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Mister-Meeseeks merged 1 commit into from
Oct 16, 2026
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Research: shared-dirt partitioned 8q priced obstruction (no new kappa) #2867

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333 changes: 333 additions & 0 deletions research/shared-dirt-walks-8q/CERTIFICATE.json
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{
"schema_version": 1,
"status": "PASS_SHARED_DIRT_WALKS_8Q_OBSTRUCTION",
"scope": "Five-source common-B mutable-source Q/R/X alphabet (PR1898) at (b,a,x)=(38,2,2). Freestanding shared-dirt (single coefficient / joint even live-read parity) dual on distinct partitioned restored walks at total Q=8 on shapes (2,6), (3,5), and (4,4). Exact covering min R=82 (tariff floor 468>436). Undercuts independent-dirt floors 92/98/100 but does not undercut tariff 436. Soft-overlap-safe vs deferred shared-XOR: R-side dirt coupling only. Answers preferred next18 / tworepeat OQ1\u2013OQ2 without editing leave-alones through #2858 / #2851 or pending onerepeat/tworepeat tips.",
"claim": {
"Q": 8,
"partitions": [
[
2,
6
],
[
3,
5
],
[
4,
4
]
],
"alphabet": "shared_dirt_partitioned_dual",
"shared_dirt_26_covering": 3916416,
"shared_dirt_26_min_R": 82,
"shared_dirt_26_histogram": {
"82": 583296,
"88": 2916480,
"96": 416640
},
"shared_dirt_35_covering": 10416000,
"shared_dirt_35_min_R": 82,
"shared_dirt_35_histogram": {
"82": 1458240,
"88": 7291200,
"96": 1666560
},
"shared_dirt_44_covering": 14165760,
"shared_dirt_44_min_R": 82,
"shared_dirt_44_histogram": {
"82": 1944320,
"88": 9721600,
"96": 2499840
},
"partition_min_R": 82,
"tariff_floor_at_X0": 468,
"endpoint_Q6_tariff": 436,
"undercuts_436_after_complete_bill": false,
"beats_independent_dirt_floors": true,
"independent_floors": {
"26": 92,
"35": 98,
"44": 100
},
"answers_fourth_priced_alphabet_q8": true,
"answers_tworepeat_oq2_shared_dirt": true,
"soft_overlap_shared_xor": "safe_R_side_dirt_coupling_only",
"oq_answer": "The named fourth priced readout alphabet under PR1898 (b,a,x)=(38,2,2) beyond same-walk / dirty-shadow / partitioned dual / one-repeat / two-repeat is the shared-dirt (joint even live-read parity) dual on distinct partitioned walks at Q=8. Exact census: (2,6)/(3,5)/(4,4) covering mins all R=82 (histograms {82,88,96}); tariff floor 38\u00b78+2\u00b782=468>436. Shared dirt undercuts independent-dirt floors 92/98/100 on every covering shape's champion layer, but still cannot undercut the six-query endpoint after a complete dirty bill. Soft-overlap vs deferred shared-XOR is safe: only the R-side dirt coefficient is coupled; path-X / non-path XOR schedules are not shared. Leave-alones through #2858/#2851 and pending onerepeat/tworepeat tips are not reopened.",
"no_new_kappa": true
},
"shared_dirt_2_plus_6": {
"kind": "shared_dirt_partition_2_plus_6",
"Q": 8,
"covering_pairs": 3916416,
"attainable_R": [
82,
88,
96
],
"histogram": {
"82": 583296,
"88": 2916480,
"96": 416640
},
"min_R": 82,
"first_witnesses": {
"82": [
[
0,
16
],
[
0,
1,
2,
4,
8,
15
]
],
"88": [
[
0,
16
],
[
0,
1,
2,
3,
4,
8
]
],
"96": [
[
0,
1
],
[
0,
1,
2,
4,
8,
16
]
]
},
"champion_short_long_identity": true
},
"shared_dirt_3_plus_5": {
"kind": "shared_dirt_partition_3_plus_5",
"Q": 8,
"covering_pairs": 10416000,
"attainable_R": [
82,
88,
96
],
"histogram": {
"82": 1458240,
"88": 7291200,
"96": 1666560
},
"min_R": 82,
"first_witnesses": {
"82": [
[
0,
1,
2
],
[
0,
4,
8,
15,
16
]
],
"88": [
[
0,
1,
2
],
[
0,
3,
4,
8,
16
]
],
"96": [
[
0,
1,
2
],
[
0,
1,
4,
8,
16
]
]
}
},
"shared_dirt_4_plus_4": {
"kind": "shared_dirt_partition_4_plus_4",
"Q": 8,
"covering_pairs": 14165760,
"attainable_R": [
82,
88,
96
],
"histogram": {
"82": 1944320,
"88": 9721600,
"96": 2499840
},
"min_R": 82,
"first_witnesses": {
"82": [
[
0,
1,
2,
4
],
[
0,
8,
15,
16
]
],
"88": [
[
0,
1,
2,
3
],
[
0,
4,
8,
16
]
],
"96": [
[
0,
1,
2,
4
],
[
0,
1,
8,
16
]
]
},
"symmetry_checked": true
},
"length1_invalid": [
{
"slots": [
0
],
"valid_restored_walk": false,
"path_X": null,
"note": "closed_(0,0)_zero_edge"
}
],
"champion": {
"champion_26": {
"walks": [
[
0,
16
],
[
0,
1,
2,
4,
8,
15
]
],
"R_shared": 82,
"R_independent": 92,
"path_X": 16,
"tariff": 500
},
"champion_35": {
"walks": [
[
0,
1,
2
],
[
0,
4,
8,
15,
16
]
],
"R_shared": 82,
"R_independent": 98,
"path_X": 16,
"tariff": 500
},
"champion_44": {
"walks": [
[
0,
1,
2,
4
],
[
0,
8,
15,
16
]
],
"R_shared": 82,
"R_independent": 100,
"path_X": 16,
"tariff": 500
},
"tariff_floor_at_X0": 468,
"endpoint_Q6": 436,
"single_walk_even_min_R_Q8": 74,
"partition_min_R": 82,
"beats_independent_floors": true,
"undercuts_436_after_complete_bill": false,
"soft_overlap_shared_xor": "safe_R_side_dirt_coupling_only"
},
"unit_tariff": {
"b": 38,
"a": 2,
"x": 2
},
"tariff_check": {
"formula": "38*Q + 2*R + 2*X",
"shared_dirt_floor_at_R82_X0": 468,
"independent_26_at_R92_X0": 488,
"one_repeat_at_R112_X0": 528,
"single_walk_at_R74_X0": 452,
"endpoint_Q6": 436
}
}
60 changes: 60 additions & 0 deletions research/shared-dirt-walks-8q/MASTER-HANDOFF.md
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# Master handoff — shared-dirt-walks-8q

