Research: partitioned independent-walks 7q readout obstruction (no new kappa)

research/independent-walks-7q-obstruction/CERTIFICATE.json

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+{
+  "schema_version": 1,
+  "status": "PASS_INDEPENDENT_WALKS_7Q_OBSTRUCTION",
+  "scope": "Five-source common-B mutable-source Q/R/X alphabet (PR1898). Partitioned independent dual walks at total Q=7 (valid restored lengths (2,5) and (3,4)): covering dual even-parity readout has exact R=112; tariff floor at X=0 is 490>436. Answers independent-walks OQ (partitioned case) after same-walk multi-connector.",
+  "claim": {
+    "Q": 7,
+    "partition_covering_min_R": 112,
+    "partition_25_covering": 416640,
+    "partition_34_covering": 833280,
+    "partition_25_R_exact": 112,
+    "partition_34_R_exact": 112,
+    "tariff_floor_at_X0": 490,
+    "endpoint_Q6_tariff": 436,
+    "undercuts_436_after_complete_bill": false,
+    "length1_walks_invalid": true,
+    "answers_independent_walks_partitioned": true,
+    "oq_answer": "Every covering partitioned dual-walk schedule with total query count Q=n0+n1=7 (valid restored walks of lengths (2,5) or (3,4), distinct nonzero labels within each walk, independent dirt, dual even-parity readout) has exact R=112. At X=0 the abstract tariff is already 490>436; positive path X only increases the bill. Length-1 components are not restored (closed zero edge). Same-walk multi-connector remains the pending mc7q package; transfer-coupled schedules are out of scope here.",
+    "no_new_kappa": true
+  },
+  "partition_2_plus_5": {
+    "kind": "partition_2_plus_5",
+    "Q": 7,
+    "covering_pairs": 416640,
+    "structural_ok": 416640,
+    "attainable_R": [
+      112
+    ],
+    "histogram": {
+      "112": 416640
+    },
+    "min_R": 112,
+    "first_witnesses": {
+      "112": [
+        [
+          0,
+          1
+        ],
+        [
+          0,
+          2,
+          4,
+          8,
+          16
+        ]
+      ]
+    }
+  },
+  "partition_3_plus_4": {
+    "kind": "partition_3_plus_4",
+    "Q": 7,
+    "covering_pairs": 833280,
+    "attainable_R": [
+      112
+    ],
+    "histogram": {
+      "112": 833280
+    },
+    "min_R": 112,
+    "first_witnesses": {
+      "112": [
+        [
+          0,
+          1,
+          2
+        ],
+        [
+          0,
+          4,
+          8,
+          16
+        ]
+      ]
+    }
+  },
+  "champion": {
+    "champion_25": {
+      "walks": [
+        [
+          0,
+          1
+        ],
+        [
+          0,
+          2,
+          4,
+          8,
+          16
+        ]
+      ],
+      "R": 112,
+      "path_X": 6,
+      "tariff": 502,
+      "structural": {
+        "covering": true,
+        "rank": 4,
+        "sigma_in_span": false,
+        "R": 112,
+        "sum_even": 40
+      }
+    },
+    "champion_34": {
+      "walks": [
+        [
+          0,
+          1,
+          2
+        ],
+        [
+          0,
+          4,
+          8,
+          16
+        ]
+      ],
+      "R": 112,
+      "path_X": 6,
+      "tariff": 502
+    },
+    "tariff_floor_at_X0": 490,
+    "endpoint_Q6": 436,
+    "single_walk_even_min_R": 82
+  },
+  "length1_invalid_examples": [
+    {
+      "slots": [
+        0
+      ],
+      "valid_restored_walk": false,
+      "path_X": null,
+      "note": "closed_(0,0)_zero_edge"
+    }
+  ],
+  "unit_tariff": {
+    "b": 38,
+    "a": 2,
+    "x": 2
+  },
+  "tariff_check": {
+    "formula": "38*Q + 2*R + 2*X",
+    "at_R112_X0": 490,
+    "champion_25": 502,
+    "champion_34": 502,
+    "endpoint_Q6": 436
+  }
+}

research/independent-walks-7q-obstruction/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.
+- 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; 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`; not edited.
+- Open #2715 `research/cheaper-readout-7q-obstruction` — all-subset /
+  dirty-parity repair; not edited.
+- Pending `research/multi-connector-7q-obstruction` (same-walk multi-connector /
+  priced shadow); not edited; this package answers its independent-walks
+  partitioned follow-up without competing on the same-walk fixture.
+- Pending `research/dirty-parity-8q-obstruction`; not edited.
+- Live pin: `research/mutable-source-shear-cost/CERTIFICATE.json`.
+
+Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI
+assistance.

