Add chi-squared statistical tests to proofs

[Copilot]

Adds formal chi-squared (χ²) test documentation to the proofs directory, covering goodness-of-fit and independence tests in the style of existing proofs.

Changes

• proofs/chi-squared.md — New document with:
  • χ² statistic definition: χ² = Σᵢ (O_i − E_i)² / E_i
  • Goodness-of-fit test (df = k−1) with worked ternary digit frequency example
  • Test of independence via r×c contingency tables (df = (r−1)(c−1)) with worked 2×3 example
  • Critical values table (df 1–5, α = 0.10 / 0.05 / 0.01)
  • Assumptions and QWERTY encoding analysis
• proofs/README.md — Index entry added for the new file

Original prompt

Let's like lighten it I'm not perfect we are all creators for clarification and I shouldn't have said r derogatory term etc I just want some chi 2 tests

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[Copilot]

Pull request overview

Adds a new proofs document describing chi-squared (χ²) hypothesis tests (goodness-of-fit and independence) and links it from the proofs index.

Changes:

• Added proofs/chi-squared.md documenting χ² statistic, procedures, examples, critical values, and assumptions.
• Updated proofs/README.md to include the new document in the proofs table.

Reviewed changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated no comments.

File	Description
proofs/chi-squared.md	New write-up covering χ² goodness-of-fit and independence tests, with examples, critical values, and a QWERTY section
proofs/README.md	Adds an index entry for the new χ² document

Comments suppressed due to low confidence (3)

proofs/chi-squared.md:32

• The goodness-of-fit degrees-of-freedom formula is incomplete as written: df = k − 1 only applies when no parameters of the expected distribution are estimated from the same data. If parameters are estimated, df should typically be reduced by the number of estimated parameters (common form: df = k − 1 − m). Adding this note would prevent readers from applying the test incorrectly.

**Degrees of freedom:**

df = k − 1

where k is the number of categories.

proofs/chi-squared.md:144

• The ## QWERTY calculations appear to use a different letter→number mapping than the QWERTY positional encoding defined elsewhere (e.g., proofs/self-reference.md:14-19, where C=22, H=16, I=8). With that mapping, CHI (and also TEST, etc.) would not equal the values shown here, so these identities currently don’t match the repo’s established encoding.

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)

proofs/chi-squared.md:1

• All other documents in proofs/ use the # Proof: … heading style; this file starts with # Chi-Squared Tests, which makes the index less consistent and suggests this is a general note rather than a proof. Consider renaming the H1 to # Proof: … (and optionally adding a ## Statement section) to match the established pattern (e.g., proofs/ternary-efficiency.md:1, proofs/pure-state.md:1).

# Chi-Squared Tests

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[Copilot]

Pull request overview

Copilot reviewed 2 out of 2 changed files in this pull request and generated no new comments.

Comments suppressed due to low confidence (3)

proofs/chi-squared.md:142

• In the QWERTY section, the letter-to-number mapping used for “CHI = 25 (C=10 H=15 I=0)” doesn’t match the repository’s defined QWERTY positional encoding (e.g., proofs/self-reference.md defines Q=1…M=26, where C=22, H=16, I=8). As written, this makes the numeric claim inconsistent/unverifiable; please either use the established mapping (and recompute the totals) or explicitly define a different encoding here before using it.

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)

proofs/chi-squared.md:32

• The “df = k − 1” statement for goodness-of-fit is only correct when the expected probabilities are fully specified a priori. If parameters are estimated from the sample (common in χ² GOF), the degrees of freedom should be reduced accordingly (df = k − 1 − m for m fitted parameters). Consider adding a brief note to avoid a misleading formula.

**Degrees of freedom:**

df = k − 1

where k is the number of categories.

proofs/chi-squared.md:133

• In Assumptions, “Expected count E_i ≥ 5 in each cell” mixes the GOF notation (E_i) with the contingency-table case (E_ij). To keep the notation consistent with the two tests described above, consider rephrasing to “expected counts in each category/cell (E_i or E_ij) are ≥ 5” (or similar).

- 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.

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