Mister-Meeseeks · Oct 16, 2026 · Oct 15, 2026
diff --git a/research/independent-walks-onerepeat-35-44/CERTIFICATE.json b/research/independent-walks-onerepeat-35-44/CERTIFICATE.json
@@ -0,0 +1,161 @@
+{
+  "schema_version": 1,
+  "status": "PASS_INDEPENDENT_WALKS_ONEREPEAT_35_44_OBSTRUCTION",
+  "scope": "Five-source common-B mutable-source Q/R/X alphabet (PR1898) at (b,a,x)=(38,2,2). Freestanding one-repeat layers inside partitioned independent dual walks at total Q=8 on shapes (3,5) and (4,4): every covering dual has exact R=112; tariff floor at X=0 is 528>436. Answers independent-walks-onerepeat-8q OQ1 without editing leave-alones through #2851 or pending onerepeat-(2,6).",
+  "claim": {
+    "Q": 8,
+    "partitions": [
+      [
+        3,
+        5
+      ],
+      [
+        4,
+        4
+      ]
+    ],
+    "one_repeat_35_covering": 2499840,
+    "one_repeat_35_min_R": 112,
+    "one_repeat_44_covering": 1666560,
+    "one_repeat_44_min_R": 112,
+    "one_repeat_44_both_or_covering": 0,
+    "distinct_35_min_R": 98,
+    "distinct_44_min_R": 100,
+    "tariff_floor_at_X0": 528,
+    "endpoint_Q6_tariff": 436,
+    "undercuts_436_after_complete_bill": false,
+    "beats_distinct_98": false,
+    "beats_distinct_100": false,
+    "answers_independent_walks_onerepeat_8q_oq1": true,
+    "oq_answer": "One-repeat layers on partitions (3,5) and (4,4) at Q=8 do not improve on distinct-label floors R=98 / R=100. Every covering distinct-L3 \u00d7 one-repeat-L5 dual has R=112 (covering 2499840; spectrum {112}); length-3 cannot itself host a one-repeat (closed zero edge). Every covering one-repeat-L4 \u00d7 distinct-L4 dual has R=112 (covering 1666560; spectrum {112}); both-sides one-repeat never covers. At X=0 the abstract tariff is already 528>436 under (b,a,x)=(38,2,2). Leave-alones through #2851 and pending independent-walks-onerepeat-8q are not reopened.",
+    "no_new_kappa": true
+  },
+  "one_repeat_3_plus_5": {
+    "kind": "one_repeat_partition_3_plus_5",
+    "Q": 8,
+    "short_tables": 465,
+    "long_tables": 13485,
+    "covering_pairs": 2499840,
+    "attainable_R": [
+      112
+    ],
+    "histogram": {
+      "112": 2499840
+    },
+    "min_R": 112,
+    "sum_even_rank3_full_reach": 16,
+    "length3_one_repeat_valid": 0,
+    "first_witnesses": {
+      "112": [
+        [
+          0,
+          1,
+          2
+        ],
+        [
+          0,
+          4,
+          8,
+          4,
+          16
+        ]
+      ]
+    }
+  },
+  "one_repeat_4_plus_4": {
+    "kind": "one_repeat_partition_4_plus_4",
+    "Q": 8,
+    "one_repeat_tables": 930,
+    "distinct_tables": 4495,
+    "covering_pairs": 1666560,
+    "attainable_R": [
+      112
+    ],
+    "histogram": {
+      "112": 1666560
+    },
+    "min_R": 112,
+    "both_one_repeat_covering": 0,
+    "first_witnesses": {
+      "112": [
+        [
+          0,
+          1,
+          2,
+          1
+        ],
+        [
+          0,
+          4,
+          8,
+          16
+        ]
+      ]
+    }
+  },
+  "champion": {
+    "champion_35": {
+      "walks": [
+        [
+          0,
+          1,
+          2
+        ],
+        [
+          0,
+          8,
+          4,
+          16,
+          4
+        ]
+      ],
+      "R": 112,
+      "path_X": 8,
+      "tariff": 544,
+      "one_repeat_long": true,
+      "long_rank": 3,
+      "long_sum_even": 16
+    },
+    "champion_44": {
+      "walks": [
+        [
+          0,
+          1,
+          2,
+          1
+        ],
+        [
+          0,
+          4,
+          8,
+          16
+        ]
+      ],
+      "R": 112,
+      "path_X": 8,
+      "tariff": 544,
+      "one_repeat_short": true,
+      "short_rank": 2
+    },
+    "tariff_floor_at_X0": 528,
+    "endpoint_Q6": 436,
+    "distinct_35_min_R": 98,
+    "distinct_44_min_R": 100,
+    "one_repeat_35_min_R": 112,
+    "one_repeat_44_min_R": 112
+  },
+  "unit_tariff": {
+    "b": 38,
+    "a": 2,
+    "x": 2
+  },
+  "tariff_check": {
+    "formula": "38*Q + 2*R + 2*X",
+    "at_R112_X0": 528,
+    "champion_35": 544,
+    "champion_44": 544,
+    "endpoint_Q6": 436,
+    "distinct_35_floor_at_R98_X0": 500,
+    "distinct_44_floor_at_R100_X0": 504
+  }
+}
diff --git a/research/independent-walks-onerepeat-35-44/NOTICE.md b/research/independent-walks-onerepeat-35-44/NOTICE.md
@@ -0,0 +1,30 @@
+# 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 `(b,a,x)=(38,2,2)`.
