Correct the propensity-score identification proof

[AliFozooni]

Summary

• distinguish propensity-score covariate balance from conditional ignorability of the potential outcomes
• prove balance first, then prove preservation of ignorability under conditional exchangeability
• state that positivity is a separate requirement and that propensity scores cannot repair unmeasured confounding
• correct the adjacent IPTW derivation to use P(T=1 | X)=e(X), not the marginal treatment probability
• derive the conditional IPW contrast and its ATE identity explicitly
• correct the standardized-outcome interpretation and relabel the later 0.47 benchmark as an unadjusted difference in means

Why this matters

The previous proof treated potential-outcome ignorability given the propensity score as equivalent to a statement that establishes only covariate balance. The revised derivation separates the two results: it obtains balance from the propensity definition, then invokes conditional exchangeability and iterated expectations to establish ignorability given e(X).

The neighboring weighting derivation also multiplied a conditional expectation by the marginal P(T). That equality is false when treatment probability varies with X. It now uses the conditional probabilities e(X) and 1-e(X), shows the weighted conditional identity, and averages it over X to recover the ATE under the stated assumptions.

The introductory regression interpretation is corrected at the same time: the intercept is the untreated mean, while the intervention coefficient is the treated-minus-untreated difference.

Validation

• built the complete Jupyter Book successfully after the proof and weighting corrections
• checked the rendered math and surrounding narrative
• validated notebook JSON and checked the diff for whitespace errors
• modified only markdown cells 8, 12, and 24

The full build retains one unrelated pre-existing Chapter 21 lexer warning.

Closes #236
Closes #470