Mister-Meeseeks · Oct 16, 2026 · Oct 15, 2026
diff --git a/research/dirty-parity-8q-obstruction/CERTIFICATE.json b/research/dirty-parity-8q-obstruction/CERTIFICATE.json
@@ -0,0 +1,233 @@
+{
+  "schema_version": 1,
+  "status": "PASS_DIRTY_PARITY_8Q_OBSTRUCTION",
+  "scope": "Five-source common-B mutable-source Q/R/X alphabet (PR1898). All-subset vs even-subset vs zero-slot dirty-parity repair on restored length-8 slot lists. Answers #2715 OQ2.",
+  "claim": {
+    "Q": 8,
+    "all_subset_min_R_distinct7": 61,
+    "even_subset_min_R_distinct7": 74,
+    "parity_repair_min_R_distinct7": 74,
+    "one_dup_all_subset_min_R": 66,
+    "one_dup_parity_repair_min_R": 82,
+    "undercuts_436_if_all_subset_free": true,
+    "undercuts_436_after_parity_repair": false,
+    "hypothetical_all_subset_tariff_at_X0": 426,
+    "parity_repair_tariff_at_X0": 452,
+    "endpoint_Q6_tariff": 436,
+    "matches_2677_even_floor": true,
+    "answers_cr7q_oq2": true,
+    "oq2_answer": "Free all-subset readout at Q=8 reaches R=61 on distinct-7 covers (hypothetical tariff 426<436 even at X=0), but odd live-connector parity leaves connector dirt. Zero-slot parity repair collapses every covering distinct-7 schedule to even-subset cost R>=74 (tariff >=452), matching #2677's distinct-slot floor; one-duplicate covers repair to R>=82. The dirty-parity collapse therefore extends from Q=7 to Q=8.",
+    "no_new_kappa": true
+  },
+  "champion": {
+    "slots": [
+      0,
+      1,
+      2,
+      3,
+      4,
+      8,
+      16,
+      29
+    ],
+    "all_subset_R": 61,
+    "even_subset_R": 74,
+    "parity_repair_R": 74,
+    "odd_dirt_mask_count": 13,
+    "hypothetical_tariff_X0": 426,
+    "repair_tariff_X0": 452
+  },
+  "dirt_obstruction_examples": [
+    {
+      "mask": 1,
+      "subset_idxs": [
+        1
+      ],
+      "values": [
+        1
+      ],
+      "size": 1,
+      "live_connector_parity": "odd"
+    },
+    {
+      "mask": 2,
+      "subset_idxs": [
+        2
+      ],
+      "values": [
+        2
+      ],
+      "size": 1,
+      "live_connector_parity": "odd"
+    },
+    {
+      "mask": 3,
+      "subset_idxs": [
+        3
+      ],
+      "values": [
+        3
+      ],
+      "size": 1,
+      "live_connector_parity": "odd"
+    },
+    {
+      "mask": 4,
+      "subset_idxs": [
+        4
+      ],
+      "values": [
+        4
+      ],
+      "size": 1,
+      "live_connector_parity": "odd"
+    },
+    {
+      "mask": 8,
+      "subset_idxs": [
+        5
+      ],
+      "values": [
+        8
+      ],
+      "size": 1,
+      "live_connector_parity": "odd"
+    }
+  ],
+  "distinct7": {
+    "covering_sets": 2430400,
+    "repair_equals_even_count": 2430400,
+    "all_subset": {
+      "attainable_R": [
+        61,
+        64,
+        68
+      ],
+      "histogram": {
+        "61": 416640,
+        "64": 1180480,
+        "68": 833280
+      },
+      "min_R": 61,
+      "first_witness": [
+        0,
+        1,
+        2,
+        3,
+        4,
+        8,
+        16,
+        29
+      ]
+    },
+    "even_subset": {
+      "attainable_R": [
+        74,
+        80,
+        84
+      ],
+      "histogram": {
+        "74": 416640,
+        "80": 1180480,
+        "84": 833280
+      },
+      "min_R": 74,
+      "first_witness": [
+        0,
+        1,
+        2,
+        3,
+        4,
+        8,
+        16,
+        29
+      ]
+    },
+    "parity_repair": {
+      "attainable_R": [
+        74,
+        80,
+        84
+      ],
+      "histogram": {
+        "74": 416640,
+        "80": 1180480,
+        "84": 833280
+      },
+      "min_R": 74,
+      "first_witness": [
+        0,
+        1,
+        2,
+        3,
+        4,
+        8,
+        16,
+        29
+      ]
+    }
+  },
+  "one_duplicate": {
