Turn a mathematical claim into an explicit model
A shared equation does not make two physical or computational systems equivalent. Specify the inputs, assumptions and output first, then compare the calculation with the claim.
Scaled attention
tool: scaled_attention {"queries": [[1, 0]], "keys": [[1, 0], [0, 1]], "values": [[10], [20]]}softmax(QKᵀ/√d)V; supplied Q/K/V, not a trained transformer.
Boltzmann distribution
tool: boltzmann_distribution {"energies": [0, 1, 2], "temperature": 1}Dimensionless energy, k_B = 1; this is a supplied energy model, not evidence about LLM physics.
Gaussian pdf
tool: gaussian_pdf {"x": 0, "mean": 0, "sd": 1}A probability density can exceed 1; interval probability requires integration.
Double integral polynomial
tool: double_integral_polynomial {"terms": [{"coefficient": 1, "xPower": 1, "yPower": 1}], "xBounds": [0, 2], "yBounds": [0, 3]}Analytical polynomial integral over a rectangle, evaluated in floating point.
Quadratic descent
tool: quadratic_descent {"a": 1, "b": 0, "start": 10, "rate": 0.1, "steps": 50}Gradient descent on supplied scalar quadratic ax²+bx+c.
Hopfield recall
tool: hopfield_recall {"patterns": [[1, 1, -1, -1]], "state": [1, 1, 1, -1]}Hebbian symmetric weights, zero diagonal, asynchronous deterministic updates. Spurious attractors are possible.