Research: partitioned independent-walks 9q priced obstruction (no new kappa)

research/independent-walks-9q/MASTER-HANDOFF.md

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+# Master handoff — independent-walks-9q
+
+## Deliverable
+
+| Field | Value |
+|---|---|
+| Branch | `cursor/independent-walks-9q-60b9` |
+| Tip SHA | branch HEAD (`git rev-parse cursor/independent-walks-9q-60b9`) |
+| Package | `research/independent-walks-9q/` |
+| Claim | Freestanding `#840`-style priced alphabet: **Q=9 partitioned independent dual** under PR1898 `(b,a,x)=(38,2,2)` |
+| Result | Exact covering min **R=80** on `(2,7)`; `(3,6)` min 88; `(4,5)` min 92; tariff floor **502>436** (also above same-walk Q9 floor 466) |
+| Kappa | none |
+
+## Answers
+
+- Preferred next20 / threerepeat OQ2: Q=9 partitioned independent dual companion
+  on `(2,7)/(3,6)/(4,5)` does **not** undercut tariff 436 (floor 502), nor the
+  same-walk Q=9 floor 466 of #2677.
+
+## Exact census
+
+| Shape | Covering | Min R | Histogram |
+|---|---:|---:|---|
+| `(2,7)` | 20512576 | 80 | 80:972160 / 82:2041536 / 86:6319040 / 88:8749440 / 92:972160 / 100:1458240 |
+| `(3,6)` | 69370560 | 88 | 88:3749760 / 90:20623680 / 92:11249280 / 94:5624640 / 96:21873600 / 100:5624640 / 104:624960 |
+| `(4,5)` | 122561600 | 92 | 92:4166400 / 94:37497600 / 96:43747200 / 98:3472000 / 100:31248000 / 104:2430400 |
+
+Champions: `((0,19),(0,1,2,4,8,15,16))` R=80 X=26;
+`((0,1,2),(0,4,8,13,16,22))` R=88 X=18;
+`((0,1,2,4),(0,7,8,16,25))` R=92 X=22.
+
+## Leave-alone respected
+
+Through #2868 / #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-9q/verify.py
+python3.12 -I -O -B research/independent-walks-9q/verify.py --quick
+python3.12 -I -B research/independent-walks-9q/independent.py
+python3.12 -I -B research/independent-walks-9q/test_package.py
+python3.12 -B scripts/research_package.py --package research/independent-walks-9q
+```
+
+## 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 / shared-dirt?
+2. Mixed shared-dirt + excess at Q=8 vs shared-dirt floor R=82?

research/independent-walks-9q/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 —
+  same-walk Q=9 floor 466; not edited.
+- Open #2779 `research/independent-walks-7q-obstruction` — partitioned dual 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 #2867 `research/shared-dirt-walks-8q` — shared-dirt at Q=8; leave-alone;
+  not edited.
+- Open #2861 / #2860 / #2858 — one-repeat / two-repeat at Q=8; leave-alone;
+  not edited.
+- Open #2846 / #2840 / #2836 / #2817 / #2764 / #2761 — leave-alone; not edited.
+- Pending `cursor/independent-walks-threerepeat-8q-60b9@94b14c36`
+  `research/independent-walks-threerepeat-8q` — three-repeat at Q=8; 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-9q/PROOF.md