+- 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 spectrum and champion path at Q=8; not edited.
+- Open #2779 `research/independent-walks-7q-obstruction` — partitioned dual at
+  Q=7; not edited.
+- Open #2846 `research/transfer-coupled-onerepeat-7q` — transfer-coupled
+  one-repeat at Q=7; leave-alone; not edited.
+- Open #2851 `research/independent-walks-8q-obstruction` — distinct partitioned
+  dual at Q=8; leave-alone; not edited.
+- Open #2840 / #2836 / #2817 / #2764 / #2761 / #2715 / #2686 / #2643 —
+  leave-alone through #2851; not edited.
+- Pending `cursor/independent-walks-onerepeat-8q-60b9@5ee2550c`
+  `research/independent-walks-onerepeat-8q` — one-repeat (2,6) at Q=8; OQ1
+  ((3,5)/(4,4) companions) answered here; not edited or competed.
+- Live pin: `research/mutable-source-shear-cost/CERTIFICATE.json`.
+
+Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI
+assistance.
diff --git a/research/independent-walks-onerepeat-35-44/PROOF.md b/research/independent-walks-onerepeat-35-44/PROOF.md
@@ -0,0 +1,102 @@
+# One-repeat (3,5)/(4,4) cannot improve on R=98 / R=100 at Q=8
+
+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) at unit
+tariff `(b,a,x)=(38,2,2)`. It is a freestanding `#840`-style priced alphabet at
+eight queries: the **one-repeat** companions of the partitioned independent duals
+on shapes `(3,5)` and `(4,4)`. It answers open question 1 of pending
+`cursor/independent-walks-onerepeat-8q-60b9`: whether allowing one repeated
+nonzero label on those partitions improves on distinct-label `R=98` / `R=100`.
+
+No new kappa is claimed. Leave-alone packages through #2851 are not edited.
+Pending independent-walks-onerepeat-8q is not edited or competed.
+
+## Setup
+
+Retain queries `Q_j`, readouts `R_m: y_m += A_m z`, and paid source shears
+`X_(s,t)`. A **restored walk** of length `n≥2` is an open slot list starting at
+`0` whose closed completion has only nonzero edges.
+
+A **one-repeat (3,5) dual** is a pair
+
+- short walk `(0,a,b)` with distinct nonzero labels (distinct-label length-3),
+- long walk whose four nonzero slots have multiset shape `{d:2,o₁,o₂}`,
+
+with independent dirt and dual even-parity cost
+
+```text
+R = Σ_{m=1}^{31} min_α ( e_short[α] + e_long[α ⊕ m] ).
+```
+
+A **one-repeat (4,4) dual** is a pair with one side a restored one-repeat
+length-4 walk (only form `(0,d,o,d)` restores) and the other a distinct-label
+length-4 walk. Both-sides one-repeat is checked separately.
+
+The abstract complete-call tariff at `(b,a,x)=(38,2,2)` is `38·Q + 2·R + 2·X`.
+
+## Length-3 exclusion
+
+A length-3 one-repeat candidate is `(0,d,d)`. Closing gives edge `d→d` of XOR
+`0`, so it is not a restored walk. The short component of `(3,5)` therefore
+cannot itself host a one-repeat layer (machine-checked: `0` valid walks).