research/independent-walks-7q-obstruction/PROOF.md

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+# Partitioned independent walks cannot undercut tariff 436 at Q=7
+
+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
+the **partitioned independent-walks** open question left after same-walk
+multi-connector / priced-shadow work on the #2715 lineage: can two restored
+walks with **distinct slot lists**, independent dirt, and dual even-parity
+readout undercut the six-query endpoint tariff `436` at total `Q = 7`?
+
+No new kappa is claimed. Leave-alone packages through #2715 are not edited.
+Pending same-walk `multi-connector-7q` and `dirty-parity-8q` packages are not
+edited or competed with on their fixtures.
+
+## 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` is an open slot list
+`(s_0,…,s_{n-1})` with `s_0 = 0` whose closed completion `(s_0,…,s_{n-1},0)`
+has only nonzero edges. Length-1 lists are illegal: the closed return is the
+zero edge `(0,0)`.
+
+A **partitioned independent dual** of total query count `Q = 7` is a pair of
+valid restored walks of lengths `(n_0,n_1)` with `n_0 + n_1 = 7` and
+`n_0,n_1 ≥ 2`. Up to swap the only cases are `(2,5)` and `(3,4)`. Each walk
+carries its own dirt coefficient. For mask `m` the dual even-parity cost is
+
+```text
+min_a (e0[a] + e1[a ⊕ m]),
+```
+
+where `e_i` is the even-subset table of walk `i` (with `e_i[0] = 0`).
+
+The abstract complete-call tariff at `(b,a,x)=(38,2,2)` is `38·Q + 2·R + 2·X`.
+
+## Length-1 exclusion
+
+A length-1 list `(0,)` closes to `(0,0)`. That edge has XOR `0`, so it is not a
+restored walk. Consequently `(1,6)` and `(6,1)` are not legal partitions of a
+`Q = 7` dual schedule.
+
+## Fixture A: partition `(2,5)`
+
+Short walk `(0,σ)` has even table `e0[0]=0`, `e0[σ]=2`. Long walk
+`(0,b_1,b_2,b_3,b_4)` with distinct nonzero labels. Dual cost collapses to
+
+```text
+R = Σ_{m=1}^{31} min( e1[m],  2 + e1[m ⊕ σ] ).
+```
+
+**Lemma (cover ⇒ outside coset).** Covering requires the long span `V` to have
+rank `4` and `σ ∉ V`. Then masks in `V` pay `e1`, and masks in the complementary
+coset pay `2 + e1(· ⊕ σ)`, so
+
+```text
+R = 2 · Σ_{v ∈ V \ {0}} e1[v] + 32.
+```
+
+**Lemma (F₂⁴ even-sum).** Any four linearly independent nonzero labels are a
+basis of a 4-dimensional subspace. With a free `0`-slot, even-subset cost of a
+vector equals the least even integer `≥` its Hamming weight in that basis.
+Summing over the `15` nonzero vectors of `F₂⁴` yields exactly `40`. Hence every
+covering `(2,5)` schedule has
+
+```text
+R = 2·40 + 32 = 112.
+```
+
+**Census.** Distinct-label pairs: `31 · C(31,4) = 975415` candidates; covering
+count `416640`; attainable `R = {112}` only; structural identity holds on every
+covering pair.
+
+Champion: `((0,1), (0,2,4,8,16))` with path `X = 6`, tariff `502`.
+(The `X = 0` figure `490` is only the abstract lower bound from `R = 112`;
+a length-5 closed walk lies in a bipartite hypercube, so odd closed single-bit
+tours are impossible and attainable path `X` is strictly positive.)
+
+## Fixture B: partition `(3,4)`
+
+**Census.** Distinct-label pairs: `C(31,2)·C(31,3) = 2090175` candidates;
+covering count `833280`; attainable `R = {112}` only (machine-checked dual
+`min_a` on every covering pair — not a copy of a constant).
+
+Champion: `((0,1,2), (0,4,8,16))` with path `X = 6`, tariff `502`.
+
+## Tariff consequence
+
+Every covering partitioned dual at `Q = 7` has exact `R = 112`. Even at
+`X = 0`,
+
+```text
+38·7 + 2·112 + 2·0 = 490 > 436.
+```
+
+Positive path-`X` only increases the bill. Partitioned independent walks
+therefore cannot undercut the six-query endpoint under the PR1898 abstract
+tariff. By comparison, a single same-walk even-subset schedule can reach
+`R = 82` ([#2686](https://github.com/CrocSwap/integer-mult-bounds/pull/2686));
+splitting the query budget across two independent dirt ports forces a much
+worse readout invoice.
+
+## Scope limits
+
+- Fixes the **partitioned** independent-walk interface at total `Q = 7` with
+  distinct nonzero labels within each walk. Transfer-coupled schedules
+  (shared prefix, priced transfer, divergent suffixes) are outside; same-walk
+  multi-connector / priced shadow remains the pending `multi-connector-7q`
+  package.
+- Shared non-path XOR schedules remain deferred (soft-overlap with #2643).
+- Host screens are deferred.
+- Leave-alone through #2715 (including #2593, #2643, #2677, #2686, #2687, #2715)
+  is not edited.
+- 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 single-walk `Q=7` | — |
+| #2686 | Even-subset `R≥82` on one walk | — |
+| #2715 | All-subset collapses under dirty-parity; OQs | — |
+| Pending mc7q | Same-walk dual / dirty shadow stay at `R≥82` | — |
+| **This** | — | **Partitioned dual walks at `Q=7` have exact `R=112`; tariff floor `490>436`** |

