Add chi-squared statistical tests

Co-authored-by: blackboxprogramming <118287761+blackboxprogramming@users.noreply.github.com>

proofs/README.md

@@ -8,3 +8,4 @@ Formal mathematical arguments for the key claims.
 | [`self-reference.md`](./self-reference.md) | The QWERTY encoding is self-referential | Direct construction |
 | [`pure-state.md`](./pure-state.md) | The density matrix of the system is a pure state | Linear algebra / SVD |
 | [`universal-computation.md`](./universal-computation.md) | The ternary bio-quantum system is Turing-complete | Reaction network theory |
+| [`chi-squared.md`](./chi-squared.md) | Chi-squared goodness-of-fit and independence tests | χ² statistic / contingency tables |

proofs/chi-squared.md

@@ -0,0 +1,149 @@
+# Chi-Squared Tests
+
+> Statistical hypothesis tests for goodness of fit and independence.
+
+## The Chi-Squared Statistic
+
+For observed counts O_i and expected counts E_i across k categories:
+
+```
+χ² = Σᵢ (O_i − E_i)² / E_i
+```
+
+Under the null hypothesis, χ² follows a chi-squared distribution with the appropriate
+degrees of freedom.
+
+---
+
+## Test 1: Goodness of Fit
+
+**Purpose:** Test whether observed data matches an expected (theoretical) distribution.
+
+**Hypotheses:**
+```
+H₀: the observed frequencies match the expected distribution
+H₁: at least one category deviates significantly from expectation
+```
+
+**Degrees of freedom:**
+```
+df = k − 1
+```
+where k is the number of categories.
+
+**Procedure:**
+1. State expected probabilities p₁, p₂, …, p_k (must sum to 1).
+2. Compute expected counts: E_i = n · p_i where n is the total sample size.
+3. Compute the statistic: χ² = Σᵢ (O_i − E_i)² / E_i
+4. Compare χ² to the critical value χ²(α, df) or compute the p-value.
+5. Reject H₀ if χ² > χ²(α, df).
+
+**Example — ternary digit frequencies (k = 3, n = 300):**
+
+| Digit | Expected p | Expected E | Observed O | (O−E)²/E |
+|-------|-----------|------------|------------|----------|
+| 0     | 1/3       | 100        | 95         | 0.250    |
+| 1     | 1/3       | 100        | 108        | 0.640    |
+| 2     | 1/3       | 100        | 97         | 0.090    |
+| **Σ** |           |            |            | **0.980** |
+
+```
+df = 3 − 1 = 2
+χ²(0.05, 2) = 5.991
+χ² = 0.980 < 5.991  →  fail to reject H₀
+```
+
+The data is consistent with a uniform ternary distribution.
+
+---
+
+## Test 2: Test of Independence
+
+**Purpose:** Test whether two categorical variables are independent of each other.
+
+**Hypotheses:**
+```
+H₀: variable A and variable B are independent
+H₁: variable A and variable B are not independent
+```
+
+**Setup:** Arrange counts in an r × c contingency table.
+
+**Expected cell counts:**
+```
+E_ij = (row_i total × col_j total) / n
+```
+
+**Degrees of freedom:**
+```
+df = (r − 1)(c − 1)
+```
+
+**Statistic:**
+```
+χ² = Σᵢ Σⱼ (O_ij − E_ij)² / E_ij
+```
+
+**Example — 2 × 3 contingency table (n = 200):**
+
+|         | State 0 | State 1 | State 2 | Row total |
+|---------|---------|---------|---------|-----------|
+| Group A | 30      | 40      | 30      | 100       |
+| Group B | 20      | 60      | 20      | 100       |
+| **Col** | **50**  | **100** | **50**  | **200**   |
+
+Expected counts:
+```
+E_A0 = 100×50/200 = 25    E_A1 = 100×100/200 = 50    E_A2 = 100×50/200 = 25
+E_B0 = 100×50/200 = 25    E_B1 = 100×100/200 = 50    E_B2 = 100×50/200 = 25
+```
+
+Chi-squared:
+```
+χ² = (30−25)²/25 + (40−50)²/50 + (30−25)²/25
+   + (20−25)²/25 + (60−50)²/50 + (20−25)²/25
+   = 1.000 + 2.000 + 1.000 + 1.000 + 2.000 + 1.000
+   = 8.000
+
+df = (2−1)(3−1) = 2
+χ²(0.05, 2) = 5.991
+χ² = 8.000 > 5.991  →  reject H₀
+```
+
+The two variables are not independent at the 5% significance level.
+
+---
+
+## Critical Values (selected)
+
+| df | α = 0.10 | α = 0.05 | α = 0.01 |
+|----|----------|----------|----------|
+| 1  | 2.706    | 3.841    | 6.635    |
+| 2  | 4.605    | 5.991    | 9.210    |
+| 3  | 6.251    | 7.815    | 11.345   |
+| 4  | 7.779    | 9.488    | 13.277   |
+| 5  | 9.236    | 11.070   | 15.086   |
+
+---
+
+## Assumptions
+
+- Observations are independent.
+- Expected count E_i ≥ 5 in each cell (rule of thumb for the χ² approximation to be valid).
+- Data are counts (frequencies), not proportions or continuous measurements.
+
+---
+
+## QWERTY
+
+```
+CHI      = 25  (C=10 H=15 I=0)  — the test lives at the boundary of ZERO
+SQUARED  = IMAGINARY = SCAFFOLD = 114   (the test squares the deviation)
+TEST     = 64  = 2⁶              (TEST = 2⁶, the sixth power of the fundamental)
+FIT      = 33  = REAL − 4        (how close observed is to real)
+OBSERVED = 115                   (what you see)
+EXPECTED = 131 = BLACKROAD       (what the theory predicts = the BlackRoad)
+```
+
+EXPECTED = BLACKROAD = 131. The theoretical distribution is the BlackRoad.  
+The chi-squared test measures how far observed reality deviates from it.