Research: three-repeat partitioned 8q priced obstruction (no new kappa)

research/independent-walks-threerepeat-8q/CERTIFICATE.json

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+{
+  "schema_version": 1,
+  "status": "PASS_INDEPENDENT_WALKS_THREEREPEAT_8Q_OBSTRUCTION",
+  "scope": "Five-source common-B mutable-source Q/R/X alphabet (PR1898) at (b,a,x)=(38,2,2). Freestanding three-repeat (excess-3) layers inside partitioned independent dual walks at total Q=8 on shapes (2,6), (3,5), and (4,4): every dual has covering count 0; no dirt-legal covering schedule exists, so none undercuts tariff 436. Answers preferred next19 / shared-dirt OQ1 (fifth priced alphabet) without editing leave-alones through #2861/#2860/#2858/#2851 or pending shared-dirt tip.",
+  "claim": {
+    "Q": 8,
+    "partitions": [
+      [
+        2,
+        6
+      ],
+      [
+        3,
+        5
+      ],
+      [
+        4,
+        4
+      ]
+    ],
+    "alphabet": "three_repeat_partitioned_independent_dual",
+    "three_repeat_26_covering": 0,
+    "three_repeat_26_valid_walks": 930,
+    "three_repeat_26_shape_32": 930,
+    "three_repeat_26_shape_41": 0,
+    "three_repeat_35_covering": 0,
+    "three_repeat_35_valid_multisets": 0,
+    "three_repeat_44_covering": 0,
+    "three_repeat_44_valid_walks": 0,
+    "covering_min_R": null,
+    "tariff_floor_at_X0": null,
+    "endpoint_Q6_tariff": 436,
+    "undercuts_436_after_complete_bill": false,
+    "beats_distinct_floors": false,
+    "answers_fifth_priced_alphabet_q8": true,
+    "answers_shared_dirt_oq1_fifth_alphabet": true,
+    "soft_overlap_shared_dirt": "safe_independent_excess_axis_not_dirt_coupling",
+    "oq_answer": "The named fifth priced readout alphabet under PR1898 (b,a,x)=(38,2,2) beyond same-walk / dirty-shadow / partitioned dual / one-repeat / two-repeat / shared-dirt is the three-repeat (excess-3) companion of partitioned independent duals at Q=8. Exact covering census: (2,6) covering 0 on 930 valid longs (only shape {3,2}; {4,1} never restores; span \u22644 so coset union \u22648); (3,5) covering 0 because shape {d:4} never restores; (4,4) covering 0 because excess 3 is impossible on length-4 (max excess 2). No covering R exists, so the alphabet cannot undercut tariff 436 after a complete dirty bill. Leave-alones through #2861/#2860/#2858/#2851 and pending shared-dirt tip are not reopened; soft-overlap with shared-dirt is avoided (excess axis, independent dirt).",
+    "no_new_kappa": true
+  },
+  "three_repeat_2_plus_6": {
+    "kind": "three_repeat_partition_2_plus_6",
+    "Q": 8,
+    "valid_walks": 930,
+    "shape_32": 930,
+    "shape_41": 0,
+    "covering_pairs": 0,
+    "attainable_R": [],
+    "histogram": {},
+    "min_R": null,
+    "structural_max_span": 4,
+    "structural_max_coset_union": 8,
+    "first_witnesses": {}
+  },
+  "three_repeat_3_plus_5": {
+    "kind": "three_repeat_partition_3_plus_5",
+    "Q": 8,
+    "short_tables": 465,
+    "long_multisets": 0,
+    "covering_pairs": 0,
+    "attainable_R": [],
+    "histogram": {},
+    "min_R": null,
+    "note": "shape_{d:4}_never_restores",
+    "structural_max_label_count_long": 1,
+    "first_witnesses": {}
+  },
+  "three_repeat_4_plus_4": {
+    "kind": "three_repeat_partition_4_plus_4",
+    "Q": 8,
+    "valid_three_repeat_l4_walks": 0,
+    "covering_pairs": 0,
+    "attainable_R": [],
+    "histogram": {},
+    "min_R": null,
+    "note": "excess_3_impossible_on_length4_max_excess_2",
+    "first_witnesses": {}
+  },
+  "champion": {
+    "covering_any_partition": false,
+    "tariff_floor_at_X0": null,
+    "endpoint_Q6": 436,
+    "undercuts_436_after_complete_bill": false,
+    "distinct_26_min_R": 92,
+    "distinct_35_min_R": 98,
+    "distinct_44_min_R": 100,
+    "one_repeat_26_min_R": 112,
+    "one_repeat_35_min_R": 112,
+    "one_repeat_44_min_R": 112,
+    "two_repeat_26_covering": 0,
+    "shared_dirt_partition_min_R": 82,
+    "three_repeat_covering_min_R": null,
+    "comparison_tariffs": {
+      "distinct_26_at_R92_X0": 488,
+      "one_repeat_at_R112_X0": 528,
+      "shared_dirt_at_R82_X0": 468,
+      "endpoint_Q6": 436
+    }
+  },
+  "unit_tariff": {
+    "b": 38,
+    "a": 2,
+    "x": 2
+  },
+  "tariff_check": {
+    "formula": "38*Q + 2*R + 2*X",
+    "three_repeat_covering_floor": null,
+    "distinct_26_at_R92_X0": 488,
+    "one_repeat_at_R112_X0": 528,
+    "shared_dirt_at_R82_X0": 468,
+    "endpoint_Q6": 436
+  }
+}