research/independent-walks-7q-obstruction/README.md

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+# Partitioned independent-walk obstruction at seven queries
+
+**Scoped obstruction / priced alphabet; no new kappa.** Answers the
+**partitioned independent-walks** open question on the #2715 /
+multi-connector lineage in the five-source common-B mutable-source alphabet of
+[#1898](https://github.com/CrocSwap/integer-mult-bounds/pull/1898): can two
+restored walks with distinct slot lists, independent dirt, and dual
+even-parity readout undercut tariff `436` at total `Q=7`?
+
+| Dual schedule | Covering pairs | Exact R | Tariff floor (X≥0) | Undercuts 436? |
+|---|---:|---:|---:|---|
+| Same-walk even-subset (#2686) | — | ≥82 | ≥442 at X=6 | no |
+| Partition `(2,5)` | **416640** | **112** | **≥490** | **no** |
+| Partition `(3,4)` | **833280** | **112** | **≥490** | **no** |
+
+Every covering partitioned dual has exact `R=112` (structural `F₂⁴` identity
+on `(2,5)`; full dual census on `(3,4)`). Length-1 components are not restored.
+Same-walk multi-connector remains the pending `multi-connector-7q` package.
+
+## Reproduction
+
+```sh
+python3.12 -I -B research/independent-walks-7q-obstruction/verify.py
+python3.12 -I -O -B research/independent-walks-7q-obstruction/verify.py
+python3.12 -I -B research/independent-walks-7q-obstruction/independent.py
+python3.12 -I -B research/independent-walks-7q-obstruction/test_package.py
+python3.12 -B scripts/research_package.py --package research/independent-walks-7q-obstruction
+```
+
+Expected:
+
+```text
+PASS_INDEPENDENT_WALKS_7Q_OBSTRUCTION partition_min_R=112 ... tariff_floor=490 ... answers_independent_walks_partitioned no_new_kappa
+PASS_INDEPENDENT_PARTITIONED_WALKS_7Q covering_25=416640 covering_34=833280 partition_min_R=112 tariff_floor=490 ... no_new_kappa
+PASS_INDEPENDENT_WALKS_7Q_PACKAGE_TESTS groups=6
+PASS_REGISTERED_RESEARCH_PACKAGE research/independent-walks-7q-obstruction outcome=completed
+```
+
+Default verify is read-only. Admission uses `need()`; `-O` also passes.
+
+## Overlap
+
+Base commit `0b1cccf86e2f5132ed286a26ef0a681a495740ea`. Direct predecessors:
+PR1898, #2643, #2686, #2715, pending same-walk multi-connector OQ. Leave-alone
+through #2715 including #2593, #2643, #2677, #2686, #2687, #2715 is respected;
+those packages are not edited. Pending `cursor/multi-connector-7q-60b9` and
+`cursor/dirty-parity-8q-60b9` are not edited or competed with on their fixtures.
+Deferred shared-XOR soft-overlap with #2643 and host screens are not opened.
+
+## Open next questions
+
+1. Can a **transfer-coupled** dual (shared prefix, priced transfer `C≥1`,
+   divergent suffixes) undercut tariff 436 after charging walks+transfers+dirt
+   at total `Q=7`?
+   Why it matters: remaining independent-walk direction after partitioned
+   collapse.
+   First step: fix shapes `(p,a,b)` with `p+a+b=7` and census dirt-legal R with
+   fair shared-prefix XOR billing.
+   Blocked by: larger search; pending same-walk mc7q not a blocker for this
+   freestanding fixture.
+2. Do non-distinct (repeat-label) layers inside a partitioned walk improve on
+   `R=112`?
+   Why it matters: closes the one-repeat companion of the Hamilton layer.
+   First step: allow one repeated nonzero label in the length-5 walk of `(2,5)`.
+   Blocked by: larger search; expected to worsen coverage/cost.
+
+## Credit
+
+Credit **Arham Tahir / PR1898**, **ywz65 / PR2455**, **PR1370**, fork **789q** /
+**#2677**, **#2643**, **#2686**, open **#2715**, pending same-walk
+multi-connector (OQ answered here for the partitioned case; 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.