C9.4: Catalog +1 injection boundary effects and prove O(1/n) decorrelation bias bound

[Copilot]

The decorrelation lemma's validity near window boundaries and in the small-n regime was unverified — specifically, whether the "+1" in 3n+1 introduces systematic bias. This adds the analysis infrastructure to answer that question rigorously.

New: scripts/collatz_c9_4_boundary.py

Core analysis module establishing:

• +1 injection bias Δ(n) = 1/(3n+1) for odd n — the fractional contribution of the additive correction to T(n)
• O(1/n) bound (proven + computationally verified for n ∈ [1, 1024]):

  Δ(n) · n = n/(3n+1)  <  1/3  for all odd n
  

• Per-window average bound: avg Δ(n) ≤ 1/(3·n_min + 1) — ratio always ≤ 1, verified via average_bias_in_window()
• Small-n catalog: 6/32 odd values in [1, 64] exit the window (18.75%), all at n ≡ 3 (mod 4) with v₂(3n+1) = 1 where T(n) ≈ 3n/2
• Boundary window detection (identify_boundary_windows) and exclusion-aware bias estimation (decorrelation_bias_estimate): excluding the low-n boundary raises n_min, reducing avg_bias

from scripts.collatz_c9_4_boundary import decorrelation_bias_estimate, catalog_small_n_exceptions

# Show bias drops after excluding boundary windows
full = decorrelation_bias_estimate(1, 1024, exclude_boundary_windows=False)
excl = decorrelation_bias_estimate(1, 1024, exclude_boundary_windows=True, window_size=32)
# excl['avg_bias'] < full['avg_bias'], excl['n_min'] > full['n_min']

# Enumerate small-n exceptions
rows = catalog_small_n_exceptions(64)  # exit_dir in {'none','above','below'}

New: docs/analysis_ns/c9_4_boundary_analysis.md

Documents definitions, the full small-n table, both propositions (per-step and per-window O(1/n)), and a checklist for invoking the decorrelation lemma safely — including when to apply uniform vs. adaptive boundary exclusion.

New: scripts/tests/test_c9_4_boundary.py

43 unit tests covering all public functions, including edge cases (n=0, empty windows, exact known values of T(n)).