Check 1 — Is The Lottery Actually Random?

Chi-square goodness-of-fit, against the real draw history

Before anything else: does the raw draw history itself show any statistical departure from true randomness? This is the correct tool for that specific question — a real chi-square test comparing how often each number has actually come up against how often it should, under pure chance. (Kolmogorov-Smirnov and Anderson-Darling, tried first, gave a 77-97% false-positive rate on data KNOWN to be random in testing — they're built for continuous distributions, and 35/20-category discrete draws aren't one. Caught that before it ever reached this page.)

Check 2 — The Free Energy Upper Bound

F = Surprise + KL Divergence

Friston's proof: Free Energy (F) always sits at or above Surprise (−ln P(y), how unexpected an outcome was) — because the gap between them is a KL divergence, and a KL divergence between two real probability distributions can never be negative. That's not an estimate. It's a theorem.

What can actually break that guarantee isn't the theorem — it's a bug. If this engine's belief (μ, what Friston calls q(s)) ever stopped being a real probability distribution, the guarantee would quietly fail. So rather than assert it, we replayed this engine's own real functions — unmodified — from flat priors across every real historical draw, and checked the promise held every single time.

The Gap Itself — Complexity (F − Surprise)

F and Surprise track each other closely at this scale, so here's the gap between them on its own axis — the exact quantity the theorem says can never cross below the zero line.

Check 3 — Markov Blanket Separation

p(Internal | Blanket, External) = p(Internal | Blanket)

Friston's Markov blanket says an organism's internal states should only ever hear from the outside world through its own senses — never directly. Translated to this engine: the belief (μ) should only ever update through the sensory layer (φ, the 9 engineered features, and the prediction error they produce) — never by reaching past that layer to touch raw history directly.

This one isn't a statistic — it's an architecture check, verified directly against the live source: Brain.prototype.step() only ever receives phi and predError as inputs to its belief update — both entirely downstream of the sensory layer. No line of that function reaches around φ to read raw draw history directly. The separation the theory requires is structural, not assumed.

Check 4 — Predictive Calibration

Hosmer-Lemeshow test, on this engine's own real probabilities

A real, meaningful gap found in this same audit: this engine's raw probability output forces itself to average ~50% no matter what, when the true base rate is ~20% (main) or ~5% (Powerball) — a genuine miscalibration, not a display bug. Fixed with Platt scaling (a standard 2-parameter recalibration, fit on real historical draws via the same machinery validated above) and re-tested with the Hosmer-Lemeshow test — the correct tool for "do these predicted probabilities match real observed frequencies."

The honest reading of the fitted slope (A ≈ 0 for both ball types): there is no real signal in this engine's 9 features for predicting an independent random draw — exactly as the randomness check above says there shouldn't be. Once corrected, this engine's real probabilities sit close to the true base rate for every ball, which is what genuinely calibrated honesty looks like here.