Built & Deployed
Live Tools
These are scientific software tools that have been built and deployed. Each one started as a request from a researcher who needed it. The tool is free for the requestor.
Optimal Income Tax Simulator
Public Finance / EconomicsComputes the revenue-maximising or welfare-maximising top marginal tax rate using the Saez (2001) formula. Given an elasticity of taxable income, a Pareto parameter for the income distribution, and a social welfare weight, the simulator outputs the optimal top rate with a full Laffer curve and sensitivity analysis table. Empirical presets for the US, France, UK, and Denmark are included.
Target Audience
Economists, policy researchers, public finance academics, and anyone working with optimal tax theory
Highlights
- Implements the Saez (2001) formula with full Laffer curve visualisation
- Empirical presets for US, France, UK, and Denmark based on published literature
- Sensitivity analysis table across ranges of ε and a
- Shareable links — parameters encoded in the URL hash
- Citable output with a built-in citation generator
Visit: https://optimaltaxrate.co
Unknotting Number Calculator
Computational Knot TheoryGiven a knot in any standard notation (PD code, braid word, DT notation, torus knot, or Gauss code), determine its unknotting number u(K) — the minimum number of crossing changes required to produce the unknot. Implements a three-layer algorithm combining the τ invariant from Heegaard Floer homology, the knot determinant via the Fox coloring matrix, and a diagram search using Reidemeister moves.
Target Audience
Researchers, mathematicians, and academics in topology and knot theory
Highlights
- Handles knots up to 100 crossings in under 1 second
- Verified against 5,997 knots from the KnotInfo database with 99.97% accuracy
- Supports five input formats: PD code, braid word, DT notation, torus knot, Gauss code
- Developed as a solution to the Epoch FrontierMath Open Problem
Visit: https://unknottingnumber.com
Structural Connectivity Classifier
Spectral Graph Theory · Multi-DomainComputes the Fiedler value λ₂ (algebraic connectivity) and Fiedler vector for any user-supplied network. Classifies the network's structural regime (disconnected / fragile / moderate / robust), shows the exact distance to the fragmentation boundary, and provides proved Cheeger inequality bounds. Domain-aware interpretation for neuroscience, ecology, epidemiology, economics, social networks, and power grids.
Highlights
- Fiedler value λ₂ — proved algebraic connectivity (Fiedler 1973)
- Fiedler vector — natural network partition and fragmentation direction
- Cheeger inequality bounds — proved two-sided geometric verification (Alon 1986)
- 8 example networks from 7 domains with known analytical values
- Full eigenspectrum visualisation for all n eigenvalues
- Domain interpretation: neuroscience, ecology, epidemiology, economics, social, power
Neuroscience–Economics Geometric Structure
Stochastic Processes · Decision Theory · Game TheoryFive formally proven equations in neuroscience and economics are the same equations. The Drift-Diffusion Model and Real Options Theory share identical stochastic differential equations. Wald’s Identity and the Optional Stopping Theorem are the same conservation law. Replicator Dynamics and Evolutionary Game Theory are formally equivalent. This tool visualises the unified geometric structure and computes from the proven mathematics.
Highlights
- DDM / SPRT — identical to Real Options investment threshold (Bogacz 2006, Dixit & Pindyck 1994)
- Wald’s Identity — same conservation law as the Optional Stopping Theorem (Wald 1945, Doob 1953)
- Replicator Dynamics — formally equivalent to Evolutionary Game Theory (Taylor & Jonker 1978)
- Cramér-Rao bound — Fisher information as Riemannian metric (Rao 1945, Cramér 1946)
- Fokker-Planck gradient flow — structural isomorphism with Black-Scholes (JKO 1998)
- Calculator: DDM, Wald Identity, Replicator Dynamics, Fisher Information — peer-reviewed math only
Synaptic Plasticity — Geometric Structure
Information Geometry · BCM Theory · LTP/LTDSynaptic plasticity (LTP/LTD) in hippocampal CA1 synapses is a statistical manifold equipped with the Fisher information metric, with a stratified boundary defined by the BCM modification threshold. This identification was confirmed by M4 lateral convergence across three independent research traditions. Explore all 10 anatomy layers, 15 domain faces, 14 mathematical structures, 9 derivations, and 6 open problems.
Highlights
- Fisher information metric — the unique Riemannian metric on the space of synaptic distributions (Rao 1945)
- BCM threshold stratification — LTP/LTD boundary formally proven (Bienenstock, Cooper & Munro 1982)
- Natural gradient learning — parametrization-invariant synaptic rule (Kreutzer et al. 2022, eLife)
- Energy-precision trade-off — Fisher information and memory lifetime at bistability onset (Karbowski 2021)
- BCM equivalence to projection pursuit — formally proven (Intrator & Cooper 1992)
- Calculator: Fisher information, Cramér-Rao bound, BCM threshold dynamics — peer-reviewed math only
Electrical Engineering Tools
Electrical Safety & Power SystemsA growing suite of IEEE-compliant electrical engineering calculators. Each tool implements the full published standard — not a simplified approximation. Currently live: Arc Flash Energy Calculator (IEEE 1584-2018) and Fault Current Calculator (IEEE 551-2006). Coming soon: transformer K-factor, voltage drop, and relay coordination.
Live Now
- Arc Flash Energy Calculator — full IEEE 1584-2018 empirical model
- 5 electrode configurations: VCB, VCBB, HCB, VOA, HOA
- 3-D incident energy surface, sensitivity table, and citable output
- PPE categories 1–4 per NFPA 70E-2021
- Fault Current Calculator — ANSI/IEEE 551-2006 point-to-point method
- 3-phase, SLG, LL & DLG faults — X/R ratio, Kp factor, motor contribution
- Impedance breakdown, sensitivity table, NEC Table 9 cable data
Your Tool Could Be Next
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