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

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

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+{
+  "schema_version": 1,
+  "status": "PASS_INDEPENDENT_WALKS_TWOREPEAT_8Q_OBSTRUCTION",
+  "scope": "Five-source common-B mutable-source Q/R/X alphabet (PR1898) at (b,a,x)=(38,2,2). Freestanding two-repeat (excess-2) 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 next17 / onerepeat-35-44 OQ1\u2013OQ2 without editing leave-alones through #2851 or pending onerepeat tips.",
+  "claim": {
+    "Q": 8,
+    "partitions": [
+      [
+        2,
+        6
+      ],
+      [
+        3,
+        5
+      ],
+      [
+        4,
+        4
+      ]
+    ],
+    "alphabet": "two_repeat_partitioned_independent_dual",
+    "two_repeat_26_covering": 0,
+    "two_repeat_26_valid_walks": 26970,
+    "two_repeat_26_shape_221": 13485,
+    "two_repeat_26_shape_311": 13485,
+    "two_repeat_35_covering": 0,
+    "two_repeat_35_valid_multisets": 465,
+    "two_repeat_44_covering": 0,
+    "two_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_third_priced_alphabet_q8": true,
+    "answers_onerepeat_35_44_oq2_tworepeat": true,
+    "oq_answer": "The named third priced readout alphabet under PR1898 (b,a,x)=(38,2,2) beyond same-walk / dirty-shadow / partitioned dual / one-repeat is the two-repeat (excess-2) companion of partitioned independent duals at Q=8. Exact covering census: (2,6) covering 0 on 26970 valid longs (shapes {2,2,1}:13485 and {3,1,1}:13485; span \u22648 so coset union \u226416); (3,5) covering 0 on 465\u00d7465 distinct-L3 \u00d7 {2,2}-L5 pairs (sumset \u226416); (4,4) covering 0 because shape {d:3} never restores. No covering R exists, so the alphabet cannot undercut tariff 436 after a complete dirty bill. Leave-alones through #2851 and pending onerepeat tips are not reopened.",
+    "no_new_kappa": true
+  },
+  "two_repeat_2_plus_6": {
+    "kind": "two_repeat_partition_2_plus_6",
+    "Q": 8,
+    "valid_walks": 26970,
+    "shape_221": 13485,
+    "shape_311": 13485,
+    "covering_pairs": 0,
+    "attainable_R": [],
+    "histogram": {},
+    "min_R": null,
+    "structural_max_span": 8,
+    "structural_max_coset_union": 16,
+    "first_witnesses": {}
+  },
+  "two_repeat_3_plus_5": {
+    "kind": "two_repeat_partition_3_plus_5",
+    "Q": 8,
+    "short_tables": 465,
+    "long_multisets": 465,
+    "covering_pairs": 0,
+    "attainable_R": [],
+    "histogram": {},
+    "min_R": null,
+    "structural_max_label_count_long": 2,
+    "structural_max_sumset": 16,
+    "first_witnesses": {}
+  },
+  "two_repeat_4_plus_4": {
+    "kind": "two_repeat_partition_4_plus_4",
+    "Q": 8,
+    "valid_two_repeat_l4_walks": 0,
+    "covering_pairs": 0,
+    "attainable_R": [],
+    "histogram": {},
+    "min_R": null,
+    "note": "shape_{d:3}_never_restores",
+    "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_covering_min_R": null,
+    "comparison_tariffs": {
+      "distinct_26_at_R92_X0": 488,
+      "one_repeat_at_R112_X0": 528,
+      "endpoint_Q6": 436
+    }
+  },
+  "unit_tariff": {
+    "b": 38,
+    "a": 2,
+    "x": 2
+  },
+  "tariff_check": {
+    "formula": "38*Q + 2*R + 2*X",
+    "two_repeat_covering_floor": null,
+    "distinct_26_at_R92_X0": 488,
+    "one_repeat_at_R112_X0": 528,
+    "endpoint_Q6": 436
+  }
+}