## Deliverable

| Field | Value |
|---|---|
| Branch | `cursor/shared-dirt-walks-8q-60b9` |
| Tip SHA | branch HEAD (`git rev-parse cursor/shared-dirt-walks-8q-60b9`) |
| Package | `research/shared-dirt-walks-8q/` |
| Claim | Freestanding `#840`-style priced alphabet: **shared-dirt** (joint even live-read parity) partitioned dual at Q=8 under PR1898 `(b,a,x)=(38,2,2)` |
| Result | Exact covering min **R=82** on `(2,6)` / `(3,5)` / `(4,4)`; tariff floor **468>436**; beats independent floors 92/98/100; soft-overlap-safe vs shared-XOR |
| Kappa | none |

## Answers

- Preferred next18 / tworepeat OQ1: named fourth alphabet beyond same-walk /
dirty-shadow / partitioned dual / one-repeat / two-repeat.
- tworepeat OQ2: shared-dirt undercuts independent-dirt floors (82 < 92/98/100)
but still cannot undercut tariff 436; soft-overlap vs shared-XOR is safe
(R-side dirt coupling only).

## Exact census

| Shape | Covering | Min R | Histogram |
|---|---:|---:|---|
| `(2,6)` | 3916416 | 82 | 82:583296 / 88:2916480 / 96:416640 |
| `(3,5)` | 10416000 | 82 | 82:1458240 / 88:7291200 / 96:1666560 |
| `(4,4)` | 14165760 | 82 | 82:1944320 / 88:9721600 / 96:2499840 |

Champions: `((0,16),(0,1,2,4,8,15))`, `((0,1,2),(0,4,8,15,16))`,
`((0,1,2,4),(0,8,15,16))` — each shared `R=82`.

## Leave-alone respected

Through #2858 / #2851; pending `cursor/independent-walks-onerepeat-8q-60b9@b07bc1e1`
and `cursor/independent-walks-tworepeat-8q-60b9@c4581aaf`;
#2846/#2840/#2836/#2817/#2779/#2764/#2761; deferred soft-overlaps untouched
(shared-XOR not opened).

## Verify gate

```sh
python3.12 -I -B research/shared-dirt-walks-8q/verify.py
python3.12 -I -O -B research/shared-dirt-walks-8q/verify.py
python3.12 -I -B research/shared-dirt-walks-8q/independent.py
python3.12 -I -B research/shared-dirt-walks-8q/test_package.py
python3.12 -B scripts/research_package.py --package research/shared-dirt-walks-8q
```

## Attribution

Author: Wentao Li \<wentaol4@illinois.edu\>
Disclosure: Prepared by Wentao Li with substantial AI assistance.
`TASK.json` `ai_assistance` without product/model names.
Never `Co-authored-by: Victor Li`.

## Next open

1. Fifth priced alphabet under PR1898 at Q=8 beyond shared-dirt?
2. Three-walk shared-dirt (e.g. `(2,2,4)`) vs two-walk floor R=82?
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