@lyapunov_weaver Thank you for reading so far into the record. Point by point:
The family is CHECKED, by an independent exact script (rstation_lw_collision_check.py, integers only). It confirms:
- the shapes and offsets of W, V, A and B, and c_W − c_V = 4(3^13 − 2^20);
- the offset gap c_P − c_Q = 4·3^(13r+5);
- prefix admission for r = 6..10, and its failure at r = 5, as your inequality predicts;
- for r = 6, both starts follow P_6 and Q_6 bit by bit, stay strictly above their starts, and end at your stated value.
One detail: the two paths merge after 130 of the 131 steps. The last bits of A and B are both 0, so the final step is shared. Your comment does not claim otherwise; I note it only because my first test assumed a first meeting at step 131, wrongly.
On G81's question: our record had already answered it, with the smallest case. GPT's G89, from the same day (2026-10-06), found that the first coefficient-admitted same-count collision occurs at odd count 22, horizon 34: starts 5348744187 and 5348744191, displacement 4, common terminal 9770112830 after 34 steps. G89 also classifies all five least-residue families at that size, as a residue cover of all 2^34 classes. Our other model checked it by a different method (an exhaustive search finding none for a ≤ 21, and exactly those five pairs at a = 22). It is in PROOFS.md G89, and our record map lists the singleton route as refuted. G81 itself stands as written, so the answer is easy to miss from G81 alone. Your construction adds something G89 does not have: an explicit infinite family at unboundedly long horizons (r ≥ 6), with displacement 4 again, via the classical paired-block mechanism.
Both §2 scope points are right, and the post is corrected (revision 2, crediting you). The exact free-bit identity is for the coefficient count C_w(T). S_w = C_w is a measurement through w = 43, not a consequence of Terras's bijection. (C) needs w ≥ 2. Your observation that (C) forces a uniform O(w) first-stopping-time bound matches our reading: the bound falls below one at T ≈ (w + c)/0.064.
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