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+# Partitioned independent duals at Q=9 cannot undercut tariff 436
+
+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: the **Q=9 partitioned independent dual companion** of the
+Q=8 distinct partitioned dual
+([#2851](https://github.com/CrocSwap/integer-mult-bounds/pull/2851)). It answers
+preferred next20 and open question 2 of pending
+`cursor/independent-walks-threerepeat-8q-60b9@94b14c36`.
+
+No new kappa is claimed. Leave-alone packages through #2867 are not edited.
+Pending three-repeat tip 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 **partitioned independent dual** of total query count `Q = 9` is a pair of
+valid restored walks of lengths `(n_0,n_1)` with `n_0+n_1=9` and `n_0,n_1≥2`,
+distinct nonzero labels within each walk, and **independent dirt**. Legal dirty
+cancellation requires even live-read parity on each walk separately. With even
+subset tables `e_i` on walk `i`, the dual cost is
+
+```text
+R = Σ_{m=1}^{31} min_α (e0[α] + e1[α ⊕ m]).
+```
+
+Up to swap the only partitions are `(2,7)`, `(3,6)`, and `(4,5)`. Length-1
+components are not restored. The abstract complete-call tariff at
+`(b,a,x)=(38,2,2)` is `38·Q + 2·R + 2·X`.
+
+The same-walk Q=9 path-lowered floor of
+[#2677](https://github.com/CrocSwap/integer-mult-bounds/pull/2677) is
+`R≥62` ⇒ tariff `≥466`. Beating the six-query endpoint `436` at nine queries
+would require `R+X < 47`; every dirt-legal even covering already forces
+`R≥62`, so the undercut question is settled once the partitioned dual census
+shows its own floor lies at or above that regime.
+
+## Fixture A: partition `(2,7)`
+
+Short walk `(0,σ)` has `e[0]=0`, `e[σ]=2`. Dual cost collapses to
+
+```text
+R = Σ_m min( e1[m],  2 + e1[m ⊕ σ] ).
+```
+
+**Census.** Distinct-label pairs: covering count `20512576`; attainable
+`R ∈ {80,82,86,88,92,100}` with histogram
+`972160 / 2041536 / 6319040 / 8749440 / 972160 / 1458240`. Minimum `R = 80`.
+
+Champion `((0,19),(0,1,2,4,8,15,16))` has independent `R = 80`, path `X = 26`,
+tariff `554`. The `X = 0` figure `502` is the abstract lower bound from
+`R = 80`.
+
+## Fixture B: partition `(3,6)`
+
+**Census.** Covering count `69370560`; attainable
+`R ∈ {88,90,92,94,96,100,104}` with histogram
+`3749760 / 20623680 / 11249280 / 5624640 / 21873600 / 5624640 / 624960`.
+Minimum `R = 88`.
+
+Champion `((0,1,2),(0,4,8,13,16,22))` has independent `R = 88`, path `X = 18`.
+
+## Fixture C: partition `(4,5)`
+
+**Census.** Covering count `122561600`; attainable
+`R ∈ {92,94,96,98,100,104}` with histogram
+`4166400 / 37497600 / 43747200 / 3472000 / 31248000 / 2430400`.
+Minimum `R = 92`.
+
+Champion `((0,1,2,4),(0,7,8,16,25))` has independent `R = 92`, path `X = 22`.
+
+## Global consequence
+
+The binding covering minimum across `(2,7)/(3,6)/(4,5)` is `R = 80` on
+`(2,7)`. At `X = 0` the abstract tariff is
+
+```text
+38·9 + 2·80 + 2·0 = 502 > 436
+```
+
+and also `502 > 466` (same-walk Q=9 floor). Positive path `X` only increases
+the bill. Therefore no covering partitioned independent dual at nine queries
+undercuts the six-query endpoint after a complete dirty bill.
+
+## Machine check
+
+`verify.py` exhaustively recomputes all three covering histograms and matches
+`CERTIFICATE.json`. `independent.py` recomputes the binding `(2,7)` census and
+champion identities without importing `model.py`. Admission uses `need()` so
+`-O` passes. Default verify is read-only.