+
+## Structural facts (machine-checked)
+
+**Lemma (Σe = 16 on covering one-repeat L5 champions).** Every covering
+one-repeat length-5 long in the `(3,5)` census that appears as a first witness
+has long span of rank `3` with
+
+```text
+Σ_{v ∈ V \ {0}} e_long[v] = 16.
+```
+
+**Lemma ((3,5) covering spectrum).** Over all `465` distinct length-3 even
+tables and `13485` one-repeat length-5 even tables, every covering dual has
+`R=112`. Covering count `2499840`; histogram `{112: 2499840}`.
+
+**Lemma ((4,4) covering spectrum).** Over all `930` one-repeat length-4 even
+tables and `4495` distinct length-4 even tables, every covering dual has
+`R=112`. Covering count `1666560`; histogram `{112: 1666560}`.
+
+**Lemma (both-sides one-repeat never covers).** The `930×930` both-one-repeat
+`(4,4)` census has covering count `0`.
+
+## Census
+
+| Dual schedule | Covering pairs | Exact min R | Beats distinct? |
+|---|---:|---:|---|
+| Distinct `(3,5)` (iw8q) | 10416000 | 98 | — |
+| One-repeat `(3,5)` (this) | **2499840** | **112** | **no** |
+| Distinct `(4,4)` (iw8q) | 14165760 | 100 | — |
+| One-repeat×distinct `(4,4)` (this) | **1666560** | **112** | **no** |
+| One-repeat×one-repeat `(4,4)` | **0** | — | — |
+
+Champions:
+
+- `(3,5)`: `((0,1,2),(0,8,4,16,4))` with path `X=8`, tariff `544`.
+- `(4,4)`: `((0,1,2,1),(0,4,8,16))` with path `X=8`, tariff `544`.
+
+The `X=0` figure `528` is only the abstract lower bound from `R=112`.
+
+## Tariff consequence
+
+Every covering one-repeat `(3,5)` or `(4,4)` dual has `R=112`. Even at `X=0`,
+
+```text
+38·8 + 2·112 + 2·0 = 528 > 436.
+```
+
+Positive path-`X` only increases the bill. Distinct-label floors `R=98` /
+`R=100` (pending iw8q / published #2851) are strictly better than one-repeat
+`R=112`, so one-repeat does not reopen an undercut of the six-query endpoint
+under the PR1898 abstract tariff.
+
+## Scope
+
+This is a priced-alphabet obstruction, not a multiplication theorem and not a
+new kappa. Leave-alone packages through #2851 and pending
+`cursor/independent-walks-onerepeat-8q-60b9@5ee2550c` are not edited. Deferred
+soft-overlaps (transfer-coupled-dual/shapes; shared-xor; host screens) are not
+opened.
+
+Prepared by Wentao Li with substantial AI assistance.
diff --git a/research/independent-walks-onerepeat-35-44/README.md b/research/independent-walks-onerepeat-35-44/README.md
@@ -0,0 +1,84 @@
+# One-repeat partitioned independent-walk obstruction on (3,5)/(4,4) at eight queries
+
+**Scoped obstruction / priced alphabet; no new kappa.** Freestanding
+`#840`-style priced obstruction under
+[#1898](https://github.com/CrocSwap/integer-mult-bounds/pull/1898) at
+`(b,a,x)=(38,2,2)`. Names the one-repeat companions of the partitioned
+`(3,5)` and `(4,4)` Hamilton layers at `Q=8` outside leave-alones through
+#2851 and pending `cursor/independent-walks-onerepeat-8q-60b9`. Answers open
+question 1 of that pending package: do non-distinct (one-repeat) label layers
+on `(3,5)` / `(4,4)` improve on distinct-label `R=98` / `R=100`?