+    "covering_multisets": 4083072,
+    "all_subset": {
+      "attainable_R": [
+        66,
+        72
+      ],
+      "histogram": {
+        "66": 680512,
+        "72": 3402560
+      },
+      "min_R": 66,
+      "first_witness": [
+        0,
+        0,
+        1,
+        2,
+        4,
+        8,
+        15,
+        16
+      ]
+    },
+    "parity_repair": {
+      "attainable_R": [
+        82,
+        88
+      ],
+      "histogram": {
+        "82": 680512,
+        "88": 3402560
+      },
+      "min_R": 82,
+      "first_witness": [
+        0,
+        0,
+        1,
+        2,
+        4,
+        8,
+        15,
+        16
+      ]
+    }
+  },
+  "leave_alone": {
+    "through_pr": 2715,
+    "not_edited": [
+      1898,
+      2455,
+      2593,
+      2643,
+      2677,
+      2686,
+      2687,
+      2715
+    ],
+    "deferred_pending": [
+      "cursor/multi-connector-7q-60b9 dual-census DEFER",
+      "mutable-source-7q-shared-xor soft-overlap #2643",
+      "host screens"
+    ]
+  }
+}
diff --git a/research/dirty-parity-8q-obstruction/NOTICE.md b/research/dirty-parity-8q-obstruction/NOTICE.md
@@ -0,0 +1,25 @@
+# Notice
+
+New material in this directory is Apache-2.0.
+
+Predecessor credit (do not alter historical notices elsewhere):
+
+- Arham Tahir / PR1898 `research/mutable-source-minimum-queries` — five-source
+  alphabet, minimum-query endpoint, abstract complete-call tariff.
+- ywz65 / open PR2455 `research/mutable-source-six-query-xor` — component lemma
+  and restored interaction-graph census at six queries.
+- Arham Tahir / PR1370 — path-local shear lowering.
+- TennisBowling / PR1524 — equal-price theorem and unequal-price handoff.
+- Ross Richardson / PR943/934 — prefix framework.
+- Fork path-lowered `research/mutable-source-789q-invoice` / open #2677 —
+  distinct-slot Q=8 floor R≥74 and champion path; not edited.
+- Open #2643 `research/seven-query-xor-floor` — restored `X ≥ 6` at `Q = 7`;
+  not edited.
+- Open #2686 `research/seven-query-r-floor` — even-subset `R ≥ 82` at `Q = 7`;
+  not edited.
+- Open #2715 `research/cheaper-readout-7q-obstruction` — Q=7 all-subset /
+  dirty-parity collapse; OQ2 answered here; not edited.
+- Live pin: `research/mutable-source-shear-cost/CERTIFICATE.json`.
+
+Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI
+assistance. Disclosure: Prepared by Wentao Li with substantial AI assistance.
diff --git a/research/dirty-parity-8q-obstruction/PROOF.md b/research/dirty-parity-8q-obstruction/PROOF.md
@@ -0,0 +1,110 @@
+# Dirty-parity collapse extends to eight queries
+
+This is a scoped obstruction in the five-source common-B mutable-source alphabet
+of [PR1898](https://github.com/CrocSwap/integer-mult-bounds/pull/1898). It answers
+open question 2 of
+[#2715](https://github.com/CrocSwap/integer-mult-bounds/pull/2715)
+(`research/cheaper-readout-7q-obstruction`): whether the same dirty-parity
+collapse applies at eight queries for all-subset versus even-subset floors.
+
+No new kappa is claimed. Leave-alone packages through #2715 are not edited.
+Pending `multi-connector-7q` dual-census repair, deferred shared-XOR soft-overlap
+with #2643, and host screens are not opened.
+
+## Setup
+
+Retain queries `Q_j`, readouts `R_m: y_m += A_m z`, and paid source shears
+`X_(s,t)`. Connector prefixes of a restored length-8 walk are slots
+`(s_0,…,s_7)` with `s_0 = 0`. The **even-subset** score is
+
+```text
+R_even = Σ_{m=1}^{31} min{|T| : T ⊂ {0,…,7}, |T| even, ⊕_{i∈T} s_i = m}.
+```
+
+[#2677](https://github.com/CrocSwap/integer-mult-bounds/pull/2677) / the path-lowered
+789q invoice records that on distinct-slot covers the minimum is `R_even = 74`
+(witness `(0,1,2,3,4,8,16,29)`), so even at `X = 0` the abstract tariff
+`(b,a,x)=(38,2,2)` is at least `452 > 436`.
