# Empirical random-trace recovery and fiber decay

Instances: 13800

| n | p | instances | censored | mean | median | p90 | max | mean dim(P) |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| 8 | 0.1 | 1000 | 0 | 9.830 | 8.000 | 15 | 53 | 2.000 |
| 8 | 0.25 | 1000 | 0 | 15.102 | 12.000 | 26 | 128 | 2.019 |
| 8 | 0.5 | 1000 | 0 | 23.267 | 19.000 | 43 | 133 | 2.262 |
| 8 | 0.75 | 1000 | 0 | 25.457 | 20.000 | 50 | 135 | 2.305 |
| 10 | 0.1 | 500 | 0 | 12.886 | 11.000 | 21 | 82 | n/a |
| 10 | 0.25 | 500 | 0 | 23.018 | 19.000 | 42 | 210 | n/a |
| 10 | 0.5 | 500 | 0 | 37.616 | 31.000 | 68 | 203 | n/a |
| 10 | 0.75 | 500 | 0 | 43.110 | 37.000 | 76 | 183 | n/a |
| 12 | 0.1 | 500 | 0 | 16.432 | 14.000 | 28 | 70 | n/a |
| 12 | 0.25 | 500 | 0 | 35.508 | 29.000 | 62 | 251 | n/a |
| 12 | 0.5 | 500 | 0 | 59.550 | 52.000 | 110 | 236 | n/a |
| 12 | 0.75 | 500 | 0 | 70.026 | 61.000 | 114 | 367 | n/a |
| 14 | 0.1 | 400 | 0 | 21.810 | 17.000 | 36 | 279 | n/a |
| 14 | 0.25 | 400 | 0 | 50.242 | 41.000 | 87 | 486 | n/a |
| 14 | 0.5 | 400 | 0 | 78.940 | 70.000 | 133 | 303 | n/a |
| 14 | 0.75 | 400 | 0 | 105.502 | 92.000 | 179 | 539 | n/a |
| 16 | 0.1 | 250 | 0 | 26.784 | 22.000 | 46 | 132 | n/a |
| 16 | 0.25 | 250 | 0 | 62.264 | 52.000 | 109 | 225 | n/a |
| 16 | 0.5 | 250 | 0 | 105.220 | 98.000 | 160 | 471 | n/a |
| 16 | 0.75 | 250 | 0 | 138.220 | 129.000 | 214 | 318 | n/a |
| 18 | 0.1 | 300 | 0 | 31.827 | 26.000 | 55 | 152 | n/a |
| 18 | 0.25 | 300 | 0 | 84.717 | 69.500 | 140 | 727 | n/a |
| 18 | 0.5 | 300 | 0 | 143.687 | 124.500 | 229 | 451 | n/a |
| 18 | 0.75 | 300 | 0 | 186.470 | 176.000 | 288 | 527 | n/a |
| 20 | 0.1 | 500 | 0 | 40.292 | 34.000 | 69 | 242 | n/a |
| 20 | 0.25 | 500 | 0 | 106.952 | 86.000 | 186 | 1679 | n/a |
| 20 | 0.5 | 500 | 0 | 178.586 | 160.000 | 287 | 679 | n/a |
| 20 | 0.75 | 500 | 0 | 244.078 | 222.000 | 386 | 873 | n/a |

## Findings

- All 13,800 instances recovered before `k_max`; there were no censored observations. The largest recovery time was 1,679.
- Plot A quantifies the paper's Section 7 dimension connection and Section 8 multiple-trace continuation: recovery increases with both n and p, and the right-skewed tails produce a measurable mean/median gap.
- Plot B shows rapid agreement-order contraction followed by slower decay to the irreducible subposet floor. This is the empirical multiple-trace analogue of the paper's inverse fiber.
- Exact dimensions were computed for all 4,000 n=8 instances. Every recovery time respected `recovered_at >= dim(P)` (0 violations).
- Across 6,000 independently diagnosed instances, the hardest-pair median proxy has Spearman rank correlation 0.859 with observed recovery.

## Scope

Fiber values are exact through n=8 and whenever the agreement order has height at most two; otherwise `r_pairs` is the reported upper-bound proxy. The generic exact counter was benchmarked at about 5 seconds for the n=9 identity order and 88 seconds at n=10, so claiming exact n=12 or n=16 curves would not be supportable with this backend.

The optional single-trace asymptotic-constant fit was not run. The repository paper proves only the 1/4-to-1/2 bounds, and the local benchmark did not justify an exact n~20 computation.