. Fri, 10 Jul 2026 00:00:00 +0000 Physics https://thalens.org/papers/phy_nbody_latent_solution/ https://thalens.org/papers/phy_nbody_latent_solution/ We extend the finite Latent representation of the gravitational three-body problem [Nagy 2026g] to the general $N$-body case. Fri, 10 Jul 2026 00:00:00 +0000 Physics https://thalens.org/papers/fin_harvestability/ https://thalens.org/papers/fin_harvestability/ This paper studies **harvestability** as a horizon object for portfolio allocation within a CRRA investor model facing Ornstein-Uhlenbeck eigenmodes. Tue, 07 Jul 2026 00:00:00 +0000 Formal Verification Lean-Verified https://thalens.org/papers/fin_equity_premium/ https://thalens.org/papers/fin_equity_premium/ The equity premium puzzle of Mehra and Prescott (1985) is the observation that the standard consumption-based asset pricing model, calibrated to plausible risk aversion and the observed smoothness of aggregate consumption, predicts an equity premium Sun, 05 Jul 2026 00:00:00 +0000 Formal Verification Lean-Verified https://thalens.org/papers/nt_rh_path1_fourier_euler/ https://thalens.org/papers/nt_rh_path1_fourier_euler/ We give a short conditional reduction of the Riemann Hypothesis to three classical inputs — Kronecker-Weyl equidistribution, the Bessel I₀ product identity, and Mertens' divergence theorem — plus a cited pair-correlation step. Sat, 27 Jun 2026 00:00:00 +0000 Mathematics Lean-Verified https://thalens.org/papers/nt_rh_path2_gue/ https://thalens.org/papers/nt_rh_path2_gue/ We prove that 100% of the nontrivial zeros of $\zeta(s)$ lie on the critical line in the density sense: $N_0(T)/N(T) \to 1$ as $T \to \infty$. The proof combines two results. Sat, 27 Jun 2026 00:00:00 +0000 Mathematics https://thalens.org/papers/fin_projected_generator_method/ https://thalens.org/papers/fin_projected_generator_method/ Derivative pricing has a clear mathematical target: compute the discounted risk-neutral value of a payoff under a specified model. For European vanilla contracts this target is often analytic or nearly analytic. Wed, 17 Jun 2026 00:00:00 +0000 Quantitative Finance Lean-Verified https://thalens.org/papers/ml_icl_architecture/ https://thalens.org/papers/ml_icl_architecture/ The companion core paper establishes, and machine-checks, a single identity: a transformer's forward pass can implement one gradient-descent step on an implicit least-squares objective (the ICL=GD mechanism). Mon, 15 Jun 2026 00:00:00 +0000 machine_learning Lean-Verified https://thalens.org/papers/ml_icl_capacity_scaling/ https://thalens.org/papers/ml_icl_capacity_scaling/ The companion core paper establishes, and machine-checks, a single identity: a transformer's forward pass can implement one gradient-descent step on an implicit least-squares objective (the ICL=GD mechanism). This satellite asks what that verified identity forces to be true about *representational capacity and scaling*. Mon, 15 Jun 2026 00:00:00 +0000 machine_learning Lean-Verified https://thalens.org/papers/ml_icl_dynamics_inference/ https://thalens.org/papers/ml_icl_dynamics_inference/ The companion core paper establishes, and machine-checks, a single identity: a transformer's forward pass can implement one gradient-descent step on an implicit least-squares objective (the ICL=GD mechanism). Mon, 15 Jun 2026 00:00:00 +0000 machine_learning Lean-Verified https://thalens.org/papers/ml_in_context_gradient_descent/ https://thalens.org/papers/ml_in_context_gradient_descent/ We turn the gradient-descent account of in-context learning (ICL) into machine-checked mathematics and falsifiable predictions about real transformers. The formal target is the linear-attention regression identity: a forward pass can implement one gradient-descent step on an implicit least-squares objective. Mon, 15 Jun 2026 00:00:00 +0000 machine_learning Lean-Verified https://thalens.org/papers/fin_dimension_collapse/ https://thalens.org/papers/fin_dimension_collapse/ The collapse of a market's effective dimensionality during financial crises is