We gave our system five fundamental physics problems. Each was computed exactly from known micro-physics, each with a target bounded between 0 and 1. Two of the five came back as exact recoveries of known physics. One of them found a mathematical identity that physicists express in a completely different form.
Bose-Einstein Condensate Fraction: Recovered Exactly
The problem: Below a critical temperature, a fraction of bosons in a gas condense into the quantum ground state. What fraction remains condensed as temperature rises?
The input: 100 data points of temperature ratio versus condensate fraction.
The result:
R² = 1.0000. Both coefficients are exactly 1.000000 to six decimal places. The system recovered the exact BEC condensation law from numerical data alone. This is the same formula Albert Einstein derived in 1924 from first principles of quantum statistical mechanics.
It was not told about Bose-Einstein statistics. It was not told about quantum mechanics. It looked at numbers and found the law.
Quantum Tunneling: A Hidden Hyperbolic Identity
The problem: A quantum particle encounters a rectangular potential barrier. The transmission probability depends on the barrier parameter κa.
The input: 100 data points computed from the exact quantum mechanical transmission formula.
The result:
R² = 1.0000. This is mathematically identical to the exact formula. The textbook expression uses sinh² in a denominator. Our system independently discovered the hyperbolic identity cosh²x - sinh²x = 1, which transforms the standard form into this more compact equivalent.
The system did not approximate the tunneling probability. It found an algebraically equivalent form that humans write differently.
Three Near-Misses That Reveal the Boundary
Not every problem yielded an exact recovery. The Lorentz factor, the quantum partition function ratio, and electron specific heat all came back as high-accuracy approximations (R² > 0.997) but not exact laws.
The pattern is telling. The two exact recoveries both had targets bounded between 0 and 1. When the target stays bounded, every data point matters equally. There is no "tail" to sacrifice for a better fit elsewhere. The physics is constrained, and the constraint forces precision.
The Lorentz factor γ - 1 grows without bound as velocity approaches the speed of light. The system achieved R² = 0.997, capturing the correct qualitative behavior, but the unbounded growth allowed it to trade accuracy in the extreme regime for simplicity.
What This Demonstrates
When our system is given data generated by exact physical laws, it recovers those laws. Not approximations. Not fits. The actual algebraic expressions, sometimes in forms that are mathematically equivalent but expressed differently than what appears in textbooks.
When the underlying physics is inherently more complex than a compact formula can capture, the system stops at the boundary of what algebra can explain. It does not overfit. It does not hallucinate structure that is not there.
The BEC recovery is particularly striking. Einstein derived that formula from the full machinery of quantum statistical mechanics: Bose-Einstein distribution functions, density of states, thermodynamic limits. Our system arrived at the same destination from 100 numbers in a table.
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