research/independent-walks-threerepeat-8q/MASTER-HANDOFF.md

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+# Master handoff — independent-walks-threerepeat-8q
+
+## Deliverable
+
+| Field | Value |
+|---|---|
+| Branch | `cursor/independent-walks-threerepeat-8q-60b9` |
+| Tip SHA | branch HEAD (`git rev-parse cursor/independent-walks-threerepeat-8q-60b9`) |
+| Package | `research/independent-walks-threerepeat-8q/` |
+| Claim | Freestanding `#840`-style priced alphabet: **three-repeat** (excess-3) partitioned independent dual at Q=8 under PR1898 `(b,a,x)=(38,2,2)` |
+| Result | Exact covering census **0** on `(2,6)` / `(3,5)` / `(4,4)`; cannot undercut tariff **436** |
+| Kappa | none |
+
+## Answers
+
+- Preferred next19 / shared-dirt OQ1: named fifth alphabet beyond same-walk /
+  dirty-shadow / partitioned dual / one-repeat / two-repeat / shared-dirt.
+- Soft-overlap with pending shared-dirt: safe (independent excess axis; no
+  dirt coupling; package path and census disjoint).
+
+## Exact census
+
+| Shape | Valid | Covering | Structural bound |
+|---|---:|---:|---|
+| `(2,6)` | 930 (`{3,2}` only; `{4,1}` never restores) | 0 | span ≤4 ⇒ coset ∪ ≤8 |
+| `(3,5)` | 0 (`{d:4}` never restores) | 0 | — |
+| `(4,4)` | 0 (excess 3 impossible on L4) | 0 | — |
+
+## Leave-alone respected
+
+Through #2867 / #2861 / #2860 / #2858 / #2851; #2846/#2840/#2836/#2817/#2779/
+#2764/#2761; deferred soft-overlaps untouched.
+
+## Verify gate
+
+```sh
+python3.12 -I -B research/independent-walks-threerepeat-8q/verify.py
+python3.12 -I -O -B research/independent-walks-threerepeat-8q/verify.py
+python3.12 -I -B research/independent-walks-threerepeat-8q/independent.py
+python3.12 -I -B research/independent-walks-threerepeat-8q/test_package.py
+python3.12 -B scripts/research_package.py --package research/independent-walks-threerepeat-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. Sixth priced alphabet under PR1898 at Q=8 beyond three-repeat?
+2. Q=9 partitioned independent dual companion (vs #2677 same-walk floor 466)?

research/independent-walks-threerepeat-8q/NOTICE.md

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+# 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 #2851 `research/independent-walks-8q-obstruction` — distinct partitioned
+  dual at Q=8; leave-alone; not edited.
+- Open #2861 `research/independent-walks-onerepeat-8q` — one-repeat (2,6) at
+  Q=8; leave-alone; not edited.
+- Open #2860 `research/independent-walks-tworepeat-8q` — two-repeat at Q=8;
+  leave-alone; not edited.
+- Open #2858 `research/independent-walks-onerepeat-35-44` — one-repeat
+  (3,5)/(4,4) at Q=8; leave-alone; not edited.
+- Pending `cursor/shared-dirt-walks-8q-60b9@fc3a2cb3`
+  `research/shared-dirt-walks-8q` — shared-dirt at Q=8; OQ1 answered here;
+  leave-alone; not edited or competed.
+- Open #2846 / #2840 / #2836 / #2817 / #2764 / #2761 / #2715 / #2686 / #2643 —
+  leave-alone through #2851; not edited.
+- Deferred soft-overlaps (transfer-coupled-dual/shapes; shared-XOR; host
+  screens) — not opened.
+- Live pin: `research/mutable-source-shear-cost/CERTIFICATE.json`.
+
+Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI
+assistance.