an **endogenous phase transition** rather than a smooth response to exogenous shocks. Specifically, it is a discontinuous, history-dependent jump. Tue, 02 Jun 2026 00:00:00 +0000 Quantitative Finance https://thalens.org/papers/prob_ips_scp_palm/ https://thalens.org/papers/prob_ips_scp_palm/ Tue, 19 May 2026 00:00:00 +0000 Formal Verification Lean-Verified https://thalens.org/papers/core_theory_universal_spectral_representation/ https://thalens.org/papers/core_theory_universal_spectral_representation/ How many parameters does it take to represent a smooth probability density on a bounded domain to accuracy $\varepsilon$? We prove that for densities extending holomorphically to the Bernstein ellipse with parameter $\rho > 1$, the answer is $N = \Th Mon, 27 Apr 2026 00:00:00 +0000 Quantitative Finance https://thalens.org/papers/phy_yang_mills_mass_gap/ https://thalens.org/papers/phy_yang_mills_mass_gap/ **Theorem A (Main result, conditional).** Conditional on the 20 named Tier A–D hypotheses of §7.1 — in particular the three Tier-D perturbative-QFT inputs (`tomboulis_formula`, `b_zero_from_feynman`, `beta_1_rge_def`) — we establish that Yang-Mills t Mon, 27 Apr 2026 00:00:00 +0000 Formal Verification Lean-Verified https://thalens.org/papers/core_latent_of_latents/ https://thalens.org/papers/core_latent_of_latents/ The Latent Theorem guarantees that any smooth system has a finite representation whose size depends on regularity and accuracy, not on ambient dimensionality. We extend this result to **families** of smooth systems. Sat, 25 Apr 2026 00:00:00 +0000 Machine Learning Lean-Verified https://thalens.org/papers/fin_basket_options/ https://thalens.org/papers/fin_basket_options/ Pricing European basket options requires the distribution of a weighted sum of correlated lognormals — the Fenton Distribution (Nagy, 2026a) — which has no closed form. Sat, 25 Apr 2026 00:00:00 +0000 Quantitative Finance https://thalens.org/papers/fin_es_backtest_exact/ https://thalens.org/papers/fin_es_backtest_exact/ Expected Shortfall backtesting under Basel III/IV suffers from an unmeasured structural weakness: Monte Carlo estimation of ES injects computational noise into the Acerbi-Székely (2014) test statistic, but the magnitude of this contamination has not Sat, 25 Apr 2026 00:00:00 +0000 Quantitative Finance Lean-Verified https://thalens.org/papers/fin_exact_var_computational/ https://thalens.org/papers/fin_exact_var_computational/ We present the Eigen-COS method, a deterministic algorithm that computes exact Value-at-Risk, closed-form Expected Shortfall, and the full CDF/PDF for weighted sums of correlated lognormal assets — without Monte Carlo simulation. Sat, 25 Apr 2026 00:00:00 +0000 Quantitative Finance Lean-Verified https://thalens.org/papers/fin_fenton_practical/ https://thalens.org/papers/fin_fenton_practical/ We present a deterministic, semi-analytical framework for computing the complete distribution of a portfolio's terminal value at horizon $T$ for correlated lognormal assets. Unlike traditional approaches, this method requires no Monte Carlo simulation. Sat, 25 Apr 2026 00:00:00 +0000 Quantitative Finance Lean-Verified https://thalens.org/papers/fin_return_paradox/ https://thalens.org/papers/fin_return_paradox/ Every formula in quantitative finance — CAPM, Markowitz, VaR, Sharpe ratio, GARCH — takes returns as input. Yet the standard definitions of return fail when prices cross zero: log-returns are undefined, and simple returns produce sign errors. Sat, 25 Apr 2026 00:00:00 +0000 Quantitative Finance Lean-Verified https://thalens.org/papers/fin_spectral_importance_sampling/ https://thalens.org/papers/fin_spectral_importance_sampling/ We develop a variance reduction framework for simulating rare events in correlated portfolios by exploiting the eigenvalue decomposition of the correlation matrix. The central observation is that the eigenvalue modes $Z_k$ — projections of the asset vector onto the eigenvectors of the correlation matrix — are mutually independent. Sat, 25 Apr 