+
+## Candidate cheaper alphabet: all-subset / odd-cardinality
+
+Allow every nonempty realizing subset (odd cardinalities included):
+
+```text
+R_all = Σ_{m=1}^{31} min{|T| : T ⊂ {0,…,7}, |T|≥1, ⊕_{i∈T} s_i = m}.
+```
+
+**Census (distinct nonzero 7-sets).** Covering sets: `2430400`. Attainable
+totals `R_all ∈ {61, 64, 68}` with histogram `416640 / 1180480 / 833280`.
+Minimum `61`.
+
+**One-duplicate layer.** Length-8 multisets with exactly seven distinct labels
+including `0`: covering count `4083072`, attainable `R_all ∈ {66, 72}`
+(minimum `66`).
+
+The distinct-7 champion `(0,1,2,3,4,8,16,29)` attains `R_all = 61`. The
+hypothetical tariff `38·8 + 2·61 + 2·0 = 426` is **strictly below** the
+six-query endpoint `436` even with free XOR budget.
+
+## Dirty-connector obstruction
+
+Each live readout `R_m` on connector `z` adds one copy of `A_m · z_dirt` to
+output `y_m`. For arbitrary connector dirt to cancel on every output, the number
+of live `R_m` events for each mask `m` must be even. Therefore every
+odd-cardinality realizing set is **illegal** in the PR1898 dirty-restoration
+contract unless repaired.
+
+Finite check: a minimum all-subset schedule on the champion leaves a nonempty
+set of masks with odd live-connector parity (13 odd min-masks).
+
+## Priced repair: zero-slot parity completion
+
+Every restored walk here has a 0-valued slot (the start). For an odd realizing
+set `S`:
+
+- if `S` already contains a 0-valued index, drop one such index (`|S|-1`, same XOR);
+- otherwise adjoin one 0-valued slot readout (`|S|+1`, same XOR).
+
+Both moves restore even live-connector parity without changing the pure mask.
+Define `R_repair` as the minimum cost over even sets and such repaired odd sets.
+
+**Lemma.** On any slot list containing a 0-valued index, `R_repair = R_even`.
+Every repaired odd set is an even set with the same XOR, and every even set is
+already admitted.
+
+**Census.** On all `2430400` covering distinct-7 sets, `R_repair` matches
+`R_even` exactly (spectrum `{74,80,84}`; `repair_equals_even_count = 2430400`).
+On the one-duplicate layer, `R_repair ∈ {82,88}` with minimum `82`.
+
+## Tariff consequence
+
+After dirty-parity repair, every covering distinct-7 Q=8 schedule still has
+`R ≥ 74`, matching the #2677 even-subset distinct-slot floor. Even at `X = 0`,
+tariff `≥ 452 > 436`. One-duplicate covers repair to `R ≥ 82` (tariff `≥ 468`).
+The free all-subset undercut (`426 < 436`) is not realizable under the PR1898
+dirty-connector contract; the minimal priced repair returns to the even-subset
+floor.
+
+## Scope limits
+
+- This prices the natural odd-cardinality alphabet (all-subset) plus zero-slot
+  parity repair at `Q = 8`. Multi-connector alphabets (including independent
+  walks), dirty-shadow fixtures, output-mixing readouts, and host screens are
+  outside; the pending `multi-connector-7q` dual-census repair is not edited.
+- Shared non-path XOR schedules remain deferred (soft-overlap with #2643).
+- Leave-alone through #2715 (including #2593, #2643, #2677, #2686, #2687, #2715)
+  is respected.
+- Exact XOR floors at `Q = 8` and non-distinct multi-repeat layers beyond one
+  duplicate are not claimed.
+- No new kappa, physical supplier, or multiplication theorem.
+
+## Relation to predecessors
+
+| Prior | Already proves | This package adds |
+|---|---|---|
+| PR1898 | Min `Q=6`, endpoint tariff 436 | — |
+| #2643 | `X≥6` at `Q=7` | — |
+| #2677 / 789q | Path-lowered invoices; distinct-slot `R≥74` at `Q=8` | — |
+| #2686 | Even-subset `R≥82` at `Q=7` | — |
+| #2715 | All-subset→repair collapse at `Q=7` (OQ2: does it extend to Q=8?) | — |
+| **This** | — | **All-subset would hit R=61/tariff 426 at Q=8, but dirty-parity repair collapses to R≥74/tariff 452** |