+
+| Dual schedule | Covering pairs | Exact min R | Tariff floor (X≥0) | Undercuts 436? | Beats distinct? |
+|---|---:|---:|---:|---|---|
+| Distinct-label `(3,5)` (iw8q) | 10416000 | **98** | ≥500 | no | — |
+| One-repeat `(3,5)` (this) | **2499840** | **112** | **≥528** | **no** | **no** |
+| Distinct-label `(4,4)` (iw8q) | 14165760 | **100** | ≥504 | no | — |
+| One-repeat×distinct `(4,4)` (this) | **1666560** | **112** | **≥528** | **no** | **no** |
+| One-repeat×one-repeat `(4,4)` | **0** | — | — | — | — |
+
+Every covering one-repeat dual on these shapes has exact `R=112`. Length-3
+cannot host a one-repeat (closed zero edge). Abstract tariff
+`38·8 + 2·112 + 2·0 = 528 > 436`. Distinct-label floors `98` / `100` are not
+improved. Transfer-coupled / onerepeat-(2,6) / dirty-shadow / iw8q fixtures
+are not edited.
+
+## Reproduction
+
+```sh
+python3.12 -I -B research/independent-walks-onerepeat-35-44/verify.py
+python3.12 -I -O -B research/independent-walks-onerepeat-35-44/verify.py
+python3.12 -I -B research/independent-walks-onerepeat-35-44/independent.py
+python3.12 -I -B research/independent-walks-onerepeat-35-44/test_package.py
+python3.12 -B scripts/research_package.py --package research/independent-walks-onerepeat-35-44
+```
+
+Expected:
+
+```text
+PASS_INDEPENDENT_WALKS_ONEREPEAT_35_44_OBSTRUCTION one_repeat_35_min_R=112 ... tariff_floor=528 ... answers_independent_walks_onerepeat_8q_oq1 no_new_kappa
+PASS_INDEPENDENT_WALKS_ONEREPEAT_35_44 covering_35=2499840 covering_44=1666560 ... tariff_floor=528 ... no_new_kappa
+PASS_INDEPENDENT_WALKS_ONEREPEAT_35_44_PACKAGE_TESTS groups=6
+PASS_REGISTERED_RESEARCH_PACKAGE research/independent-walks-onerepeat-35-44 outcome=completed
+```
+
+Default verify is read-only. Admission uses `need()`; `-O` also passes.
+VERIFY.json timeout sum `180+180+180+600=1140≤1200`.
+
+## Overlap
+
+Base commit `0b1cccf86e2f5132ed286a26ef0a681a495740ea`. Direct predecessors:
+PR1898, #2851 / pending iw8q (distinct partitioned dual at Q=8), pending
+iwor8q (one-repeat `(2,6)`; OQ1 answered here), #2779, #2846. Leave-alone
+through #2851 including #2846, #2840, #2836, #2817, #2779, #2764, #2761,
+#2715, #2686, #2677, #2643 is respected; those packages are not edited.
+Pending `cursor/independent-walks-onerepeat-8q-60b9@5ee2550c` is not edited
+or competed. Deferred soft-overlaps (transfer-coupled-dual/shapes;
+shared-xor; host screens) are not opened.
+
+## Open next questions
+
+1. Does any third priced readout alphabet under PR1898 `(b,a,x)=(38,2,2)`
+   (beyond same-walk / dirty-shadow / partitioned dual / one-repeat) undercut
+   `436` at `Q=8`?
+   Why it matters: remaining unequal-price handoff after one-repeat companions.
+   First step: name one new alphabet outside leave-alones and census dirt-legal R.
+   Blocked by: leave-alone packages through #2851; pending onerepeat-(2,6).
+2. Does allowing two repeated nonzero labels (or a full non-distinct layer) on
+   any `Q=8` partition undercut the distinct floors?
+   Why it matters: closes the next relaxation after one-repeat.
+   First step: census two-repeat length-5 multisets against distinct length-3.
+   Blocked by: larger search; one-repeat already worsens.
+
+## Credit
+
+Credit **Arham Tahir / PR1898**, **ywz65 / PR2455**, **PR1370**, fork **789q** /
+**#2677**, **#2643**, **#2686**, **#2715**, **#2761**, **#2779**, **#2846**,
+**#2851** / pending **independent-walks-8q** (not edited), pending
+**independent-walks-onerepeat-8q** (OQ1 answered here; not edited),
+**TennisBowling / PR1524**, **Ross Richardson / PR943**, and the retained
+mutable-source lineage. Live input pin:
+`research/mutable-source-shear-cost/CERTIFICATE.json`. Prepared by Wentao Li
+([VictorLiwentao](https://github.com/VictorLiwentao),
+[@Is__Victor](https://x.com/Is__Victor)) with substantial AI assistance.
+Apache-2.0. No independent human review or multiplication theorem is claimed.