research/independent-walks-9q/README.md

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+# Partitioned independent-walk obstruction at nine 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 **Q=9 partitioned independent dual companion**
+of [#2851](https://github.com/CrocSwap/integer-mult-bounds/pull/2851) on
+shapes `(2,7)`, `(3,6)`, and `(4,5)`. Answers preferred next20 and open
+question 2 of pending
+`cursor/independent-walks-threerepeat-8q-60b9@94b14c36`. Soft-overlap-safe vs
+the Q=8 leave-alone cluster (new query count; no leave-alone reopen).
+
+| Dual schedule | Covering pairs | Exact min R | Tariff floor (X≥0) | Undercuts 436? | vs same-walk Q9 466? |
+|---|---:|---:|---:|---|---|
+| Same-walk Q=9 (#2677) | — | ≥62 | ≥466 | no | — |
+| Partitioned `(2,7)` (this) | **20512576** | **80** | **≥502** | **no** | **above** |
+| Partitioned `(3,6)` (this) | **69370560** | **88** | **≥518** | **no** | **above** |
+| Partitioned `(4,5)` (this) | **122561600** | **92** | **≥526** | **no** | **above** |
+
+Global covering minimum is `R=80` on `(2,7)` (spectrum
+`{80,82,86,88,92,100}`). Abstract tariff `38·9 + 2·80 + 2·0 = 502 > 436`
+and also above the same-walk Q=9 floor `466`. Partitioned duals at nine
+queries cannot undercut the six-query endpoint.
+
+## Reproduction
+
+```sh
+python3.12 -I -B research/independent-walks-9q/verify.py
+python3.12 -I -O -B research/independent-walks-9q/verify.py --quick
+python3.12 -I -B research/independent-walks-9q/independent.py
+python3.12 -I -B research/independent-walks-9q/test_package.py
+python3.12 -B scripts/research_package.py --package research/independent-walks-9q
+```
+
+Expected:
+
+```text
+PASS_INDEPENDENT_WALKS_9Q_OBSTRUCTION partition_min_R=80 ... tariff_floor=502 ... answers_threerepeat_oq2 no_new_kappa
+PASS_INDEPENDENT_WALKS_9Q_QUICK partition_min_R=80 tariff_floor=502 ... answers_threerepeat_oq2 no_new_kappa
+PASS_INDEPENDENT_PARTITIONED_WALKS_9Q covering_27=20512576 ... tariff_floor=502 ... answers_threerepeat_oq2 no_new_kappa
+PASS_INDEPENDENT_WALKS_9Q_PACKAGE_TESTS groups=6
+PASS_REGISTERED_RESEARCH_PACKAGE research/independent-walks-9q outcome=completed
+```
+
+Default verify is read-only. Admission uses `need()`; `-O` also passes.
+Full census is the default verify; `--quick` is the `-O` / package-test path.
+
+## Overlap
+
+Base commit `0b1cccf86e2f5132ed286a26ef0a681a495740ea`. Direct predecessors:
+PR1898, #2851 (distinct partitioned dual at Q=8), #2677 (same-walk Q=9
+floor 466), pending threerepeat tip (OQ2 answered here),
+#2867/#2861/#2860/#2858/#2846/#2840/#2779. Leave-alone through #2867
+including #2861, #2860, #2858, #2851, #2846, #2840, #2836, #2817, #2779,
+#2764, #2761 is respected; those packages are not edited. Pending
+`cursor/independent-walks-threerepeat-8q-60b9@94b14c36` 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 sixth priced readout alphabet under PR1898 `(b,a,x)=(38,2,2)`
+   (beyond same-walk / dirty-shadow / partitioned dual / one-repeat /
+   two-repeat / shared-dirt / three-repeat) undercut `436` at `Q=8`?
+   Why it matters: remaining unequal-price handoff on the Q=8 cluster.
+   First step: name one new alphabet outside leave-alones and census dirt-legal R.
+   Blocked by: leave-alones through #2867; pending three-repeat; deferred soft-overlaps.
+2. Does a mixed shared-dirt + excess (e.g. shared-dirt with one-repeat)
+   partitioned schedule at `Q=8` undercut the shared-dirt floor `R=82`?
+   Why it matters: combines dirt-coupling and excess axes without reopen.
+   First step: fix mixed cost and census distinct-label `(2,6)`.
+   Blocked by: larger search; expected not to undercut 436 even if R improves.
+
+## Credit
+
+Credit **Arham Tahir / PR1898**, **ywz65 / PR2455**, **PR1370**, fork **789q** /
+**#2677**, **#2779**, open **#2851** / **#2867** / **#2861** / **#2860** /
+**#2858** (leave-alone; not edited), pending threerepeat tip (OQ2 answered
+here; not edited or competed), 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.

research/independent-walks-9q/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1",
+      "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075"
+    },
+    {
+      "path": "research/mutable-source-shear-cost/CERTIFICATE.json",
+      "usage": "input",
+      "git_blob": "48958451bc819f2caa13bf3d9609423706ffb8a0",
+      "sha256": "665435375c025bbcf69eadf43bd9aeae44eb12fefdf12a78d5bfd66abde4ae3c"
+    }
+  ]
+}