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

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+# Master handoff — independent-walks-tworepeat-8q
+
+## Deliverable
+
+| Field | Value |
+|---|---|
+| Branch | `cursor/independent-walks-tworepeat-8q-60b9` |
+| Tip SHA | branch HEAD (`git rev-parse cursor/independent-walks-tworepeat-8q-60b9`) |
+| Package | `research/independent-walks-tworepeat-8q/` |
+| Claim | Freestanding `#840`-style priced alphabet: **two-repeat** (excess-2) 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 next17 / onerepeat-35-44 OQ1: named third alphabet beyond same-walk /
+  dirty-shadow / partitioned dual / one-repeat.
+- onerepeat-35-44 OQ2: two-repeat does not undercut distinct floors (in fact
+  never covers).
+
+## Exact census
+
+| Shape | Valid | Covering | Structural bound |
+|---|---:|---:|---|
+| `(2,6)` | 26970 (`{2,2,1}` 13485 + `{3,1,1}` 13485) | 0 | span ≤8 ⇒ coset ∪ ≤16 |
+| `(3,5)` | 465 `{2,2}` multisets | 0 | sumset ≤16 |
+| `(4,4)` | 0 (`{d:3}` never restores) | 0 | — |
+
+## Leave-alone respected
+
+Through #2851; pending `cursor/independent-walks-onerepeat-8q-60b9@b07bc1e1` and
+`cursor/independent-walks-onerepeat-35-44-60b9@8bd3d298`; #2846/#2840/#2836/#2817/
+#2779/#2764/#2761; deferred soft-overlaps untouched.
+
+## Verify gate
+
+```sh
+python3.12 -I -B research/independent-walks-tworepeat-8q/verify.py
+python3.12 -I -O -B research/independent-walks-tworepeat-8q/verify.py
+python3.12 -I -B research/independent-walks-tworepeat-8q/independent.py
+python3.12 -I -B research/independent-walks-tworepeat-8q/test_package.py
+python3.12 -B scripts/research_package.py --package research/independent-walks-tworepeat-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. Fourth priced alphabet under PR1898 at Q=8 beyond two-repeat?
+2. Shared-dirt dual on distinct partitioned walks (soft-overlap check vs shared-XOR)?

research/independent-walks-tworepeat-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 #2846 `research/transfer-coupled-onerepeat-7q` — transfer-coupled
+  one-repeat at Q=7; 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@b07bc1e1`
+  `research/independent-walks-onerepeat-8q` — one-repeat (2,6) at Q=8;
+  leave-alone; not edited or competed.
+- Pending `cursor/independent-walks-onerepeat-35-44-60b9@8bd3d298`
+  `research/independent-walks-onerepeat-35-44` — one-repeat (3,5)/(4,4) at Q=8;
+  OQ1–OQ2 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.

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

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+# Two-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: the **two-repeat** (excess-2) companion of partitioned
+independent duals. It answers preferred next17 and open questions 1–2 of pending
+`cursor/independent-walks-onerepeat-35-44-60b9`.
+
+No new kappa is claimed. Leave-alone packages through #2851 are not edited.
+Pending onerepeat tips are 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.
+
+The **excess** of a walk is `Σ (count(ℓ)−1)` over nonzero labels. One-repeat
+has excess `1`. **Two-repeat** means excess exactly `2`.
+
+A **two-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 two-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 2 on five nonzero slots
+forces exactly three distinct nonzero labels. Restoring shapes:
+
+| Shape | Count |
+|---|---:|
+| `{d:2,e:2,o}` | **13485** |
+| `{d:3,o₁,o₂}` | **13485** |
+| Total valid | **26970** |
+
+**Lemma (span ≤ 8).** Three labels span a subspace of rank ≤ 3, hence size ≤ 8.
+
+**Lemma (coset union ≤ 16).** Pairing with short `(0,σ)` reaches at most
+`span ∪ (span ⊕ σ)`, size ≤ 16 < 32. Dual covering is therefore impossible.
+
+**Census.** Exact covering count **0** (machine-checked on all `26970·31`
+pairs).
+
+## Fixture B: partition `(3,5)`
+
+Short walk: distinct-label length-3 (`C(31,2)=465` even tables). Long walk:
+excess 2 on four nonzero slots.
+
+| Shape | Restores? | Count |
+|---|---|---:|
+| `{d:2,e:2}` | yes | **465** |
+| `{d:3,o}` | no | 0 |
+| `{d:4}` | no | 0 |
+
+**Lemma (sumset ≤ 16).** Long `{d,e}` has span size ≤ 4; short length-3 has
+span size ≤ 4; the XOR sumset has size ≤ 16 < 32. Dual covering is impossible.
+
+**Census.** Exact covering count **0** on all `465·465` pairs.
+
+## Fixture C: partition `(4,4)`
+
+Two-repeat on length-4 means shape `{d:3}` (three identical nonzero labels).
+Any ordering has an adjacent zero-XOR edge on the closed walk.
+
+**Census.** Valid two-repeat length-4 walks: **0**. Covering count **0**.
+
+## Tariff consequence
+
+No covering two-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 (this)** | **— (covering 0)** | **—** | **no** |
+
+## Scope limits
+
+- Fixes the **two-repeat** (excess-2) partitioned independent-walk interface at
+  total `Q=8`. Same-walk, dirty-shadow, distinct partitioned, and one-repeat
+  leave-alones / pending tips are outside and not edited.
+- 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.

