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A rigorous mathematical companion to The Reality Equation: complex division, logarithms, inference, learning, attention, coherence, and intelligence, with proofs, counterexamples, and twenty worked problems. … readThe Reality Equation: Mathematical Companion
To be stuck, every direction around you has to point up. In a million dimensions that almost never happens except near the bottom, which is why AI training works at all. … readEvery Trap Is Near the Floor
Surprise is one number made from two, and the feeling does not come labelled with which term moved: a world that fell short and an expectation that had grown produce readings identical to the third decimal. A single reading is a line of position, not a fix — which is why the numerator gets convicted, why disagreement is the only gauge of a denominator, and why a forecast written before the world arrives is the one artifact that makes the next reading identifiable. … readOne Reading, Two Unknowns
E = mc² is the first term of a series that never ends, and so are F = ma, Hooke’s law and the ideal gas. Not because the true law is infinitely long: because every smooth structure looks like its tangent from wherever you stand, and an expansion point is a frame. Part Eleven, for philosophers and theologians: Taylor’s theorem, why the series dies where it does, and what no series can see. … readEvery Formula Is a First Term
The Reality Equation reads surprise as the natural log of Actual over Expectation. That log was never a choice — any readout of a ratio that adds must be a logarithm — and taken on an expectation that has an angle it splits into a real part you feel and an imaginary part that is the angle itself, which you cannot feel at all. … readSurprise Has an Imaginary Part
Eleven experiments and an epilogue in the order that breaks things properly, starting with the diagonal that broke the only numbers anyone believed in. Every crack ever found in mathematics has ended as a theorem. … readJohn Rector’s Math Lab
Classical mechanics forbids the crossing for an exact reason: inside the barrier there is no real number left to be the momentum. The Schrödinger equation carries on — one coefficient changes sign, oscillation becomes decay, and the faint exponential tail on the far side is why the Sun shines and how the first microscope to image atoms worked. Part Ten, for philosophers and theologians: tunnelling without the folklore. … readThe Wall Is a Change of Sign
One condition — there shall be no undoing — forces a turn; a turn forces exactly four directions; and the derivative ladder hands the Four Cardinal Ideas their bearings in order. When the walk comes home at 2π it has added nothing to the circle: everything creation made is the difference between 0 and 2π, which only the logarithm can see. … readCreation Ends Where It Began
If “I Am” is a position and “I want” is a velocity, then desire is the derivative of identity: every want splits into a component that grows you and a component that turns you, a want is felt only when something resists it, and whether knowing can bend wanting turns out to be the oldest fight in moral psychology. … read“I Want” Is the Derivative of “I Am”
Quantum mechanics contains exactly two ways a state can change: a rotation that is reversible, deterministic and always running, and a projection that keeps one component and deletes the rest. A lesson for philosophers and theologians on what the geometry fixes about wave function collapse — its shape, its weights, its basis, even the impossibility of dating it — and the one question it leaves open. … readCollapse Is Not a Rotation