2026 00:00:00 +0000 Quantitative Finance https://thalens.org/papers/nt_damped_vinogradov/ https://thalens.org/papers/nt_damped_vinogradov/ Sat, 25 Apr 2026 00:00:00 +0000 Mathematics https://thalens.org/papers/nt_euler_product_smoothness/ https://thalens.org/papers/nt_euler_product_smoothness/ We prove that the distribution of values of random Euler products on the critical line possesses a stable Latent — a finite rational approximation with exponential convergence — and provide a **complete structural proof** of the Euler Product Smoothn Sat, 25 Apr 2026 00:00:00 +0000 Mathematics https://thalens.org/papers/phy_navier_stokes_latent/ https://thalens.org/papers/phy_navier_stokes_latent/ We introduce a grade decomposition of the Gevrey energy balance for the incompressible Navier-Stokes equations. The physically correct model uses $\mathbb{C}$-valued Fourier coefficients with a factor of $i$ in the advection; the real-coefficient model trivializes all grade-3 terms. Sat, 25 Apr 2026 00:00:00 +0000 Formal Verification Lean-Verified https://thalens.org/papers/phy_navier_stokes_regularity/ https://thalens.org/papers/phy_navier_stokes_regularity/ We reduce the global regularity problem for the three-dimensional incompressible Navier-Stokes equations to a single quantitative estimate: the integrability of the adiabatic error in the Gallay-Wayne convergence of vortex cross-sections to the Burgers profile. Sat, 25 Apr 2026 00:00:00 +0000 mathematical physics https://thalens.org/papers/phy_practical_pade_3body/ https://thalens.org/papers/phy_practical_pade_3body/ We demonstrate that Padé resummation of Taylor-series solutions provides a practical, machine-precision representation of the full gravitational three-body problem. Sat, 25 Apr 2026 00:00:00 +0000 Physics Lean-Verified https://thalens.org/papers/phy_quantum_tuna9_parity/ https://thalens.org/papers/phy_quantum_tuna9_parity/ We pre-register and test a spectral-error-mitigation prediction for the two-qubit gate fidelity of Quantum Inspire's Tuna-9 9-qubit transmon processor and execute it in four cryptographically timestamped stages. Sat, 25 Apr 2026 00:00:00 +0000 Physics https://thalens.org/papers/nt_rh_latent_existence/ https://thalens.org/papers/nt_rh_latent_existence/ We propose a conditional moment-based diagnostic for the Riemann Hypothesis based on stable finite rational approximations — *Latents* — for the distribution of $|\zeta(1/2+it)|$ on the critical line. Thu, 19 Mar 2026 00:00:00 +0000 Mathematics https://thalens.org/papers/fin_tensor_spectral/ https://thalens.org/papers/fin_tensor_spectral/ We prove that every Itô diffusion $dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t$ admits a complete, finite-dimensional representation through its Fokker--Planck generator discretized in cosine basis. Sat, 07 Mar 2026 00:00:00 +0000 Quantitative Finance https://thalens.org/papers/fin_fenton_spectral/ https://thalens.org/papers/fin_fenton_spectral/ The CDF of a weighted sum of correlated lognormal random variables has lacked a tractable characterization since Fenton (1960). We show that eigenvalue conditioning of the correlation matrix, followed by Fourier-cosine inversion, yields an analytic, grid-free $N$-term spectral representation of that CDF: the **Spectral Lognormal Distribution**. Quantitative Finance Lean-Verified https://thalens.org/papers/ml_knowledge_artifacts_algebra/ https://thalens.org/papers/ml_knowledge_artifacts_algebra/ A 200-tree Random Forest has 126,074 parameters. Its knowledge? Three numbers and a basis. We introduce the **Knowledge Artifact** — a portable representation of what any ML model has learned — and the **Knowledge Algebra** — provably exact arithmetic on these artifacts. Machine Learning https://thalens.org/papers/phy_fundamental_constants_as_grade_ratios/ https://thalens.org/papers/phy_fundamental_constants_as_grade_ratios/ A speculative grade-ratio model for fundamental constants, with formalized arithmetic components and explicit physical assumptions. Physics Lean-Verified