research/independent-walks-threerepeat-8q/PROOF.md

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+# Three-repeat partitioned walks cannot cover 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 names one freestanding `#840`-style priced
+readout alphabet at eight queries beyond same-walk / dirty-shadow / partitioned
+dual / one-repeat / two-repeat / shared-dirt: the **three-repeat** (excess-3)
+companion of partitioned independent duals. It answers preferred next19 and
+open question 1 of pending `cursor/shared-dirt-walks-8q-60b9@fc3a2cb3`.
+
+No new kappa is claimed. Leave-alone packages through #2861 / #2860 / #2858 /
+#2851 are not edited. Pending shared-dirt tip is not edited or competed.
+Soft-overlap with shared-dirt is avoided: this package stays on the
+independent-dirt excess axis and does not couple the dirt coefficient.
+
+## 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.
+
+The **excess** of a walk is `Σ (count(ℓ)−1)` over nonzero labels. One-repeat
+has excess `1`. Two-repeat has excess `2`. **Three-repeat** means excess
+exactly `3`.
+
+A **three-repeat partitioned dual** of total query count `Q = 8` is a pair of
+valid restored walks of lengths `(n_0,n_1)` with `n_0+n_1=8` and `n_0,n_1≥2`,
+where at least one side is three-repeat, with independent dirt and dual
+even-parity cost
+
+```text
+R = Σ_{m=1}^{31} min_α ( e0[α] + e1[α ⊕ m] ).
+```
+
+Up to swap the only partitions are `(2,6)`, `(3,5)`, and `(4,4)`.
+
+The abstract complete-call tariff at `(b,a,x)=(38,2,2)` is `38·Q + 2·R + 2·X`.
+
+## Fixture A: partition `(2,6)`
+
+Short walk `(0,σ)`. Long walk of length 6 with excess 3 on five nonzero slots
+forces exactly two distinct nonzero labels. Restoring shapes:
+
+| Shape | Restores? | Count |
+|---|---|---:|
+| `{d:3,e:2}` | yes | **930** |
+| `{d:4,o}` | no | 0 |
+| Total valid | — | **930** |
+
+**Lemma (span ≤ 4).** Two labels span a subspace of rank ≤ 2, hence size ≤ 4.
+
+**Lemma (coset union ≤ 8).** Pairing with short `(0,σ)` reaches at most
+`span ∪ (span ⊕ σ)`, size ≤ 8 < 32. Dual covering is therefore impossible.
+
+**Census.** Exact covering count **0** (machine-checked on all `930·31`
+pairs). Shape `{4,1}` is exhaustively checked never to restore.
+
+## Fixture B: partition `(3,5)`
+
+Short walk: distinct-label length-3 (`C(31,2)=465` even tables). Long walk:
+excess 3 on four nonzero slots forces shape `{d:4}`.
+
+Any ordering of `(0,d,d,d,d)` has an adjacent zero-XOR edge on the closed walk.
+
+**Census.** Valid three-repeat length-5 walks: **0**. Covering count **0**.
+
+## Fixture C: partition `(4,4)`
+
+Three nonzero slots admit at most excess `2` (shape `{d:3}`). Excess `3` is
+impossible on length-4.
+
+**Census.** Valid three-repeat length-4 walks: **0**. Covering count **0**.
+
+## Tariff consequence
+
+No covering three-repeat partitioned dual exists at `Q=8`. There is therefore
+no dirt-legal `R` and no complete-call tariff in this alphabet that can undercut
+the six-query endpoint `436`. By comparison:
+
+| Alphabet | Covering min R | Tariff floor (X=0) | Undercuts 436? |
+|---|---:|---:|---|
+| Distinct partitioned (`#2851`) | 92 | 488 | no |
+| One-repeat partitioned (pending) | 112 | 528 | no |
+| Two-repeat partitioned (pending) | — (covering 0) | — | no |
+| Shared-dirt partitioned (pending) | 82 | 468 | no |
+| **Three-repeat partitioned (this)** | **— (covering 0)** | **—** | **no** |
+
+## Scope limits
+
+- Fixes the **three-repeat** (excess-3) partitioned independent-walk interface at
+  total `Q=8`. Same-walk, dirty-shadow, distinct partitioned, one-repeat,
+  two-repeat, and shared-dirt leave-alones / pending tips are outside and not
+  edited.
+- Soft-overlap with pending shared-dirt is avoided: this package does not couple
+  dirt coefficients (R-side shared-dirt) and stays on the independent excess
+  axis.
+- Transfer-coupled dual/shapes, shared-XOR, and host screens remain deferred
+  soft-overlaps and are not opened.
+- No new kappa and no multiplication theorem are claimed.
+
+## Relation to predecessors
+
+| Prior | Already proves | This package adds |
+|---|---|---|
+| PR1898 | Min `Q=6`, endpoint tariff 436, `(b,a,x)=(38,2,2)` | — |
+| #2851 | Partitioned independent dual at `Q=8` has `R≥92` | — |
+| pending onerepeat / tworepeat | Excess-1 / excess-2 companions | — |
+| pending shared-dirt | Shared-dirt `R≥82`; OQ1 asks for fifth alphabet | — |
+| **This** | — | **Three-repeat dual at `Q=8` has covering 0 on all shapes; cannot undercut 436; answers next19 / shared-dirt OQ1 without competing with shared-dirt** |