research/independent-walks-tworepeat-8q/README.md

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+# Two-repeat partitioned-walk obstruction 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 one new dirt-legal readout alphabet at `Q=8` beyond
+same-walk / dirty-shadow / partitioned dual / one-repeat: **two-repeat**
+(excess-2) layers inside partitioned independent duals on shapes `(2,6)`,
+`(3,5)`, and `(4,4)`. Answers preferred next17 and open questions 1–2 of pending
+`cursor/independent-walks-onerepeat-35-44-60b9`.
+
+| Dual schedule | Valid objects | Exact covering | Undercuts 436? |
+|---|---:|---:|---|
+| Distinct partitioned (#2851) | — | min R=92 | no |
+| One-repeat (pending tips) | — | min R=112 | no |
+| Two-repeat `(2,6)` | **26970** walks | **0** | **no** |
+| Two-repeat `(3,5)` | **465** multisets | **0** | **no** |
+| Two-repeat `(4,4)` | **0** walks | **0** | **no** |
+
+Span/sumset lemmas: `(2,6)` long span ≤8 so coset union ≤16; `(3,5)` sumset ≤16;
+`(4,4)` shape `{d:3}` never restores. No covering `R` exists, so the alphabet
+cannot undercut tariff `436`.
+
+## Reproduction
+
+```sh
+python3.12 -I -B research/independent-walks-tworepeat-8q/verify.py
+python3.12 -I -O -B research/independent-walks-tworepeat-8q/verify.py
+python3.12 -I -B research/independent-walks-tworepeat-8q/independent.py
+python3.12 -I -B research/independent-walks-tworepeat-8q/test_package.py
+python3.12 -B scripts/research_package.py --package research/independent-walks-tworepeat-8q
+```
+
+Expected:
+
+```text
+PASS_INDEPENDENT_WALKS_TWOREPEAT_8Q_OBSTRUCTION covering_26=0 covering_35=0 covering_44=0 ... answers_third_priced_alphabet_q8 no_undercut_436 no_new_kappa
+PASS_INDEPENDENT_WALKS_TWOREPEAT_8Q covering_26=0 covering_35=0 ... no_undercut_436 no_new_kappa
+PASS_INDEPENDENT_WALKS_TWOREPEAT_8Q_PACKAGE_TESTS groups=6
+PASS_REGISTERED_RESEARCH_PACKAGE research/independent-walks-tworepeat-8q outcome=completed
+```
+
+Default verify is read-only. Admission uses `need()`; `-O` also passes.
+
+## Overlap
+
+Base commit `0b1cccf86e2f5132ed286a26ef0a681a495740ea`. Direct predecessors:
+PR1898, #2851 (distinct partitioned dual at Q=8), pending onerepeat tips,
+#2846/#2840/#2779/#2677. Leave-alone through #2851 including #2846, #2840,
+#2836, #2817, #2779, #2764, #2761 is respected; those packages are not edited.
+Pending `cursor/independent-walks-onerepeat-8q-60b9@b07bc1e1` and
+`cursor/independent-walks-onerepeat-35-44-60b9@8bd3d298` are not edited or
+competed. Deferred soft-overlaps (transfer-coupled-dual/shapes; shared-xor;
+host screens) are not opened.
+
+## Open next questions
+
+1. Does any fourth priced readout alphabet under PR1898 `(b,a,x)=(38,2,2)`
+   (beyond same-walk / dirty-shadow / partitioned dual / one-repeat / two-repeat)
+   undercut `436` at `Q=8`?
+   Why it matters: remaining unequal-price handoff after two-repeat is closed.
+   First step: name one new alphabet outside leave-alones and census dirt-legal R.
+   Blocked by: leave-alones through #2851; pending onerepeat; deferred soft-overlaps.
+2. Does a shared-dirt (single coefficient) dual on distinct partitioned walks
+   undercut the independent-dirt floors?
+   Why it matters: changes the dual cost from independent `min_a` to a coupled
+   readout.
+   First step: fix shared-dirt dual cost and census `(2,6)` distinct labels.
+   Blocked by: soft-overlap risk with shared-XOR leave-alone.
+
+## Credit
+
+Credit **Arham Tahir / PR1898**, **ywz65 / PR2455**, **PR1370**, fork **789q** /
+**#2677**, **#2779**, open **#2851** (distinct partitioned; leave-alone; not
+edited), pending onerepeat tips (OQ answered here; not edited), open **#2846** /
+**#2840** / **#2836** / **#2817** / **#2764** / **#2761** (leave-alone),
+**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.